text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
Q: Trying to make a random speed generator Its on Roblox studio
Every 5 seconds the players speed changes
Im kind of new to coding
I tryed to use the math.random
I probably did it wrong
Tried to like use the output
I dont think it works like that
I do not know
PLEASE HELP
A: First insert a LocalScript in the PlayerGui then open the script for editing
in the scrpt type:
local MinimalSpeed = your minimal speed in numbers
local MaximalSpeed = your maximal speed in numbers
local LocalPlayer = game:GetService("Players").LocalPlayer
local Humanoid = LocalPlayer.Character:WaitForChild("Humanoid")
function makespeed()
Humanoid.WalkSpeed = math.random(MinimalSpeed,MaximalSpeed)
wait(5)
end
while true do
makespeed()
end
the default speed is 16 so type in what speed you want
| {
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{"url":"https:\/\/socratic.org\/questions\/the-concentration-of-the-appetite-regulating-hormone-1-3-times-10-10-m-in-a-fast","text":"# The concentration of the appetite-regulating hormone 1.3 times 10^-10 M in a fasting person. How many molecules of ghrelin are in 1 L of blood?\n\n##### 1 Answer\nSep 4, 2017\n\nApproximately $7.8 \\cdot {10}^{13}$ ghrelin molecules are present in a fasting person's blood.\n\n#### Explanation:\n\n$M$ is molarity, units of molarity are $\\frac{m o {l}_{s o l u t e}}{{L}_{s o l n}}$.\n\nThus there are $1.3 \\cdot {10}^{-} 10 m o l$ per liter of blood.\n\nFurthermore, using Avogadro's number, we'll arrive at the solution:\n\n1.3*10^-10mol* (6.023*10^23 \" molecules\")\/(mol) approx 7.8*10^13\" molecules\"\n\nGhrelin is a protein that functions as a hormone:","date":"2021-09-17 12:11:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 5, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8820593953132629, \"perplexity\": 12145.549539505526}, \"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-2021-39\/segments\/1631780055645.75\/warc\/CC-MAIN-20210917120628-20210917150628-00081.warc.gz\"}"} | null | null |
Large Mirrors For Bathrooms Vintage Full Length Mirror Bedroom Vintage Floor Length Mirror Uploaded by admin in category Floor Mirror.
See also The Red Trousers March 2013 Vintage Floor Length Mirror from Floor Mirror Topic.
Here we have another image H O L L Y W O O D Vintage Leaning Mirror Floor Mirror Regency Vintage Floor Length Mirror featured under Large Mirrors For Bathrooms Vintage Full Length Mirror Bedroom Vintage Floor Length Mirror. We hope you enjoyed it and if you want to download the pictures in high quality, simply right click the image and choose "Save As". Thanks for reading Large Mirrors For Bathrooms Vintage Full Length Mirror Bedroom Vintage Floor Length Mirror. | {
"redpajama_set_name": "RedPajamaC4"
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package cryodex.modules.runewars;
public class RunewarsConstants {
public static int WIN_POINTS = 1;
public static int BYE_POINTS = 1;
public static int LOSS_POINTS = 1;
}
| {
"redpajama_set_name": "RedPajamaGithub"
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User has a Intel Mac G5 running OS 10.7.5 with Fetch 5.5.3. When you open an ftp site and enter password, if you check the box to save the password, the following error appears.
"The action could not be completed because the server is unreachable. Check your network connection and settings in the Network pane of System Preferences and try again."
If you do not check the box, you can login to the ftp site without a problem. Everything else network related is working fine. I have created a default keychain for the user but still getting the same error and the keychain is working fine with all other applications. Any ideas why we are getting this error?
You can email it to bugs@fetchsoftworks.com . | {
"redpajama_set_name": "RedPajamaC4"
} | 5,487 |
Amiri King Wiki, Age, Wife, Net Worth & More
Tony Donavan Schork, a.k.a. Amiri King is an all-time multitasker. He started his career in 2007 by posting comedic videos including parodies on his YouTube channel – RoyalMediaMafia, where he has over 794K subscribers, Currently, he is a motivational speaker for the youth and also writes content for local comedians.
King's transformation from a life of crime to the prosperity he enjoys now is a shining example of how one may go from a tumultuous past to a bright future. He's also starred in a number of films, including Rick and Ed's Surviving the Zombie Apocalypse. King has also appeared on the front cover of Readers Digest magazine three times and his YouTube channel has a whopping 218M+ views.
Amiri King Profile
Birthname Tony Donavan Schork
Profession Social Media Influencer
Amiri King Personal Life Info & More
King was born in Louisville, Kentucky, and was raised by his mother, Sharon Brown Colvin, as a single parent and faced a lot of problems all throughout his childhood. He was born on 10 July 1979 and is currently 42 years old. He is American by nationality and his zodiac sign is Cancer.
Date of Birth/Birthday 10 July 1979
Age (as in 2021) 42
Birthplace Louisville, Kentucky
Hometown Louisville, Kentucky
Zodiac Sign/Sun Sign Cancer
Amiri King Height, Weight, Physical Stats & More
Amiri King stands at a height of 6 feet (1.82m). He has a good muscular build, however, his weight is not known to us. He has light brown and brown hair. He is a health and fitness freak and has kept himself extremely fit.
Read About: Fahad Asad - Rising Social Media Influencer from United Arab Emirates
Height (approx.) in meters – 1.82 m
in feet-inches – 6 feet
Eye Color Light Brown
Amiri King Family Members Names & Information
Amiri King was raised single-handedly by his mother, Sharon Brown Colvin. He had an extremely tough time growing up with his father and even ran away from his house after a huge fight with his father where he beat Amiri as well.
Mother Sharon Brown Colvin
Amiri King Marital Status & More
Amiri is currently happily married to Sara Ruminski, who is of Irish descent. They also have three daughters- Kennedy, Tilly, and Mery.
Wife/Spouse Sara Ruminski
Children's 3 Daughters: Kennedy, Tilly, and Mery
Amiri King Education Qualification, School & College
Any details about Amiri's educational qualifications are not known to the public.
Amiri King Money Factor
Following Amiri's YouTube career, he has managed to make a good living for himself as his net worth is estimated to be $250,000.
Net Worth $250,000
Amiri King Biography & Some Other Lesser Known Facts About Amiri King:
He had an extremely difficult childhood, and when he was 11 years old, he ran away from home after being violently assaulted by his father. To survive, he swept the parking lots of gas stations while waiting for leftover hot dogs and doughnuts, and he moved from one shelter to another, including Phoenix House, Boys Haven, and Liberty Shelter House.
When King was 16, he was accused of armed robbery, which changed his life forever as he was tried as an adult and found guilty, spending three years in prison before being released on good behaviour. King then chose to change his life and became an inspiring speaker in his hometown, sharing his experience with various youth organizations and high schools.
Read About: Jentzen Ramirez Wiki, Age, Girlfriends, Net Worth & More
He's also appeared in a number of movies, including Rick and Ed's Surviving the Zombie Apocalypse and Rick and Ed's Surviving the Zombie Apocalypse. Three times, King has graced the cover of Readers Digest magazine. He also has a silver play button from YouTube. King's popularity and celebrity as a YouTube phenomenon have earned him a slew of awards and honours.
Amiri King Social Media Accounts & Channels
YouTube: Check Now
This was the complete latest information about "Amiri King" that we curated through the internet. We at FameImpact hope you enjoyed this article and got satisfactory answers to your questions. The information provided above will be updated regularly. Any mistake that you might've noticed or any edit that you want to suggest is always welcome in the comment box below.
Valentine Michael Manson Wiki, Age, Wife, Net Worth & More
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Elizabeth Vega is a well-known Social Media Influencer from The United States. With her amazing fashion sense, she has garnered…
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Hamid Saif (Food Blogger) Wiki, Age, Biography & More
Hamid Saif Ismail is a well-known Food Blogger, Entrepreneur and Social Media Influencer. Hamid Saif Ismail was born in UAE…
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Lowe Abloh Wiki, Age, Bio, Wife, Net Worth & More
Lowe Abloh is the little daughter of the American stylist and creator, Virgil Abloh. Virgil Abloh has donned the hats…
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Actor | Influencer | Model | TikTok
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Tayler Holder is a popular Model, Actor, TikTok star, and social media influencer. Tayler Holder Profile/Introduction Birthname Tayler Holder Nickname…
Read More Tyler Holder Wiki, Age, Girlfriends, Net Worth & MoreContinue | {
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homesubmissionsabout dhqdhq peoplecontact
2018: 12.4
Preview Issue
2015: 9.4
Call for Reviewers
DHQ: Digital Humanities Quarterly
2017 11.3 | XML | Discuss ( Comments )
The Digital Classicist: building a Digital Humanities Community
Simon Mahony <s_dot_mahony_at_ucl_dot_ac_dot_uk>, University College London
There has been much discussion about digital humanities (DH) both as a discipline and as a community of practice.[1] Whatever the balance of opinion, the emergence of digital scholarship in the humanities has undoubtedly had considerable impact on many disciplines; one such discipline is Classics and the study of the ancient world more generally. This article uses the Digital Classicist (DC) as an example of a DH community in a case study which traces its development and growth to examine what might be learned. As a community the DC joins together practitioners interested in the application of innovative digital methods and technology to the study of the ancient world (in its widest sense). How has this come about and perhaps more importantly, how has it been sustained and indeed provided the inspiration for other affiliated communities? What do we understand by a community and the association of individual practitioners separated by distance? It is important that members feel that they are stakeholders, that they have a sense of ownership and derive value from participation and contribution. It is argued here that a community could be seen as a symbolic and intellectual construct, one of perception rather than physicality to facilitate the exchange of ideas and so effect growth and strengthen the discipline.
The Digital Classicist is used here as a case study for looking at the development of and more importantly how we might sustain a Digital Humanities (DH) community. This paper examines the background to the foundation of the Digital Classicist (DC), how it all came about; its development, why it evolved in the way that it did; presents some reflections on what was learned along the way, and, looking ahead, considers where, as a community, we might go from here. Putting this in the wider context, it asks the question why, how and when does a community become a community and how do we recognise one as such?
What is the motivation that is needed to start a community of this type? Firstly, it needs a critical mass of people coming together with similar interests and then the necessary spark of an idea combined with the will to make it happen. The DC certainly had its antecedents both in Classics and cognate disciplines. The Digital Medievalist [1] was established in 2003 and indeed many people (scholars, practitioners, and students) are members of both communities as we have many common interests and concerns; the DC however looks back to Ross Scaife and the Stoa Consortium for Electronic Publication in the Humanities which was established in 1997 for inspiration and with the introduction of the Stoa blog in December 2003 being a mobilising catalyst.[2]
The DC was established in 2004 and set up as a community of users to provide a central hub to draw together practitioners interested in the application of innovative digital methods and technology to the study of the ancient world (in its widest sense). The launch of the DC into the wider world could be said to have been at the presentation of the prize-winning poster at the Digital Resources in the Humanities (DRH) conference held at Lancaster in September 2005.[3] This was at a time when we called what we did Humanities Computing as the proto-Digital Humanities and hence: "Humanities Computing applied to the study of the Ancient World" was the poster strapline (see Fig. 1).
The prize-winning DC poster at DRH Lancaster 2005. Wikimedia Commons http://en.wikipedia.org/wiki/File:Digitalclassicistposter.jpg
Establishment means creating a presence and clearly an online one was needed with the setting up of the website which was quickly followed by a wiki (see Figs 2 and 3).[4] Collaboration and cooperation are central to the DC philosophy and from Fig. 2 it is possible to see the links to the many partnerships that were set up with other projects. Setting up in competition with any of these was never the intention but rather to provide a central web-based focus for research in this rich, diverse, and multi-national field of scholarship [Bodard and Mahony 2008].[5] One of the stated aims of the DC is to bring scholars together and to address head-on the issues of collaborative working; hence the additional use of a wiki:
[…] as well as sharing information about themselves and their own work, members collaboratively compile, review and comment upon articles on digital projects, tools and research questions of particular relevance to the ancient world. They also list guides to practice, introduce the discussion forum and, most importantly, list events. It is these events that more than anything else define the Digital Classicist community by providing a showcase for our members' research and a venue for discussion, introductions, and inspiration for new collaborative relationships and projects. [Mahony and Bodard 2010, 2]
The Digital Classicist webpage: http://www.digitalclassicist.org (17/08/2014)
The Digital Classicist Wiki: http://wiki.digitalclassicist.org (23/08/2014)
In 2006 the Digital Classicist Wiki was presented and discussed in the context of openness and collaborative working by this author at the 7th Computers, Literature and Philology (CLiP) conference: "Literatures, Languages and Cultural Heritage in a digital world", held at King's College London.[6]
Importantly, and in the same year, the Digital Classicist Summer seminar series was launched at the Institute of Classical Studies (ICS), Senate House London. This too had an antecedent in the form of an earlier series run by the organisers. This was effectively the proto-DC seminar series and (ironically) named The Summer Ersatz WiP (Work-in-Progress) seminar as it occupied the same slot as the Postgraduate Work-in-Progress routinely run by students (with staff excluded except by specific invitation) on a Friday afternoon with the added bonus to finish early and to socialise.[7] The Ersatz series ran during the Summer of 2004 as the vibrant and comprehensive seminar programme in Classics supported by the ICS ran in Term time only. We ran our early seminars (and later the DC series) in the Summer because nobody else did; the precedent was set and proved to be valuable experience in setting up and running a successful seminar series.
The sole surviving promotional flyer from the Ersatz series (with thanks to Kim Shahabudin and used with permission).
Running a seminar series once is a "one off"; running it for two consecutive years and it is possibly only a "follow on" (fitting in the papers that could not be included in the first round); run it for the third consecutive year and it is then established with support in place and every expectation that it will grow and continue to thrive.[8] Pulling together a nucleus of people willing to give their time for the organising and also members of the community willing to present research papers suggests the possibility of further endeavours. For the DC, this gave the impetus and inspiration for two panels presented at the Classical Association (CA) Annual Conference 2007 held in Birmingham: Research into people and places and Interdisciplinary approaches to research and pedagogy .[9] The DC was presenting Digital Humanities research at a mainstream Classics conference under the aegis of the largest Classical organisation in the UK as well as at the ICS which is arguably the foremost Classics Institute in the UK, if not the world. Further DC research has been presented at later CA conferences: Glasgow (2009) Ancient World and e-Science [10]; Cardiff (2010) Linked data for archaeology and geography [11]; Nottingham (2014) Open Educational Resources and their place in teaching and research for Classics .[12] The CA conference in Durham in 2011 saw not only two DC panels: Teaching and Publication of Classics in the Internet Age and Ancient Space, Linked Data and Digital Research but also a dedicated Digital Classicist Training Day featuring "Generic Web Tools" and the "Papyrological editor".[13] DC research papers have not been limited to the CA but have also been presented at the American Philological Association (APA), Computer Applications in Archaeology Conference (CAA), Digital Resources for the Humanities and Arts (DRHA), and Digital Humanities; DC takes digital humanities to the Classics and classics to the Digital Humanities.[14]
The Digital Classicist was always considered by the founders to be a "network" to link together people and organisations; a community of users set up by and for practitioners interested in the application of DH methodologies to the study of the ancient world [Mahony and Bodard 2010, 2]. Putting out information gave it a more public voice and a clearer focus with seminars and conference panels giving our members a forum as well as a voice. There is a Jiscmail-hosted mailing list for dissemination of information, making connections and starting conversations and discussions.[15] However, just as with the website and the wiki these are virtual ways in which we connect and communicate asynchronously with our fellows; it is primarily the seminars and also secondarily the conferences that give a locus, a physical presence to the DC.
As well as the mailing list, other communication channels were set up with the obligatory blog being one. However, after an initial period it was incorporated with the Stoa to avoid repetition and to keep one central focus.[16]
The Stoa Consortium blog http://www.stoa.org/archives/1568 (23/08/13)
As previously mentioned, the DC was always conceived of as a community and a network of users. The wiki was set up as a collaborative medium to allow members to compile, review and comment on digital tools, projects and research questions that would be of interest to them. As with all wikis this had no pre-set design structure but has grown organically over the years in particular in response to DC members as they are the authors of the content.[17] In this way it should follow the interests of the members as well as opening up the opportunities for collaborative working which again reinforces the community aspect of the DC. Indeed, the content of the Guides to Good Practice, FAQs, Tools, and Projects pages should be considered research output in their own right. As the wiki platform is more than simple static webpages it allows for additions and amendments not only by the author but also by other members.[18] The wiki format, set up in this way, can be considered to provide rolling and ongoing peer review and again provide another focus for a community.[19] Moreover, all the original content published in the DC Wiki is released under a Creative Commons Attribution License[20] allowing it to be shared, distributed and adapted so long as it is appropriately attributed and not subject to any further restrictions.
Following the successful establishment of the DC seminar series as an annual event, they have been helped each year with an ever-widening programme that seeks to include research students as well as early career researchers and established practitioners. The emphasis throughout has been on new innovative techniques and methodologies which advance the research interests of classicists (in the widest sense to include all cognate areas of interest such as historians, palaeographers, epigraphists and archaeologists) as well as information specialists or digital humanists. Indeed, the annual "call for papers" specifies that proposals for papers should have an academic research agenda relevant to at least one of these fields.
One aspect of the DC seminars that is often commented on by speakers and visitors alike is the relaxed atmosphere and particularly so after the formal presentation is concluded which then allows and indeed encourages further discussion in an informal setting. This is where the networking and discussions that result in the exchange of ideas and the plans for collaboration take place. This is a throwback to the original proto-DC seminars of the Ersatz series where the point was to have fun and be a foil to the oh-so formal seminars that were held in term time; an excuse to finish study early on a Friday and come together with other postgraduate students.
That said, there have been developments over the years. A growing reputation attracts international speakers and audience; the DC seminars have worldwide appeal and hence in response to requests we started making audio recordings and posting them online together with the presentation slides.[21] For the first time in 2013, and with the help of a professional videographer and some funding from the UCL Centre for Digital Humanities and the Department of Digital Humanities at King's, there are video recordings as well as audio and slides to help to create a more permanent record as well as an archive.[22]
The seminars and conference panels create a nucleus of research and so publications became the next appropriate step to make the DC more than a transient entity. Selected papers from the inaugural DC seminar series in 2006 along with one from a conference panel and another specially commissioned were published in a special collaborative issue of the Digital Medievalist [Bodard and Mahony 2008]. This seemed a natural venue for the DC's first publication as the Digital Medievalist already had an established and robust publications platform and peer review mechanism; additionally, both Classics and Medieval projects have a long history of pushing forward the digital agenda through the works of Roberto Busa and IBM on the Index Thomisticus [23], through the Thesaurus Linguae Graecae [24] and a host of other innovative projects. A first DC print publication appeared in 2010 as part of the Ashgate series, Digital Research in the Arts and Humanities , with peer-reviewed contributions based on presentations from seminars and conference panels in 2007 and 2008 [Bodard and Mahony 2010]; this was followed in 2013 with another peer-reviewed publication this time in the ICS in-house journal the Bulletin of the Institute of Classical Studies (BICS) [Dunn and Mahony 2013]. During the revision of this article we have seen another important and innovative DC publication edited by organisers of the seminars in London and Berlin focussing on collaborative and public-facing digital research that engages non-academic and other broader audiences [Bodard and Romanello 2016]. These print publications ironically have given the "Digital" Classicist space on library shelves as well as online.
The seminars have become central to our activities and give a focus for the DC as a community. They promote the research activities of our members; they allow the promotion of the DC; they raise the profile of our speakers. This final point is important for, as well as publishing the presentation online, the DC has an extensive promotions network which as well as announcing the programme, pushes out weekly notifications of speaker and abstract. This is further supported by announcements on the Stoa, the Institute of Classical Studies and the University of London School of Advanced Studies networks. Both the DC and our members take advantage of social media and particularly Twitter. Using the hashtag #DigiClass, blog announcements are automatically "tweeted" by the Stoa and then individually circulated by the organisers and members to their own followers. "Live tweeting" is also encouraged at the seminars and any other DC event. The emphasis is changing from static to dynamic but as well as this the seminars create a focus through the Summer months both online and in the ICS. As well as in "time" the DC now has a presence in "space", a physical as well as virtual location.
What is it that makes a community and particularly a scholarly one? As discussed earlier, as well as sharing information about themselves and their own work, DC members collaboratively compile, review and comment upon articles on digital projects, tools and research questions of particular relevance to the ancient world. They also list guides to practice, introduce the discussion forum and, most importantly, list events. It is argued here that it is these events more than anything else that define the DC community by providing a showcase for our members' research and a venue (both physical and virtual) for discussion, introductions, and the inspiration for new collaborative relationships and projects. The traditional scholarly community would be centred in an institution but here we have a virtual one with "a group of people who share the same interests, pursuits, or occupation […]" (OED Online s.v. community). Distance is no longer an obstacle and the spatial dimension with a common physical institutional location is no longer needed [Kenyon 2000, 22]; although of course the institutional support is as important as ever. Consequently, it is possible to conceive of a community as more of a symbolic and intellectual construct [Cohen 1989]. The spatial dimension becomes less important and the time factor more so (in the sense of when these seminars occur as they attract a virtual audience as well as a physical one) as the seminars and panels become representative of the DC and the community element becomes a mental construct and one of perception rather than physicality. However, the seminars (and to a lesser extent the conferences) now substitute for that physical presence and give a locational as well as a temporal point of focus. This sense of "community at a distance" is of great importance and studied much in the area of distance education.[25]
Indeed, "community" is one of the foundational terms in Social Science and too many attempts to theorise about it in those terms will be avoided here. However, it would be remiss not to at least include a brief mention of Gemeinschaft and Gesellschaft, with the former being the strong ties that have become associated with "community" and the latter the somewhat weaker ties of what could be termed "association", and the tension between what could be described as the physical community and one based on occupation or interest; further discussion on this topic should be saved for another paper on Ferdinand Tönnies. However, the point here is that the DC community falls between the two and is neither one nor the other but has some attributes of both. It is fundamentally a community of interest but the seminars and conference panels give the locus and (virtual) physicality and so a strengthening of the bonds. In the sociological sense, a scholarly community might be described as a group who share a common professional interest, communicate and collaborate with each other, but also and importantly identify themselves with the group's goals and values, and experience the feeling of belonging [Kenyon 2000, 22]. It is these factors that hold the "community" together.
Examples of this in practice can be seen in many DC publications and research activities. The Introduction to the 2010 volume, mentioned above, self-consciously uses the term "community of users" to describe the DC as it "has become defined by what we (as a community) do" and indeed that the "unifying agenda of the volume" does not depend on any particular technical, methodological or philosophical approach but rather as a "community of expertise and practice" [Mahony and Bodard 2010, 1–6]. This sense of community is a common theme expressed throughout this first print collection of DC papers, whether that be concerning material culture (Heath ch.2), EpiDoc (Tupman ch.4), or the survival of texts (Cayless ch.8). Perhaps the most pertinent is the concluding chapter (Terras ch.10) where Melissa Terras draws together many of the volume's central themes with a focus on interdisciplinary research (and the problems that arise at both an individual and institutional level) which by their nature require participants coming together in communities of practice to achieve their research goals. In the broad spectrum of DC interests (just as in the wider field of DH where we position ourselves) no single person has all the skills that are needed for interdisciplinary working; cooperation and collaboration is needed.
This vision of the DC as a "community" goes beyond theorising and manifests itself in practice. This can clearly be seen in initiatives such as EpiDoc (a sub-set of the TEI and a set of XML standards for "the representation of texts in digital form").[26] Again, users of EpiDoc self-identify as a "community of practitioners" by those who both make use of and teach encoding methods for inscriptions [Dee et al. 2016, 15]. The same is true of many other DC projects such Perseids (a collaborative editing platform), EAGLE (the Electronic Archive of Greek and Latin Epigraphy) and those involved in documenting cultural heritage; these also self-identify and represent themselves as communities [Almas and Beaulieu 2016, 178] [Orlandi 2016, 209] [Vitale 2016, 147]. Just as the wider DC community, these too can be typified as virtual ones as they also exist without a physical location; yes, there is an institutional site where the server is hosted and maintained but the users and contributors are spread far and wide. This is also true of the wider DH community. This author is Associate Director of the UCL Centre for Digital Humanities (UCLDH) and we are very much a "virtual centre"; we have signs on our office doors to indicate our affiliation but members of the Management Team are spread across various department in the Arts and Humanities, as well as Computer Science and beyond.[27] We are a cross-faculty research centre; we share a vision and a taught programme; we put on seminars and other events but we do not have a physical space; we have posters on display but no bricks-and-mortar evidence of our existence. Indeed, we are ourselves part of the wider virtual community of DH centres that is CentreNet.[28]
Sustaining the community
How might a community such as DC become sustainable when all the members undoubtedly have great pressure on their limited time? As above, the community is mostly identified by what it does and, for the DC, the events more than anything else define the community and provide, along with the wiki, a showcase for members research and a point of focus. For a community, and particularly a virtual one, to flourish members need interaction with the rest of that community and to have a sense of belonging: of being a part of that community. Communication via the Jisc discussion list, coming together for seminars and conferences, both virtually by social media and in person, creates that sense. The common focus supports the sense of belonging and connects with the shared interests and values of members. Debate and discussion follow and the asynchronous medium of the mailing list and wiki allow a democratisation of the discussion process where everyone potentially gets to have their say and is equally valued [Mahony 2007]. This communication opens up possibilities for the sharing of information, knowledge, tools, and advice; the scholarly exchange of expertise and resources given freely here becomes part of a socially organised form of reciprocity such as described by Marcel Mauss who sees this type of sharing as a mechanism to maintain and preserve the social relations within a society and by extension a community [Mauss 1990]. Thus the community is sustained by engaging in the activities that attracted the members to begin with. Members need to feel that they are part of the community, that they have a voice, and most importantly that they are able to have a positive and valuable interaction with other members of the community [Millan and Chavis 1986]. In other words this might be reduced down to the simple question: what do the members get out of being part of the community? To be sustained a community must continue to engage its members.
In addition, the institutional affiliation needs to be maintained and the DC has been generously funded and supported since its inception by the ICS at Senate House, London (part of the University of London, School of Advanced Study) and they in turn are accountable to their funding bodies. It is therefore important to satisfy basic funding requirements to ensure further support. Our seminars are open and everyone is welcome, from specialist to those with a casual interest; we are part of the ICS's programme of "outreach" and "public engagement"; we increase their "impact" as well as participating in those other activities favoured by funding bodies, "knowledge transfer" and "knowledge exchange". However, the DC goes much further than this as we are part of and participate in networks much wider than the DC such as those of the Digital Humanities and e-Science communities in both the UK and internationally.[29] Our collective looks outward rather than inward and engages with the wider international community. Indeed, each year our seminars have speakers from outside the UK and Europe as well as international visitors in the audiences.[30]
The emergence of digital scholarship in the humanities has had considerable impact on disciplines such as Classics and the study of the ancient world. The example of the DC is used here to demonstrate the possibilities for collaborative authorship, the creation of reusable research output, the opportunities to add thoughts and comments in the form of annotation, and for bringing people together (both physically and virtually) to facilitate the exchange of ideas. These are all central to building communities of learning and scholarship, but the most important is the exchange of ideas. It is in this way that knowledge grows and we are able to push the boundaries of scholarship. To be clear, this was never, to my knowledge as one of the founders, planned at the outset. Communities grow organically and in response to their members and their members' interests. It is necessary, of course, to have people who are prepared to commit time and effort in the planning, organisation, development, contribution, and participation. To flourish such a community needs to be community driven and to give members a sense of ownership, where users become contributors and so stakeholders, to be welcoming and inclusive rather than exclusive. This is particularly true of a community that exists for most part at a distance and falls between Gemeinschaft and Gesellschaft.
Using the DC as a case study for the development and sustainability of a DH community, what can be learned from these experiences and what could usefully guide other communities? As mentioned above, there was no specific plan for the development of the DC but rather an idea: to form a central hub and bring together researchers and practitioners with common interests. The DC grew in a pragmatic way creating partnerships with an inclusive rather than exclusive vision and this inclusivity allowed for diversity and a widening of the sphere of interests. This is turn brought together a greater variety of people, researchers, practitioners and students to allow a much greater facilitation of cross-disciplinary discussion and possibility of collaborations. This inclusivity is important and something that is fostered for us at DC and within DH more generally. For a virtual community to survive it needs to be outward facing and have a focus; one that is recognisable with branding and outreach, importantly outreach beyond the immediate and obvious community. The DC takes digital humanities to Classics and classics to the Digital Humanities with papers and panels at major classics conferences as well as at digital humanities ones.[31] The established seminar series gives the opportunity for this outreach and the promotion of the organisation, the speakers, their research, the centre or whatever as well as a relaxed opportunity for networking. The inclusivity also encourages the breadth of contributions evidenced by the wide-ranging topics found on the seminar listings.[32] The DC has robust and effective mechanisms for promotion via mailing lists, discussion fora and social media with an archive of events, presentations and videos on the website. Showcasing publications from the community afford the same opportunities for outreach and promotion. Engaging with other communities facilitates knowledge exchange and transfer.
The community begins with a nucleus of willing people which the activities described above allow to develop and grow. More is needed, however, for a virtual community to be sustained and to understand this we need to go back to Gemeinschaft and Gesellschaft. With no physical location, the community begins with the weaker ties of interest and association; the locus and point of contact afforded by the seminars and conference participation strengthens those ties but they are not enough. Members need to be involved and have a sense of ownership; they need to be stakeholders – this is what strengthens the ties. The DC Wiki is one showcase for members' research and a point of focus for our virtual community. A new initiative to draw members to this sense of ownership and closer ties is the monthly wiki "sprint".[33] This involves participants coming together for an hour or two to work on improving the content of the DC Wiki; these can be themed sprints or consist simply of going through the pages updating links and content as well as deleting redundant pages. This gives another level of participation for willing members and another opportunity to meet and work together in a virtual environment; there is a Google doc to monitor edits being made and a IRC (Internet Relay Chat) channel for any discussion or questions. This is particularly useful for drawing in members who consider their specific area of interest to be in the minority or under represented amongst the publications, such as Arabic.[34] Members need to be engaged by the community and feel that they are part of it; they must associate themselves and self-identify with the community. It starts with shared interests and values but the successful community needs members (a critical mass) to have the commitment to ensure it is sustained; having a sense of ownership and a stake in this ensures that it will still be around in years to come.
In any discussion on sustainability, the importance of institutional support cannot be underestimated. The DC has always benefited from the generous support of the ICS along with that of the institutions that employ the organisers. With the ever-increasing pressures of academic and research commitments, for staff, students and practitioners, organisations such as the DC allow the opportunities for outreach and public engagement looked for by promotion and interview panels. Moreover, they go some way to combating the lack of institutional memory by establishing a record of activities, involvement and value in the long term which can then be pointed to should the need arise to justify the time and expense involved.
The DC model seems a robust and effective one as, at the time of writing, it has now passed its first decade. But what of the future? We have seminars, conference papers and panels, we have publications, we are developing a substantial archive of contributions, and now our web is spreading further. Presenters at our seminars are now setting up their own networks within which bringing people together both physically and virtually also plays an important role. There is now the Digital Classicist Germany which plans to "function as a hub for Digital Classics-related initiatives in Germany".[35] Moreover, the Digital Classicist Berlin launched its own seminar series complete with lecture videos in October 2012 and was they claim, "inspired by the ICS London Seminar", with the inaugural keynote presentation given by Gabriel Bodard.[36] The keynote presentation for the second Berlin series (2013-2014) was given by this author[37] and the 2016-2017 series is (at the time of writing) in progress;[38] they too are now established and will no doubt grow and continue to thrive. This is indeed an honour and a reflection of the regard in which the DC is held by members (as the Berlin organisers are also DC members) of the international community. In addition, 2012 saw the launch of e-humanities.net at the University of Leipzig and their eHumanities seminars, again modelled on the DC seminar format, where it was a great honour to be invited to give the inaugural talk which is the subject of this paper.[39]
Writing up this talk for publication has allowed much reflection on the past and on the organic development of the DC as a community; it has allowed an evaluation of what is meant by a community and how that might be understood and sustained. Looking to the future it is pleasing to see that the DC has provided inspiration for the development of new seminar series both at Berlin and Leipzig. This is perhaps how we might measure a community and recognise one as such by the extent to which it is regarded by others.
[1] This paper is based on my talk given to open the Leipzig eHumanities Seminar series and is presented here with many thanks to the organisers for their kind and generous invitation. Thanks also to the DHQ reviewers whose valuable comments have helped me to strengthen this paper.
[1] The Digital Medievalist website and online journal: http://www.digitalmedievalist.org
[2] The Stoa Consortium blog, "Serving news, projects, and links for digital classicists everywhere." http://www.stoa.org See also: In memoriam Ross Scaife (1960-2008) http://www.stoa.org/archives/786. Quatenus nobis denegatur diu vivere, relinquamus aliquid, quo nos vixisse testemur (Pliny EP.3.7.14)
[3] DRH Lancaster 2005 http://projects.oucs.ox.ac.uk/DRHA/2005/drh2005-programme.pdf
[4] Digital Classicist webpage http://www.digitalclassicist.org and wiki http://wiki.digitalclassicist.org
[5] For examples of the breadth and diversity of the field of digital classics see: Crane, G. and Terras, M. eds [Crane and Terras 2009] and Babeu [Babeu 2011].
[6] The only surviving webpage with details of this event is at the Digital Medievalist https://digitalmedievalist.wordpress.com/2007/03/31/clip-2006/
[7] This proto-seminar series was entirely the idea (and arguably the first foundation of the DC) of Gabriel Bodard who solicited this author's assistance simply to help with the logistics and to deal with room bookings.
[8] The most important support throughout has been from the ICS who have generously provided the rooms, equipment and general expenses. Over the years, additional support has come from the Centre for Computing in the Humanities (now the Department of Digital Humanities (DDH)) and the Centre for e-Research (CeRch) both at King's College London. More recent years have seen the co-sponsorship of video recording of the seminars by the UCL Centre for Digital Humanities (UCLDH), DDH and the Department of Classics, Kings College London.
[9] University of Birmingham Programme CA2007: http://www.classicalassociation.org/pastconferences/Birmingham 2007.pdf; DC Wiki CA2007 abstracts http://wiki.digitalclassicist.org/CA2007_abstracts
[10] The Stoa Consortium blog: http://www.stoa.org/?p=889
[11] 2010 Classical Association Annual Conference programme http://www.classicalassociation.org/pastconferences/Cardiff%202010.pdf
[12] 2014 Classical Association Annual Conference programme http://www.nottingham.ac.uk/classics/documents/classical-association/ca-conf-programme.pdf
[13] 2011 Classical Association Annual Conference https://www.dur.ac.uk/classics/events/ca_conference2011 and the DC Training Day programme http://www.dur.ac.uk/resources/classics/DigitalClassicistWorkshop.doc
[14] For a comprehensive list of DC presentations and the wide range of conferences see the Stoa and search the category "conferences": http://www.stoa.org/archives/category/events/event-conferences
[15] Digital Classicist mailing list http://wiki.digitalclassicist.org/Discussion
[16] The Stoa Consortium: "Serving news, projects, and links for digital classicists everywhere". http://www.stoa.org
[17] For an in-depth discussion of the DC Wiki and its use as an example of open collaboration see [Mahony 2011].
[18] The wiki is open and so anyone can view and download the material but only members with editing rights are able to upload and edit or otherwise make changes to the content. See the members list at: http://wiki.digitalclassicist.org/Members
[19] For a wider discussion and further bibliography regarding wikis in the context of building and sustaining online communities see [Mahony 2011].
[20] More precisely this is a Creative Commons Attribution 3.0 Unported License (CC BY 3.0) http://creativecommons.org/licenses/by/3.0/, which allows for sharing and adaptation to help redistribution and building upon the material.
[21] A full archive of all slides, audio and video recordings is maintained on the DC website seminar page http://www.digitalclassicist.org/wip. This has now been supplemented with the addition of the Digital Classicist London Seminars YouTube channel https://www.youtube.com/channel/UCIamtu1Z62wL5XRk2mE8HKw and SlideShare collection http://www.slideshare.net/DigitalClassicistLondon
[22] With thanks and acknowledgement for the videos to: the Department of Digital Humanities at King's College London and the UCL Centre for Digital Humanities. All the seminar listings from 2006 to the present along with slides and recordings (where available) are on the DC seminar webpage: http://www.digitalclassicist.org/wip
[23] Index Thomisticus : http://www.corpusthomisticum.org/it/index.age;jsessionid=6C24FC710B5C992C12A04D5FC2080486
[24] Thesaurus Linguae Graecae : http://www.tlg.uci.edu
[25] See for example: [Rovai 2002].
[26] EpiDoc Sourceforge page http://epidoc.sourceforge.net
[27] UCLDH Management Team: http://www.ucl.ac.uk/dh/people
[28] CentreNet: "an international network of digital humanities centers": http://dhcenternet.org
[29] As evidenced by our Administrators, Partner Institutions, and full DC Community listings http://wiki.digitalclassicist.org/Members
[30] See the seminar listings: http://www.digitalclassicist.org/wip. Full statistics on seminar attendance and website views and downloads from the DC site are collected on behalf of the ICS to allow impact to be measured. These are not currently available although the views and downloads for the DC YouTube channel and SlideShare account (see n.24) are public. Combining these for a longitudinal study of membership and impact is planned.
[31] Examples of DC papers and panels at Classics conferences are given above and for examples of DC at DH conferences, see the papers on Perseids (Beaulieu et al.), Natural Language Processing (Buchler et al.), Orosius' Histories (Franzini et al.), Recogito 2 (Isaksen et al.) at DH2017 (https://dh2017.adho.org/program/abstracts/); the panel on Linked Ancient World Data (Bodard et al.) as well as papers on EpiDoc (Cummings et al.), Latin dictionary (Litta et al.) (http://dh2016.adho.org/abstracts/).
[32] The Digital Classicist seminar page listing seminars, events and peer-reviewed volumes. http://www.digitalclassicist.org/wip/
[33] The name is derived from the more common 'book sprint' and used as a means of collaboratively working towards publication within a short period of time. Digital Classicist Wiki: Wiki editing sprints https://wiki.digitalclassicist.org/Wiki_editing#Wiki_editing_sprints
[34] Dar al-Kutub: Collection of the Egyptian National Library https://wiki.digitalclassicist.org/Dar_al-Kutub:_Collection_of_the_Egyptian_National_Library
[35] Digital Classicist Germany http://de.digitalclassicist.org/about.html
[36] Digital Classicist Berlin 2012-2013 seminar programme http://de.digitalclassicist.org/berlin/seminar2012 (accessed 24/01/2015)
[39] 2012 Leipzig eHumanities Seminar Schedule https://eadh.org/news/2012/05/02/leipzig-ehumanities-seminar. Again, many thanks to Marco Büchler and the seminar organisers for the kind and generous invitation.
Almas and Beaulieu 2016 Almas B. and Beaulieu M-C. (2016). "The Perseids Platform: Scholarship for all!" in Bodard, G. and Romanello, M. (eds) 171-186
Babeu 2011 Babeu, A (2011) ""Rome Wasn't Digitized in a Day": Building a Cyberinfrastructure for Digital Classicists", Council on Library and Information Resources . http://www.clir.org/pubs/reports/reports/pub150/pub150.pdf
Bodard and Mahony 2008 Bodard, G. and Mahony, S. (eds) (2008) ""Though much is taken, much abides": Recovering antiquity through innovative digital methodologies", Digital Medievalist 4. https://journal.digitalmedievalist.org/4/volume/4/issue/0/
Bodard and Mahony 2010 Bodard, G. and Mahony, S. (eds) (2010) Digital Research in the Study of Classical Antiquity , Ashgate.
Bodard and Romanello 2016 Bodard G. and Romanello M. (eds) (2016). Digital Classics Outside the Echo-Chamber: Teaching, Knowledge Exchange & Public Engagement . Ubiquity Press. https://www.ubiquitypress.com/site/books/10.5334/bat
Bodard et al. 2016 Bodard, G., Broux, Y. and Tarte, S. (eds) (2016) "Digital Approaches and the Ancient World", Bulletin of the Institute of Classical Studies BICS 59(2), Institute of Classical Studies.
Cohen 1989 Cohen, A. (1989), The Symbolic Construction of Community , Routledge.
Crane and Terras 2009 Crane, G. and Terras, M. (eds) (2009) "Changing the Centre of Gravity: Transforming Classical Studies Through Cyberinfrastructure", Digital Humanities Quarterly 3.1. http://digitalhumanities.org/dhq/vol/3/1/
Dee et al. 2016 Dee, S., Foradi, M., and Šarić, F. (2016) "Learning By Doing: Learning to Implement the TEI Guidelines Through Digital Classics Publication", in Bodard, G. and Romanello, M. (eds) 15–32.
Dunn and Mahony 2013 Dunn, S. and Mahony, S. eds, (2013) The Digital Classicist 2013, BICS Supplement 122, Institute of Classical Studies.
Kenyon 2000 Kenyon, E. (2000), "Time, Temporality and the Dynamics of Community" Time Society 9.1.
Mahony 2007 Mahony, S. (2007) "Using digital resources in building and sustaining learning communities", Body, Space & Technology Journal , Vol 07/02. http://people.brunel.ac.uk/bst/vol0702/simonmahony/
Mahony 2011 Mahony, S. (2011), "Research communities and open collaboration: the example of the Digital Classicist wiki", Digital Medievalist 6. https://journal.digitalmedievalist.org/articles/10.16995/dm.26/
Mahony and Bodard 2010 Mahony, S. and Bodard, G. (2010) "Introduction" in Bodard, G. and Mahony, S. (eds) Digital research in the study of Classical Antiquity , Ashgate.
Mauss 1990 Mauss, M. (1990), The Gift: Forms and Functions of Exchange in Archaic Societies . Routledge.
Millan and Chavis 1986 Millan, D and Chavis, D. (1986), "Sense of Community: A Definition and Theory", Journal of Community Psychology 14, 1-23.
Orlandi 2016 Orlandi, S. (2016), "Ancient Inscriptions between Citizens and Scholars: The Double Soul of the EAGLE Project", in Bodard, G. and Romanello, M. (eds) 205-221.
Rovai 2002 Rovai, A. (2002), "Building Sense of Community at a Distance", The International Review of Research in Open and Distance Leaning, 3,1.
Vitale 2016 Vitale, V. (2016) "Transparent, Multivocal, Cross-disciplinary: The Use of Linked Open Data and a Community-developed RDF Ontology to Document and Enrich 3D Visualisation for Cultural Heritage", in Bodard, G. and Romanello, M. (eds) 147–168
2017 11.3 | XML | Print Article
URL: http://www.digitalhumanities.org/dhq/vol/11/3/000335/000335.html
Comments: dhqinfo@digitalhumanities.org
Published by: The Alliance of Digital Humanities Organizations
Affiliated with: Literary and Linguistic Computing
Copyright 2005 - | {
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Red Carpet is one famous Champagne bar located at Jl. Oberoi / Kayu Aya, Seminyak. On June 18, they held a big party and chose us to document every moment at that time. Yeah, we've got very nice shoots and apparently the owner really love the photos. | {
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Q: Email said to be null is not with Rails 3.2 and Devise Using Rails 3.2 and Devise, I have overridden the Registrations Controller with a custom one. I did not change the code in the create method at all, it is the original one in Devise.
But strangely enough, I keep getting this error whenever I'm trying to create/sign up a new User.
PG::Error: ERROR: null value in column "email" violates not-null constraint
: INSERT INTO "users" ("created_at", "current_sign_in_at", "current_sign_in_ip", "email", "last_sign_in_at", "last_sign_in_ip", "payroll", "sign_in_count", "updated_at", "username") VALUES ($1, $2, $3, $4, $5, $6, $7, $8, $9, $10) RETURNING "id"
But below it, everything seems to be ok as the User is getting a valid email.
{"utf8"=>"✓",
"authenticity_token"=>"8/ZKW11lF22WYKV2K36zBkSk6DSU36/1/zU54a2IRmM=",
"user"=>{"username"=>"someguy",
"email"=>"email@yahoo.com",
"password"=>"[FILTERED]",
"password_confirmation"=>"[FILTERED]"},
"commit"=>"Sign up",
"format"=>"user"}
I am not quite sure what other code to paste in here as everything is obviously right for me. So please ask for anything which might help.
This is my Registrations controller, but create basically has the same code its super. The error occurs at the line which has resource.save.
class UserRegistrationsController < Devise::RegistrationsController
def new
super
end
def create
build_resource
if resource.save
if resource.active_for_authentication?
set_flash_message :notice, :signed_up if is_navigational_format?
sign_up(resource_name, resource)
respond_with resource, :location => after_sign_up_path_for(resource)
else
set_flash_message :notice, :"signed_up_but_#{resource.inactive_message}" if is_navigational_format?
expire_session_data_after_sign_in!
respond_with resource, :location => after_inactive_sign_up_path_for(resource)
end
else
clean_up_passwords resource
respond_with resource
end
end
def update
super
end
end
Apparently, all of the fields are nil. This is the query that it executes:
INSERT INTO "users" ("created_at", "current_sign_in_at", "current_sign_in_ip", "email", "first_name", "last_name", "last_sign_in_at", "last_sign_in_ip", "payroll", "sign_in_count", "updated_at", "username") VALUES ($1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12) RETURNING "id" [["created_at", Wed, 17 Apr 2013 19:34:22 BST +01:00], ["current_sign_in_at", nil], ["current_sign_in_ip", nil], ["email", nil], ["first_name", nil], ["last_name", nil], ["last_sign_in_at", nil], ["last_sign_in_ip", nil], ["payroll", nil], ["sign_in_count", 0], ["updated_at", Wed, 17 Apr 2013 19:34:22 BST +01:00], ["username", ""]]
A: Is email a mass-assignable on the User class? Try declaring attr_accessible :email in the User class.
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package org.gradle.internal.operations;
import javax.annotation.Nullable;
/**
* Runs build operations: the pieces of work that make up a build.
* Build operations can be nested inside other build operations.
*/
public interface BuildOperationRunner {
/**
* Runs the given build operation.
*
* <p>Rethrows any exception thrown by the action.
* Runtime exceptions are rethrown as is.
* Checked exceptions are wrapped in {@link BuildOperationInvocationException}.</p>
*/
void run(RunnableBuildOperation buildOperation);
/**
* Calls the given build operation, returns the result.
*
* <p>Rethrows any exception thrown by the action.
* Runtime exceptions are rethrown as is.
* Checked exceptions are wrapped in {@link BuildOperationInvocationException}.</p>
*/
<T> T call(CallableBuildOperation<T> buildOperation);
/**
* Starts an operation that can be finished later.
*
* When a parent operation is finished any unfinished child operations will be failed.
*/
BuildOperationContext start(BuildOperationDescriptor.Builder descriptor);
/**
* Executes the given build operation with the given worker, returns the result.
*/
<O extends BuildOperation> void execute(O buildOperation, BuildOperationWorker<O> worker, @Nullable BuildOperationState defaultParent);
}
| {
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package com.hazelcast.config;
import com.hazelcast.quorum.QuorumListener;
/**
* Confiquration class for {@link QuorumListener}
*/
public class QuorumListenerConfig extends ListenerConfig {
public QuorumListenerConfig() {
}
public QuorumListenerConfig(String className) {
super(className);
}
public QuorumListenerConfig(QuorumListener implementation) {
super(implementation);
}
@Override
public QuorumListener getImplementation() {
return (QuorumListener) implementation;
}
public ListenerConfig setImplementation(QuorumListener implementation) {
return super.setImplementation(implementation);
}
@Override
public boolean isIncludeValue() {
return false;
}
@Override
public boolean isLocal() {
return true;
}
@Override
public int getId() {
return ConfigDataSerializerHook.QUORUM_LISTENER_CONFIG;
}
}
| {
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Table of Contents
Title Page
Copyright Page
Dedication
Chapter 1 - We meet the Enthusiast ~ True Love beckons ~ a Mysterious Stranger ...
Chapter 2 - I seek Counsel ~ a Domestic scene ~ Dancing Lessons.
Chapter 3 - Ball Gowns ~ Footwear ~ Barns ~ A Masked Man.
Chapter 4 - Tenth Grade ~ Extracurriculars ~ A Sonnet.
Chapter 5 - A ride through the Dark ~ A menacing adder ~ A gallant rescue ~ A ...
Chapter 6 - More adders ~ Ginger ale ~ untimely Flushing ~ we dance the Sir ...
Chapter 7 - An unglass slipper ~ A Farewell to Forefield ~ I eat the Pancakes ...
Chapter 8 - I Renounce my Dream ~ I maintain my Dignity ~ I carry boxes ~ I E-mail.
Chapter 9 - Rumors of rivals ~ I withdraw ~ I join up ~ a Surprising ...
Chapter 10 - Et tu, Samantha? ~ An Encounter with a Pirate ~ We prepare ...
Chapter 11 - Parts ~ scripts ~ rhymes ~ songs ~ an igsome Moth ~ an Artistic ...
Chapter 12 - I keep up my grades ~ My father grouses ~ A Turkey again ~ Rehearsals.
Chapter 13 - My mother gives up ~ Thanksgiving ~ yet another Turkey ~ an ...
Chapter 14 - Musings about the Inscrutable Gender ~ A Date ~ Ashleigh to the ...
Chapter 15 - Holiday cheer ~ The baby's birthday ~ Sweet Sixteen and Never ...
Chapter 16 - Paperwhites ~ Hothouse flowers ~ The Great White Way ~ Parr's ...
Chapter 17 - A Limited Junior License ~ A disastrous Mocharetto ~ Mint Sauce ~ ...
Chapter 18 - My first appearance in Print ~ Ashleigh interferes ~ A Midnight ...
Chapter 19 - A song ~ an Unspeakable Scandal ~ my Mother takes a new Job ~ the ...
Chapter 20 - My Fifth Kiss ~ Mom to the rescue again ~ Midwinter Insomnia ~ ...
Chapter 21 - A Nonstatic Screen Wipe ~ Ashleigh's new Craze.
Chapter 22 - The B-word ~ Seth vanquished ~ a Ring ~ my Sixth Kiss ~ an Acrostic.
Chapter 23 - Bliss ~ Farewell.
Acknowledgements
Enthusiasm
_In which Ashleigh discovers Jane Austen,
and her wardrobe, to the embarrassment of Julie._
"Listen, Ash," I said. "You're not planning to go to school wearing that, are you? No guy will even _look_ at you." Me neither if they see me with you, I added inwardly. "Couldn't you please, please, please wear jeans?"
As always, my plea fell on deaf ears.
"I see not the necessity of discussing with _you_ , Miss Lefkowitz, the propriety of a young lady wearing Trousers. As you know, modesty forbids us to reveal the shape of the Lower Limbs."
"If you do get a boyfriend, he's going to want to see a lot more than just the shape of your Lower Limbs," I argued.
Fortunately, I reflected, the school year wouldn't start for another week—enough time, I hoped, to make her see reason.
OTHER BOOKS YOU MAY ENJOY
SPEAK
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First published in the United States of America by G. P. Putnam's Sons, a division of Penguin Young Readers Group, 2006
Published by Speak, an imprint of Penguin Group (USA) Inc., 2007
Copyright © Polly Shulman, 2006 All rights reserved
THE LIBRARY OF CONGRESS HAS CATALOGED THE G. P. PUTNAM'S SONS EDITION AS FOLLOWS:
Shulman, Polly. Enthusiasm / Polly Shulman. p. cm.
Summary: Julie and Ashleigh, high school sophomores and Jane Austen fans, seem to fall for the same Mr. Darcy-like boy and struggle to hide their true feelings from one another while rehearsing for a school musical.
eISBN : 978-1-101-11882-5
The publisher does not have any control over and does not assume any responsibility for author or third-party Web sites or their content.
<http://us.penguingroup.com>
_For Anna Christina and Andrew_
**_Chapter 1_**
_We meet the Enthusiast_ ~ _True Love beckons_ ~ _a Mysterious Stranger_ ~ _Ashleigh's Plan._
_**T**_ here is little more likely to exasperate a person of sense than finding herself tied by affection and habit to an Enthusiast. I speak from bitter experience. My best friend and next-door neighbor, Ashleigh Marie Rossi, is an Enthusiast.
All last summer, Ashleigh was mad for the Wet Blankets. On the day they released their new album, she insisted that I accompany her to Outer Music, where they had advertised free tickets for a Blankets concert in the city. We started at ten o'clock in the morning, the break of dawn, Ashleigh time. "Ash," I objected, "they said they won't give out tickets till midnight. What are we going to do for fourteen hours?"
"You don't want to be stuck at the end of the line, do you? Don't worry, I packed lunch. Here, take one of these," she said, hauling a large woolen blanket out of her closet and dumping it in my arms.
"What's this for? It's about a million degrees out there. We'll be sitting in the sun."
"That's why we're bringing these!" She flourished two five-liter bottles of mineral water. "Wet Blankets, get it? We'll be appropriately dressed, and they'll keep us cool through the process of evaporation." She opened one of the bottles and reached out to splash me with the water.
"Ash, you freak, get away! Stop it! I'm not sitting around in the middle of town with my clothes soaked!"
With difficulty, I persuaded her to recap the water bottle, but nothing would convince her to leave the blankets behind. At Outer Music, we spread them on the sidewalk and sat down to wait. People looked at us strangely as they went in and out of the store, and I hid my face in my book, Jane Austen's _Pride and Prejudice_. At six o'clock, just as I reached the exciting proposal scene, Ashleigh's dad arrived with sandwiches. At nine, other Wet Blankets fans began lining up behind us.
Ashleigh's blankets came in handy after all, when the skies opened in a cloudburst around eleven. Overjoyed to find that Fate had cooperated with her planned pun, she invited her co-fans to seek shelter with us under the blankets.
This sort of behavior was nothing new for Ashleigh. All through elementary school, her crazes kept me in a constant flame of embarrassment. After she read the Little House books, it was all I could do to stop her from wearing her flowered flannel nightgown to the mall. During her Harriet-the-Spy period, which coincided with my parents' breakup when we were eleven, I had to confiscate two of her notebooks to keep them from falling into the hands of the divorce lawyers. When King Arthur ruled supreme in her heart, she thee'd and thou'd everyone from teachers to bus drivers.
But although our classmates considered Ashleigh weird back then, they respected her for her courage and inventiveness. Nothing ever embarrassed Ashleigh. Teasing her was pointless, since nobody could make her cry. Some of her reputation for oddness rubbed off on me, but so did some of her oddball prestige. Hanging out with Ashleigh in elementary school stopped just short of social suicide.
High school, though, was another matter. By then her ability to ignore giggles and stares had become less an asset than a liability. Oh, we still had plenty of friends—girls like Emily Mehan and the Gerard twins—but if Ash pulled any more stunts like that time freshman year when she borrowed Michelle Jeffries's handbag for a juggling trick and spilled the contents, including a selection of feminine hygiene products, I feared for our social standing among the girls. And as for guys—well, _that_ was too painful to bear thinking about.
One hot afternoon about a month after the Wet Blankets incident, I sat by my window peeling my sunburn and considering the coming school year. Although Ashleigh did tend to get carried away, the Wet Blankets was a perfectly respectable interest for a Byzantium High sophomore. If only it would last through the vital first few weeks of school! Could I possibly be so lucky?
Evidently not. A rap on my windowpane interrupted me in the middle of removing a satisfyingly large patch of skin. Looking up, I saw the Enthusiast herself perched outside my window. (For reasons of convenience and privacy, Ashleigh and I exchange visits by way of the oak tree whose branches graze both our bedroom windows, rather than by the doors.) She was wearing a long black garment that caught on the twigs; I recognized it as a robe from last year's Freshman Chorus.
"Miss Lefkowitz! Miss Lefkowitz! My dear Miss Lefkowitz," she called.
I hauled the window open wide. "What's all the 'Miss' stuff?" I said. "You're not starting on an etiquette craze, are you?"
Ashleigh shot me her second-favorite expression, Reproach Tinged with Disgust. (Her favorite is the Mad Gleam.) "Etiquette?" she cried. "I hope I always conduct myself as befits a young lady. But my dear Miss Lefkowitz, why did you wait so long before introducing me to the joys of Miss Austen's work? Elizabeth Bennet! Jane Bennet! The incomparable Mr. Darcy!" She waved my copy of _Pride and Prejudice_ at me, dislodging baby acorns and a leaf or two.
My heart sank. How many weeks of antiquated grammar were we in for now? And it was my own fault too. While Ashleigh bounced around the room, knocking things over with her skirts and raving about Austen's heroines and the gentlemen they loved, I considered my situation. Always before, Ashleigh had started a craze, and I had followed. Now, for the first time, I had taken the lead, introducing her to an interest of my own. But how long would it be before her passion overshadowed mine? Would she take over my favorite books, leaving nothing for me? I was convinced that I felt as strongly about Jane Austen's books as Ashleigh had ever felt about any of her crazes, but my love was deep and silent—and therefore easily overshadowed. I would never, for example, speak Jane Austen's language. That would be undignified and unworthy of the writer I adored.
Rescuing my clock radio, which had tumbled off my night-stand and was hanging by its cord, I told myself sternly not to be so ungenerous. Ashleigh never hesitated to share _her_ interests with _me_. If only! No, she always insisted on dragging me in, however boring or unpleasant I might find them. (Military strategy? Ballet? Ig, no thanks!—Although I did rather enjoy candy making and reptiles.) The only time Ash let me wiggle out of a craze was when she knew I couldn't afford it—and when that happened, she gave it up herself, generous girl that she was. She squelched a growing passion for horses, for example, because my mother couldn't pay for riding lessons after my parents' divorce. And Ashleigh's generosity didn't stop there. Whenever her crazes got me in trouble—like the time I ruined my father's barbecue tools digging military trenches in the lawn—she devoted her savings and countless Saturdays to repairing the damage.
As I contemplated my pettiness, Ashleigh startled me with an emphatic bounce. (She's always bouncing with excitement, and when she bounces, she _bounces_ —particularly in the past year or so. For my part, I barely jiggle, no matter how vigorously I move.)
"And I believe I know where to find them!" she cried.
"Where to find what?" I asked.
Ashleigh gave me her you're-not-listening look, a variant on the ever popular Disgusted Reproach.
"Not what—who. Our heroes. What good is a heroine without a hero? From what I remember of freshman year, we will be hard-pressed to find even a single gallant at Byzantium High. I despair of finding a pair of them there! But fortunately, I have discovered the answer."
Clearly Ashleigh had finished the research portion of her fad and moved on to the active stage. Now that she had decided to enact a 200-year-old love story with us as the heroines, I was afraid the results would be mortifying.
Without much hope, I tried to head her off. "I thought you despised boy-crazy girls like Michelle Jeffries and those people. You always said crushes were for noodleheads."
Ashleigh drew herself up to her full height, which I couldn't have done in her position—standing on my bed—since my head would have hit the sloping roof; her figure may be more mature than mine, but she's six inches shorter.
"I speak not of crushes, Miss Lefkowitz," she replied, "but of True Love."
True Love! What girl hasn't dreamed of _that_? Even the shyest among us longs for a soul mate—someone who will understand our hopes and fears, laugh at our jokes, offer us his coat when the afternoon turns cold, charm our parents, and admire us, flaws and all (such as a sharp chin, perhaps, and a marked lack of jiggle).
Although I had never discussed it with anyone, not even Ashleigh, I shared that dream. My ideal hero borrowed his appearance from a guy I thought of as the Mysterious Stranger. I had seen him just five times. The first was by the swimming hole on a windy Saturday in late spring. A woman's hat blew off her head and flew straight for the water, when the stranger snatched a fallen branch from the ground and, with a daring leap, caught it. I had seen him twice since then in the state park, on foot and on horseback. Once I glimpsed him through the window of the Java Jail drinking what looked like a Magna Mocharetto with a bevy of guys. And once we crossed paths as he left the public library, trailing a cloud of air-conditioned calm. I was on my way in; he held the door for me.
The man I might someday come to admire would, I hoped, share this stranger's poise, his grace, and his deep vertical dimple.
With such secret thoughts, I shouldn't be surprised to hear my friend talk of Heroes. Yet if Ashleigh cherished a similar dream, I feared for her peace of mind. For is True Love likely to come to a high school sophomore who dresses in a chorus robe and ballet slippers?
"Okay, but listen, Ash," I said. "You're not planning to go to school wearing that, are you? No guy will even _look_ at you." Me neither if they see me with you, I added inwardly. "Couldn't you please, please, please wear jeans?"
As always, my plea fell on deaf ears. "I see not the necessity of discussing with _you_ , Miss Lefkowitz, the propriety of a young lady wearing Trousers. As you know, modesty forbids us to reveal the shape of the Lower Limbs."
If you do get a boyfriend, he's going to want to see a lot more than just the shape of your Lower Limbs, I argued silently. Fortunately, I reflected, the school year wouldn't start for another week—enough time, I hoped, to make her see reason.
"And don't you think you could call me Julie?" I continued. "We've known each other long enough, surely."
"My dearest Julia, you are right, indeed you are right. After all, in _Pride and Prejudice_ Miss Elizabeth Bennet addresses her bosom friend, Miss Lucas, by the name of Charlotte, and they are no more affectionately attached than the two of us. But please, my dear friend, allow me to continue. As I said, I believe I have the solution to our puzzle of where to find our heroes."
" _Our_ puzzle? It's not _my_ puzzle," I put in.
Ashleigh shook me by the arm, letting her language slip a bit in her impatience. "Will you listen already? In _Pride and Prejudice_ , where do the younger Bennet girls turn for lively masculine company? Why, to the regiment of soldiers quartered near their home. Were we to follow their lead, where better to seek suitors than among our neighboring young men in uniform?"
Could she be referring to the West Point cadets? The U.S. Military Academy at West Point sits high on a cliff overlooking the Hudson, hidden from Byzantium by the curve of the river. There brave and disciplined students train to lead our country's great army. Last year the center of the Byzantium Bullfrogs turned down Harvard to become a West Point cadet.
"Oh, Ashleigh, you've got to be kidding! You want us to go chasing after West Pointers? They're way too old! They've got crew cuts! You'll get us court-martialed!"
My friend held up her hand. "Hear me out, Julia," she said. "Hear me out. As you so rightly observe, the officers in training are not perfectly suited to ladies of our tender years. I propose instead another population of gentlemen in uniform—gentlemen younger than the cadets—I speak, in short, of the students at the Forefield Academy."
This suggestion was better, but only slightly. Forefield, an exclusive boys' prep school, rises above the town of Byzantium both geographically and socially. Its main building, once the mansion of the Forefield family, can be seen from most of the town, including my attic window. As a little girl I thought it was an enchanted castle, the home of a witch or a princess. I now considered it the home of gawky boys with crests embroidered on their blazer pockets—that is, of snobs, dorks, adders, or (most likely) snobbish, dorky adders.
"Forefield, huh? What's your plan? Are we going to dress up as boys and sneak in? Watch out—they'll see our lower limbs."
Flashing me a look of reproach and triumph, Ashleigh reached into her robe pocket and produced a piece of paper, which she silently handed to me. It appeared to be a page Xeroxed from a newspaper.
" 'Library Renovations,' " I read. " 'An extensive overhaul of Forefield's Robert Rive Science Library and Media Resource Center is on schedule for completion in time for the—' "
"No, not that. Underneath."
"What, this announcement? 'Forefield Fall Formal. The president and faculty of the Forefield Academy look forward to welcoming students, alumni, and their guests to the 97th annual Columbus Cotillion, at 8:00 P.M., Saturday, October 12. Formal attire.' Well, what good is that? We're not Forefield students, we're not alumni, and we're not their guests. We're not invited."
"Oh, that won't matter." Catching herself slipping into ordinary speech, Ashleigh began again. "I mean, That will be of no importance. With the crush of guests, two more will surely pass unnoticed."
"You want to crash the Snoot School Dork Dance? Are you out of your candy wrapper? What could that possibly have to do with Jane Austen?"
"Surely, Miss Lefkowitz, you can see that a gathering of young gentlemen dressed in formal attire, well practiced in time-honored dance steps, and unaccustomed to the company of young ladies—and therefore bound to treat us with modesty and respect—is the ideal place to meet our matches. Can you be blind to the perfection of the plan?"
Perfection! If the plan had any, I certainly was blind to it. In my experience, at least, boys who hadn't spent a lot of time around girls were less likely, not more, to behave themselves.
The sound of a maternal voice came faintly up the stairs.
"Is that Mrs. Lefkowitz calling?" said Ashleigh.
"Look—whatever you call me, you're _not_ calling my mother Mrs. Lefkowitz. She didn't like it even when she was married to my father. If Helen isn't formal enough for you, call her Ms. Gould."
"I shall call her Madam."
Before I could make any further objections, the person in question knocked on the door.
"Come in," I called.
"Honey, I—oh, hi there, Ashleigh, you're here too. I was wondering. I thought I heard bouncing, but I didn't see you come in."
It astonishes Ashleigh and me that our parents have failed for so many years to notice how we use our tree.
"Good afternoon, Madam," answered Ashleigh.
"Excuseth me, Sir Ashleigh. I bid thee, too, the fairest of afternoons. Evenings, actually. Killed any dragons today?"
Ashleigh appeared too pained to reply. I took up her cause. "Mom, you're years out of date."
"Oho, years out of date, ameth I?" she said agreeably. "How canst thou tell—the grammar? I pray thee, forgive thine old, antiquated mommeth. Keeping up with the latest in teendom beeth too difficult for me. Dinnertime, honey. Ashleigh, you're welcome to join us."
"I thank you, Madam, but I knew not how far the day had advanced. My parents await me. Farewell." Curtsying to my mother, the Enthusiast tripped lightly down the stairs and took her leave.
**_Chapter 2_**
_I seek Counsel_ ~ _a Domestic scene_ ~ _Dancing Lessons._
_**T**_ he next day was Tuesday, known in the Lefkowitz and Gould households as the Day of the Dad. I was glad. Not from any eagerness to spend time with my father, of course; relations between us have been strained ever since he left my mother and me for the Irresistible Accountant (or "Amy," as he prefers me to call her). Rather, I needed the advice of his next-door neighbor, the savviest person I know: Samantha Liu.
A visit to Dad and "Amy" often includes Samantha. Our fathers, both pediatricians, share a practice and a backyard hedge; Samantha's mother, an allergist, also shares the hedge, of course, but she has a separate medical practice. The long bicycle ride over gave me plenty of time to consider Ashleigh's wild plan. Though my father and stepmother didn't expect me until dinnertime, I started early, hoping to find Samantha free for a chat. Some subjects are best discussed in person—particularly those subjects that live a mere tree's breadth away from me. From time to time, Ashleigh and I overhear each other's phone conversations.
I was in luck: Samantha was home. Once we had installed ourselves comfortably in the Lius' hammock with a pair of ginger ales, I opened my heart.
"Okay, Sam, warm up the advice generator," I entreated.
"What's up? Stepmother trouble again?"
"No—at least, not yet. I haven't seen 'Amy' for a week. Right now it's Ashleigh."
At that name, Samantha gave an affectionate, twinkly grimace. It always surprises me that the two of them like each other as much as they do. Samantha is more than a year older and at least a millennium more mature. Ashleigh's reputation for eccentricity prevents her from rising to the upper circles of our high school world, even if she wanted to. Samantha, on the other hand, enjoys the status of a gymnast, a beauty, last year's president of the sophomore class, and the sister of the famously hot Zach Liu, still an object of near-universal fantasy even though he graduated last year. In fact, it's a measure of Samantha's social standing that she can afford to be gracious to someone as odd as Ashleigh (or, I often think, me). Yet Sam is loyal and, though skilled at manipulation, essentially kind.
"What is it this time?" she asked. "Let me guess: Ashleigh's taken over your bathtub for her starfish collection? She's excavating an emerald mine under your basement and you're afraid your house will collapse? No, wait, I know—she's decided to go to school every day dressed as Martha Washington."
With such intuition, is it any wonder Sam is so successful?
"You got the first problem right on the nose," I said. "Well, the left nostril. Not Martha Washington, Jane Austen—close enough, though. Jane Austen doesn't involve a white wig, which makes her a bit better, I guess. Ash is refusing to wear anything but a long skirt. No jeans, no pants—she doesn't want her 'lower limbs' to show."
"Oh, dear. I take it you've tried reasoning with her? Told her no one will sit next to her in homeroom and so on?"
"When did Ashleigh ever listen to reason? Besides, she knows _I'll_ sit next to her."
"Well, you might not be in her homeroom this year, but I see your point. And begging didn't work either, right?"
"Of course not."
"Could you move her on to a new fad—get her interested in rooting for the football team or something?"
I paused to smile at the thought of Ashleigh rooting for the football team. "This _is_ a new fad, unfortunately. It started yesterday. It replaces the Wet Blankets."
"Then you'll just have to bargain with her. Refuse to do something she wants unless she agrees to wear normal clothes. In the meantime, maybe you can find some sort of in-between outfit that would work for either era. A long black skirt or an Indian print, something like that. That way, even though she might not look crisp, at least she won't look insane. Tell you what, if you don't find an effective threat by the weekend, let me know and I'll get a few of the girls on the gymnastics team to come to school in long skirts on Monday, so Ashleigh won't be the only one."
A generous offer! No one, however fond of gossip, could blame a sophomore for dressing like the gymnasts, the most successful athletes at Byz High. Last year the girls' gymnastics team placed first in twice as many meets as the football and basketball teams combined won games. They were the pride of the school and the leaders of fashion.
"But I didn't tell you the worst part yet," I continued. "Ashleigh's planning to crash a dance at the Forefield Academy, and she wants me to come with her."
Samantha gave a thumbs-up with her ginger-ale can. "There you go! Tell her you'll go only if she gives up the weird wardrobe."
"No way, Sam! You want me to crash a dance at Snoot School? I'll die of embarrassment."
"Face it, Julie, you know you're going to end up going anyway. You can never say no to Ashleigh. You might as well get something out of it this time."
Much as I hated to admit it, she had a point. I finished my ginger ale, thanked her, and made my way to my father's house.
After we finished our dinner of grilled chicken breast marinated in pomegranate juice and spinach-walnut salad with orange vinaigrette—my stepmother is a skillful if fussy cook—Ashleigh called. She was buzzing with news of some sort.
I told her I would discuss it tomorrow and hung up hastily. The Irresistible Accountant disapproves of teenagers talking on the telephone. She especially disapproves of Ashleigh. Having extracted my father from his messy life, she wants to simplify and straighten out the only aspect that he couldn't entirely leave behind: me. In her view, Ashleigh belongs to the world of mess. Amy much prefers Samantha, holding her up as an example of ideal girlhood. Although the feeling isn't mutual, Samantha always advises me to keep the peace with my stepmother. Ash, in contrast, speaks her mind and encourages me to speak mine.
"Was that Ashleigh on the phone, sweetie?" asked Amy. "I wish you would ask that girl not to call during family time. Your father and I only get to see you for a few precious hours a week. I think it's very inconsiderate of her not to respect that."
Yeah, you wouldn't want to waste a precious minute that you could be spending picking at me and criticizing my friends. This is what I did _not_ say to Amy. If I had a dollar for every sharp remark I keep to myself, I would be able to fund the Stepfamily Peace Prize, my dream version of the Nobel, to be awarded annually to the person who displays the greatest familial restraint. I considered it especially unfair that, having voluntarily given up the pleasure of talking to Ashleigh, I should still have to listen to my stepmother's complaints, just as if Ash and I had yapped away on the phone for hours.
"Don't you agree, Steve?" said my stepmother.
"Hmm? Yes, of course," said my father, getting up and heading to his study. He stopped on the way to kiss his wife on the top of her head. "I love to see my two favorite girls together. You two sit here, relax, and catch up."
I waited until the door had swung shut behind him, then took my bag and headed upstairs to "my" room, which I share with the I.A.'s sewing machine. The knowledge that she has an excuse to enter at any moment makes the room feel less than comfortable and far from private. Still, at least I would be alone there.
"Where are you going, sweetie?" she asked.
"Homework."
"How can you have homework when school doesn't start for another week?"
"Summer reading," I lied. (I had already finished the assigned book— _Lord of the Flies_ —back in June.)
"Really? What are they having you read? Let's see," she said, reaching for my book. "Oh, you lucky girl—Jane Austen, _Sense and Sensibility_! I loved that book. She's my favorite writer. The romance between your father and me was straight out of an Austen novel."
Right, like Jane Austen wrote about igsome schemers who steal other people's husbands, I didn't say. I awarded myself an imaginary dollar for refraining, bringing the total for the evening up to two dollars, and escaped upstairs.
When I got home the next morning, I slipped next door in search of news. Finding Ashleigh still asleep, I pounced on her toes.
She gave a most satisfactory squeal. "Juniper, get _off_!" she cried, mistaking me for her kitten.
Mewing and pinching, I attempted to keep her in the dark as to my true identity. Pretty soon, however, she stopped thrashing blindly, opened her eyes, and identified me as a member of the Human Race.
"Oh, it's you. What are you doing up so early? What time is it?"
"Long past a bat's bedtime. Get up, lazybones!"
Ashleigh buried her head under her pillow.
I like morning. It's the only time when _my_ enthusiasm outstrips _hers_.
"Fine, I guess you don't want to tell me your news then," I said, making a feint toward the window. That roused her. She sprang out of bed—or, more accurately, she rose with a speed somewhat greater than that of a daffodil emerging from the moist earth in March. In a mere twenty minutes she had put on a selection of interesting clothes and her Austenesque manner.
"Let us repair to your abode, where there is more room, my dear Miss Lefkowitz—"
"You mean your dear Julie—"
"Quite right, my dearest Julia—Let us repair to your abode, where there is more room."
"More room for what?" I asked, retreating through the window. Ashleigh followed me out. Our rooms are more or less the same size, but in the course of her enthusiasms Ashleigh has accumulated far more stuff than I have. In particular, one large papier-mâché dragon hanging from the ceiling tends to bump you on the head if you try to walk across her room without looking both ways first. Well, it bumps _me_ , anyway.
"Dancing lessons, Twinkle-Toes," she explained, diving through my window and landing on my bed. "If we are to discharge our duties as party guests with the dignity that befits our position as Ladies, we must learn to perform the required steps. I have here a book"—she thumped the pillow with it—"that promises to instruct us in the Art of Terpsichore."
"Whoa there! You want us to learn how to dance by reading a book called _The Art of Sick Twerpery_? Have you lost your lemon drops?"
"Not sick twerpery—Terpsichore. Terpsichore, the Muse of Dance. It is she who breathes life into the Limbs of the Dancers as they perform their graceful movements." She flung her arms out to demonstrate; a series of crashes followed. Luckily, nothing broke.
"Very graceful," I said, putting my desk lamp back on my desk. "Okay, so you want us to learn to dance by reading a book called _The Art of Terpsichore_?"
"No, no, 'the Art of Terpsichore' is merely a description of the book's contents. The volume itself is called _Dancing_. To be precise, _Dancing and Its Relations to Education and Social Life, with a New Method of Instruction, Including a Complete Guide to the Cotillion (German) with 250 Figures_. By Allen Dodworth. Published in New York, 1888. New and Enlarged Edition."
"I don't care what it's called, you can't learn to dance by reading a book."
"Yes, you can. It has 250 figures. See?" She opened the book to show me.
" 'Number Forty-eight. The Inconstants,' " I read. "'Three couples.—They arrange themselves in a phalanx behind the conducting couple; the first gentleman turns round, giving his left arm, crossed at the elbow, to the left arm of the gentleman behind him, with whom he changes places and partners; he goes on without interruption to the last lady; when he reaches the last, the second gentleman, who is then at the head of the phalanx, executes the same figure, and so on for the rest, until every one has regained his place; general waltz follows.' Well, that's as clear as crumb cake."
"Nonsense, it's perfectly simple. The first gentleman goes like this, then the second gentleman goes like that, then he takes the lady's arm, and they go like this, and meanwhile the third gentleman goes like that, and the lady goes like this, and the other couple does the same thing, and then they all dance. Come on, try it, it'll be fun."
I picked up my desk lamp and put it back on my desk again. "That's a dance for three couples," I said. "Maybe we should start with something for one couple." I paged through the book. " 'The Galop, the Racket, the Esmeralda (or Three-Slide Polka), the Minuet, the Lancers, the Quadrille.' You realize, don't you, that nobody dances these things nowadays? We'd be better off learning the fox trot or the twist. Or just figuring out how to wiggle with dignity."
"Fox Trot? Faugh! Elizabeth Bennet and Fitzwilliam Darcy never danced any such thing. That's a Twentieth-Century dance."
"Well, what makes you think they danced the Esmeralda? This book was published in 1888—that's a good seventy-five years after _Pride and Prejudice_."
"Okay, perhaps not the Esmeralda," said Ashleigh, "but they're always talking about the Lancers. Let's learn that."
"Right, the Lancers. Ever popular with the crisp crowd. Kids were trampling each other to get to the dance floor when they started to play it at last year's prom."
Ash shot me her Scornful Look. Taking Mr. Dodworth's volume in one hand, she began reading minuet instructions aloud while prancing up and down the room. For a third time I returned my lamp to its place on the desk. It was a bit dented, but the lightbulb was intact.
My mother put her head in the door. "I take it Ashleigh's here?" she said. "Hello, Ashleigh. Funny, I never hear you come in. Broken anything?"
"Nothing important," I assured her.
"Ah, Ms. Gould! Just whom I most wished to see! Do us the kindness to step over here, Madam," cried Ashleigh, taking my mother by the elbow. "We are short a couple. Would you rather be a Gentleman dancing with an imaginary Lady, or a Lady with an imaginary Gentleman?"
"The lady with the imaginary gentleman, I think—that's pretty much how things stand anyway, so it won't take much imagination," said Mom. "What are we dancing?"
"The minuet," answered Ashleigh, showing Mom the book.
"Don't you think it would be easier with music?" suggested my mother. She turned to me. "Run and get the Mozart string quartets from the dining room, honey—oh, and grab some Strauss too, in case we want to waltz. Ashleigh, want to give me a hand with the desk? If we push it under the eaves, you might stop running into it."
My mother was right to take charge. She turned out to be an excellent dancer. Who would have guessed? Ducking from time to time when a turn took us too close to the slanting roof, we mastered the three essential movements—the walk, the slide, and the balancé. Then we practiced combining them into various figures of the quadrille. Another surprise: this turns out to be nothing but a slightly-less-dorky square dance. We made a stab at the minuet, which is a bit more complicated, and rounded off the lesson with a waltz.
"There," said Ashleigh, after my mother had left. "Now we will indeed be ready for the Forefield ball, where I, like Elizabeth Bennet, will find my heart's mate—my Mr. Darcy. And you, like Jane Bennet, will find your Mr. Bingley."
I bristled inwardly. I could understand how Ashleigh might identify with Elizabeth Bennet, that lively-minded young lady, though to me she seems more like one of Elizabeth's flighty younger sisters. But I was a little offended at her equating me with dull, good Jane, who falls in love with the second-rate Mr. Bingley. Though well meaning and well off, Bingley doesn't have half the brains or one-tenth the inner strength (not to mention the income) of proud, handsome Mr. Darcy, Elizabeth's suitor. It seemed doubly unfair that Ashleigh should intend to hog Mr. Darcy when I was the one who gave her the book in the first place. But I suppressed my annoyance and opened negotiations to crash the dance with Ash, in return for her promise to dress like a normal human being during school hours.
**_Chapter 3_**
_Ball Gowns_ ~ _Footwear_ ~ _Barns_ ~ _A Masked Man._
_**O**_ ur next task was to find suitable clothing for the dance. Fortunately, help was close at hand: my mother's shop, where she sells dainty objects such as greeting cards, teacups, beeswax candles, and—most important—vintage clothes.
Before my father left, Helen's Treasures occupied our front parlor and served as a hobby for my mother, offering its antique footstools and potpourri to weekend visitors with noses of steel. Now the shop's tentacles reach back through the entire ground floor. Even in the kitchen, I have to push aside boxes of scented soap whenever I need an onion.
Despite the expansion, however, Helen's Treasures is not a financial success. A painter, art teacher, and waitress before she married my father, Mom has more artistic talent than business smarts. Helen's Treasures makes just barely enough for our expenses—that is, if you factor in alimony too.
Ashleigh and I spent the last two weekdays before school started helping out in the shop: sorting, taking inventory, and replacing the unsold summer merchandise with fall items. In return, Mom offered us our choice of the vintage clothing.
Ashleigh rummaged through the dresses and pulled out a pair of matching fluffy pink things that some long-ago bride had inflicted on her bridesmaids. "Aren't these perfect?" she cried.
I was incensed. "Are you missing your macaroons? Crashing the dance is bad enough. I draw the line at crashing in a clown suit."
"Although perhaps a little more pink than is strictly desirable, these dresses are modest and seemly," countered Ash. "Do you have a better idea?"
"How about a couple of blouses and our long black skirts from chorus?"
"Unthinkable! Insufficiently formal for a formal!"
As we argued, I remembered that we still had a trunk to unpack from one of Mom's estate liquidations. We opened it and hit pay dirt—dresses from long ago in many hues and sizes.
I chose a sleeveless gown of silver-gray satin that matched my eyes and brought out the golden highlights in my hair. It fell in pools and folds like the drapery of a classical statue; for once I looked graceful instead of bony. A line of glittering rhinestones accentuated my collarbones. The gray went well with the black onyx of my lucky thumb ring, I thought.
Ashleigh picked a frock of deep crimson silk, the rustly kind, whose tight waist, low V-neck, and full skirt showed off her figure. The red suited her curly black hair and dark eyes. We each added a wrap—mine satin to match my dress, hers cashmere trimmed in black mink, shedding a little.
One problem still remained: what to wear on our feet. I had grown so much over the summer that even my sneakers had started to pinch. My old dress flats, scuffed and childish, were fortunately far too small. As for Ashleigh, a tomboy till last Monday, she had never owned any feminine footwear. Although we found a number of vintage shoes among my mother's things, none of them fit us.
Clearly we both needed new shoes. "Let's get your mom to take us to the mall on Saturday," I suggested. "Mine can't leave Treasures on the weekend—that's when she makes most of her sales."
To my surprise, Ashleigh objected. "We will be wearing long dresses, as befits ladies," she said. "Nobody will see our shoes."
"They will if we're dancing," I pointed out.
"Not if we dance with dignity," she answered.
Dignity? Ashleigh? A laughable concept. Ashleigh's attitude puzzled me. Shopping may not be our favorite activity in the world, but we like it fine, and it certainly seemed necessary now. After some probing, however, I discovered what held her back: empathy. Ashleigh's easygoing, indulgent, and well-to-do parents kept their only child supplied with everything she needed, and a great many things she merely wanted. For me, it was more complicated. My mother would give me what I needed if I asked for it, but I hated to ask, knowing how our budget would suffer. My father, on the other hand, had plenty of money, but I couldn't stand to ask him for it. Not only was I too proud, but I wanted to protect my mother from looking incapable. As for the money I'd earned over the summer scooping ice cream at Conehead's, I'd earmarked that for more urgent wardrobe needs.
Out of loyalty, Ashleigh hesitated to buy new shoes when I couldn't. I was touched, but I urged her to reconsider. Ashleigh being Ashleigh, she wouldn't budge.
"The problem is clearly beyond us," she said. "We must consult a Higher Authority. Call Sam."
"I know you're not going to like this," Sam told me when I reached her on her cell phone, "but the answer is Amy. She has plenty of money and perfect taste. Don't waste perfectly good parental guilt. If my father left my mother for a younger woman, I'd have a pair of Manolos for each toe. Get Amy to take you shopping for shoes, and tell her it was Ashleigh's idea to ask her for help. She'll be flattered—maybe she'll like Ashleigh better. Two good results for the price of one. I'm going to the mall myself tomorrow, to shop for uniforms with the gymnastics team. If you go then, you can give me a ride home afterward."
I asked Amy to pick us up behind Ashleigh's house on Saturday morning, not wanting my mother to see my father's SUV. I hadn't mentioned my stepmother's involvement to Mom, explaining simply that I was going shopping with Ashleigh.
My stepmother and my friend were on their best behavior. Amy called us both sweetie and warned Ashleigh only twice to keep her feet off the seats. Ash, for her part, took pains to support my claim that she admired Amy's taste, by complimenting her on her haircut, her handbag, her shoes, and her sunglasses. She stopped only after I kicked her hard.
Conversation was strained at first. Topic after topic fizzled after a sentence or two. Things perked up a bit when Ashleigh hit on bird watching, pointing out what she claimed were red-tailed hawks, a pair of falcons, and a bald eagle—which I maintained were crows, ducks, and a seagull.
"A seagull? Faugh! What would a seagull be doing so far from the sea?"
"They come up the Hudson. You've seen them a million times."
"We have left the Hudson far behind us. Do you think me unable to recognize the National Bird of Our Great Nation? Look, there it is again!"
Amy refused to be drawn into the dispute. "I couldn't say, girls," she said when we appealed to her. "I didn't see it. I have to keep my eyes on the road."
We were all relieved to arrive at the mall, where we began our search at the Teen Barn. (All right, it's not really called that, but why should I provide them with free publicity after the way the last three things I bought there fell apart in the wash? And why, by the way, must every shop bill itself as a Barn, a Warehouse, a Depot, or a Garage? Who'd want to buy their clothes in a garage?)
Shopping with Amy was nothing like shopping with Ashleigh or my mother. _We_ prefer to linger and laugh, leaving at last with at least one ill-judged purchase to be returned later, when we come to our senses. Still, much as it chilled me, I admired Amy's efficiency. The Irresistible One plucked shoes from the racks as a magician does rabbits. From her suggestions, I chose a pair of silver-gray pumps that I knew would go beautifully with my silver gown. Ashleigh wanted a red pair, but agreed to black.
Amy also bought me several pairs of new pants, pointing out with disapproval that my legs had grown several inches, as if it was something I could help. She seemed to know what would fit and flatter without so much as a glance at the tags. Her narrow heels clicked on the linoleum like fingernails on a keyboard. In record time she had me outfitted for the winter.
"Okay, girls, I have a manicure appointment," she said when we were done. "Come and get me in an hour. Remember we promised to give Samantha a ride home too. Call my cell phone if you need me. Is your phone on, sweetie? Have fun," she said, kissing me on the forehead. I suppressed a flinch and waved good-bye as the Demon of Efficiency clicked her way out of sight behind the fountain.
Ashleigh and I spent the next hour wandering happily from Barn to Barn. In the Candy Barn we played Sherlock Nose, a game Ashleigh invented years ago, in which one player blindfolds the other and takes her on a tour of jellybean bins until the blindfolded one has correctly identified seven flavors in a row.
After they kicked us out, we visited the Game Barn, where Ashleigh pestered the staff with requests for the official rules to Loo, Vingt-et-un, Casino, Lottery Tickets, Picquet, Whist, and Fish, which the alert Jane Austen reader will recognize as the names of card games played by characters in Miss Austen's novels.
The Game Barn staff, evidently, were not Jane Austen readers.
After they kicked us out, we retreated to the Book Barn, where, for a change, they let us browse our fill. Ashleigh bought her own copy of _Pride and Prejudice_ , as well as _Love and Freindship_ , Jane Austen's very first novel, written when she was just our age and not very good at spelling.
Then Ash wanted to play video games. Fearing for my nerves, I went to see if Sam had arrived yet at our rendezvous point, the Sports Barn.
I found her just as she left her fellow gymnasts. I saw their supple backs as they strode away.
"Oh, hey," she greeted me. "Where's Ashleigh and Amy?"
"Ash is in the Arcade Barn shooting alien starships, or enemy soldiers, or fish in a barrel, or something. She'll be here when she runs out of quarters," I replied. "The I.A. is having her talons painted red. I think she wanted to get away from Ashleigh."
"Oh, dear. Friction?"
"Not so bad, really. They kept it polite."
"I can just imagine."
"On the bright side, if Ash hadn't come along, I'd probably still be stuck with the I.A., having some sort of horrible just-us-girls version of Family Time. What is it with that?"
Sam made a sympathetic face. "Right, the maternal thing. You know why she does that, don't you?"
"Not really," I said. "She's not stupid—can't she tell I don't like her? And it's not like _she_ likes _me_ , either. Does she think it'll please my father, or is she just trying to torture me?"
"Maybe a little. But mostly I think her deal is, she really wants children and she's afraid she can't have any. She thinks you're all she'll get."
"Ig," I said. I couldn't decide which was worse, being stuck as Amy's substitute daughter or having a little half-sibling, with Amy contributing the other half.
Sam changed the subject: "Hey, speaking of ig, want to see something funny?" She steered me past aisles of fleece and spandex. "They've got sample team uniforms in here that must go back to at least the 1920s. There—just look at that!" She held out a little pleated dress with puff sleeves. Pinned to the bottom of the skirt was a pair of bloomers. "What kind of sado-coach would make a team wear _that_?"
We spent some moments urging each other to try on the worst of the samples. When a saleslady headed our way, however, with "Can I help you?" on her lips and murder in her eye, we made for the calmer waters of the Aquatics Department.
But before we could reach the kayaks, we found our way blocked by a figure in a close-fitting white jacket, smooth across the chest. Matching trousers fit snugly as well, showing off his powerful thighs. His face was hidden under what looked like a wire colander. In his hand he held a sword, which he was using to menace a large inflatable frog.
Samantha cleared her throat.
The warrior sprang to attention. With one graceful movement he brought his sword down and touched the blade to his forehead. Then, sweeping off his mask, he stood aside and bowed silently.
As he rose from his courtesy, I found myself staring transfixed, eyes locked with the blazing turquoise eyes of my Mysterious Stranger.
For a moment I stood and stared. Then Sam said, "Oh, hi," breaking the spell. The Stranger smiled at her, showing the tips of his beautiful white teeth. He bowed again slightly and withdrew.
Feeling weak and trembly, I breathed, "Samantha, who was that Masked Man?"
"Some guy from Zach's dojo," she answered. "I don't remember his name. I think they might have had kendo together."
"Kendo?"
"Japanese sword fighting. I was considering trying it myself, but Zach thinks I'd like aikido better."
Shyness prevented me from asking Samantha any more questions about the Stranger. She continued to weigh the relative merits of the various martial arts, but I can't tell you what she said. Indeed, the rest of the afternoon passed as in a dream, those turquoise eyes always before my inner eyes. All the way home Ashleigh called seagulls eagles to her heart's content while Sam entertained my stepmother with details of team uniforms, without any interference from me.
**_Chapter 4_**
_Tenth Grade_ ~ _Extracurriculars_ ~ _A Sonnet._
_**T**_ he dream faded soon enough, however, and I awoke to the cold, hard knowledge that summer was over. I speak metaphorically, of course. Actually it was still pretty hot out, especially in my attic bedroom. Mom is always promising to redo the insulation if business ever picks up.
Monday morning Mom made me my traditional back-to-school breakfast: whole-wheat buttermilk waffles with maple syrup and homemade sour cherry jam. (Ashleigh's candy-making phase had a strong jam component.) For an extra treat, Mom set the table downstairs in the front parlor, at a claw-footed oak table that's been on sale for several years. If it ever does sell, I'll miss it terribly.
I brushed my hair quickly, put on my lucky thumb ring, and came downstairs. I was wearing some of the clothes my stepmother had provided, and I worried a little that my mother would notice and ask where they had come from. Indeed, she looked me up and down appraisingly. All she said, though, was, "How nice you look, honey."
At school, Ash and I were disappointed to find we had different homerooms. She drew Frau Riechstoff-Murphy, the German teacher, and I landed the notorious math teacher Mr. Klamp.
Mr. Klamp laid down the law. No tardiness, no talking above 40 decibels, no untied shoelaces, no visible undergarments, no eating, no chewing gum, no chewing tobacco, no chewing betel nuts, no chewing coca leaves, no chewing out students (unless Mr. Klamp was doing the chewing out), no chewing out teachers (unless ditto), no unnecessary displays of temper (unless ditto), no unnecessary displays of affection (no exceptions), no pets over one ounce or under one ton, and no singing, except in Bulgarian. I began to think Mr. Klamp wouldn't be so bad—which was lucky, since I had him for math as well.
This year, the social highlights of my homeroom included three of the grade's five Seths; Tall Alex and Mad Alex; Michelle Jeffries; Cordelia Nixon; and one of the Gerard twins—Yolanda or Yvette.
The Y girls are identical twins: the same light-footed roundness, tapered fingers, smooth, dark skin, and elegantly swooping nostrils. Like many identical twins, they like to confuse people by playing games with their clothes and hairstyles. One favorite trick involves gradually trading the colored beads at the ends of their braids, so that, for example, Yolanda will start off with nothing but yellow and Yvette with nothing but green. By the end of the week they'll both be fifty-fifty yellow and green. Then comes the tricky part. One twin will gradually acquire all the yellow beads and the other all the green—but is the green twin Yolanda, taking over her sister's look, or is it Yvette, returning to her original color?
Fortunately—or unfortunately, I guess, if you're a Gerard twin—there's a simple way for those in the know to tell whether someone is Yolanda or Yvette. Just stand near her and wait. If the twin in question starts to talk, there's a good chance it's Yolanda. If she goes on talking for three or four sentences, the good chance becomes a certainty. Yolanda once told me in confidence that in her elementary school, they used to call her Yoyo Mouth.
"Julie Lefkowitz! Look at you! You got so tall! Are you taking physics this year? Let's see your schedule. Look, we're in gym together. And English—Ms. Nettleton, ig. No fair! I heard we were supposed to get Ms. Muchnick, everybody says she's loudly crisp, but she had to go get pregnant. Why would she want a baby when she could have had _us_? Hey, did you do the summer reading? They love _Lord of the Flies_ , don't they? We had it in eighth grade at Sacred Heart, and the next summer in Enrichment. If I have to read it one more time, I'm going to go throw myself off a cliff. They call that book realistic? If you ask me, not even boys would act that way. Speaking of boys, here comes Seth Young! Hey, Seth Young! Where've you been all summer? Let's see your schedule. Did you hear about Muchnick?" Diagnosis: Yolanda.
For the first few days, school had an air of embarrassed festivity. Everyone had come back from their vacations taller, stronger, gawkier, slimmed down or curvier, with their hair grown past their shoulders or newly cropped and sticking out funny. The lawyers' sons had deep tans from their wilderness adventures, the hippie farmers' daughters from their long days working outdoors. The cliques shimmered like a mirage, and for a moment it seemed as if a former nerd might cross unharmed into the crisp crowd. Then the walls firmed up again and the moment passed.
"Julie, it's time for you to start thinking seriously about college," said my father one Tuesday evening. "Your grades are good, but that's not enough. Admissions officers are going to want to see strong extracurriculars too. I know you like to write. Have you thought about joining the school newspaper? Or what about the literary magazine?"
I groaned silently. The editors of the _Byzantine Bugle_ publish enthusiastic little stories about pep rallies and food drives. Everything has to pass the scrutiny of the administration; the result is loudly dull. The literary magazine, _Sailing to Byzantium_ , isn't so bad—at least, it wasn't so bad last year, when Ms. Muchnick was the adviser. With the Much on maternity leave, though, Ms. Nettleton had taken over. Three periods a week of _her_ was quite enough for _me_.
"I don't know, Dad," I said. "I'm pretty busy with school, plus there's my job at Conehead's." (I decided not to tell him that, due to a weather-linked decrease in the demand for frozen treats, Conehead's had let me go for the winter.) "Anyway, I'm just not into the whole newspaper/magazine thing so much."
"You know I wish you'd give up that job," said Dad. "Cone-head's isn't exactly the most impressive credential to have on your record. What about student government or science club? That might be even better than the newspaper. The admissions officers like to see well-rounded students."
Well rounded! I glanced ruefully at my bony knees. Which, I wondered, would be worse: to tell my father that a midlevel nobody like me had no chance whatsoever of winning a school government election—essentially a popularity contest—or to express distaste for science, his favorite subject and the basis of his career? For the thousandth time, I envied girls whose fathers had a clue about their interests and personalities. Banking two imaginary dollars in the Familial Restraint Fund, I told Dad I would think about it.
And I did. What I thought was this: If there were justice in the world, the hours I spent with Ashleigh would count as an extracurricular activity. Science Club, History Club, and Future Candy Makers of America, all smushed together and laid out to dry like a Fruit Roll-Up.
Autumn blew in cold and clear the next week. As the days grew shorter and their hours grew longer, we settled down in earnest to tenth grade.
In Mrs. Marlin's class, Charlemagne advanced across Europe (or do I mean retreated?), followed (or perhaps preceded) by his ancestors, descendants, henchmen, or enemies, Clovis, Childeric, and Pepin the Short. (History was never my thing. Unlike English, where you can make things up, or math, where you can figure things out, history depends on happening to know what happens to have happened. Where's the sense in _that_?)
In English, the only class I had with Ashleigh, the vicious children of Summer Reading—I refer to the characters in _Lord of the Flies_ , not my classmates—made way for Shakespeare's unfortunate lovers.
"How do we know that Romeo and Juliet are in love?" asked Ms. Nettleton one rainy third period. "Yes, Julie?"
"Shakespeare tells us in the prologue. He calls them 'A pair of star-cross'd lovers' and talks about 'The fearful passage of their death-mark'd love,' " I said.
You'd think any teacher would be thrilled to have evidence that a student had read and understood the homework. Not Ms. Nettleton. When she asks a question, she doesn't want just any answer; she's interested only in the one _she'd_ give.
"Yes, but what clues does Shakespeare give in the dialogue itself? Anyone? Not you, Seth, I know you know. Peter?"
"When Juliet goes, 'Romeo, Romeo, oh, wherefore art thou, Romeo?' " said Peter the Short.
Ms. Nettleton squinted at him mistrustfully. That line doesn't appear until the next week's reading, and Peter is not the type to read ahead. She clearly suspected him of winging it. "Before that. At the dance—the Capulet party, where Romeo and Juliet meet. Did anyone notice anything special about the first words they say to each other?"
"They're kind of flirting," said Yolanda. "They're kissing each other's hands and things."
"Yes, but what about the _form_ of the lines? Did anything look familiar from our unit last year on poetry? Anyone? All right, Seth, tell the class."
"They speak in rhyme and meter," said Seth Young. "In fact, the first part of their conversation takes the form of a sonnet."
"Thank you, Seth," said Ms. Nettleton. She wrote _sonnet_ on the blackboard and started explaining in words as dull as they were informative. From rhyme schemes she proceeded to iambic pentameter, fourteen lines, final couplets. The bell rang before we got back to Romeo's feelings for Juliet, or vice versa.
"Who _cares_ if it's a sonnet?" said Yolanda as we made our way to the cafeteria. "That whole love-at-first-sight thing is a pile of crock, anyway. Okay, it's better than _Lord of the Flies_ , but not much. Romeo's already in love before he meets Juliet—with that Rosalind person, who's her _cousin_ , mega-ig. Then he sees Juliet and he's all 'Let me kiss your hand, I really mean it this time, you know I do 'cause I'm telling you in a _sonnet_.' And Juliet's not even _fourteen_ yet. He's going to kill himself over an eighth grader? Yeah, right."
Ashleigh disagreed. While not every pair of lovers understood the true nature of their attachment at the moment of their first meeting, she maintained, some did. She gave as an example of the former type, Elizabeth Bennet and her Mr. Darcy; of the latter type, Elizabeth's sister Jane and her Mr. Bingley. Finding that Yolanda had not yet read _Pride and Prejudice_ , she jumped at the chance to describe it.
In the meantime I chewed my egg salad in silence, thinking about the nature of love.
Two people could know each other for years, I reflected, and promise to love each other forever, yet find their hearts and interests at odds. That was certainly the case with my parents. However, the example of the Drs. Liu suggested that lasting love did sometimes blossom from the first encounter. Samantha's parents met in a singing group—Haichang has a baritone voice, Lily a sweet mezzo-soprano—and they still put their cheeks together and croon in harmony whenever they think nobody's around.
What, I wondered, would be _my_ fate in love?
If Ashleigh was right, I would find out soon enough. Only a week remained until the Forefield Columbus Cotillion. We had rehearsed our dance steps until we could confidently hop through three flavors of quadrille, a minuet, and the Sir Roger de Coverly, as well as the fox trot, the waltz, and some simple swing. We had even practiced wiggling freestyle. I felt we were as ready as we ever would be.
First, however, we had an obstacle to overcome: how to get there. Ash and I hesitated to ask our parents to drive us to the dance—we were afraid they might somehow figure out that we hadn't, in fact, been invited. In the end, we decided it would be best not to tell them about it at all. We would let them think we had gone to the movies with Sam. ("Wearing ballgowns?" objected Ashleigh.—"You've dragged me to the movies wearing far worse," I answered.) That left a choice of walking the three miles to Forefield or riding our bicycles up the long hill, catching our hems in the gears and arriving in a sweat. Our return seemed even more problematic.
Once again, Sam came to our aid—or, more precisely, her brother, Zach, home from college for the Columbus Day weekend. When I went to the Lius' house to borrow a pair of evening purses from Sam's large collection, she offered us Zach's services as a chauffeur. "I told him if he didn't drive you, you'd get tangled in your bike pedals and wind up in a ditch with a broken neck. Then your father would die of a broken heart and Dad would have to find a new partner. Zach said you were idiots, but he'd do it for the family honor. He likes an excuse to drive that car of his."
"Thanks—I guess. You sure you don't want to come?"
Samantha laughed. "Can you see me chasing boys at the Forehead Academy? The guys I already know are quite enough for me, thanks. Have a good time, and don't let Ashleigh do anything too embarrassing."
**_Chapter 5_**
_A ride through the Dark_ ~ _A menacing adder_ ~ _A gallant rescue_ ~ _A Quadrille_ ~ _A Waltz_ ~ _A second Sonnet._
_**Z**_ ach picked us up at Ashleigh's house. "Ready for the costume party, kids?" he said.
"It's not a costume party, it's the Columbus Cotillion," said Ash, getting in the front. "And we're hardly kids."
Zach headed uphill along the river. "Hmm," he said, looking her over critically. "You're right, you don't look so kidlike in that dress. Those boys at the Foreplay Academy better watch out."
Ashleigh slapped at his shoulder. He grabbed her wrist with three fingers and started to twist. "Hey! Guys! Keep your eyes on the road," I said.
My stomach fluttered as we turned off the river road and drove up the twisting approach toward the Forefield gate. In my long friendship with Ashleigh, I had become accustomed to a certain level of public attention. When your best friend goes around town dressed in armor constructed from cookware, eyes naturally turn your way. But getting thrown out of the Candy Barn for sniffing too many jellybeans is one thing; marching boldly into a nest of reputed snobs while dressed in ancient frocks that smell faintly of mothballs kicks up the potential for embarrassment to a whole new level.
"Let us off here, Zach," I said. "We'll walk the rest of the way. It'll be easier to get in if we kind of edge along behind some other people." Zach got out to open the rear door, which sticks from the inside.
"Okay, kids—ladies. Call me when you've had enough. And tell me if any of those Foureyes kids get fresh—I'll kick their asses." He demonstrated with a carefully placed karate kick that fortunately left no mark on the back of my silver dress, then drove off into the night.
Ashleigh and I gathered up our skirts and edged through the gate to the Forefield Academy, our heels sinking into the grass by the side of the drive. The air was sharp; I pulled my wrap around my shoulders. Carved lions observed us from either side of the gate, their tails curled catlike around their flanks, their noses lifted in stony disdain, as if we weren't worth the effort of a pounce. Two or three cars whished slowly by, fluttering my hair and Ashleigh's sash.
We reached the top of the hill and began to pass the school buildings, each more imposing than the last. After a minute or two we drew near enough to hear fragments of music trickling across the lawns from the old Forefield mansion, the heart of the academy. As soon as I saw it, I recognized it as the palace visible from my attic. From close up it looked at once more real and more enchanted. Light spilled out through tall windows and laughter mingled with the music. It sounded elegant and merry, utterly unlike the noisy chaos that passed for dances in the Byz High gym.
Ashleigh was all for charging up the broad marble steps to the door, but I held her back. We waited until a group of partygoers came up behind us, then hurried through the door at their heels, hoping any observers would think we were with them.
No such luck. At the entrance a red-faced man, gaunt yet jowly, sat behind a table taking tickets. "Excuse me! Excuse me, girls! Tickets?" he honked at us as we tried to sneak past.
Ashleigh opened the black beaded evening bag Sam had lent her, peered in, mimed astonishment, and patted the sides of her frock as if it had pockets. "I must have left them in my other cloak," she announced with her best innocent look.
I made my usual embarrassed attempt to hide behind her, but it never works—Ashleigh is six inches shorter than me.
Turkeyface frowned. "Where are your escorts?" he asked.
"Oh, they're around somewhere. They told us to meet them here," Ashleigh tried.
Our challenger grew sterner. "This event is for the Forefield community and their guests only," he said icily. "I'm afraid I can't admit you without a ticket or, at the very least, an escort."
Our wisest choice, I thought, would be to leave by the door and sneak back in through a window, crossing our fingers that the watchdog would keep his eye on the entrance rather than the room behind him. Well—our very wisest choice would be to go home, but I knew the Enthusiast would never agree to _that_.
"Can't we just—" she began.
"I'm afraid the rules on this point are very—" Turkeyface countered, speaking over her words.
I felt the blushing faintness so familiar from years of hanging out with Ashleigh. But as I tried to distract myself by wondering whether I had turned as red as Turkeyface, my ears caught a sound as welcome as a fire drill during finals week. It was a voice behind me speaking miraculous words: "It's all right, Mr. Waters. They're my guests."
Turning, I recognized the speaker as my Mysterious Stranger.
Turkeyface looked as astonished as I felt. "Really, Parr?" he said, raising an eyebrow. "Both of them? Two dates, all for you? My, my, you're quite the lothario."
Did my hero turn faintly pink himself, or was that an effect of the lighting? "No, just one—the other is Ned's guest—right, Ned?" He grabbed a square-set guy by the shoulder. "Got your tickets, Ned? Here, hand them over."
The boy called Ned fumbled in a vest pocket—he wore a vest!—and pulled out a clump of paper, a pack of cinnamon gum, a pencil stub, a tuning fork, and, finally, a pair of tickets. They appeared to be printed on smooth, thin cardboard like theater tickets, not Xeroxed onto colored paper like school notices. My hero contributed a pair of his own.
Turkeyface pushed his glasses down his nose to inspect the tickets. He made no further objections, however, waving us into the room and turning back to menace new arrivals.
Once we were out of his range, Ashleigh reached up and hugged the arm of the Mysterious Stranger called Parr. "Our hero," she cried. "You saved our lives! Without your aid, we would have been forced to climb in a window, endangering our Necks and Frocks. How can we ever thank you?"
"Hey, no problem," he answered. "Always glad to do what we can to foil old Wattles. Right, Ned?"
"The supreme joy of our young lives, foiling Wattles," agreed the one called Ned.
"Supreme though the joy of foiling Wattles may be, it can never compare in value to the service that you have rendered us tonight," argued Ash. "How will we ever repay you?"
"Honestly, we were happy to. But if you really want to thank us, some dances should do it," said Parr.
"With pleasure," cried Ashleigh. I inclined my head.
"And will you tell us the names of our dancing partners?" asked the handsome hero, turning to me.
I felt my blush intensify. With all the blood rushing to my cheeks, I worried that none would remain to carry oxygen to my vital organs. "I'm Julie—Julia Lefkowitz," I said, "and this is Ashleigh Rossi."
To my horror, Ashleigh curtsied. "And you, sir?" she asked.
"Charles Grandison Parr, at your service, madam," he said, sweeping an imaginary hat off his head and executing a bow worthy of Dodworth. "Allow me to introduce my companion, Edgar Downing, aka Ned the Noodle."
"Don't listen to old Granddad. Nobody calls me that," put in Ned, kicking at his friend.
"He's a dreamer, old Noodles. A fine intelligence, but a dreamer," countered Parr, dodging neatly.
"But what, pray, did Mr. Turkeyface have against us?" asked Ashleigh. "Did he think we were going to steal the ancestors off the wall?"
The suggestion seemed almost reasonable. The walls of the room in which we were standing—a sort of medieval hall, complete with suits of armor, presumably empty, guarding the doors at each end—were covered from ear-height to the rafters with paintings of sour-looking men in dark suits.
"Oh, I doubt it—that's just old Wattles acting wattly," said Parr. "Of course, there _was_ the time the Emerson House seniors sneaked in the night before Founder's Day and turned all the pictures upside down. But I can't imagine he would blame _you_ for that."
"Oh, yes he would, Gramps. He'd blame them for anything that crossed his mind. They're girls, aren't they? He's a dirty-minded old Puritan. He probably thinks dancing is the devil's work," said Ned.
"But nobody's actually dancing," Ashleigh pointed out.
Indeed, the room was full of people milling around in knots like ours. Although a small chamber orchestra stationed overhead on a minstrels' balcony was pouring forth music, not a single couple was dancing to it. The young musicians were even pretty good too, if you like Mozart and can ignore acne.
"Everyone's waiting for the headmaster and his wife to open the dance. It's a Forefield tradition," said Parr. "But," he continued after a pause, "how did you wind up here without tickets? Did you lose them? Or do you actually have escorts who stood you up?"
When we hesitated, wondering how to answer, Ned added, "Don't tell me you really crashed! Somebody dared you, right? You can't have come of your own free will."
Ashleigh and I looked at each other, but before she could open her mouth, a fanfare sounded from the musicians' gallery. A hush fell across the grand hall. A tuxedoed teenage trumpeter put down his instrument and announced in a voice as brassy as the horn: "Ladies and gentlemen, please take your places for the Founder's Quadrille!"
In the bustle that followed, a silver-haired gentleman emerged from the crowd and led a stout but handsome lady to the far end of the room, by the empty knights. Couples, mostly older, arranged themselves geometrically down the grand hall. Music struck up, and the lead couple began the elegant ritual of walk, slide, and balancé as the others looked on.
Ashleigh took Parr by the arm. "Well, sir? Didn't you want to dance?" she cried.
"Yes, but we have a while. They always play one or two really weird antique dances before the waltzes. It takes at least an hour after that before they get to the normal stuff. We have a long wait ahead."
"But why wait—don't you know the quadrille?" persisted Ash.
"Well, _we_ do—they teach us in phys. ed. when we're first formers—but I can't imagine _you_ do. Unless—you're not from Miss Wharton's, are you?" Parr gave us a doubtful look.
"No, but we do know our quadrilles. Which one is this? The Coquette? The Polo? The Basket Dance? Well, we can wing it—I mean, we will endeavor to improvise. Come on, the rest of the couples are starting to dance!" said Ashleigh, stuffing her purse and wrap behind the nearest suit of armor. Parr let her pull him into position at the bottom of the room.
"Are you ready?" said Ned, turning to me. "I'm glad you showed up—for once I get someone good to dance with."
I tucked my things in beside Ashleigh's, took the arm he offered, and followed him off to join our friends.
The Founder's Quadrille couldn't be more unlike twenty-first century dancing. Today, a couple or small group stands together, rhythmically contorting their arms, shoulders, torso, hips, and lower limbs. The point is to wiggle in harmony with one's companions while distinguishing oneself from the crowd by imaginatively displaying one's attractive parts, all the while avoiding—as far as possible—looking like a dork.
Not so the Founder's Quadrille. Looking like a dork seems to be required. Another difference is the miles that the quadrille dancers cover. They step forward and back, spin, approach the opposite corners, return to the first spot. Often throughout the dance, Ned took my hand, walked me and turned me, bowed to me and acknowledged my curtsy. But often, too, I found myself face-to-face with some other gentleman, or arm-in-arm with a lady. With all this to-ing and fro-ing, it was hard to carry on a conversation.
I fell to musing about the voice of my Mysterious Stranger, Charles Grandison Parr. In the six times I'd seen him before, I had never once heard him speak. Even at the Sports Barn, he had merely bowed in silence. His voice, now that I heard it, was nothing like what I had imagined: not a rumbling bass, but a strong, smooth tenor, full of caressing vowels that seemed to reach to my very toes and fingertips. It vibrated through me as he talked disjointedly with Ashleigh, leading her through the steps of the dance.
"You never answered Grandison, you know," said Ned, giving me his hand. "Why _did_ you crash the dance?" Funny, he had the rattling bass voice I had imagined for Parr. Hearing it now, it seemed unsubtle. Why did I imagine Parr would sound like that?
The quadrille separated us for a minute, leaving me time to think. I decided to tell the truth—or part of it, at least.
"It was Ashleigh," I said as Ned and I slid to the right, then left. We were well matched as dancers—the same height, so that our eyes were exactly level. "Ash gets these ideas in her head. Last year it was marine biology, the year before it was candy making. There's no way to stop her, short of locking all the doors and windows from the outside."
The dance carried Ned to Ashleigh's corner, where he turned her by the arm; for a breathless moment I felt Parr's hand on my own arm and looked up into his eyes. Then Ned was back. "Ashleigh's amazing," he said. "I've never seen anyone dance the Founder's Quadrille with actual enthusiasm! Not that I've seen all that many people dance it," he added after a foray into opposite corners, during which I gave my hand not to my hero but to a middle-aged gentleman with a potbelly, from the next set of couples, "mostly just the older teachers and the girls from Miss Wharton's. And us, of course, when we can't avoid it."
"Sorry to put you through this," I said, feeling a little hurt.
"No, no, I didn't mean that," said Ned. "This is surprisingly fun." He grinned at me. "You're not so bad, for a girl."
The music drew to a close, and we made our bows and curtsies. As the four of us stood in an awkward square, wondering what to do next, the band struck up again. Hidden within what sounded like a waltz by Strauss was the tune to "Take It Back," by the Wet Blankets—a favorite of Ashleigh's for most of the summer, and not yet fully abandoned despite her new interests.
"Hey, Noodles, they're playing your tune!" said Parr.
"Oh, are you a Wet Blankets fan?" asked Ashleigh eagerly.
"Yeah, I pretty much like them a lot," Ned said.
"Don't be so modest," said Parr. "He wrote this arrangement. Ned's our school composer."
"Really? I _love_ this song," cried Ashleigh.
"In that case, would you like to dance?" Ned asked her.
This time it was Parr's turn to ask me, "Shall we?" and offer his hand.
Once the two of us were alone together—that is, as alone as a pair can be in a room full of dancing couples—my hero seemed to lose his suave. As for me, I had been tongue-tied from the start. Standing beside the Magnet of My Yearning—touching his hand—frantically sending signals to my toes to keep them from tangling with his: none of this was likely to make me articulate. I worried that my nervousness had infected my partner.
The silence stretched out. Clearly one of us had to say something.
Parr began.
"Haven't I seen you around town before?"
"I think so—at least, I know I've noticed _you_."
"The Sports Barn, was it? Or the candy store?
Didn't I see you with Samantha Liu?"
"That's right. She says you know her brother, Zach.
Are you another black belt in Nintendo?"
"I only wish! I'm blue, three down from black—
that is, assuming you refer to kendo."
"Kendo! Right! Kendo! Well, I'm clearly not
a black belt in talking. At least, not tonight."
"Hey, don't you think it's getting kind of hot?
After this dance, you want to grab a bite?
They've got a bar set up by the parterre.
Just chips and soda, but it's cooler there."
I nodded my agreement and fell silent again, internally kicking myself for my comments. Would Parr realize I had stored up every sighting like a treasure in my heart? Uncool, uncool. And that Nintendo gaffe—what sort of marshmallow head would he take me for? Yet how handsome he looked as he was laughing at me! His smile crinkled his turquoise eyes and stamped a single dimple in his left cheek. And it _was_ kind of him to change the subject and suggest going for food, as if he had sensed my discomfort. Or was he merely looking for an excuse to ditch me?
With these thoughts, I waltzed myself dizzy in his arms.
The dance ended, the couples pattered their applause, I retrieved my things from behind the armor, and Parr and I stepped out into the welcome coolness of the October night.
**_Chapter 6_**
_More adders_ ~ _Ginger ale_ ~ _untimely Flushing_ ~ _we dance the Sir Roger de Coverly._
_**A**_ re you imagining a romantic scene of distant music, softly scented breezes, and twinkly lanterns, with moonlight falling over everything? Do you picture me beginning to shiver, while Parr wraps me tenderly in my shawl? In your vision, does he leave his arms casually around me as we lean against the balustrade, gazing at the stars?
Happy dream!
It was crowded on the long brick terrace overlooking the parterre. (A parterre, in case you were wondering—I was—turns out to be a chessboard arrangement of flower beds.) Now past the season of prime bloom, the Forefield parterre minced down to a long lawn, which swooped down to the river. Staircases and gravel paths threaded the flower beds, punctuated by large stone urns spilling over with late grasses and vines.
On the terrace where we stood, boys in blazers, now and then with a date in a pale dress, jostled one another to get at the food and drinks. Spilled pretzels crunched underfoot. From time to time clumps of muttering youths burst into wild chortles, as if to celebrate some successful act of wickedness. Released from his guard table, Turkeyface stalked along the edge of the terrace, sniffing the air for illicit smoke.
Grandison Parr led me to a sheltered spot by a pair of planters. "What would you like to drink?" he asked.
"Hmm—ginger ale?" I hoped the choice wouldn't sound too babyish.
"Right. Be right back." He pushed his way into the crowd.
At first I kept his golden head in sight, but after he turned around and glanced at me twice, I looked away, embarrassed to be caught staring. When I looked again, he was gone.
A long time went by.
I played with the fringes of my wrap, braiding and unbraiding them.
I wondered whether Ashleigh was still dancing. I considered going to find her, but decided to stay put, in case Parr came back.
Three or four gangly boys nearby nudged and punched one another. They ejected one and gave him a little push in my direction. He approached hesitantly.
"So, um, you wanna dance?" he muttered, addressing an area a little below my collarbones. He couldn't have been more than fourteen.
"Sorry, I can't—I'm waiting for my . . . escort . . . to come back with a drink," I answered.
With a little gulping noise, the boy skittered back to the safety of his companions.
A girl in green leaned against my planter, glancing sideways at me. I considered speaking, but decided against it. The girl's escort soon appeared and carried her away to the ballroom.
A handsome guy with the look of a large and powerful cat—a junior, I thought, or possibly even a senior—presently took her place. He plucked a pair of cigarettes from his blazer pocket and held them out. "Trade you a smoke for a light," he offered.
I shook my head. "Sorry, I don't have any matches."
He tucked the cigarettes away and leaned against my planter, his arm touching mine. I edged away, but he relaxed closer to me, keeping his arm in contact with mine.
"Sounds like they're finally done with the ancient music," he said after a minute. "Let's go dance."
"I can't. I'm waiting for someone."
He raised an eyebrow. "You've been waiting a long time. Are you sure he's coming back?"
I hesitated, considering what to say. I was beginning to have doubts.
The cat-guy pressed his advantage. "You seem pretty bored. If you don't want to dance, I'm sure we can find other things to do." He raised the eyebrow even higher. A hundred years ago, I thought, would he have twirled a moustache instead? I gripped the planter behind me, wondering how to get rid of him.
To my relief, rescue came running up, in the form of the person ultimately responsible for my trouble: Ashleigh.
"There you are," she cried. She turned her head and called behind her, "See, I told you she'd still be here!"
Parr and Ned followed more slowly, their hands full of drinks they were trying not to spill.
"Sorry that took so long," said Parr. "They didn't have ginger ale at the bar. I had to try three different vending machines." With a flourish, he presented me with a cold can. Glad to have something to do with my hands, I busied myself with it, snapping it open and sipping; the bubbles got up my nose.
The feline guy gave me what I imagined he must consider an intimate look, then turned to the newcomers. "Hey, Parr, your girlfriend here was about to give up on you," he said. "What?" he added, "no drink for me?"
"Hello, Chris," said Parr coldly. "I hear Wattles is looking for you. Oh, look—there he is now."
Indeed, Turkeyface appeared to be heading toward us.
The cat-guy brushed my arm with his hand. "Catch you later," he said, and melted away in the opposite direction.
"Ig, who was _that_?" said Ashleigh.
"I don't know, W-, maybe?" I suggested, naming a seductive creep in a Jane Austen novel.
"You don't know Chris? Chris Stevens?" said Parr. "He looked as though he knew you pretty well—or wanted to, anyway." Parr paused, as if deciding whether to say more, then added, "I hope he wasn't bothering you. I really am sorry I was gone so long."
"Grand Parr is a stubborn old thing," rumbled Ned. "Ashleigh told him you'd be just as happy with Sprite or Coke, but he wouldn't believe it. He had to drag us all the way out to the new science library."
"I said I'd bring ginger ale," said Parr. "A promise is a promise. If I had known Chris would come sniffing around—Was he being obnoxious?"
"Nothing terrible, just asking me to dance," I said.
"Sounds like a plan," said Ned. "The waltzes are finally over. I mean, you waltz beautifully," he added hastily, addressing Ashleigh, "but now we won't be the only ones dancing . . . kids, I mean, not teachers . . ." He trailed off.
"I'm up for it," I said, eager to leave my planter. Gulping down ginger ale, I followed my companions back into the ballroom.
On the dance floor, both the music and the crowd had increased in volume. A DJ had taken over from the band.
At first I felt more self-conscious than ever, dancing without the prescribed steps of the quadrille or waltz. There's something especially awkward about free-form wiggling in a ball gown, and to make matters worse, the Guy of My Dreams was watching. But the rhythm of the music quickly took over, and with it the release that comes from vigorous physical activity.
Dancing in a group of four was a far cry from quadrilling or waltzing in couples. For one thing, Parr no longer touched me (except the occasional accidental, electrical brush). It was too loud to talk, beyond a shouted word or two. Soon several friends of Parr and Ned's caught sight of Ashleigh and me and joined us.
After a number of songs I found myself at the center of a circle of guys, detached from my friend and our rescuers. The ginger ale began to make its presence known. I excused myself—hoping that the boys would have the guts to go on dancing without the presence of a girl to give them an excuse—and went off to find a ladies' room.
Ladies' rooms, it turns out, don't flourish in boys' schools. Each likely-looking door seemed to taunt me. I discovered a coat closet, a broom closet, a conservatory dripping with greenery, and wood-paneled, book-lined chambers of various shapes and sizes—but no restroom. At last I found a chaperone to ask. She directed me to a boys' bathroom, temporarily reassigned to meet the needs of female guests. "Boys: STOP! Girls: GO!" read a laser-printed sign taped to the door—not, I thought, the most tactful way to put it.
The uneasy sense of trespass that I'd felt all evening intensified when I went in. What most unnerved me were the urinals. With their exposed position, unprotected by so much as a door-less stall; with their long, jutting necks and their intense smell—of ammonia, strong detergent, and something else—is it any wonder I slunk past with a shudder?
I chose a stall at the end of the long room. As I sat resting my feet and watching the rows and columns of blue tiles dance a quadrille before my eyes, I heard the door swing open. I froze—boys?! No, thank God—girls. Just girls. Prep-school girls, judging by their accents. Perhaps girls from Miss Wharton's?
I decided to wait them out.
There seemed to be four or five of them. Some made for the toilet stalls while their friends stood by the sinks. A couple of them compared and exchanged lipsticks; another requested a comb. ("Promise you don't have nits?"—"Will you get over that? Fourth grade was six years ago!")
They praised each other's shoes and disparaged various boys, mostly unknown to me, although I did hear the name of Chris Stevens. "Unthinkable creep, keep him away from me!" commented one girl with a melodious voice that seemed to curl musically around my ears.
"Oh, I don't know, he has a sort of viscous charm," disagreed another.
"I _guess_ , if you like a guy to _ooze_ at you," answered her friend.
Their conversation went on for so long that the toilet seat began to dig uncomfortably into the upper half of my lower limbs. I was considering making a break for it when I heard another familiar name.
"Anyone at all? My choice, the whole school? Okay, give me Parr," said the curly-voiced anti-oozer.
"Grandison Parr? The junior—the fencer?"
"That's the one. Mmmm! Rich, firm goodness."
"Really? You've experienced this firsthand?"
"Oh, don't I wish! I'm not _that_ lucky."
"Parr? Isn't he taken?" objected one of the urinators from her stall. "He seems to have a date, anyway. That tall—"
An ill-timed flush, echoing in the tiled, high-ceilinged room, cut off the rest of the sentence. Considering all the ill-timed flushing I'd been doing myself that evening, I reflected—flushing of the skin, not of the toilet—(the water gurgled to a stop before I could finish the thought; I turned my attention back to the deeply interesting conversation—)
"—and the little one in red? Where did they _come_ from? Where did they _get_ those _dresses_?"
"I think the tall one's his sister. She kind of looks like him. She was dancing more with that dorky guy, the one in the three-piece suit."
"No, but would you let your sister dance with the dorky guy in the three-piece suit?"
"Would you let your girlfriend?"
"Well, they were all in the same set, anyway, early on. Did you see the little red one bouncing away? No way she learned _that_ from the quadrille sergeant!"
"The guy in the funny suit is Parr's roommate. I still think the girlfriend is the tall one. She—"
As if to mock me, the last urinator finished her business and drowned out the end of another interesting sentence. By the time her toilet ceased its gurgling, the girls had clattered out of bathroom, leaving me alone to stare dizzily at the blue tiles.
When I rejoined my party, Ned and Ashleigh were dancing vigorously to the last few bars of "Take It Back"—the Wet Blankets version, not Ned's waltz—while Parr looked on with an amused smile.
"There you are," he said. "I was afraid I'd lost you again."
"It took me a while to find the ladies' room. They hid it behind a sort of greenhouse thing and a room full of silver cups in glass cases."
"Oh, you found the trophy room? Good place to take a nap when you're supposed to be in study hall. There's a big, puffy sofa behind the cabinets, and nobody ever goes in there."
The song ended, and the trumpeter blew a fanfare. I saw that the band had reassembled in the musicians' gallery. The room fell silent. Parr leaned close so he could whisper in my ear, "I'm sorry to say this is it—the last dance." His breath tickled my neck. The sensation made me my heart pound so loudly, I was afraid he'd hear it.
"Ladies and gentlemen, please take your places for the Virginia reel!" announced the trumpeter to moans and cries of "Already?!" In Jane Austen's time—or her novels, at any rate—this dance, known to Miss Austen and her characters as the Sir Roger de Coverly, signals the end of a ball.
"I guess it's over," I said to Ashleigh. "Better call Zach."
"Zach?" asked Ned.
"Our ride home," I explained.
Ashleigh retrieved her purse from behind the knight, fished out her cell phone, and handed it to me. "Here, you do it—just hit redial," she said, grabbing Parr's arm. Ned offered me his again.
The Sir Roger de Coverly is a complicated and vigorous dance: no easy thing to get through while talking on a cell phone. Still, I managed somehow, and by the time the dance brought my friend and me back within talking distance, I was able to report that Zach was on his way.
**_Chapter 7_**
_An unglass slipper_ ~ _A Farewell to Forefield_ ~ _I eat the Pancakes of Anguish._
_**T**_ he boys insisted on walking us to the gate, where we had told Zach to meet us. Clumsy in my silver pumps, I stumbled going down the steps. My right shoe flew off. Parr caught me by the elbow. "Careful, Cinderella," he said, retrieving the shoe. I stretched out my hand for it, but he held it back for a moment. "Should I keep this, in case I need to find you again?" he said.
"If you do, you'll have to carry me to my pumpkin."
"Don't tempt me," he said. Kneeling, he held the shoe in front of my foot. I couldn't decide which he was more like: a knight in a fairy tale, or an old-fashioned shoe salesman—the kind your grandmother might have taken you to when you were little, who measured your feet with a cold metal sliding device.
The knight, I decided. The salesmen, I remembered, always had a bald patch clearly visible from above; Parr's hair shone thick and pale in the moonlight.
Parr eased my shoe over my heel with a little wiggle. He rose and took my arm again, holding me steady as I picked my way downhill along the grassy edge of the road. We caught up with Ash and Ned, who were chattering about which Wet Blankets songs would make the best waltzes.
The rest of the walk went by in an instant. "Gentlemen, we cannot thank you enough for your gallantry," said Ashleigh when we reached the stone lions.
"Hey, it was our pleasure. Next time, though, don't give Wattles the satisfaction of gobbling at you—call first or drop an e-, and I'll make sure you get official invites," said Parr. "Here—got a pen, Noodles?"
Ned selected a small felt-tip pen from the items in his pocket.
"Paper?" asked Parr, fishing around in his own pockets.
"Here," said Ashleigh, thrusting her hand into his. "Write on my palm. I always do."
Zach drove up as Parr bent over Ashleigh's hand. "Oho! Grandison Parr!" he said, leaning over to pop open the passenger-side door. "So that's how it is, is it? Are you treating my little friends right?"
"Little friends, indeed!" said Ashleigh, waving her hand to dry the ink and flouncing into the car. "Mr. Parr is treating us with a great deal more respect than _you_ do, Mr. Liu. He and Mr. Downing rescued us from a particularly nasty adder and stood up with us for a quadrille, a waltz, and the Roger de Coverly. He is entirely a gentleman."
"Yeah? Glad to hear it, because that's one ass I'd rather not have to kick. I'm not saying I _couldn't_ , but it would be a challenge. Black belt yet, Parr?"
"No, don't worry, you're still king of the hill," said Parr. "I'm glad to see the Hunkajunk is still in one piece," he added. "But if you're so worried about the girls' safety, why are you driving them around in that thing?"
I was astonished to hear him speak that way about Zach's pride and joy. Last year Haichang Liu had passed on to his son the family's old—or, as Zach prefers to call it, _vintage_ —Saab, as an early graduation present when Zach got into Cornell. Zach spent so much time tweaking, tuning, and polishing it that I was surprised he had managed to graduate afterward.
But Zach just laughed. "Jealous? Learn discipline, young lion, and someday you too may be worthy of such a car. Come on, Julie, get in."
"Hai, Sensei," said Parr, giving a little martial-arts bow, palms together. He opened the car door, helped me in, and handed me the end of my wrap, which was trailing out.
"Thanks so much for everything," I told him. "You too, Ned."
"No, thank _you_ ," said Ned, poking his head in Ashleigh's window. "I never dreamed I'd actually enjoy the Founder's Quadrille. I'm glad you two decided to crash."
"Me too," said Parr. "But once is enough for one evening. Don't let Zach wrap you around a tree—use that e-mail address to let us know you're okay, would you? Cparr@forefield.org."
"Yeah, yeah, get going before I wrap _you_ around a tree," said Zach.
Parr shut the door and gave the car's rear end a little pat, like a cowboy with his horse, to send us on our way.
"So you know Grandison Parr?" asked Ashleigh.
Zach nodded. "He's a smartass, but a pretty fair swordsman. Decent guy on the whole. More than decent, actually—he helped me push the Saab all the way uphill to the garage when she broke down near the dojo last summer. Of course, he thinks that gives him the right to call her the Hunkajunk. Smartass. But he seems to like _you_." He gave Ashleigh a penetrating look.
"How long have the two of you been acquainted?" she asked.
"Oh, three or four years, I guess."
"How did you meet?"
"He practices kendo at the dojo."
"How could he? I was under the impression that the Forefield authorities kept their students locked up on the hill," said Ashleigh.
"No, they let them out for things like that. Haven't you ever seen them rowing on the river or riding around on horses in those ridiculous outfits? Besides, his family has a weekend house not all that far north, so he's around for part of the summer."
So that was why I'd seen him in town before school was in session.
"Are there any girls at the dojo?" asked Ashleigh.
"A few. Not as many as the guys, but a couple of the teachers are women, and there's a women's self-defense class that's pretty popular. Some of the karate classes have a fair number of girls in them. Why?"
"I was thinking kendo might be fun."
I was surprised to hear it. Didn't she realize that the martial arts uniform consisted of a short, bathrobelike tunic over loose trousers? How did she expect to kick an assailant without displaying her lower limbs? Was it too much to hope that we might be in for a craze change already?
"I think aikido would be more your thing," said Zach. "It's all about turning your enemy's strength against him, so your size doesn't matter so much, and face it, you're pretty little—in most ways, anyway. The main thing is balance and discipline."
Balance and discipline, I reflected, were not chief among Ashleigh's virtues. However, the conversation having left the riveting topic of Grandison Parr, it soon ceased to hold my attention; for the rest of the brief ride home I stared out the window at the dark trees, reliving the hours just past and musing on the uncertain future.
The next morning—Saturday—I awoke to feel bouncing near my toes. I opened my eyes in wonder. Ashleigh, and so early! I could count on one hand the times she had willingly gotten up before me—and two of those times she had forgotten to turn back the clock for daylight saving time. Her enthusiasm must have reached quite a peak.
"There! Admit it. Was I not right to insist on our attending the dance? Did I not tell you that you would meet your Bingley and I my Darcy? Was he not _wonderful_? His charm, his gallantry! Come on, get up! Let's go give Samantha back her handbags and see if Zach's still there. Maybe he can tell me more about Darcy."
"Okay, okay. Ouch! I'm coming, you don't have to pull my feet off," I said. I was a little surprised to hear Ashleigh refer to Ned as Mr. Darcy. The square-set young composer seemed sweet enough, but nothing like the proud, aristocratic, icy-fiery hero of _Pride and Prejudice_. Nor did tall, teasing Parr seem in the least like the insipidly agreeable Mr. Bingley. And why should Zach be able to tell us anything about Ned, when Parr was the one he knew? I attributed Ashleigh's confusion to Love. The tender passion is not known for sharpening the intellect.
I packed up some schoolbooks and a favorite sweater—I was spending the rest of the weekend at my father's—and wheeled out my bike. Ashleigh rode beside me, chattering swoonily about the dances, the dresses, the music, the ballroom, and—most of all—the gentlemen. Mr. Darcy, she maintained, was the picture of perfection, although she generously allowed "my" Mr. Bingley to be an intelligent, lively, pleasant fellow. I smiled to myself at the thought of anyone preferring Ned to Parr, although certainly he—Ned, that is—seemed made for Ashleigh, with his musical enthusiasm and pocketful of peculiar objects. They even looked a little like each other, with the same curly hair and warm brown eyes.
When we reached the Lius', the doctors were planting bulbs in their garden. "Hello, girls," said Lily. "Samantha's in the kitchen. We just finished eating pancakes, but there's some batter left—you can have it if you're hungry."
"Mmmm! Thanks, Dr. Lily," I said.
"You better cook it first," said Haichang.
"You don't think pancake batter would make a good drink?" I asked.
"Is Zach around?" asked Ashleigh.
"He must be," I said. "The Saab's here."
"He's still sleeping, the lazybones," said Lily. "Serve him right if you eat up his pancakes. Go on, before the griddle cools down."
Sam was putting the butter away in the fridge, but she gladly took it out again when she saw us. She spooned batter onto the griddle.
"How was the boy hunt?" she asked. "Zach says you landed a couple of live ones."
"We did indeed have the good fortune to make the acquaintance of two young gentlemen of high character and pleasing appearance," said Ashleigh.
"Hmm, not quite how Zach put it. What about you, Julie? Did you have a good time? Meet any lofty and pleasing gentlemen?" asked Samantha.
"As a matter of fact, except for the really embarrassing parts, it was surprisingly fun. The guys we met were really nice—one of them was that friend of Zach's you and I ran into in the Sports Barn. I overheard some of the girls making fun of our dresses in the bathroom, but none of the guys seemed to mind how we looked. Lots of them danced with us, anyway. On the whole, it was one of the more successful of Ashleigh's marshmallow-headed schemes."
Ashleigh gave me her Reproachful Look. "You met Grandison Parr before?" she cried. "Why did you not tell me?"
"Oh—I—There wasn't much to tell. We just passed him at the mall—he didn't even talk to us, and Sam couldn't remember his name."
"Oh, that guy? I like him," said Sam. "But be careful. If this were a real Jane Austen story, one of those guys would turn out to be a cad who's only after your money."
"Scratch that—for me, anyway," I said.
"Or your honor, maybe—or just your clothes—remember that movie _Clueless_?" added Sam.
"Yes, well, if this were _Clueless_ , we'd all fall in love with Zach," said Ashleigh scornfully, flipping the pancakes.
The person in question chose that moment to make his appearance in the kitchen, clad only in pajama bottoms and looking pleased with himself. Zach obviously shares the widespread opinion that his shirtless torso is a magnificent sight.
"Good plan! I wish you would. Then you'd be nice and give me those," he said, reaching for the pancakes with a fork.
Ashleigh fended him off with her spatula. "Keep your fork to yourself," she cried.
"I bet if I were _Grandison Parr_ you'd let me have them. No, more than that—you'd make me my own batch. In heart shapes," said Zach, easily evading her spatula like the fencer he was. He skewered a pancake and crammed it into his mouth, then followed it with a chaser of syrup, drizzled directly from the bottle, which he held a few inches above his lips.
Ash jittered with indignation. "If you were Grandison Parr, you would never rob a defenseless female in this manner! You villain! You unspeakable adder! You are not fit to speak the name of the noble Mr. Darcy!"
Busy as I was admiring Zach's syrup caper, it took me a moment to realize what Ashleigh had said. As soon as I did, an electric shock went through me.
"Darcy," I gasped weakly. "Darcy—Parr?"
I bit my tongue to stop myself from revealing any more of my feelings before I had a chance to understand them myself. It was too late, however. Every eye was upon me.
"Why, yes, of course, Parr! Who did you think?" said Ashleigh. "Ned? Ned the Noodle—you thought _he_ was Mr. Darcy?"
"No, of course not, don't be silly," I protested. "Frankly, neither of them seems much like Darcy to me."
"Really? You certainly didn't say so earlier this morning. I seem to recall you agreeing with me when I asserted that Darcy was wonderful. Are you not protesting just a teeny, tiny bit too much? Methinks?"
Zach took up the cry. "Look, she's blushing! Oho! Sensible Julie isn't so sensible today, is she, now? Who would have thought those Foreskin boys would break _two_ hearts!"
"Stop it, you guys! I mean it! Ig—Ned—emphatic ig! I _really_ don't like him. I mean, I like him fine, but I don't _like_ him."
In my agony, it seemed, I had turned into a second grader.
Ashleigh gave me a look of happy condescension. "Now, now, my dearest Julia, I cannot see why you refuse to admit it. Ned is a very agreeable fellow indeed—almost as handsome as my Parr. The two of you are a perfect match, exactly the same height. And he likes you, Julie—you know he does. He danced the last dance with you, and the first dance. He tried to talk Parr into bringing you a Sprite instead of a ginger ale, so we could get back to you. And he even asked me for your e-mail address—well, he asked for both of our addresses, but I gave him yours. I could tell that was what he really wanted."
Could she be right? Could Ned have developed feelings for me like mine for Parr?
Samantha saw my discomfort and tried to help by turning the conversation from my affairs to Ashleigh's. " _Your_ Parr? Are you admitting _you're_ in love?"
Alas, Ashleigh's answer pained me more than all the previous conversation.
"In love?" said Ashleigh. "How can I answer _that_? If you believe—like our English instructress, Miss Nettleton—that true love comes only to those who, upon first meeting, speak together in rhyme and meter, so that their conversation produces a sonnet, then no. But I confess that never before have I encountered so gallant, so courageous, so handsome a gentleman as Grandison Parr. If ever there was a man born to capture my heart, then that man is Grandison Parr. And although modesty warns me to discount them, I believe I saw signs that he returned my regard. He danced the quadrille with me. He drew me apart from the others as he searched the campus from end to end for ginger ale, thus affording us quiet time together, accompanied only by Ned. He queried me most particularly about my childhood, my abode, and the society I keep, showing a keen interest in all my doings. And he took my hand in his to write his e-mail address on my palm—writing I preserve to this day, and will as long as hygiene permits it!" She held her hand up triumphantly, palm out.
"Yup, I saw that part," agreed Zach. "Well, aren't you the lucky girl! Won't you please, please give me another pancake? Surely I deserve a booby prize."
She shot him a look of scorn and handed the pancakes to me instead. But although I tried to eat as if nothing had happened, they stuck in my throat. As soon as I could, I escaped to my father's house to brood over my troubles.
**_Chapter 8_**
_I Renounce my Dream_ ~ _I maintain my Dignity_ ~ _I carry boxes_ ~ _I E-mail_.
_**W**_ as Ashleigh right? Had Grandison Parr, over the course of the previous evening, developed feelings for Ashleigh?
There could be no doubt about _her_ feelings for _him_. I knew that enthusiastic gleam in her eye all too well. Had I been deluding myself, daring to imagine that he might like _me_? Sitting on the bed in the room I shared with Amy's sewing machine, I went over the events of the previous evening in my mind, just as I had through the night. What a difference there was this time! Every clue that had raised my hopes could equally well dash them.
At first, Parr's promptness in rescuing us from the turkey-faced doorkeeper had seemed like evidence that my hero had noticed me, and maybe even liked me. But was that just wishful thinking? Wouldn't the gallant fencer have sprung to the aid of anyone in distress? Or maybe—I shuddered at the thought, then shuddered at myself for shuddering—maybe it was Ashleigh's daring and charm that had persuaded him to help us. After all, her liveliness, along with her rapidly developing maturity of looks, seemed to appeal to guys—especially in that crimson dress. Even Zach had noticed it. Why not Parr?
Then, Parr danced the first quadrille with her. I had put that down to her energy—she had pulled him onto the dance floor. But he certainly hadn't tried to resist, and they seemed to be enjoying it, chatting away. When he and I waltzed, our conversation seemed stilted and awkward. (Remembering the waltz, I felt his hand once again on my mind's waist and shivered with pleasure and distress.) The night before, when I looked back on our first conversation, I hoped its awkwardness might be due to our mutual attraction. Maybe he felt shy with me at first, just as I felt with him. But maybe not—maybe he merely found me dull.
Nobody could ever find Ashleigh dull.
Then there was Parr's long disappearance during the ginger-ale quest. At the time, I wondered whether he had been trying to abandon me entirely, but when he showed up with the elusive soft drink, I was touched. What a lot of trouble he'd taken for me, I thought. Now, though, Ashleigh's theory seemed equally likely: that he was trying to spin out his time with her.
The other apparent signs of Parr's feelings toward me—his friendly teasing, his disapproval when creepy Chris got too close, and his Cinderella remarks, which put him in the role of the prince—also melted away on closer inspection. I bit my lip to keep from crying with jealousy. Why did Ashleigh always get _everything_? Not only had she taken over my enthusiasm for Jane Austen, but now she seemed hell-bent on stealing my secret love!
For a long time I struggled with myself, feeling bitter resentment and condemning myself for it. After all, I could not question Ashleigh's generosity or the purity of her motives. When she fell for Parr, she had no idea that I had gotten there first. You could even say the whole thing was my own fault for not taking her into my confidence from the start. Ash would never have looked twice at a boy she knew I liked. She was too loyal. For my sake, she had even given up her plans to become a nun at age eight, when she learned that Jewish girls couldn't enter a Catholic sisterhood. If she had known my feelings, I believed she would have tried to suppress her own.
No, if somebody had to suppress her feelings, it should be me. After all, I was much better at it than Ashleigh. I would prove to myself, if it killed me, that I could be as generous as my friend.
Still, if Parr didn't see me in a romantic light, it didn't necessarily follow that he had chosen Ashleigh. She and I were far from the only ones who admired the handsome fencer. I remembered the Wharton girl in the bathroom with the crush on Parr. She considered him beyond her reach. She thought he was already taken. Well, perhaps he was—not by me or Ashleigh, as she seemed to assume, but by someone else!
And even if Parr's heart _was_ free, did Ash or I stand a chance with him? Impossible to say. For, as I gradually realized, he was stuck in Forefield, and we would have no chance to get to know him better.
Hopeless, hopeless, all of it. The world that had seemed so bright and sharp faded to gray. Even the leaves blazing outside the window looked washed out, as if fall no longer mattered. I lay back on the bed, closed my eyes, and let tears leak into my ears.
"Jul—Oh, napping?" said my stepmother disapprovingly, coming into the room with a perfunctory knock. "Would you mind helping me downstairs, sweetie? I'm not supposed to lift anything."
I awarded myself half a dozen imaginary dollars: one for not answering snappishly that Amy had interrupted a period of quiet, mindful contemplation; two more for not telling her she could perfectly well carry her own groceries; and the rest for not smashing the furniture in my despair.
I spent the afternoon stowing bulk packages of toilet paper, diet soda, and other scintillating commodities in the laundry room and hauling junk from the other basement room up to the attic. Although I didn't have the emotional strength to ask what it all meant, I hoped the I.A. was preparing a new home for her sewing machine, so I wouldn't have to live with it in my room. She had been using it quite a bit over the last few weeks; the table next to it was covered with pastel-colored fabric scraps.
I worked obediently, the physical activity distracting and soothing me. Still, my sorrow preyed on my mind, killing any urge to socialize; when Ashleigh called on Sunday, I even let Amy tell her I was too busy to talk.
It wasn't until Monday evening—Columbus Day—that I summoned the strength to check my e-mail. I found this message waiting:
From: Downing, Ned <edowning@forefield.org>
To: Julielefk@hotmail.com
Sent: Sunday, 2:21 P.M.
Subject: upsidedown headmasters
hi julie,
hey it was fun dancign with you and ashley. if you guys snuck into the great hall and turned the headmasters upside down would that make them feetmasters? if you hung them on teh stairs would they be stairmasters? i hope you'll come help i have a plan but i'm not sure it'll work. pleaes say hi to ashley for me. do you have her email address?
best wishes
ned
Oh, great, I thought. The first time a boy ever invites me to hang out with him (or, more precisely, hang pictures with him), it's (a) the wrong boy, who (b) can't type, and (c) has the world's least romantic ulterior motive—a practical joke.
For a painful minute I considered going along with his plan, whatever it was: since it would have to take place on the Forefield campus, there was a chance I would see Parr again. But such pleasures, I told myself sternly, were not for me.
How, then, should I answer Ned's message? Sending him Ashleigh's e-mail address could only lead to more excruciating escapades. Given half a chance, the Enthusiast would surely insist on flipping the portraits, not only from her love of mischief, but from the same motive I was resisting: the hope of seeing Parr. However, it seemed cruel not to respond to Ned at all—what if I was right that he had fallen for Ash? And if he had, I caught myself thinking, perhaps he could win her away from Parr, leaving the field open for me. Hastily I squelched the thought.
After some hesitation, I wrote back:
Hi Ned,
Thank you for your message. I had a great time at the dance too. Overturning the Forefield headmasters sounds a little beyond me, though. Ashleigh might be up for it—but please don't let her get mauled to death by fierce turkeys!!!! Her address is sirashleigh@hotmail.com. (Remember to spell her name with an eigh, or it won't get there.)
I added, "Please give my regards to Parr," deleted it, undeleted it, deleted it again and added, "Hi to Parr," deleted and undeleted that a few times, reinstated "Please give my regards to Parr," signed the message, and hit <send>.
By Tuesday morning, while no more cheerful, I was at least calm enough to meet Ashleigh and pretend things were normal. Our first chance to speak came during lunch. "There you are," she said excitedly, slipping into the seat next to me. "Why didn't you call me back? Didn't Amy give you my message?"
"What message?" I lied.
"Oh, that certified public adder! Of course she wouldn't give you my message, even though I told her it was important. I was calling to say I received an e-mail from Him."
"Him who—Ned?"
"Ned! Faugh! How your mind does run on Ned! From Parr, of course."
My sandwich—one of Amy's ordinarily delicious pesto-and-roasted-vegetable specials—turned to leaf mold in my mouth.
"Really? What did he have to say?"
"He was glad we reached home in safety—I e-mailed him right away, as he had requested. He described his experience with kendo and recommended it as a sport well suited to an active young lady. He passed along greetings from several of the gentlemen with whom we danced. He also praised my dancing—he said he had enjoyed my 'unique approach to the quadrille.' What do you think it means? Do you think he _likes_ me?"
On the whole, I did tend to think so. Yet I feared that in her enthusiasm Ashleigh might have mistaken a mild regard for a more intense emotion.
Or did I merely _hope_ so?
"Sounds like it might be a good sign," I said cautiously.
"Doesn't it? I really think it is. Oh," added Ash, "and he said to tell you hi."
Yvette and Yolanda joined us, and the conversation turned to more general topics, such as the impossibility of getting parts in the school production of _West Side Story_ when competing against people like Michelle Jeffries and Cordelia Nixon, who ran the show like a popularity contest.
After school Ashleigh wanted to delve deeper into the subtleties of Parr's message, but I begged off, pleading homework. I hurried to my father's house. For once the I.A. was in a subdued, even glum mood that matched my own, and she left me alone. For several hours I balanced chemical equations, memorized French verbs, and tried to anticipate Ms. Nettleton's opinion on the death of Romeo's cousin Tybalt. When it came to history, however—a chapter on European weapons and military strategy in the Middle Ages—I couldn't concentrate. The subject reminded me too much of Ashleigh and Parr.
I checked my e-mail again, and there it was:
From: Parr, Grandison <cparr@forefield.org>
To: julielefk@hotmail.com
Sent: Tuesday, 9:45 P.M.
Subject: Help me stop them
Dear Julia,
Your friend Ashleigh gave me your e-mail address—I hope that's okay?
I was relieved to hear that Zach Liu got you home in one piece. Or I guess I should say two pieces, since there are two of you.
I take it all the shoes arrived safely too?
Which brings me to the topic of this message: safety.
I assume Ashleigh told you about the plan she and Ned are cooking up to rearrange the portraits in the Great Hall. Is there any way you can talk her out of it? Ned's already in trouble for miking the stalls in the faculty bathroom and wiring them to the PA system. Wattles has it in for him. I'm worried that if Ned goes through with the portrait thing, he could lose his scholarship.
I tried to talk him out of it, but he says he doesn't want to disappoint Ashleigh. Can you stop her? She's obviously strong willed and high-spirited, but you seem like a sensible person, someone she might listen to.
I'm glad you came to the dance last weekend. I've never had such a good time at a Forefield social event. I wouldn't have thought it was possible. If only you'd crash our classes, I'd even look forward to trigonometry.
Sincerely yours,
C. Grandison Parr
My pulse beat hard in my throat as I read this message, especially the last paragraph. He was glad I had come to the dance! He'd look forward to trig if I were there! The first two times I read the message, I hardly took in the main subject, Ashleigh and Ned's dangerous plan.
The third time through, however, I was struck by a painful thought: In English, _you_ can be plural as well as singular. Perhaps Parr meant that he was glad to have danced with me _and Ashleigh_ —that the presence of me _and Ashleigh_ would make trig bearable. After all, that was how he used _you_ in the second sentence, the one about Zach getting us home in two pieces. And Ash was the topic of the message, its entire purpose. Probably the _you_ in question included me only as an afterthought: probably it was meant to express more strongly Parr's admiration for my lively friend, whom he had already praised.
To him, I was nothing but a sensible person.
All right, then, Julia Lefkowitz, I told myself: BE sensible.
Dear Grandison,
I wish I could help, I really do. But in the 10 years I've known Ashleigh, I've only been able to talk her out of one scheme—the time she wanted to jump off the roof wearing papier-mâché wings. I convinced her to try it with her doll first. After Arabella's (the doll's) head cracked open, Ash didn't speak to me for a week and a half. The first thing she said to me afterward was that she'd never listen to me again. And she pretty much never has.
Did you try to talk her out of the plan yourself? I think she'd be more likely to listen to you than to me. I know she admires and respects you.
Or maybe you could somehow get Wattles to lock the hall extra carefully for a while?
I'm sorry I can't help more, especially since Ash and I owe you so much for saving us from Wattles last week.
Sincerely yours,
Julia Lefkowitz
I read the message through, deleted the sentence about Ashleigh admiring Parr for fear it might be betraying her trust (or, muttered a little voice inside me, for fear it might give him ideas), and clicked <send>.
**_Chapter 9_**
_Rumors of rivals_ ~ _I withdraw_ ~ _I join up_ ~ _a Surprising communication from my Mother_ ~ _a Shocking communication from my Stepmother_ ~ _Ashleigh too_ ~ _I Endeavor to come to my Senses._
_**H**_ ave you ever tried to avoid your best friend: the girl who knows all your secrets (or all but _one_ ), the girl who up to now has spent every free moment by your side and is liable to appear at your window at any hour of the day or night with acorns in her hair, expecting to be admitted?
If so, you know how difficult the days that followed were for me.
Ashleigh wanted me to read and interpret all her e-mail messages from Parr—and there were a ton of them. She never tired of combing through them for clues to his feelings, or of dreaming up schemes to see him in person.
"Julia! Come read this—I need your advice," she said one afternoon as I sat on her bed doing my math homework.
"What is it?"
"A disturbing message indeed. I need your help interpreting it."
"Is it another e-mail from Parr? I don't know, Ash, aren't those sort of private?"
"My dearest Julia! You know I have no secrets from you! Anyway, it's from Samantha Liu."
"Oh, okay," I said. Looking over her shoulder at her computer screen, I read:
Hi Ashleigh—
I asked around for you; see below. Look how you're ruining my reputation! I'll let you know if I hear anything else.
—Sam
> Oh, so your "friend" wants to know about Grandison Parr, does
> she? Really, Sam, I wouldn't have thought he'd be your type—
> isn't he a little romantic for you? I mean, he writes *poetry*!
> Well, you're in good company, according to my sources at Miss
> Wharton's. He's—what's that thing they all chase after in
> Quidditch? The Golden Snitch? Unfortunately, they say he's
> going out with a tall blonde. I haven't been able to pinpoint
> which one, but it might be Emily Wardwell or Kayla Thwaite—
> they were both seen with him at the recent Forefield dance.
> Sorry! But don't despair. I'd back you against any Wart, no
> matter how tall and blonde. Do I get a reward for this? Maybe
> you could get me a date with that yummy brother
> of yours?
> Just kidding, sort of . . .
"Well?" I said.
"Well, what do you think?" said Ashleigh. "Do you think it's true?"
"I don't know, Ash. What do _you_ think?" I selfishly hoped it was. I would far rather have a Wart girl as a rival than my beloved Ashleigh. Maybe then Ash and I could even share a companionable gloom.
But Ashleigh quickly cheered herself up: her enthusiasm comes with a hearty dose of optimism. "I don't know!" she said. "My impulse is to believe Miss Liu, whose judgment is remarkably sound. However, we have no information about her friend's judgment—we don't even know who her friend _is_. It's just like Samantha to be so discreet! Perhaps those 'sources' may be mistaken. Parr's last message sounded very encouraging. Listen to this: 'Sounds like you and Julia had a great time apple picking last weekend. I wish I'd been there.' What do you think that means? Do you think he _likes_ me?"
These discussions made me so miserable that I tried my best to avoid them, and when I found that impossible, I began to avoid Ashleigh.
At least one good thing came from her devotion to Parr: she agreed to give up the plan to monkey with the headmasters. I heard this first in an e-mail from Ned, who blamed me for her change of heart. Later Parr made the same mistake and thanked me. I answered Ned politely but briefly, and Parr not at all, although I found it hard to keep my cursor away from the <send> button. I couldn't let myself carry on a correspondence, however innocent, with Ashleigh's beloved. After a few unanswered messages, Parr stopped writing.
I threw myself into my schoolwork, the best way I knew to numb my mind. I took dawn bike rides and hikes in the hills among the ever-barer trees, sneaking out while Ash was still in bed. I spent long hours working in the storeroom and on the computer, attempting to straighten out the inventory and accounts of Helen's Treasures. And holding my inner nose, I joined the staff—or, as Ms. Nettleton called it, the _crew_ —of _Sailing to Byzantium_ , our high school literary magazine.
The Nettle gave me the position of assistant editor and began smiling at me in class. My father was overjoyed. He rained down little pellets of smugness, like a squirrel shelling nuts overhead. "I'm glad you've started taking a real interest in your academic career," he beamed. "Amy will be so proud."
When Ashleigh expressed her astonishment at my extracurricular activity, I lied to her (a painful new habit), explaining that Dad had threatened to withhold my allowance unless I joined up. She generously offered to keep me company, but I told her it would only make the whole thing worse if she suffered too.
Even without her, I suffered. The worst was the loneliness. Though surrounded by people, I felt utterly isolated. I missed my friend, the only person who really understood me, yet in her company I felt lonelier than ever.
Early one evening, as I sat on my bed staring out the window at the pattern made by the oak branches—a lattice of bars between me and heaven—I heard a knock at my door.
"You in there, honey?" asked my mother.
"Uh-huh—come in."
"It's so dark in here! Why don't you turn on a light? No, leave it if you like this better—listen, I want to talk to you." She sat down on the end of my bed and curled her legs under her. "I've noticed that you haven't been yourself for the last few weeks, and I think I know why."
"You do?" I said. Had my secret somehow gotten out? A tingling alarm swept over me, accompanied by a soft cascade of relief, as if something tight had loosened. I felt tears well up in my eyes.
Mom put her arms around me and stroked my hair. "I'm sorry, honeybear. I'm so sorry. I know it's been tough on you with your father gone and money being tight. You take it so hard—you're such a grown-up kid. But it's not your job to take care of everything. That's up to me, I'm the mom here. And honey, I promise you it will all be all right. I'm not going to let us starve. And I'm not going to stay dependent on your father, either. I realize Helen's Treasures isn't working out the way I hoped, so I've been looking for a job. No, just listen. I've had a few offers I could have taken, but I held off because I was waiting to hear from the one I really want, teaching art. But even if I don't get it, there are other things I can do, so you don't have to take everything on your big little grown-up freckled shoulders. Okay, honey? Shh, shh—there, there. Better now? I was going to wait to tell you until I heard about this job for sure, but you've seemed so stressed that I thought I'd better talk to you now."
As she spoke, I felt the relief and tension swirl around within me, trading places like a couple in a quadrille. My secret was safe! A reprieve!—yet a disappointment too, to find myself once again deeply alone.
I wiped my eyes and pulled myself together. "That's great, Mom," I said.
Next it was Amy's turn. On Tuesday she cornered me behind the sewing machine, where I was doing my math homework. "I know why you've been so sad lately, sweetie, and I'm touched, I really am," she said. "I know how disappointed you must be after all these years alone, and especially with all the help you've been giving me getting the room ready. I wish I had good news for you now. But I promise, your father and I are doing everything we can, and I'm sure we'll be successful sooner or later."
"You are?" I asked, not sure what she was talking about. I had a strong hunch it wasn't anything good, though.
"Oh, yes, we're doing everything we can. After we lost the baby, we went to see a new fertility specialist in New York who has an excellent track record with couples in our situation."
I stared at her. What was she talking about? What baby?
"We've been following his instructions carefully—which, I must say, we've both enjoyed," she continued, with a coy smirk that made my stomach lurch. "And on the plus side, at least until I get pregnant again, I can help you carry your things down to your new room in the basement. Have you chosen what color you want yet? I thought I'd paint this room a nice pastel yellow, since we don't know whether it'll be a boy or a girl. I always think yellow goes with everything. What do you say, should we stencil a border of ducks just under the ceiling, so your little brother or sister will have something to look at? Or stars on the ceiling?"
For a long time I was speechless. The I.A. didn't notice—she was too busy planning where she would put the changing table and the bassinet. To lose in one stroke my status as an Only Child and my airy (if sewing-machine-ridden) bedroom! To be banished to the basement! So that was why she'd been emptying out that dark little room downstairs—not to hide away her sewing machine, but to hide away _me_!
And what could my father possibly want with another child, when he hardly bothered to talk to the one he already had?
Ashleigh caught me Thursday morning as I was exiting the window for an early run. "Hang on, Jules," she said, climbing down to meet me at the tree's roots. "I need to talk to you." (Oh, no, I thought, Ashleigh too!) "Is something the matter?" she asked. "Are you okay? I almost get the feeling you've been avoiding me. Did I do something wrong? Is there any way I can make it better?"
I was overcome with guilt. My best friend had taken the trouble to get out of bed before her alarm went off just to express her concern about me. She had even used ordinary speech, rather than her high-flown Austenish. She heaped blame on herself—blame that belonged to me. I sternly resolved to take myself in hand. My period of pouting must cease. What were my feelings for a guy I had spoken to only one night, compared to the chief friendship of my entire life?
"I'm sorry I've been such a pill," I said. "Of course it's not you. Family things, and other stuff like that. I didn't mean to take it out on you."
Ashleigh looked at me keenly. "Other stuff like that, hmm? I think I know what's wrong. It's Ned, right? You're depressed because you can't see him. I know exactly how you feel. I wish I could see Parr too. E-mail helps, but it isn't enough. I wish we could be together in person! Under that dignity of his—that beautiful athletic bearing—he has such depths of kindness and strength, such good humor, such true manliness . . ." For a long time she continued in that vein, brushing twigs from her pajamas, as I forced myself to listen, and even to smile.
**_Chapter 10_**
_Et tu, Samantha?_ ~ _An Encounter with a Pirate_ ~ _We prepare Speeches_ ~ _Forefield again_ ~ _Disaster_.
_**A**_ nd where was Samantha during my period of grief? Not around very much, especially after evening gymnastics practice got moved to Tuesdays, the time when I was most likely to see her. In my worst moments I considered hunting her down and laying my troubles at her feet, but in the end I always balked. My pain felt too raw.
So when she firmly straightened the drooping feather on my flapper hat as we were helping set up for our fathers' Halloween party and said, "Cheer up, Julie, I know how you feel, but it's not worth breaking your heart over," I almost dropped my bowl of gummi syringes.
Halloween may seem like a grisly theme for a pair of pediatricians to choose for their annual party, but it's very popular with their young patients. The greatest draw, I think, comes from the possibility that if they chose, Dad and Dr. Liu could spike the tomato juice with real blood.
"What's not worth what?" I stammered, thinking, Not Samantha too! Was everyone in my life hell-bent on interpreting my pain to suit their own needs?
"Cute blond princes up on a hill. Not worth crying over. The world's a big place—even Byzantium's a big place, comparatively—it's crawling with guys if you really want one. You don't need to get stuck on one particular unavailable guy. Unless you enjoy the melancholy, of course."
Sam is uncanny. It's as if she reads minds.
Ashleigh arrived before I could answer and dragged me off to help her arrange the jack-o'-lanterns to mimic the lighting effects of early-nineteenth-century candelabras. Then other guests arrived and kept her busy explaining that she was Jane Austen—Jane _Austen_ , the writer—not a witch, a ghost, or Martha Washington.
Was Sam right? I wondered. Did I enjoy the melancholy? This was certainly a good time of year for it, with gusts of autumn wind blowing the storm clouds around and slapping the fallen leaves wetly against one's knees. I decided to take Sam's remarks to heart. When Ashleigh and I went to Emily Mehan's Halloween party the next night, I even tried flirting with Seth Young from my English class, the managing editor of _Sailing to B_. He was wearing a pirate costume, which made him look almost palatable. The red bandana he wore on his head gave his olive skin an appealing glow, and his blousy pirate shirt made him look lanky instead of skinny. An eye patch completed the romantic picture; I noticed for the first time that he had a nice nose. But his self-importance kept popping out from beneath the dangerous swagger he affected, and when he put his arm around me in the Mehans' backyard, I shrugged it off. Sam's advice might be good, but my heart just wasn't in it.
Ashleigh's mother came to pick us up before Seth could make any further moves, so I was spared having to reject him definitively and make future _Sailing_ meetings awkward. At the lunchtime meeting the next day, he sat next to me but would not meet my eye. His face retained traces of pirate makeup, principally eyeliner, which I found obscurely embarrassing. When the fourth-period bell rang, I left quickly to avoid any possibility of conversation. He got up as if to follow, but evidently changed his mind when he saw Ashleigh waiting for me outside Ms. Nettleton's room.
"You've got to see this," she cried, grabbing my elbow and pulling me downstairs to the announcement board, which the Gerard twins were inspecting in postures of excitement (Yolanda, I assumed) and mild interest (Yvette).
"Look!" commanded Ashleigh with a sweeping gesture.
"What?" I asked. The twins' beaded heads blocked my view.
"Auditions," answered Ashleigh joyfully.
"Auditions?" Why would Ash care about auditions?
"At Forefield, for their musical," she elaborated.
"Forefield, get it?" said Yolanda. "The boys' school. That means not a lot of people auditioning for girls' parts. Wholly crisp—no Cordelia Nixon or Michelle Jeffries, 'cause they're in _West Side Story,_ and who else from here is going to bother? I bet if we just show up, we can get parts, and if you can carry a tune, you can be a star. How about it, you want to be the heroine?" she asked her sister.
Yvette shook her head. "You can be the heroine. I'm playing the most important part," she said.
"What part's that?" said Yolanda. "They're not going to let you be the hero, silly, they have plenty of boys. And it says here, 'Directed by Benjamin Seward.' "
"No, silly, the audience."
I thought Yvette had the right idea. Acting in a play—a musical, no less—was a frightening thought for someone as shy as me, not to mention the danger of a painful meeting with Parr. How much easier it would be, if Ash would only let me, to stay home and brood.
But _that_ , I told myself, I must not do. No, seeing Parr and Ash together might be good for me, like cauterizing a wound to make it stop bleeding. In fact, I found myself almost hoping that I _would_ see Parr: surely, a tempting little voice whispered, it would help me get over my troubles.
The next problem Ash and I faced was finding suitable monologues for our auditions.
Ash naturally first thought of Darcy's proposal in _Pride and Prejudice_ —the speech that begins, "In vain have I struggled. It will not do. My feelings will not be repressed. You must allow me to tell you how ardently I admire and love you."
Unfortunately, as we found when we consulted the book, that's also where the speech ends. Jane Austen tells us that "the avowal of all that he felt and had long felt for her immediately followed," but she doesn't specify what he says. The rest of the scene takes the form of a dialogue between the proud hero and offended heroine—deeply interesting to readers, but useless to auditioners.
We considered and rejected various alternatives, such as Mr. Collins's letter announcing his visit to the Bennet family and Lady Catherine de Bourgh's howl of disapproval at the thought that Elizabeth might become her niece by marrying Darcy. They were all either too brief or too deeply embedded in the novel's plot to stand alone.
"The problem is, it's a novel," I argued. "Don't you think we'd have better luck finding monologues if we looked at plays instead of books? Or movies, even."
"No drama could be more dramatic than the works of the great Miss Austen," said Ashleigh dismissively.
"Let's at least go down to the video store and see if we get any ideas," I urged.
She shot me the Mad Gleam. "My dear Julia, I believe you may have hit upon the solution! Perhaps some playwright or screenwriter may have supplied Miss Austen's missing words!" We rented three different _Pride and Prejudice_ s. After some discussion and much poking at the rewind button, Ash picked the Colin Firth version of Darcy's proposal and scribbled out a transcription.
For my audition piece, I chose Mercutio's Queen Mab speech in _Romeo and Juliet_. It's part of a scene in which Mercutio, my favorite character, mercilessly teases his cousin Romeo about being in love. He attributes Romeo's mooniness to a visit from Queen Mab, the fairy responsible for dreams. I chose it because I knew it practically by heart, having written a paper about it for the Nettle. Still, I tended to agree with Yolanda that the play was at least as silly as it was beautiful. The whole tragedy was so unnecessary! If Romeo and Juliet had just _talked_ to each other, nobody would have had to die.
Besides being easy for me to memorize, the speech also had the advantage of being by Shakespeare, and therefore tough for a modern girl to deliver and even tougher for a modern listener to follow. Although I refused to let myself flub the audition on purpose, I secretly hoped that the difficulty of the material would keep me from getting a part. Then I'd be spared the pain of watching my best friend's budding relationship with my lost love.
Mrs. Gerard drove Yolanda, Ashleigh, and me to Forefield for our auditions. As the car wound up the drive toward the school on the hill, I felt my insides quadrilling in a way that couldn't be explained by mere motion sickness.
"Break a leg, girls," said Mrs. Gerard, dropping us in front of the R. McNichol Robbins Theater Arts Center, behind the main classroom building. We pulled open the heavy bronze doors and followed signs into the theater, where a group of people clustered near the stage.
A spotlight reflected brightly off the hair of a slim, tallish figure, transforming my inner quadrille into a gymnastics meet. When he stepped aside, however, I saw that he was not the person I half hoped, half dreaded to see, but a brown-haired boy about the same height.
"Ashleigh! Julie!" called a male voice from the other side of the room. It set the trampolines going again briefly until I recognized it a split second later as Ned's bass. He bounded up the aisle to meet us. "You made it! Come meet Benjo and Ms. Wilson."
Ashleigh introduced Yolanda, and we followed Ned down to the front of the theater. Aside from one pale creature in a Sacred Heart uniform, we three were the only girls. "Hey, it's Erin from Sacred Heart," said Yolanda, running up to greet her. Chris Stevens—the boy who had shared my planter at the Columbus Cotillion—lounged beside Erin. He winked at me. Boys of various sizes punched each other and squirmed, or sat apart reviewing their monologues; some stared at us out of the corners of their eyes.
Benjo turned out to be the tallish, brown-haired guy who had so alarmed me. After a few minutes, during which a bell rang somewhere and additional aspiring actors arrived—including another Sacred Heart girl, this one quite young—he called for silence and addressed us. "Okay, let's get started. I'm Benjamin Seward, and I'll be directing _Midwinter Insomnia_ , an original musical by Barry Davison, with music by Ned Downing and lyrics by Grandison Parr. That's Barry over there, and Ned's next to him, and Parr—where's Parr?—oh, I guess he's still at fencing practice. Anyway, most of you know Mr. Barnaby, our faculty adviser, and Ms. Wilson, our musical adviser." He indicated a bald, bearded man with a barrel chest and prominent ears and a slender, petite woman with straightened hair pulled back into a knot at her neck. Benjo continued, "When I call your name, please come up onstage and give your music to Tyler at the piano. All right? Alcott Fish."
A small boy presented himself, cleared his throat, sang "You're a Good Man, Charlie Brown" in a pretty soprano, recited a speech from the same play, and sat down again. The four directors whispered together, then called the next boy.
During the auditions that followed I had time to imagine various dire scenarios in which I fell off the stage, forgot my lines, changed key halfway through my song, fainted, laughed hysterically, or compulsively shouted _fire_ ; at last I decided to dull my thoughts by running through my speech over and over in my head.
When Erin's turn came, I stopped and paid close attention. By then Shakespeare's words in my head were beginning to sound dangerously like nonsense. She sang "My Favorite Things" with all the corn-syrup sweetness it deserves; her speech, from _The Glass Menagerie_ , was similarly well articulated, sincere, and over-sweet.
Next came a striking boy with a dark complexion and a beautiful baritone. Then, after a few so-so singers and two pretty good younger boys, it was Yolanda's turn. Her rich alto, surprisingly sultry in someone so young, made a strong showing in "Too Darn Hot," from _Kiss Me, Kate_ , and her speech from _Raisin in the Sun_ moved me almost to tears.
Ashleigh, too, acquitted herself well, with a loud and tuneful rendition of "Take It Back" and a loud and passionate rendition of the Darcy proposal.
Then it was my turn. I made it onto the stage without falling over and handed my music to the boy at the piano. Things started out well enough, but I began to have second thoughts as I sang "It's All Right with Me." "It's the wrong time and the wrong place," the song begins (How true, how painfully true! I thought). But when I reached the part about trying to get over someone by throwing myself into someone else's arms, I felt Chris Stevens watching me slyly. By that time I wished I had chosen something else—anything else.
Still, despite my embarrassment, I managed to pronounce the words clearly and stay in tune. Relieved, I started in on my Queen Mab speech—but that too felt far more problematic on stage than it had in the safety of my attic bedroom. "She is the fairies' midwife, and she comes," said my mouth, while my mind, racing, chided me: What made you think it was a good idea to give a speech about fairies at a boys' school? How's that going to go over? I glanced cautiously around the audience—another bad idea. There was Ashleigh grinning at me, which had the perverse effect of making me more self-conscious; there was Chris Stevens, winking with his long cat's eyes; there was a little boy chewing the end of his pen and another sprawled out over two seats with his eyes closed, both radiating boredom; and there in the back—oh, horror! Had he been there the whole time?—stood Grandison Parr, tall and golden, looking right at me.
I panicked. My voice dropped to nothing. I rushed and mumbled my way to the end, stopping abruptly and cutting off the last three lines (which are kind of obscene anyhow). I dragged myself off the stage and sank into the dusty velvet seat beside Ashleigh's, where I wished I were dead.
The rest went by in an excruciating, slow-motion blur. Parr took the stage, and I sat, drinking in his pleasant, confident voice, with frozen limbs and cheeks that burned on and on through a thousand other speeches and meaningless songs. When the auditions were over, he came to find us. Ashleigh greeted him warmly, but I could hardly hear what she said over the pounding in my ears, nor could I choke out more than a monosyllable. All through the ride home, while Ashleigh and Yolanda eagerly reviewed the afternoon's events, I sat with my cheek pressed against the cool glass of the window, hardly blinking, hardly breathing. And the torture repeated itself all night long, first in my memory and then in my dreams, until I half hoped my blushes would set my sheets on fire, ending my misery in one magnificent blaze.
**_Chapter 11_**
_Parts_ ~ _scripts_ ~ _rhymes_ ~ _songs_ ~ _an igsome Moth_ ~ _an Artistic Rivalry_ ~ _a direly misleading Scene involving a Sofa._
_**A**_ fter the previous day's disaster, I walked the long way around to my first-period social studies class to avoid the bulletin board. I had no wish to see the cast list posted without my name. True, I had half hoped to tank at the audition; but half hoping to tank is one thing, actually tanking quite another.
In the end, my careful detour came to nothing. Ashleigh and Yolanda appeared at my lunch table waving a piece of paper.
"Good afternoon, Headmistress Lytle," cried the Enthusiast.
I frowned impatiently. I was in no mood for Ashleigh's play-acting. "What are you talking about?" I said.
"Look!" said Yolanda. She put the page down in front of me, just missing a pool of spilled mustard. "It's your part—you got a 'little' part—Headmistress Lytle—see? And there's me, I'm Tanya, president of the student body—I hope I get lots of lines—and Ashleigh's Hermia, and that's it from Byz High. We figured it was okay to take the poster down, since nobody else from here tried out. But Erin got a part too—she's Helen. And Emma Caballero, that freshman from Sacred Heart, she's Chloe."
"Is this not good news?" said Ashleigh. "Grandison Parr plays Owen, captain of the debate team, and your beloved Ned is the musical director, so you will have frequent opportunities to converse with him during rehearsals."
"Oh, are you going out with that guy Ned?" said Yolanda. "Crisp! You never told me that. He seems like a really nice guy. I kind of liked that tall guy with the nice voice—he was cute. I wonder if he got a part. He had to, he had the best voice there. Which one do you think he is? Kevin Rodriguez? Ravi Rajan? Ask your boyfriend, okay? Oh, but don't tell Adam!" Adam White, a junior, was sometimes the man in Yolanda's life.
"Ned's not my boyfriend," I protested. "I only met him twice."
"Yes, but you called him a Darcy, remember? Pay her no mind, Yolanda, she is too modest to admit her true feelings," said Ashleigh.
"Whatever," I said testily. The alternating waves of anticipation and terror, disappointment and relief, which had been sweeping over me for the past few weeks, had taken their toll on my usually even temper.
But how had I gotten a part, after such a spectacularly bad performance at the audition? Ash and Yolanda insisted that I had sung sweetly and spoken well, though softly at the end. But I knew they were just trying to make me feel better. No, the only possible explanation was lucky (or unlucky) chance. Five girls had auditioned—five had been cast. If a sixth had shown up at the tryouts, she would surely have won the part of Headmistress Lytle.
Nicole Rossi, Ashleigh's mother, picked up our scripts for us at Forefield that evening on her way home from work.
While _Midwinter Insomnia_ may not be the very silliest play I've ever read, it's up there. It takes place in a boarding school rather like Forefield, but coed. The scene opens with romantic mixups among the fifth formers, or juniors: Xander (played by Ravi Rajan) is going out with Hermia (Ashleigh); Daniel (Chris Stevens, apparently typecast) is trying to steal her away; and Helen (Erin) has a hopeless crush on Daniel. Meanwhile, Owen, the captain of the debate team (Parr), and Tanya, the president of the student body (Yolanda), are having a lovers' quarrel over a third former (that is, a freshman), formerly a member of the debate club, whom Tanya has enticed to serve on the student council, which meets at the same time, therefore forcing him to quit debate. To punish her, Owen convinces his younger brother, Rob (Alcott Fish), a science geek, to sneak into the chem lab and create a love potion that he can give Tanya, causing her to fall for the ridiculous Butthead (Kevin Rodriguez), who plays Romeo in the middle school's laughable production of _Romeo and Juliet_. When Rob mischievously taints a drinking fountain with the love potion, Xander, Hermia, Daniel, and Helen begin a game of musical partners that ends only with the grand finale.
I played the headmistress, Miss Lytle, who puts in occasional appearances calling for order, scolding mischief makers, and presiding over the happy ending. She also sings a duet with the dean of students, a cameo appearance by Forefield's actual dean, Mr. Hanson. Altogether, she has eleven lines, not counting the duet.
I was afraid they would be eleven lines too many.
"I envy Yolanda—oh! how I envy her," said Ashleigh, squeezing Juniper until he gave a reproachful kitten squeak.
"She does have more songs, but you have more lines," I pointed out.
"Faugh! Little do I care for lines and songs! It's the _kisses_ that I envy. She gets to kiss Grandison Parr!"
"Yes, but you get to kiss Ravi Rajan—isn't that the guy Yolanda thinks is so cute? Maybe you'll be so swept away, you'll forget all about Parr."
Ashleigh gave me her look of Reproach Tinged with Disgust. "Forget! Forget Grandison Parr! Ask me to forget my own name—my father and mother—my native tongue—the points of the compass—I will forget what it means to be human before I forget Grandison Parr!"
As for me (I thought with a sigh), I had better forget Grandison Parr before I forgot what it meant to be human. In the weeks that followed, I came much closer to that goal.
Not that I ever managed true forgetfulness: how could I, when I saw him at least twice a week at rehearsals? But practice made my heart grow tougher, like a blister that breaks and hardens to a callus, until I could smile at him, answer his remarks in sentences longer than a word or two, and even meet his eyes.
The hardest moment was my first rehearsal, when I felt him watching me. It took all my willpower to obey Benjo and focus on my character's quarrel with the dean—far too lenient a man, in Miss Lytle's opinion. When Benjo directed me to stamp my foot, turn my back on the dean, and face the audience, I trained my eyes on the exit sign until I could bear to look at them directly. After a week or two, though, I grew used to having an audience.
Talking to Parr took even more courage, but I found I couldn't avoid it. Although Ashleigh and I had relatively few scenes with him, he made a point of seeking out our company.
"Hey," he said, coming up behind me as I was helping Ashleigh go over her lines before the second rehearsal, "can either of you think of a better rhyme for Hermia? Barry says _germier_ is revolting, and anyway, I'm not even sure it's a word."
_"Wormier?"_ suggested Ashleigh.
_"Wormier?_ Hmm, I hadn't thought of that. It's . . . a possibility," said Parr.
"Oh, Ash, ig! That's even more revolting," I said.
"All right, _squirmier? Sp—_ no—well, _you_ think of something then, Julie, I have the highest confidence in your abilities. Julie writes poetry, you know," she told Parr.
He turned to me keenly. "Do you?"
"Oh, Ash," I moaned, feeling squirmier myself. "Not much, and it's not any good."
"What do you mean, it's not good?" cried the loyal Enthusiast. "What about that beautiful poem you wrote in seventh grade about the sunset and—"
I moved quickly to stop her, before she could recite one of my juvenile efforts; she likes my older, flowerier poems the best. "I know!" I said. "What about _hypothermia_?"
"Brilliant! That's perfect, Julia, thank you!" said Parr, making a motion as if to hug me. Startled, I drew back, and his gesture trailed off into awkwardness; but he continued to grin at me. He had very white teeth. I loved the way he called me by the formal version of my name—it made me feel like a grander version of myself.
"See, I told you she'd think of something," said Ashleigh proudly. "You can always count on Julie."
Although these moments with Parr were the shaky high points of my days, I naturally spent more time with Ned, who ran the musical rehearsals and stood in as my singing partner, the dean, for Dean Hanson, who rarely made it to rehearsals. The more I saw of Ned, the more I liked him. His tunes were so catchy that I often found myself singing them around the house, and more than once I noticed my mother humming "Who Would Want to Hook Up with Helen?" or "Oh Lord, What Fools!" And I soon came to appreciate Ned's good nature as well as his music. In a room full of big egos—Benjo, Barry, Chris, Erin in her quiet way, Ravi—Ned's was a hardworking and self-forgetful presence. He reminded me of another friend, a person of boundless energy and loyal encouragement: Ashleigh. Most musical directors would have lost their patience long ago, I was sure. Ned, though, never stopped encouraging me.
"That's great, Julie," he said. "You got a much bigger sound that time. Remember how quiet you were last week? Okay, now this time focus on the deer head across the room. You want to really make his ears curl. Great! That was great, now this time let's see if you can really concentrate on keeping from going flat on the high notes. Nice and loud! Yes! Yes! Listen to you! Okay, I think maybe I pushed you too far that time, you went a little sharp. Not 'ah,' more like 'ah.' Try it again. Good! Ashleigh, did you hear that? Did you hear how great Julie's sounding? That was really good, Julie, and you were definitely loud enough if Ashleigh heard you all the way over by the door."
At first I was so caught up in learning my lines, governing my heart, and training my voice not to slink off with its tail between my tonsils that I had no time to watch my fellow actors. But as I grew more accustomed to the scene around me (except for Parr kissing Yolanda—I never grew accustomed to that), I began to notice several dramas.
The most obvious, because it touched me personally, involved Chris Stevens. Chris tried to ooze his way into the good graces of every girl in the production, with the sole exception of Emma Caballero, who was too young even for him. He persisted like an elegant insect, gently dodging any slaps and returning to buzz and brush against you. His technique involved floating around nearby and implying that _you_ were interested in _him_.
"Sorry I didn't see you much last time, Julie," he said soon after we started rehearsals. "I was in the trophy room with Erin and couldn't get away. But I don't want you to feel I was neglecting you."
"Don't worry, Chris, I don't," I told him. "Actually, I'd prefer it if you _did_ neglect me."
As I soon learned, this was the wrong approach to take with Chris, who took resistance as a challenge. Far from keeping him away, it drew him to me as pheromones might draw a monstrous moth. I had learned all about pheromones during Ashleigh's insect craze. Chris fluttered softly near me, fanning his vast, pale wings and reaching out with his hairy feelers. Ig!
Yolanda's approach—treating his advances with friendly, offhand patience—worked far better. "Chris, how'd you get back there? Sorry, I keep stepping on you! Did I hurt your toe? Hey, didn't you call me last night? Sorry, I meant to call you back—I didn't forget about you, I swear, it's just that I had a lot of homework, and then I was talking to Ravi, and then it got late, and anyway, don't they make you turn off your phones after ten?" Something about her careless solicitude kept him, if not at arm's length, at least at elbow's.
Oblivious Ashleigh didn't respond to his attentions at all, since she didn't notice them. But Erin, poor thing, responded all too well. She developed an obvious, violent crush on him. Chris tormented her by ignoring her most of the time, giving her just enough attention to keep her going, and flirting with the rest of us whenever he saw her watching.
"Where's Chris?" asked Erin one afternoon after we'd been rehearsing for several weeks. "We're supposed to go over the scene where I give him the answers to the math test."
"I last saw him with Yolanda," said Ashleigh. "He said something about showing her the trophy room."
Erin stiffened. "I'd better go find him," she said, and she hurried off.
Kevin Rodriguez and little Alcott Fish giggled.
"What's so amusing?" asked Ashleigh.
Alcott turned pink.
"You know, the trophy room? With all the sofas and everything?" said Kevin.
"What about it?"
Alcott turned pinker, and Kevin rolled his eyes.
"Oh, grow up, guys," said Ravi. "It's supposed to be where people go to—where people go for privacy," he explained.
Of the actors, the most talented by far were Ravi, Alcott, and Kevin, who turned out to be a surprise comic genius. As Butthead, the boorish boy playing Romeo in the play within a play, he overacted with such flawless control that he never once overdid overdoing it. When Yolanda's Tanya drank Rob's love potion and fell for Butthead, Kevin turned into a parody of Chris Stevens subtle enough that Chris himself never noticed, yet broad enough to keep the rest of us choking back laughter. Yolanda deepened the impression by putting a touch of lovesick Erin into her Tanya, but only a touch. Either she didn't want to be cruel to her old friend, or she was acting unconsciously, not quite aware of her influences.
Ravi was a delight to watch and listen to: handsome, lithe, with a voice like honey and butter. I could see why Yolanda had a thing for him. Ash, I thought, was lucky to have her heart already occupied, or after their first kiss in rehearsals she would have been as bad as Erin with Chris. But although Ravi clearly knew the impression he made, there was nothing wolfish or manipulative about him. He accepted admiration as his due and repaid it with friendly attention, as if to suggest that a warm admiration for Ravi was a pleasure that you and he could share.
Tensions ran high between Benjo, the director, and Barry, the playwright, who attended every rehearsal and held strong opinions about how we should deliver his beloved lines. Young Emma—as Chloe, the middle schooler playing Juliet in the laughable play-within-a-play—had an uncontrollable tendency to giggle. Unable to curb this habit, Benjo used it as a way of showing up the silliness of the middle schoolers' production. But Barry couldn't stand it. He snapped, "Stop giggling!" at poor Emma whenever she so much as smiled, which made her giggle even more.
"I can't—I can't—I c—I can't help—!" gurgled poor Emma.
"Barry, quit messing up my actors! I mean it! You're making her choke," said Benjo.
"Your so-called actors are messing up my play. Can't you control them?" said Barry, stepping closer.
"What needs controlling is _you_ need to control _yourself_. Leave now, please!"
"Leave? Leave this mess with _you_?"
Both guys, I saw, had clenched their jaws and fists. Any minute they would come to blows. Fortunately, Parr saw it too and stepped in. "Hey, Barry, do you have a sec? I rewrote the chorus to 'Queen of the Ice,' and I want to know what you think."
Benjo was still glaring at Barry, so I decided to distract him too. "Benjo, can I ask you a quick question? Should I exit while the dean is still singing, or should I wait till he's done?" Benjo seemed annoyed at the interruption, but Parr repaid me with a grateful look.
Because my part was so small, I had plenty of time to observe all these offstage dramas. When not rehearsing my scenes, I made myself useful as a page turner, prop fetcher, and prompter. I learned Yolanda's part before she did herself.
Was I foolish to watch her scenes with Parr, exposing myself over and over to their kisses? Perhaps, but I couldn't keep away, and they seemed to find my presence useful.
One day while I was helping Ned go over the drinking-fountain song with Alcott and Kevin, Alcott threw his beaker full of love potion at the fountain with a little too much zest. It shattered.
He leaped back. "Hey, wasn't that supposed to be safety glass?"
"Good thing it broke now, not during a performance," said Ned. "We need to find something stronger. Plastic or metal."
"Like a loving cup, maybe," I joked. "The trophy room's full of them."
"A loving cup for the love potion—ha! Julie, that's brilliant. Wait there, I'll be right back." Ned ran off.
Parr, who was nearby helping Benjo choreograph the big fight scene between Daniel and Xander, saw him go. "Oh, no, Ned, don't do that! Quick, somebody, stop him," he said.
"Stop what? Where's he going?" asked Alcott.
"The trophy room, obviously. If Wattles catches him taking a trophy, that's it for his scholarship."
"I got it," I volunteered, feeling responsible.
I thought I remembered the way to the trophy room, from my bathroom adventures at the Columbus Cotillion. However, it took me longer than I expected to find it. By the time I got there, Ned was perched on tiptoe on the back of a green leather sofa, trying to pry open a trophy-case door with a protractor.
Something was clearly about to snap. I hoped it would be the protractor, but the door looked more likely.
"Ned! Stop!"
"Oh, hi, Julie—give me a hand up here, will you? I've almost got it."
"Stop it, Ned, you're going to break the case!"
"No, I'm not, I've almost got it—"
As he levered the protractor, the case begin to tilt. I scrambled up on the sofa to pull him away before everything fell. Our joint weight made the sofa tip, and we both lost our footing on the slippery leather, landing in a tangle on the seat. Fortunately, the trophy case stayed where it was.
"Julie, are you okay?"
"Ouch!"
"Sorry, was that your leg? Why'd you do that, anyway? I almost had it!"
He tried to get up, but I grabbed him around the neck and shoulders and hung on tight. "Stop, Ned! Think of your scholarship."
"But it would be so perfect," he said, squirming.
Just then the door opened and Ashleigh and Erin burst in.
"Oh! Jul—Ned—forgive me, I didn't mean to intrude," said Ashleigh, backing out and pulling Erin with her.
"Ash, wait!" I yelled, but by the time I had untangled myself from Ned and talked him into leaving the loving cups in their cases, she was long gone.
The scene in the trophy room dashed my hopes of convincing Ashleigh that my feelings toward Ned were nothing more than friendship. "I was just trying to stop him from stealing a loving cup to use as a prop," I protested, but it was no use. Even I could hear how lame it sounded. "You can ask Parr, he told me to go," I added feebly.
"Did he? Did he indeed aid Ned in planning an assignation? Clearly _Parr's_ friend confides in _him_ far more trustingly than _mine_ does in _me_ ," said Ashleigh, working up to full-blown Austenese. "No, no! Say no more! Far be it from me to pry from you a confidence that you do not willingly surrender! But if it were me, you know _I'd_ tell _you_."
"Ash, I swear, there's nothing to tell."
"Because we interrupted you."
"No, because we weren't doing anything. But what were _you_ doing there, anyway?"
"Erin was looking for Chris, and Kevin said you'd gone to the trophy room. I thought you might need protecting. Little did I imagine what scenes we would interrupt! Next time, tell me and I'll guard the door for you."
**_Chapter 12_**
_I keep up my grades_ ~ _My father grouses_ ~ _A Turkey again_ ~ _Rehearsals_.
_**H**_ ave you ever noticed how once teachers get an idea into their heads, it's easier to interrupt a bus of kids singing "100 Bottles of Beer on the Wall" than to change their minds? This is why, if you have limited time for study, it's best to apply it at the beginning of the semester. If you do, most teachers will dismiss you early on as a good student and not look too hard for mistakes.
What with _Insomnia_ and _Sailing to B_., my hours available for homework plummeted. My grades, however, rose. My B-pluses puffed up to A-minuses, my A-minuses to full-out A's. One would think such a state of affairs would please my father. But no: he considered two time-consuming extracurriculars one too many. "You want to appear well rounded, not dilettantish," he said. "If you had more time to study, you could push those minus marks up to straight A's."
Dropping _Sailing_ was tempting, but it seemed unwise. By putting me in the good graces of the Nettle, my work on the magazine probably saved me hours that I would otherwise have had to spend preparing for class by second-guessing her opinions. And although I half longed to give up _Insomnia_ , I felt I couldn't let down the other actors. I explained to Dad that I had plenty of time, really, now that the fall foliage-viewing rush was over at Helen's Treasures.
Amy rolled her eyes slightly at the mention of my mother's business, but she took my side. "Julie's learning follow-through, Steve. Colleges value that," she told my father. She even volunteered to drive me to and from rehearsals when they met on Tuesdays.
One Tuesday, then, in the middle of November, she dropped me off half an hour early on her way to meet a client. It was unseasonably warm. I unbuttoned my coat and sat on the steps of the Robbins Center to wait for Benjo or Mr. Barnaby, who both had keys.
The first person to arrive at the center that afternoon, however, was not the director or faculty adviser, but Turkeyface from the Columbus Cotillion. When he saw me, his face turned red—or rather, redder.
"What are you doing here, young woman?" he spat. "Don't you know this is a boys' school?"
"I'm just waiting for Benjo Seward," I said. "He's—"
He cut me off. "Don't go trying to implicate Benjamin Seward. He would never sneak a girl onto campus. He's a responsible young man. He knows the rules."
As Turkeyface lectured me, Grandison Parr appeared over his shoulder. "Hello, Julia," he said.
Turkeyface spun around. "I knew it!" he gloated. "Not only is your girlfriend here on campus illegally, but she was trying to blame Seward!"
"But Julia's—" began Parr.
"Not a word! One word buys you three demerits. You're both coming with me to see the dean."
He took us each by a shoulder and marched us to the administration building.
Dean Hanson's door was ajar. "What's up, Matthew?" asked the dean, looking up from his computer. "Oh, hello, Julie—Grandison. What can I do for you? I _have_ been practicing—I promise—listen:
_If you force me to be harsh, I'll_
_Try my best to be impartial,_
_But a carrot's always better than the most effective stick._
I sang back the next verse:
_My dear dean, you're much too soft—when_
_I remember just how often_
_Your supposed angels misbehave, I swear, it makes me sick!_
"Cool beans, Julie! Sounding good!" said the dean.
"You know this girl?" sputtered Turkeyface.
"Of course I do—she's Headmistress Lytle."
"Headmistress? _Headmistress_?"
"Yes, and an excellent one too. Way better than my dean. Of course, she rehearses way more. You look puzzled, Matthew. _Midwinter Insomnia_. The musical, man, the musical! What's up—is there a problem?"
"Well! No, not if you know this girl. I expect you know your own business. Forgive me. I would never interfere." Turkeyface made his exit.
The three of us waited until the door clicked shut before laughing. "Matthew obviously agrees with Miss Lytle that I'm much too soft," said Dean Hanson. "But actually, you're the ones who are too soft on _me_. Barnaby's right, I should make it to more rehearsals. Come on, let's get down to the theater."
"Are girls really not allowed on campus?" I asked as we walked back along the gravel to the Robbins Center.
"No way—did Matthew tell you that? I guess, technically, nonstudents aren't allowed except under special circumstances—which covers things like playing the headmistress in the school play, so you're okay there. And, of course, girls tend to be nonstudents at a boys' school. But there's nothing in the charter forbidding girls per se. Matthew gets a little carried away with rules. He's a—well, a—hmm . . ." Dean Hanson trailed off, evidently remembering his position as a member of the administration speaking about a member of the faculty to students (or, in this case, a student and a nonstudent). He collected himself and began again: "So, Grandison, what do you think of the Saberteeth's chances against Groton this season?"
"I'm a little worried, actually. The Teeth are facing some serious competition. Groton's got Dashwood now, so of course that gives them an edge over last year. I can't really blame him for transferring—coed's a temptation," said Parr, glancing at me. "But I wish he'd waited another year. Bloom and Coe are going to be killer once they get their footing, but they're not there yet."
The Saberteeth's prospects took us the rest of the way to the Robbins Center, where the cast was waiting for us somewhat impatiently. Ned was delighted to see the dean, whose presence kept me busy, for once; the two of us worked hard on our duet all through the rehearsal. "Nice, Julie—you're sounding very disapproving," said Ned. "You could even pump up the resentment, if you want. Go ahead and squeak on that high G. Mr. Hanson, you're doing well with the sheepish expression, but if you could find the time to rehearse more, you might remember more of the lyrics."
Some of the others came over to watch the dean and me while they waited for Ned to listen to their songs. "Oh, well done, well done, my dearest Julia," cried Ashleigh. "And you too, Dean—well done! Julia, you are indeed fortunate that the dean himself, at heart, shares those qualities which make his character so infuriating to the headmistress. It must greatly ease your task of acting stern. For my part, I find it difficult to maintain the necessary anger at Xander's coldness, since Ravi himself is the soul of kindliness."
"That's nice of you to say, but ouch!" said Ravi. "You certainly slapped me like somebody angry."
"That's just Ashleigh's natural enthusiasm," said Ned. "She gets carried away."
"It was my duty as an actress—it was the least I could do," said Ash. "If it's any comfort to you, I slapped Chris harder. Speaking of which, some charitable soul ought to go rescue Yolanda. I saw Chris follow her into the lighting booth."
"Not me," I said. "You'd just have to send someone else to rescue me next."
"I'll go," said Parr quickly. "Yolanda and I should be practicing anyway."
Amy's meeting ran late, making me the last of the girls left at Forefield after rehearsal. Parr and Ned sat with me on the Robbins Center steps to wait for her. The dregs of pink drained from the sky and a cold wind started up; Parr moved down a step to put his body between me and the wind.
"Why are you at Forefield if you'd rather go to a coed school?" I asked him.
"It's a family tradition. My father and his father and his father and _his_ father and the rest of their fathers went to Forefield, back to when it was just five pupils and a scandalous headmaster. Did you know the first head got thrown out of England for killing a horse in a duel? The man he was fighting survived, but the horse died. Apparently it was a very important horse."
"Parr's great-great-great-great-grandpa, I mean great-great-great-great-grand-Parr, is one of the boys in the frieze carved over the fireplace in the Great Hall. He's the one on the far left, with the funny ears. Hard to get a hat over them," said Ned.
"Hey, Noodles, quit putting hats on my ancestors—I mean it," said Parr, cuffing Ned gently.
"I never put a hat on your ancestor," said Ned. "Like I just said, his ears stick out too much."
"Anyway, though, Dad would be heartbroken if I didn't go to Forefield," continued Parr. "We don't exactly see eye to eye on everything, so I assume I'm going to be disappointing him enough later on—I might as well let him win what he can now, before the real battle starts." He paused, then added, "The boy-girl ratio isn't so bad at Forefield this year, though, with the play."
"What about you, Ned—do you mind the all-boys thing?" I asked.
"Oh, well, I wouldn't say I _like_ it, but I can't really complain. They're giving me a scholarship. Apparently Grandison's great-great-grandpa—or somebody's great-great-grandpa, anyway— thought the gramophone was destroying society by letting people play records instead of musical instruments. He endowed a scholarship for musicians. The only catch is that I'm not allowed to make any records while I'm at Forefield, or even listen to them."
"No records?" I exclaimed. "Does that mean no CDs? How can you stand it?"
"Fortunately, the trustees interpret that to mean ancient stuff like wax tubes and 78s—the kind of records that were around when the scholarship was started. They said it's fine for me to listen to anything digital."
"That wasn't my ancestor," said Parr. "Can you imagine anyone related to my father endowing a scholarship for music? Tin-ear Charlie himself? Although it would be almost like my grandfather to make sure a musician wasn't allowed to listen to music. He has strong ideas about what's worth spending time on. My father too, but he's not as mean about it."
"My father's kind of like that too," I said. "He's always bugging me to do more extracurriculars so I can get into college, and then telling me that my extracurriculars are bringing my grades down."
"Is he why you tried out for _Insomnia_?" asked Parr.
"Yes—sort of, pretty much," I said.
"Thank God for our fathers, then," said Parr. "Otherwise—"
Amy drove up just then and honked, so I didn't get to hear why Parr was grateful for our fathers. He opened the car door for me, extracting a sour smile of approval from Amy, who sets great store by courtesy. As we drove off, I wondered what he had been about to say.
**_Chapter 13_**
_My mother gives up_ ~ _Thanksgiving_ ~ _yet another Turkey_ ~ _an Identity Crisis_ ~ _a Comeuppance._
_**W**_ hen I got home from school the next day, my mother was packing away the Halloween merchandise and bringing out the Christmas things.
"Don't we usually do that after Thanksgiving?" I said.
Mom finished unwrapping a tin Santa and sat back on her heels. She looked up at me seriously. "Hi, honey. I thought we'd better try to catch whatever traffic there is from the Thanksgiving weekenders while we still can. I didn't get that job I was hoping for, so I'm going to work for the Nick-Nack Barn. They may not pay much, but it's steady and I get health insurance. They want me to start right after Thanksgiving."
"Oh, God, Mom, I'm sorry," I said. The Nick-Nack Barn, a heartless, tasteless chain two towns south, was my mother's ugliest rival.
"I'm not," said my mother. "Don't look so gloomy. It's just until I find something better. It won't be so bad—the manager's a nice woman, she's letting me do the window displays, and I can open Helen's Treasures on weekends. Want to give me a hand with these things?"
"Sure," I said. "I just have to send some e-mail first. I promised Eleanor—she's our editor at _Sailing_ —that I'd let her and Seth know what I think about a couple of poems we're considering."
Mom and I didn't have long to set up and sell the Christmas stock before Thanksgiving was upon us.
I won't dwell on this bitter holiday, which I spent with my stepmother and her family. I would naturally have preferred my mother's company, but I wasn't given a choice: it was Dad and Amy's turn to have me. I biked over, envying the wild turkeys that vanished into the trees in a pale whir of feathers as I passed. If they had been shot, plucked, roasted with rosemary and lemons, and set on the table to be torn to pieces by Amy's critical mother, her prune-faced brother, his cowed wife, and their four boisterous, self-satisfied little boys, would the turkeys have had a worse time than I had?
I will admit that the food was good. Of course it was: Amy made it. No soggy Brussels sprouts and cardboard stuffing for her. We had vegetables that snapped gently when you bit them, squash roasted to melting depth, fresh citrus-cranberry sauce, and turkey whose tenderness remained uncompromised by the crispness of its skin.
"Amy, when are you and Steve going to give me a grand-daughter?" asked my stepgrandmother, helping herself to the last slice of white meat. "It looks like I'm getting nothing but boys out of Mark and Susie."
Amy went pale. Taking pity on her, I spilled some lemon-rosemary gravy on her mother's blouse.
The distraction worked. Beneath Amy's scolding, I detected a wisp of gratitude. But my act of generosity put me in disgrace with the family for the rest of the weekend, so I was doubly glad to get home that Sunday, especially after a weekend in my new, dark basement room.
I found my mother on Ashleigh's roof with my friend and her father, helping install their annual Christmas display. This invariably involved Santa and his sleigh, but the Rossis relied on my mother to give each year's display a distinctive character. During Ashleigh's King Arthur phase, for example, Mom had made Santa into a knight and the reindeer into unicorns. Last year she had made Santa fly over the Manhattan skyline, which she outlined in Christmas lights. When I arrived, Joe Rossi was urging Mom to make the reindeer's antlers into menorahs, in honor of our family's heritage. She thanked him, but declined.
This year Santa was much slenderer than usual. He was wearing a top hat and a tall collar.
"Looks crisp!" I called up to them.
"Oh, honey, you're back! You look so short down there," my mother called down.
"The door's open—come on up," shouted Joe.
"No, that's okay, I'm done up here," said Mom. "Hang on, I'll be right down. Oh, Ashleigh, are you coming too?" The two of them vanished through the roof's trapdoor (I could have told her, but didn't, that the tree made a quicker and easier route), and emerged at the front door. Joe waved at us from the roof, where he stayed to admire their handiwork.
"How do you like Mr. Darcy as Santa?" said Ash as the three of us went into Helen's Treasures. "Ned suggested putting bonnets on the reindeer, but when your mother tried it, they wouldn't go over the antlers."
"Oh, that's Darcy? Do you think playing Santa is really in character for him? Seems more like something Mr. Bingley would do," I said.
"Very well, Mr. Bingley, if you prefer," said Ash. "Most people seem to think it's someone from _A Christmas Carol_ , anyway. Philistines! So how was your Thanksgiving? Were your step-cousins there? Was it utterly unsupportable?"
"Yes, did you have a nice time, honey? Aunt Ruth sends her love," said Mom. "She gave me a new coat that Molly's grown out of already. It should fit you. That girl's growing even faster than you are. Oh! and one of your friends came by the shop on Wednesday and left something for you. Wait a sec, I think I put it in the desk." She rummaged around for a while and came out with a small package.
"Who was it?" I said. "One of the Gerard twins?"
"No, a boy. Nice-looking young man. He introduced himself, but I'm sorry to say I was a little distracted and I don't remember his name. It was busy here Wednesday. I sold all the reindeer soap."
"What did he look like?" said Ashleigh. "Was he of middle stature, about Julia's height, with lightish brown hair and deep, soulful brown eyes?"
"Um, he could have been. I'm sorry—I should have noticed better. I forgot you girls are getting to the age when you need all the details you can get about boys."
Casting reproachful glances at Mom, Ashleigh and I carried the package upstairs to my attic.
"It's from Ned, I know it is! Does it have a note? Open it!" cried Ash, bouncing wildly.
"No note, but there's writing on the box." I read: " _Had enough wattles this season? If not, here's sweets for the sweet. Yours ever—_ I can't read the name."
"Let's see! That must be _E-_ something- _D—_ what's Ned's middle name?"
"Does he even have one? That looks nothing like an _E._ More like a _C._ Chris Stevens? Could that be possible? Too bad it's so smudged," I said.
"Of course it's an _E—_ well, I guess it could be an _N—_ N, E, D, maybe?"
"How do you get an _N_ from that? It's got to be a _C_ or a _G_ , or maybe a sloppy script _A_ —something open on the right—well, I guess it _could_ be a really messy capital _E_ , but for sure it's no _N._ Here, give it back, let's see what's inside."
The box contained a gorgeous chocolate turkey, its plumage delicately marked in three colors of chocolate: dark, milk, and white.
"Sweets for the sweet! Is that not a chivalrous thought? That settles it—it must be from Ned."
"Or whoever sent it could be calling me a turkey," I said.
"Nonsense, Ned would never suggest such a thing. He has too kind a heart."
"Why would Ned give me a chocolate turkey?"
"Oh, Julia! Do not pretend you do not know! Chivalrous young men courted their chosen ladies with gifts of sweetmeats even in King Arthur's time. As for the turkey—well, it _is_ Thanksgiving."
Whatever Ashleigh said, I didn't believe Ned was the turkey giver. For one thing, he couldn't spell—or at least, he couldn't type. Of course, the note was handwritten; that could explain the absence of typos. Still, it didn't sound like his style. But if not Ned, who? Chris Stevens, Mr. Igsome himself? Unappetizing thought! Seth Young? Dean Hanson, perhaps, as an apology for his turkey-faced colleague's treatment? Surely not: my mother would never call the dean a boy; and similar reasoning ruled out Zach Liu, since she would have recognized him. Grandison Parr, then? Possibly. The reference to wattles suggested, if not the dean, then Ned or Parr. Nibbling on a bit of the tail—first-rate chocolate—I felt my heart begin to beat faster. From the sugar? Or from a powerful feeling that I would not allow myself to put into words?
But would my unspoken hopes turn out to be hollow after all—as hollow as the chocolate turkey that was vanishing before my eyes?
And whoever the kind turkey giver was, how would I express my gratitude? Obviously, I couldn't just thank all the candidates, or the ones who hadn't given me any chocolate would think I had left my mental marshmallows in the microwave a bit too long.
After some deliberation, I sent e-mail to Ned, Parr, and Seth, thanking them in general terms for their recent kindness, and slipping in a reference to Thanksgiving. I hoped that the innocent guys—the ones who hadn't sent me a turkey—would conclude that the theme of the holiday had made me think grateful thoughts. As for the chocolate giver, I hoped he would interpret my message as a response to his gift.
And if the turkey had come from Chris, he could just consider me rude. He would get no thanks from me. I hadn't asked him to shower me with turkeys. He would have to do more than ply me with chocolate to worm his way into my good graces.
I received the following answers.
From Parr:
Dear Julia, What a sweet message. But it's the other way around—I'm the lucky one. CGP.
From Ned:
happy thangsgiving to you too julie. i am glad you and ashleigh are in the play its much more fun than any ohter year!
From Seth:
Hi, Julie. I was touched to receive your message. I hope you enjoyed your Thanksgiving, and I look forward to seeing you when school resumes. Yours, Seth.
Inconclusive, I thought. Nobody either acknowledged or repudiated the turkey. Well, at least if it was one of them, he wouldn't think me rude and ungrateful.
After the holiday, the pace picked up at school. Final papers and exams approached, and the deadline loomed for the winter issue of _Sailing._ Work on _Midwinter Insomnia_ slowed dramatically, however, since finals were an even bigger deal at Forefield than at Byzantium High. With rehearsal time given over to review sessions, our schedule shrank to a weekly rehearsal, "the minimum we can meet and still expect to have anything left to forget by the time we get to winter break," as Benjo put it.
My part was so small, and my partner, the dean, so rarely around during these weeks, that I had almost nothing to do but prompt the others. I spent my time watching Ashleigh (as Hermia) chase after Ravi (as Xander) and defend herself from Igsome Chris's accusations of coldness.
As Daniel, Chris sang:
_Half an hour of hanging out with Hermia_
_Would give a seal or walrus hypothermia._
_She's the Queen of the Ice._
_She doesn't know the meaning of nice._
_Turn the thermostat up and crank it!_
_I need another sweater and a blanket._
Ash/Hermia responded,
_Insinuating snake!_
_He's a man on the make,_
_Out to get what he can take,_
_And take what he can get—_
_Which is nothing . . . yet._
The _yet_ came as Hermia drank the tainted water and found herself falling under his spell.
Whenever Chris saw me watching him in his scenes with Ashleigh, he would give me a horrible, languid smile.
I naturally took frequent breaks from their rehearsals to torture myself by watching Parr and Yolanda quarrel passionately, then kiss and make up.
One day shortly before winter break, when Ashleigh's dad drove us to Forefield, I noticed that Yolanda was uncharacteristically quiet. She contributed almost nothing to our discussion of the dance number leading up to the grand finale (which I found too energetic, whereas Ashleigh considered it not energetic enough).
"You okay, Landa?" I asked.
"Fine."
"It's just, you seem subdued."
"Subdued? Oh, I—sorry, I was thinking about something else. What were you saying?"
"The finale. Too tame? Too wild?"
"I like it the way it is. It's, uh, kind of energetic but not all that energetic, if you see what I'm saying. And that's what it should be like, because it's the finale."
Although Yolanda does not always think through what she's going to say before she says it, this remark seemed especially incoherent. It made me wonder.
Arriving at Forefield, however, I turned my attention to my part and forgot about the conversation until much later in the afternoon, when I went to help Yolanda and Parr. They were rehearsing alone in the Robbins Center's dance studio upstairs, while Alcott Fish, Ashleigh, Ravi, Chris, and Erin worked on their big jealousy number on the stage. Yolanda and Parr needed me to stand in for Alcott, who had a couple of lines in their scene.
To my surprise, Yolanda seemed to be having some trouble remembering her part. Suspicious, I checked her hair. The day before, she had worn all green beads, and her sister wore all red. Today Yolanda still wore green. And the beads weren't just at the ends of her braids, but up at her scalp as well. If this was Yvette in disguise, the twins must have gone to a great deal of trouble to make her look right.
The green beads clicked as my friend leaned her head back for the big reconciliation kiss. I flinched as usual, but forced myself to watch.
Parr kissed her.
"You're not Yolanda, are you?" he said.
She made a gesture of surprise. "Who else would I be?"
"The famous identical twin, maybe?"
"Why do you say that?"
"Yolanda kisses differently. You can tell a lot from a kiss."
The Gerard twin hesitated, then took a deep breath. "You're right," she said. "I'm Yvette."
"I thought so!" I said. "You were so quiet in the car. Where's Yolanda?"
"She got grounded."
"Grounded! What for?"
"Dumb girl says she accidentally ordered some sexy underwear over the Internet, and my mom got the bill. Tell me, how do you _accidentally_ order some sexy underwear? She's grounded for two weeks, and she's afraid she'll lose the part if anybody here finds out. You won't tell, will you?"
I shook my head. "Of course not," I said. "Ashleigh's going to notice, though. You should tell her—she'd never give you away."
"Yeah, I wanted to tell you both, but Landa said to wait and see if you noticed. You won't tell either, right?" she asked Parr.
"No, of course not. You know the part—they'd have to replace Yolanda with somebody anyway, so why not you? Why _do_ you know it, anyway?"
"I learned it helping my sister practice. Thank you so much, guys. Yolanda will appreciate it."
"I don't get it. I thought you hated performing and things like that," I said.
"Yeah, I do. She owes me."
"You're a generous sister," said Parr. "Shall we take it from after the kiss?"
Except for Parr, Ashleigh, me, and Ned, whom Ash told but swore to secrecy, no one involved with _Insomnia_ noticed the new actress playing Tanya. The substitution did have one dramatic result, however. Igsome Chris followed Yvette into the prop room, where she'd gone to put away some test tubes. He yelped and came out again quickly.
"What did you do to him, Yv—Yo?" asked Ashleigh.
"Something my sister should have done weeks ago. That girl's too soft-hearted."
She refused to say more.
**_Chapter 14_**
_Musings about the Inscrutable Gender_ ~ _A Date_ ~ _Ashleigh to the rescue_ ~ _Painful Praise._
_**W**_ hen I began tenth grade, I never imagined I would become a Belle, but when Seth Young called the third December evening in a row and Mom made a comment about boys, I began to rethink my self-image.
"It's not _boys_ , Mom—it's just Seth. He wanted the math homework."
"The math homework, eh? What did he want yesterday?"
"How do you know he called yesterday? Did you go snooping in my Calls Received list?"
Mom looked hurt. "You know I wouldn't do that. But if he calls you while we're in the car, I can't help overhearing, can I?"
"Well, if you did overhear, you'd know he wanted to find out if I'd finished reading Mad Alex's story for _Sailing_ —the literary magazine."
"And the day before?"
"Oh, Mom! It's just Seth. Really—would _you_ go out with him?"
"I don't know, honeybear, I don't think I've met him. Unless, is he that nice-looking young man who gave you that chocolate turkey?"
Was he? Whatever Ashleigh might believe about Ned, I hadn't solved the chocolate-turkey mystery to my satisfaction. "I'm not sure," I said. "There's a bunch of people it could be. You didn't exactly give us a good description."
"There, what did I tell you?" said Mom triumphantly. "Boys!"
Taken one by one, I felt, Ned, Igsome Chris, Seth, and Parr added up to something less than—or at least other than—Boys. None of them seemed to be behaving like a real suitor. No matter what Ashleigh said, I couldn't believe that Ned had feelings for me. Igsome did pursue me pointedly—he was out for conquest—but as long as he didn't conquer me, you could say I'd won. However, my victory was nothing personal, as Yvette had shown; he chased after anything female. Nor did Seth fit the bill, I told myself. Nothing could be more natural than for a guy to call a girl with whom he shared several classes and endless literary duties.
As for Parr—well, what was there to say about Parr? I was afraid my conflicted feelings for him might be clouding my observations. The warm, teasing gallantry that marked our first meeting had given way to something more restrained. Now when we were together, I always felt a barrier between us, as if he were quietly holding me off or holding himself back. At moments I even imagined that he was aiming some intensity directly at me, but stopping, perfect swordsman that he was, with the point at my heart, a fraction of an inch from drawing blood. What this meant for Ashleigh and her dreams, I couldn't say. I often thought he treated her with the same courtesy he gave me, but with more freedom, more warmth.
Individually, then, none of these boys seemed to justify that remark of my mother's. Taken all together, though, there certainly were a lot of them. Was Mom right? Did they count as Boys?
There was no talking to Ashleigh about it, of course—I knew I'd just get an earful about Ned. But Sam, when I consulted her, sided with Mom.
"Why does Seth have to call you on the phone about the homework and stuff?" she asked. "If all he wants is information, he could just as easily e-mail you. With the phone, he can get you to give him one-on-one, person-to-person alone time—even if he can't actually get you alone in person. I don't know, Julie. Unless you do something definite to discourage him, I bet he'll make a move soon."
As usual, Samantha was right.
Seth sent me a text message one Thursday afternoon: _can u meet me at java jail 2 discuss page proofs?_
I should have guessed what was going on when he paid for my latte and insisted with extra-nervous pompousness that half my _which_ s should have been _that_ s, but it wasn't until he had put away all his papers and turned the subject to the movie playing at the Cinepalace that I realized what he had in mind.
"Well, if you haven't seen it either, you want to go see it now?" asked Seth.
"I—I can't—I have to—my mom needs me to help her in the shop," I garbled, taken by surprise.
"Then what about tomorrow night?"
A movie alone with Seth—on a Friday night! What would that mean—what would that make us? What would people think if they saw us?
"I promised Ashleigh," I began, meaning to finish the sentence, "that I'd go see it with her." But I realized in time that I'd already told Seth all about her reaction to the movie, which she'd seen with Emily Mehan the previous weekend, while I was at my Dad's. ". . . that I'd hang out with her and help her with her dance routine," I finished lamely.
Seth got a stubborn look in his face, centered mostly around the jaw. "What about Saturday night, then?" he said.
Saturday night—even worse. I caved. "I'll call my mom and see if I can help her tomorrow instead," I said.
His jaw relaxed and he gave me a proud smile, as if he had beaten me by seven points on an English test and I had praised him for it.
Have you ever been to the movies with a boy you most certainly don't Like? A boy whose hands you can almost feel thinking (as if they had their own separate little brains) about creeping over to your shoulder or reaching for your hands? He leans closer to whisper some sardonic comment to show he's superior to the movie. You nod abruptly, trying to fend him off with your famously pointy chin. His shoulder brushes yours, and you feel him trembling a little under his pose. You draw away to the other side of your seat, pushing against the armrest until it digs into your waist.
I escaped to the ladies' halfway through and put in a rescue call to Ashleigh. "Help," I whispered. "I seem to be on some sort of horrible date or something with Seth Young. Can you meet me accidentally at the Cinepalace in an hour, when the movie gets out?"
"A what? A _date_? With _who_? What?"
"Can't talk now—I've got to get back to my seat—please, it's important—Cinepalace, one hour."
Seth seemed relieved when I came back; I think he was afraid I'd walked out on him altogether. But he bristled when we ran into Ashleigh and the Gerard twins on our way out of the theater, just as he was reaching for my arm.
"Julie! There you are!" cried Ashleigh. "Where were you? We've been trying to call you."
"Where does it look like she was?" said Seth. "We went to see the movie."
"Oh, hi, Seth," said Ashleigh, as if she'd only just noticed him.
"How did you like the movie?" said a twin.
"What are you doing out? I thought you were grounded," I answered. Unsure which one was Yolanda, I directed my question to the space between them.
The one on the left answered. "I got a ninety on my math test from Mr. Klamp, so my mom let me out for the evening. Kind of like bail, or is it parole? We're going to the Java Jail to celebrate. Want to come?"
"We already spent hours there," said Seth. He turned to me, shutting them out with his shoulder. "Shall we go to Bennie's Burgers?" he suggested.
"Yeah, Bennie's, that sounds great," said Ashleigh. Ignoring Seth's irritated look, she took me by the arm and charged off down the street.
"What do you expect, the way you encourage him?" said Yvette later, when we were back at Ashleigh's. Seth had made an attempt to outwait my friends at the restaurant, but after Ashleigh had shown that she was prepared to out-outwait him, he had given up and gone home.
"I don't encourage him—what do you mean?" I objected.
"You're always replying to his messages right away and letting him sit next to you in the Nettle's class."
"But what am I supposed to do, without being totally rude? And how do you know where he sits? You're not even _in_ that class."
Yvette just smiled.
"Well, I am, and she's right—he _is_ always sitting next to you," said Yolanda. "Why don't you like him back, anyway? He seems like a pretty nice guy, and he's cute too. Not crisp-cute, like Adam or Ravi, but sort of cutish-cute. He's got that artistic, romantic thing going on. He's got a nice nose. He looked really good that time at Halloween when he was a pirate. You really don't like him? I would, if he liked _me_ like he likes _you_."
"Landa, your standards are so low," said her sister. "You think everybody's kind of cute, even when they're igsome. You should be slapping them yourself, so I don't have to."
"I don't know that Seth is igsome, exactly," I said. "I just don't Like him."
"The point isn't whether he's igsome," said Ashleigh. "The point is that Julie's affections are Otherwise Engaged."
"Oh, right, I forgot, you're going out with Ned, right?" said Yolanda. "But I bet you could still go out with Seth too, if you wanted. How's Ned going to find out? He doesn't exactly get out much."
That raised Ashleigh's fighting spirit. "Yolanda! How can you suggest such a thing?" she flashed out. "Julie would never be so false—she would never treat anyone with such disloyalty, especially not a noble being like Ned! Her love, like her nature, is pure and true!"
"But I keep telling you, I'm not going out with Ned," I protested feebly. I didn't press the point, though. For one thing, it was useless—I knew I would never change Ashleigh's mind. And her passionate words distracted me, filling me with guilt. Never false—incapable of disloyalty—my nature pure and true. This—from the girl whose hoped-for boyfriend I couldn't get out of my mind! Ashleigh's words would be far, far more fitting if she applied them to herself. How would I ever deserve my loyal friend's praise?
**_Chapter 15_**
_Holiday cheer_ ~ _The baby's birthday_ ~ _Sweet Sixteen and Never Been Kissed_ ~ _my First Kiss._
_**T**_ he Christmas vacation arrived in a flurry of exams and term papers. The winter issue of _Sailing to Byzantium_ went to the printer. Seth dropped off the disk; I used my last English paper as an excuse not to go with him. I rushed through my essay, repeating ideas from the previous one, but Ms. Nettleton didn't notice.
Yolanda's sentence ended, but she got regrounded for cutting physics to hang out with Adam.
Ashleigh and I exchanged our yearly Hanumas/Chrisukka presents. She gave me a CD of songs popular in nineteenth-century parlors—"What Jane Austen's heroines would have listened to instead of musicals," she explained. I made her a magic kit from unsold odds and ends in my mother's shop: a bouquet of colorful scarves, a wand cut down from a broken walking stick, a stuffed rabbit. I was particularly proud of the top hat, which I fitted out with a false bottom and a hinged trapdoor on top. I hoped the gift would spark a new craze—but no. "How Jane Austen's characters would love this!" cried Ashleigh. "Perfect for those long evenings at Pemberley. Hey, what do you think about doing a musical version of _Pride and Prejudice_? Wouldn't this hat look great on Darcy?"
The other major holiday of the season is, of course, my birthday: December 17. It fell early in the vacation, as it usually does. With his strict attention to his parental rights, my father insists on my spending alternate birthdays at each house; this year was his turn.
I awoke to the sound of footsteps on the ceiling of my basement bedroom. The Irresistible Accountant was in the kitchen directly overhead, stomping and crashing breakfast into life. I buried my head in the pillow, but sleep had fled, so I put on my bathrobe and slippers and went upstairs.
Amy handed me a plate of winter-squash frittata, herbed home fries, and sliced citrus salad. "There you are, sweetie," she said. "Happy birthday." Then she burst into tears and ran out of the room.
As I stared after her, my father gestured at my plate with his fork. "Aren't you going to eat your breakfast?" he asked. "Go on, eat it. You'll hurt Amy's feelings."
I took a halfhearted bite. "What's the matter with her? Is my birthday such a tragedy it makes a grown woman cry?"
Dad gave me a look of grave reproach. "How can you be so thoughtless? Don't you know what day it is today?" he said.
"Um, December 17th?"
"Yes, to you it's December 17th—but to Amy, it's the baby's birthday."
"What baby's birthday? She doesn't have a baby."
"That's why she's so upset," said my father patiently. "The baby was due on December 17th. If she hadn't had the miscarriage, today would be his birthday."
I did some arithmetic. "How can that possibly be?" I said. "She had the miscarriage in October. She didn't even look pregnant. The baby can't have been due for months and months."
"Not that miscarriage, Julie," said my father with a touch of irritation, as if I had missed a very easy question on a quiz. "That was only the latest one. You don't know how hard things have been for Amy. I'm talking about the first miscarriage, the one four years ago, when Amy and I first got together. After we lost that baby, she was devastated. Don't you remember? She's been very, very brave, but when we lost the new baby again two months ago, it opened the wound all over again for her. You'll be kind to her today, won't you? I know you will. It's a very sad day for her, and she's feeling very vulnerable."
With these words, my father finished his frittata, put on his coat, and went off to work.
I stared blankly at the elaborate eggs congealing on my plate. After a while, I noticed that my brain had continued to do arithmetic all by itself. The sum it produced horrified me. If my father was telling the truth—and there was no reason to think he wasn't—then Amy had already been pregnant for months before my father left my mother. All those weeks when my parents went together to marriage counseling, all those weeks when he swore to her—and me!—that he would start fresh: all of it lies. He had known all along he would leave. He hadn't meant a word of it.
Of course, I had no illusions that my father had succeeded in the new start he had promised to make. How could I? He had left us, hadn't he? But it was another thing to learn that he hadn't even tried.
And another thing entirely to learn it on my sixteenth birthday.
I abandoned my Familial Restraint Fund for good. No amount of imaginary money could ever compensate for this.
My cell phone rang. I checked the number: my mother, doubtless calling to wish me a happy birthday. I decided I couldn't deal and let the voice mail get it.
Amy came back into the kitchen. "What's the matter, sweetie?" she said. "You're not eating. Don't you like your frittata?"
With the house already dank with Amy's tears, I balked at adding mine. It was too cold out for long woodland rambles, however, so I took refuge in the Lius' greenhouse next door. The conservatory, I called it in my head. But unlike the conservatory at Forefield—an elegant structure where cast-iron frets offered up crystal panes to the sun—the Lius' version was small and practical, close cousin to a shed. Haichang had built it from plywood and sheet plastic to keep the worst of the cold off his orchids and Lily's vegetables.
It was chilly but bright in the greenhouse. The sun wavered in through the plastic and the thin winter clouds. Carefully moving aside two pots of hybrid phalaenopsis, I sat on a bench, breathed in the wet air, and gave myself over to self-pity.
The worst of it was, I felt I had no real right to feel sorry for myself. I had friends, I had parents (two sets of them) who neither beat nor neglected me, I had good grades, acceptable looks, absorbing activities—all the trappings, I told myself, of privilege. I had my heart's sister, Ashleigh. I even had a suitor, of sorts—not, alas, the guy of my dreams—just Seth. Imagine kissing Seth. Ig! But what if I couldn't escape it? I was getting older and older, and Seth was the closest I had ever come to anything approaching a boyfriend. Sixteen years old already, of all pathetic things! What if my longed-for first kiss was with Seth?
I began to cry in earnest.
The wind ruffling the plastic, the hum of the heater, the gurgle of the humidifier, and my own snuffling sobs filled my ears, so I didn't notice that I was no longer alone until I felt an arm around my shoulder.
"Stringbean! What's the matter?" said a man's voice. Zach, home from college for the vacation. He used an ancient nickname that I'd hoped everyone had forgotten long ago.
"Oh, Zach," I said. "It's my birthday." I hid my face against his shoulder and sobbed harder.
"But Beano, that's a _good_ thing. Happy birthday! Seventeen?"
I shook my head, his sweater scratchy against my cheek. "Sixteen," I said.
"Even better! Sweet sixteen."
I sobbed still harder. "Yeah, right. Sweet sixteen and never been kissed."
"Oh, is _that_ the problem? Have you really never been kissed? What's the matter with that boyfriend of yours—that Foureyes boy? They're all wimps at that place. Except my young sparring partner Parr, of course, there's a kid with a spine—I bet he's not leaving your little friend Ashleigh unkissed. Hey, easy there, Bean Cuisine, you sound like you're choking. Just because your boyfriend is too scared to make a move, that isn't any reflection on _you_."
"If you mean Ned, he's not my boyfriend," I said for the millionth time. "I know Ashleigh says so, but he isn't. I don't have a boyfriend. And I'm so tall, and I have stringy arms and stringy legs and stringy hair and a stringy face, and nobody ever wanted to kiss me except creeps and stuffy Seth Young, and even if they did, I wouldn't know how."
"Wow," said Zach. "That sounds pretty bad." He held me in his scratchy arms and patted me a little too hard between the shoulder blades, as if he were trying to dislodge a chicken bone. I started to feel a bit better.
"I could show you, if you want," he said. "You could practice on me."
"What do you mean? Practice what?"
"How to kiss. That way when the boy who isn't your boyfriend finally gets off his ass and kisses you, you won't worry about getting it wrong. And if he doesn't—well, anyway, you won't be sweet sixteen and never been kissed."
I took my face out of his sweater and looked at him.
"But not if you don't want to, of course," he said. "Sorry, I didn't mean—"
Quickly, before I could change my mind, I kissed Zach. Handsome Zach, heartthrob of the seniors, kind, vain, teasing, brotherly, out-of-my-league Zach.
The first kiss—the one I launched—landed hard and sudden, off center. I didn't quite know what to do with it. "Mmmm," said Zach tactfully when it was over, taking my face in his hands and moving in with gentle expertise.
"That's the way," he said when he was done. "Another?"
I nodded. This time our mouths came open a little. Alarmed, I felt myself fluttering. Something bumped, something seemed to tangle.
"Easy, now," said Zach, pulling back. "Relax."
I nodded again. After the next kiss, it began to feel almost natural—more like a dance, and less like two people trying to push through the same swinging door from opposite sides. I found I could even breathe while kissing; I considered opening my eyes. Before I could, however, I felt a crash judder through Zach's torso, bumping his teeth into mine. "Ow!" he said.
Simultaneously, I heard Samantha yell: "Zach! You creep! Leave her alone! What are you doing?"
I opened my eyes. Sam was hitting Zach with a bag of potting soil. I had never seen her so angry.
"Julie! Are you okay? Zach, what do you think you're doing?"
"What does it look like I'm doing?" answered Zach, brushing soil off his jeans.
"It's okay, Sam," I said, mortified. " _I_ kissed _him_."
Samantha looked at me for a moment, then turned back to Zach. "How can you be so irresponsible?" she said. "What about Jenna?"
"Calm down, Sam. You don't need to throw dirt around. I'm not cheating on Jenna, it's just a kiss. Julie and I both know what we're doing. It's her birthday and she was feeling lonely. Nobody's going to get their heart broken. Julie's in love with that Forefield boy, anyway."
Samantha put the bag of potting soil down. "Get out of here, Zach," she said. "I mean it, go on." Zach gave me a sheepish look, brushed off more dirt, and left. Sam turned to me. "I apologize for Zach—he's an idiot," she said. "Are you all right?"
Was I? I had no idea how I felt—thrilled, terrified, shaken? I needed to go away somewhere and figure it out, but first I needed to calm Sam down. "It's okay, Sam," I said. "I'm not going to do anything stupid like fall for Zach. Don't worry about it, okay? He caught me crying and he was comforting me. That's all. He wasn't taking advantage of me. _I_ kissed _him_."
"If you say so," said Sam. "But if you need me to kill him, tell me. In fact, I might just do it anyway. Oh, and happy birthday, by the way."
**_Chapter 16_**
_Paperwhites_ ~ _Hothouse flowers_ ~ _The Great White Way_ ~ _Parr's house_ ~ _Footprints in the snow_ ~ _A Third sonnet._
_**W**_ hen I got home, I found someone had slipped into my room and cleaned it up for me. My bed was made, my clothes neatly straightened, my collection of shells, stones, bones, and fascinating bits of broken china carefully dusted and arranged on their shelf. The floor gleamed, as if someone had mopped it. There wasn't a cobweb in sight, not even in the farthest reaches of the roof peak, where a displaced spider had begun to spin new tether lines. Even my desk was free of dust, the books and papers arranged in the exact order I had left them in, but with their corners straightened, all at right angles. A bowl of paperwhites bloomed on my windowsill, filling the room with their sweet, slightly gasoliny fragrance.
This, I realized at once, must be Ashleigh's birthday present to me. She knew she was the only person who could get away with touching my things.
Sure enough, her curly head popped up at the window. "Open up," she said. "It's starting to get icy out here."
Looking at my sparkling room and my grinning friend, I felt ashamed of myself. What did I have to complain of, compared with what I had to be thankful for? I pushed the window up and gave Ashleigh a hand in.
"Happy birthday," she announced with satisfaction, taking off her sneakers so as not to track bark dust on the gleaming floor. I noticed that she was wearing jeans, without any regard for the visibility of her lower limbs. "How was it? Did anything earth-shattering happen?"
"Actually, yes," I said. I was tired of keeping secrets from Ashleigh. This, at least, I could tell her. Perhaps she could help me figure out what I had done and why.
"Wait, let me get this straight," said Ashleigh when I was done. "You kissed Zach Liu? Four times?"
"Yes—well, technically, I only kissed him once. The other times _he_ kissed _me_."
"But Julie, I had no idea you felt that way about Zach. Why didn't you tell me?"
"Because I _don't_ feel 'that way' about him. If you mean am I interested in him, no, I'm not. That would be idiotic. He's way out of my league. All the girls at Byz have a crush on him, and he knows it. He's in college, for God's sake. He has a girlfriend. I'm barely sixteen."
"Then why did you kiss him? What about Ned?"
"What _about_ Ned? Ash, come on! I keep telling you. I'm not interested in Ned. I was never interested in Ned. He's a nice guy, but I'm not interested in him. And he's not interested in me, either. He sure as hell never tried to kiss me."
"I see," said Ashleigh. "I wish I hadn't interrupted you guys in the trophy room that time, before he had a chance. I wish I'd just tiptoed out without saying anything! I would have, but I thought I was rescuing you from Chris. Well, I guess I can understand it. You get tired of waiting for the one you love to kiss you, so you go and kiss someone else."
I sighed. She was right, although not the way she thought. "Do you realize I'm sixteen years old and I never kissed anyone?" I said. "Ned never wanted to kiss me. That's not what we were doing in the trophy room. Nobody ever wanted to kiss me—unless you count Seth, maybe, which I'd rather not. I didn't really know what I was doing when I kissed Zach. I wanted to see what it was like. I guess I was afraid that if I waited, Seth would somehow get me to let him kiss me, with that stubborn persistence of his, and then that would be my first kiss. At least Zach is someone I like."
"So he _is_ someone you like!"
"Not _Like_ —just like. He's a nice guy, he's really good-looking, he's Samantha's big brother, and he's a college student, away in college, where he has a girlfriend. He's not going to be after me to go out with him like Seth is. Anyway, why do I have to Like someone before I can kiss him? Are you in love with Ravi? You kiss _him_ every week—twice a week, or more."
"But that's different—I have to, for the play. I wouldn't if I didn't have to."
"Why not? Doesn't Ravi kiss well?"
"I don't know—he's fine, I guess—but I'm not interested in him."
"Well, have you ever kissed Parr, then?" I asked. As soon as I said it, I wished I hadn't.
"No, of course not, you'd know if I had. I would never keep such a thing from you," she said impatiently. "So what _was_ it like?"
"What—kissing Zach?"
"Yes."
"It was—nice. Surprisingly nice. I'd do it again in a heartbeat."
"You'd kiss Zach again?" said Ashleigh, shocked. "I thought you said he has a girlfriend!"
"Well, maybe not Zach—Samantha would kill me. Anyway, she'd kill one of us. And there _is_ that girlfriend. But somebody, yes, I'd definitely kiss somebody, if it was the right guy."
The likelihood of kissing the Right Guy, however, seemed so distant that I allowed myself to wish, for a moment or two, that Zach was unattached, that he wasn't in college, that he wasn't the son of my father's partner, wasn't as far above me as the reindeer on Ashleigh's roof, and would kiss me again, this time without a dirt-throwing sister to interrupt.
Samantha came by the next day with an armful of flowers. I recognized them from her parents' greenhouse. She made Zach drive her in the famous Saab, but she wouldn't let him get out of the car. "Happy birthday," she said. "These are from my idiot brother. I'm delivering them personally to make sure the message comes through loud and clear. They're not romantic flowers. They're happy-birthday, I'm-sorry-I-molested-you, will-you-ever-forgive-me-or-does-my-sister-have-to-kill-me flowers."
"I wanted to get you roses, but Sam wouldn't let me," called Zach from the car.
"Shut up!" said Sam, and hit him through the window.
For Christmas, Ashleigh's parents gave her a pair of tickets to see _Fascination!_ on Broadway, and she invited me. We rode down on the Metro-North train and stayed overnight with my aunt Ruth and uncle John. We spent the afternoon before the show eating dumplings in Chinatown, browsing through the giant used bookstore in the Village, and trying on false moustaches at a theatrical supply shop.
The best part of the show was the songs. Ashleigh couldn't get over the voices and the orchestration, and I thought the lyrics were almost as clever as Parr's. When the curtain fell, we clapped until our hands went numb.
We slept in Aunt Ruth and Uncle John's living room, Ashleigh on the couch and me on an inflatable mattress. I sank slowly through my dreams and woke up in the morning flat on the floor, with a crick in my back. "Oh, dear," said Aunt Ruth. "Looks like the bed needs a patch. Sorry about that."
We spent the morning at the Frick Museum, the former Fifth Avenue mansion of a nineteenth-century steel magnate that houses his art collection. We enjoyed ourselves arguing about which of the portraits matched which of the people we knew. Ashleigh was easy: she could have been the model for George Romney's portrait of Lady Hamilton, a pretty young woman in a red dress with abundant dark hair and a lively little dog under her arm. It was harder to find a picture of me, though. Ashleigh pointed to a graceful Gainsborough lady in an elaborate blue dress, but I felt more like a severe Whistler girl in black.
After lunch Ashleigh said, "Hey, doesn't Grandison Parr live in this neighborhood? Let's go check out his house. Maybe Ned will be there too."
"I don't know, Ash," I said. I felt the familiar dread of public embarrassment. "What will we say if they see us? And why would Ned be there, anyway?"
"We'll say we were in the neighborhood, which is true. And Ned told me he'd be spending some of the vacation with the Parrs. Come on." She pulled my arm over her shoulder with both her hands and used it to tug me down the street.
"Okay, okay, let go," I said. Recovering my arm, I followed her with a sigh.
Parr turned out to live in a tall, narrow town house that looked as if it had been built around the same time as the Frick Museum, a century or so ago. It had a limestone stoop leading up to a shiny red door. My heart fluttered to think that I was looking at his home, where he read, showered, slept, dreamed.
My heart fluttered even more—in a very bad way—when Ashleigh started up the stairs to ring the doorbell. I hauled her back. "No. Absolutely not. People don't just ring each other's doorbells around here." She protested, but I refused to let go. "If you do, I'm leaving without you. I'll catch the early train back. I'm serious, Ashleigh."
"Oh, all right," she said. She leaned against a tree on the sidewalk in front and looked up. "Which window do you think is his? Do you think Ned is staying in the guest room? Which window do you think is the guest room?"
The thought that Parr might be standing behind one of those windows—might look out and see us—sent scared thrills buzzing in my wrists. "I don't know," I said. "You've seen where he lives, okay? Can we go now?"
"Just hang on a minute—maybe they'll come out."
"If they do, I'll die of embarrassment. Come on, let's go. It's cold out here. I feel like an idiot."
"Well, if you let me ring the doorbell, we could go in and get warm," said Ashleigh.
"Good-bye, I'm leaving now, see you back in Byzantium," I said.
"Okay, okay, okay! Just wait a little. Maybe they'll come out."
Fortunately, they didn't.
A tall blonde girl walked by slowly, looking at the windows. "Do you think that's that girlfriend of Parr's?" I whispered.
"What girlfriend?"
"That Sam's friend was talking about—remember, in the e-mail?"
"Could be. I'll go ask her—maybe she knows where they are," said Ashleigh.
"Ashleigh, you're nuts! Don't you dare," I hissed, holding her arm as tightly as I could. The girl walked away down the block.
After half an hour of stamping in the cold, even Ashleigh admitted her feet were getting numb. We caught the 2:25 north from Grand Central.
The weather turned bitter after New Year's. Drafts slashed through my attic. They were more painful than usual because my mother and I had decided to keep the thermostat low, to save on heating oil. I piled every available blanket on the bed and took to sleeping in my warmest, ugliest pajamas, the ones with fried eggs on them. I even wore a cap to bed.
Snow fell: not enough to shut the schools, alas, but enough to add half an hour of shoveling to our mornings. Our tree grew damp and awkward, liable to dump snow down our necks. Ashleigh and I suspended our arboreal crossings until kinder weather.
School started again. In history, the French revolutionaries stormed the Bastille. In English, we began reading _Pride and Prejudice_ , to my dismay—I worried the Nettle would ruin my favorite book. Ashleigh and Yolanda did their best by raising their hands nonstop and talking as long as possible whenever she called on them. For once, though, Seth's class participation fell. He's one of those boys who consider Jane Austen silly and trivial. He did have a few nice things to say about Elizabeth's father, Mr. Bennet, whom he found witty, in contrast to the "repellent" Mr. Darcy.
The Forefielders returned to their palace on the hill. Ashleigh got e-mail from Ned saying they were back. He had indeed spent his vacation with the Parrs, but on Bermuda, not in Manhattan, so I had wasted all my anxiety on East 74th Street. _Insomnia_ rehearsals didn't start up yet, however—the boys had their finals after their vacation, poor things, and extracurriculars were suspended so they could study.
"Did you hear anything last night, Julie?" asked Ashleigh one morning as we waited for the school bus.
"What kind of anything?"
"Sort of thrashing. I thought it was a bear, or a deer eating the tree, but when I looked in the morning, I saw footprints in the snow. People feet, not deer hoofs. Unless the deer was wearing boots."
"Do you think it was a person eating the tree?"
"People don't usually like bark, do they?"
"Not unless it's almond bark."
We paused for a moment of nostalgia, remembering the many delicious pounds of almond bark we had made during Ashleigh's candy-making period.
"Funny, I wonder who it was," said Ashleigh.
There were no more footprints when we checked for the next couple of days, but three days later—a Saturday—we found the snowdrift beneath our tree kicked and dented. Not only that, but pinned to the tree with a red thumbtack was a sonnet, its edges curling from the damp. Fortunately, it had been written in ballpoint, so the ink hadn't run.
This is what it said:
_Just let me wait a little while longer_
_Under your window in the quiet snow._
_Let me stand here and shiver. I'll be stronger_
_If I can see your light before I go._
_All through the weeks I've tried to keep my balance._
_Leaves fell, then rain, then shadows. I fell, too._
_Easy restraint is not among my talents;_
_Fall turned to winter and I came to you._
_Kissed by the snow, I contemplate your face._
_O do not hide it in your pillow yet!_
_Warm rooms would never lure me from this place_
_If only I could see your silhouette._
_Turn on your light, my sun, my summer love._
_Zero degrees down here: July above._
"Wow!" said Ashleigh. "Somebody likes you!"
"Why me? It could just as easily be you."
"He tacked it to your side of the tree."
"That's the easiest side to reach."
"But it's obviously about you, Julie! He even uses your name. Look, 'July above.' Not quite Julie, but close enough. And he says, 'Easy restraint is not among my talents.' If that's not Ned describing himself, I don't know what it is. He's so sincere—so spontaneous—so unrestrained!"
"I disagree—it doesn't sound like Ned to me," I said. "That's not how he writes. He misspells like crazy, and he doesn't use punctuation—at least, in e-mail he doesn't. If I were going to guess, I'd say it's Parr. We know he likes to rhyme. Doesn't _pillow yet/silhouette_ sound like some of the _Insomnia_ lyrics? And _balance/talents_?"
"Parr? I guess that's possible. Does _Parr_ like you too? Ned _and_ Parr? Well, I don't blame them a bit!"
Parr! Writing love poems to _me_? Could it be possible? And was there pain in Ashleigh's voice as she suggested it? I hastened to reassure her. "No, no, Ash, he's clearly talking about you," I said. "Listen: 'My sun, my summer love.' That's got to be you, you're much sunnier than me."
"No, silly, that's you, you're the sunny one—I'm dark and curly. Could it be Seth? He writes for that literary magazine of yours, and we know for a fact that he likes you."
"Oh, I hope not! I don't think so, though. He thinks he's Emerson, not Shakespeare."
We debated for a while longer without resolving the question. Ashleigh, generous girl—stubborn girl—insisted that I take the sonnet home with me. I pinned it on my bulletin board beside the other mysterious note, the one from the chocolate turkey. I studied them, trying to decide whether the same person had written them both. The turkey note was cramped and messier, possibly because the writer had to fit his message on the side of a small box, but the letters seemed not dissimilar. I decided I needed a larger handwriting sample from the turkey giver before I could say for sure.
**_Chapter 17_**
_A Limited Junior License_ ~ _A disastrous Mocharetto_ ~ _Mint Sauce_ ~ _My father and stepmother Approve._
_**S**_ eth Young stopped me as I was leaving school with Ashleigh and the Gerards that Tuesday. "Oh, Julie, Ms. Nettleton says the printer called," he said. "The bound copies of _Sailing_ are ready. Eleanor asked me to pick them up. Can you come help? It's over in North Byz."
It seemed impossible to refuse without being rude. "All right—how are we getting there? Is Ms. Nettleton driving us?"
"No," he said proudly, "I am. I got my license last week."
"Don't you need an adult in the car till you're eighteen?"
"Nope—Limited Junior License: may drive alone for school course or activity. This is a school activity."
I gave Ashleigh a help-me look. "Great!" she said. "North Byz—that's where Yv and Yo live. You can take the three of us with you and drop us off."
Seth gave her a look of fake concern. "Sorry, Ashleigh, I wish I could, but it's a Limited Junior License. No more than two underage passengers."
I tried a last-ditch effort to discourage him. "I better call my dad and see if it's okay. I'm not sure he'll want me driving with someone who just got his license last week."
"Let me talk to him, then," said Seth.
Worse and worse.
Terri, Dad's receptionist, put me through. "Dad, is it okay if I come home a little late? I need to go with my friend Seth to pick up the bound copies of _Sailing to Byzantium_. The literary magazine, remember? He'll be driving—he got his driver's license last week."
Dad expressed concern, as I'd hoped he would.
"Yes, just last week. I don't know, I've never seen him drive, but I'm sure it'll be fine, he's very energetic," I said in an attempt to alarm my father further while making Seth think I was calming him down.
Seth tugged on my arm to ask for the phone. "Hang on," I told Dad, "he wants to talk to you."
"Hello, Dr. Lefkowitz?" said Seth. "Seth Young. I just wanted to assure you that I'm a very safe driver and I'll take good care of your daughter. I had sixty hours of practice before I took my road test—twice the recommended state guidelines. I got perfect scores on all the exams, including the road test. I'm certified in first aid and CPR. Not that I expect them to be necessary this afternoon, of course, but I think it speaks to my character. What? Yes, my parents' Volvo. . . . No, never. . . . Of course. . . . Oh, that sounds wonderful, thank you very much, I'll just have to ask my parents." He handed me the phone back. "He wants to talk to you again."
"Your friend sounds like a very responsible young man," said my father. "I invited him to dinner. Amy's roasting a leg of lamb."
Seth drove exactly at the speed limit the whole way to the printer, coming to a complete stop at every stop sign. He held the steering wheel with both hands and checked his mirrors five times a minute.
We handed over our paperwork in the storefront office and sat down to wait on the mahogany-red vinyl sofa. Seth draped his arm along the back, a little too close to me, but not quite close enough that I could shrug it off. I stood up and wandered around the room to look at the framed handbills hanging on the walls, samples of the printer's work. After a few minutes, we heard the thumping trundle of our order approaching on a dolly.
"Here you go, kids," said the printer. "There's your disk back, and your receipt."
Seth insisted on opening a box and inspecting a copy of the magazine. He flipped through it, snapping the pages to feel the weight of the paper, and studied the cover picture through a magnifying frame he found on the counter. I couldn't help rolling my eyes.
The printer winked at me. "Everything look okay?" he asked Seth.
Seth completed his inspection. "It's all good," he pronounced.
We rolled the boxes out to the car, Seth pulling, me steadying, the dolly doing its best to make a break for it, and loaded them into the trunk. They were heavier than they looked. "Bend from the knees," instructed Seth. "Use your legs, not your back."
On the way back to school, I made up my mind to find out whether Seth was responsible for the sonnet on the tree. I hoped not, but if so, perhaps there was more to him than I thought.
"Seth, have you ever written a sonnet?" I asked.
"Oh, yes, several. I had one in the issue before last of _Sailing_ —don't you remember? My most recent was for Ms. Nettleton's class, for the creative writing assignment in October. It was about moral responsibility. Why do you ask? Are you writing one? Would you like me to read it and give you advice? It's a tricky form, but I'm sure you can learn."
Seth, I concluded with a silent sigh of relief, could not be our secret arboreal author. Last weekend's sonnet, with its references to snow, must have been written more recently than October, and if Seth _had_ written it, he would have made sure I knew.
After we unloaded the boxes in the English Department office—Seth had Ms. Nettleton's elevator key, a sign of supreme favor—he asked, "Want to stop at the Java Jail? It's only four-thirty, so we have plenty of time before your dad expects us. Come on, let's get a mocharetto. This is cause for celebration!"
"All right," I said reluctantly. "Just one."
The Java Jail was crowded when we got there. I grabbed the only table left, a tippy, drafty one near the door, while Seth went to order.
Looking around, I saw with alarm that most of the customers were boys in Forefield uniforms. What if someone I knew saw me with Seth?
He came back with our steaming drinks. "Here you go." He moved his chair closer to mine and lifted his paper cup in its dimpled cardboard sleeve. "A toast: to _Sailing_ , the magazine that brought us together!"
As I lifted my cup in return, I felt a cold blast run down my neck. Foreboding? Air from outside? It seemed rude not to return the toast. But I couldn't quite bear to meet Seth's eyes, so I turned mine away—and met, instead, the eyes of Grandison Parr, standing at the door.
"Hello, Julia," said Parr, with a formal smile.
"Parr! What are—I thought you guys had finals."
"They ended today. We get the afternoon out to blow off steam."
Seth cleared his throat.
"Oh! I'm sorry," I said. "Seth Young—Grandison Parr. Seth—Seth and I—we work on the literary magazine—he's in my English class—we just . . ." I trailed off.
"How do you do?" said Seth stiffly, offering his hand, as if he expected a Forefield boy to have fancy manners and wanted to prove that his were just as good.
Parr took his hand and shook it. "A pleasure. Well, don't let me interrupt." He gave me another formal smile and moved on into the café.
Should I have asked him to join us? But I wouldn't have been able to bear it, having Parr see Seth at his most pompous, having Parr think that this was the sort of person I would choose to associate with. My mocharetto scorched my mouth. I edged my chair around so that I had my back to Parr. For the duration of our drinks, I felt my back burning like a sacked and fallen city.
Naturally, my father was pulling into the garage just as Seth and I drove up. He stood at the door and watched Seth perform a perfect, though unnecessary, parallel parking maneuver.
"Come on in," he said. "Seth, I presume? You're staying to dinner, right?"
"Oh, yes, thanks," said Seth, stepping out of the car and following Dad into the house.
"Amy, this is Julie's friend Seth," said my father.
"Nice to meet you, Mrs. Lefkowitz," said Seth.
"Hello, Seth. I'm glad you can stay," said the Irresistible. "There's plenty of food—I roasted a leg of lamb." She turned to me. "And I made that mint sauce you like, sweetie, with fresh mint from the Lius' greenhouse."
The thought of the Lius' greenhouse, on top of everything else, made my stomach lurch.
While I struggled to eat dinner, Seth regaled Dad and Amy with details of our _Sailing_ responsibilities and my improvement in Ms. Nettleton's eyes since I'd taken them on. I watched my father and stepmother inflating with approval, like Aunt Ruth's air mattress. (I wondered how soon, also like the air mattress, it would all leak out again.)
"Seth," I said when dinner was over, "don't you have to get going? Doesn't the Limited Junior License come with a curfew?"
"Yes, you're right: 9 P.M.," said Seth reluctantly. "Thanks for dinner, Mrs. Lefkowitz—it was delicious. Julie? Walk me to the car?"
Dinner with my folks had clearly boosted his confidence. He looked as though he might try to kiss me once my father and Amy were out of sight. "No shoes," I said, wiggling my toes in his direction. I stayed firmly seated and let Amy show him out.
"Thanks again, Mrs. Lefkowitz. See you tomorrow, Julie."
"What a nice young man! Good-looking too," said Amy after the door shut. "You sly girl, is _that_ why you joined the literary magazine! Why didn't you tell us?"
**_Chapter 18_**
_My first appearance in Print_ ~ _Ashleigh interferes_ ~ _A Midnight Visitor_ ~ _A Quatrain._
_**A**_ shleigh loyally bought four copies of _Sailing_ : one for herself, one for each of her parents, and one, she said, for Ned. Although it would have been the depths of ingratitude to ask her not to, I wished she hadn't. The editorial board had chosen a poem of mine that I now felt was perhaps a trifle too personal—too open to interpretation—too revealing. They had published it under my initials, not my full name, but I was afraid that anyone who knew me would easily figure out what they stood for. Indeed, Ashleigh already had. I'll spare you the details, but if you want to get the flavor, imagine what a girl of some sensitivity might have written in her first flush of excitement at meeting the person who was to become the Magnet of Her Thoughts.
Plus, the rhymes were pretty lame.
"I don't think it's lame at all! I think it's beautiful!" insisted Ashleigh, handing over the nineteen dollars. "A fitting tribute to—all right, all right, don't hit me. I won't say it. But I still don't see why you won't admit it. Your poem's about a million times better than Seth's three essays, anyway— _yours_ is sincere. Speaking of which, sorry I couldn't chaperone you yesterday. How did it go?"
"Oh, my God, Ash, it was awful! He somehow managed to charm Dad into inviting him to dinner, and now he thinks he's my boyfriend."
"How can he?"
"He clearly thinks it's like getting your driver's license, or A's in math, or getting the Nettle to like you. You just follow the steps right, and that's it, you're done."
"What if you told him you already have a boyfriend?"
"Oh, I don't know—I thought about it—but I _don't_ have a boyfriend. I can't quite bring myself to just lie straight out."
"It's not actually so much of a lie. At the rate you're going, you will soon."
"At the rate I'm going, if I do, it'll be Seth. But it's weird, isn't it? I really don't get it. What is it about me? If _I_ were a guy, I wouldn't look at me twice. I'm so tall and gawky."
"Don't say those things about my best friend! You're beautiful! You look like a model, only not weird. You don't have that overgrown-grasshopper thing. And you're more approachable. You have this quality of agreeableness that guys find . . . well, agreeable. You go along with things. What you need is for the right one to give you something good to go along with."
I saw the Right One quite a bit once the _Insomnia_ rehearsals began again—which they did that week, with a vengeance. There were, after all, very few weeks left until February 2, opening night. But he gave me nothing but measured politeness, with the occasional smoldering look.
"Has Ned said anything yet?" asked Ashleigh one evening, absently scratching Juniper behind the ears. He was no longer a kitten, but a rangy young cat. We were doing our homework in her room, which was far better heated than mine.
"Has Ned said anything about what?"
"You know—has he explained himself, has he declared his intentions? I thought he would have by now. I gave him a strong hint last week."
"Ashleigh! You didn't! You . . . What did you say?"
"I told him he'd better get moving if he didn't want to miss his chance, because you had a serious suitor."
"Ash! I'm going to kill you! How could you do that?"
"I'm sorry, but I just couldn't stand it anymore, watching you wait and wait. The suspense was driving _me_ nuts too."
"But Ashleigh—I keep telling you—oh, never mind, it's pointless. I think I'm going to die of embarrassment." I buried my head in my hands and moaned. "What did he say?" I asked.
"He didn't say anything. He's as shy as you are. But Parr asked if I meant that guy from the literary magazine. I said yes. How does Parr know Seth? You never told me they'd met."
The humiliation!
I wanted to kill Ashleigh, but of course it wasn't really her fault, since she didn't know how I felt about Parr. I tried to feel glad about that. After all, I had tried as hard as I could to keep it hidden from her. From Parr too. Could he really not know, when I felt so strongly? Surely he would see it in my eyes! Was he treating me with that distant politeness because he knew how I felt and didn't return my feelings? Or did he have no idea what he meant to me? Maybe he did like me, but he thought I was going out with Seth.
Horrible!
I thought about explaining to Parr next time I saw him, but what would I say? That I didn't like Seth—that he wasn't my boyfriend? Any such explanation seemed presumptuous, since it assumed that Parr would care. And anyway, there was still the question of Ashleigh.
Twice during rehearsal breaks I tried to speak, but I couldn't bring myself to do it.
Meanwhile, the _Insomnia_ production advanced at breakneck speed. Mr. Hatchek, the Forefield art teacher, set the entire sophomore (or rather, fourth form) art class to work painting backdrops. The costumes were mostly ordinary streetclothes, an exotic sight at Forefield. It took more fuss than you would think possible to get the cast outfitted in scruffy jeans. I began going over to the Gerards' to help Yvette rehearse with Yolanda so Yolanda would be up to speed on her part when her grounding was over. Yolanda even risked showing up for rehearsal once a week, leaving her sister behind to cover for her with their parents. The plan was for each of them to take the part in one of the play's two performances.
Late in January, I woke with a start in the middle of the night. It had snowed heavily the week before. Hip-deep drifts covered the roots of our oak tree, even in its sheltered position between Ashleigh's house and mine. Some large animal must have blundered into the hidden roots and branches; I could hear it crashing around in distress. Pulling the quilt around my shoulders, I opened the window to look.
Snow was falling heavily, obscuring my view, but I could see that it was no deer down there.
"Ashleigh, is that you?" I called softly.
The figure looked up. "Julia?" it said.
"Who's that?"
"It's me—Grandison. I d-didn't mean to wake you."
"Grandison! What are you doing down there?"
"I—I got locked out. I hoped—I th-thought if you w-were awake—"
His teeth were chattering so hard, he could barely talk.
"You're frozen! You'd better come up here. Do you think you can climb up? It's pretty icy. Should I come down and let you in the door?"
"No, d-don't, I've g-got it." He swung himself up from branch to branch with surprising grace. Clumps of snow fell around him and sank into the drift below.
I gave him a hand in and shut the window quickly. His gloves, his sleeves were icy wet. My room, though drier, seemed only a shade warmer than the air outside. He stood shivering by the window, dripping snow on the floor.
Grandison Parr in my room!
Parr in my room, and me in my fried-egg pajamas, my night-cap like something out of "The Night Before Christmas," my hair poking unevenly out of its braid, and my feet in fuzzy pink slippers, a gift from Amy, which I would have thrown away long ago if they weren't the only thing that could protect me from the demonic chill of the floor. I quickly took off my ridiculous night-cap and turned on the light.
We blinked at each other. His face was red and white.
"You're soaked—you better get out of those wet things," I said. I took his coat, hat, and scarf to drip in the storeroom next door. I put his boots up on the drying rack, which was built for apples, and brought him a towel.
"I'm s-s-sorry to b-b-barge in," began Parr. He could hardly talk through his chattering teeth.
"I know, it's freezing in here," I said. I felt his arm; the sleeve of his sweater was wet. "I'll find you some dry clothes." I rummaged in my dresser and came up with clean sweatpants, T-shirt, and sweatshirt. For almost the first time, I was glad to be so tall. "There, I think these should fit. Go in there and put them on."
When he came back from the storeroom, he was still shivering violently. His lips were blue. I handed him my quilt.
"Th-thanks, Julia. I'm s-sorry to burst in on you like this—I didn't mean to wake you."
"What are you doing here, anyway?"
"I—I got locked out of c-campus and it's pretty nasty out there. I didn't know where to go. You're an angel—thanks for the dry clothes. I'll just wait here a little while until it lets up a bit, if you don't mind, and then I'll go back."
"Go back? How will you get in?"
"How?—Oh. There's a place in the wall where I can sometimes get over. I—My hands were too cold when I tried to climb it before, but I'm warmer now."
"Are you serious?" I said. "All the way back to Forefield in this snow—in your wet coat? You'll freeze to death! You'll never get over the wall if you didn't before. They have to unlock the gates in the morning, right? You'd b-better stay here until then."
"Oh, no—now _you're_ shivering," he said. "Here, take this back."
He tried to put the quilt around me, but I resisted. "You need it more than I d-do," I said. I couldn't tell whether I was trembling from cold or from his nearness.
"It's big enough for two," he said, wrapping it around both of us.
Parr's icy hand grew warmer on my shoulder. He smelled beautiful—like wet hair and tree bark and strength. My cheeks burned. I thought they must be giving off enough heat to warm the room—to warm the whole house.
"Julia, I'd better go," said Parr after a while. "I can't stay here all night. You need to sleep. I'll be fine."
The insane gallantry! "No—you will _not_ be fine. You'll get frostbite. You're staying here till morning. You can take my bed, and I'll sleep downstairs on the couch."
"If anyone's sleeping on a couch, it's me."
"You can't—my mother will freak if she sees you."
"Won't she wonder why _you're_ sleeping on the couch, then?"
I considered this. If Mom caught me sleeping downstairs, she'd die of guilt for keeping the thermostat so low. She'd insist on turning it up for the rest of the winter, which we couldn't afford. But with my room so cold, I didn't have enough blankets for two.
"See? It won't work. Where did you put my boots?" he said.
"I'm not giving them back. You're not going anywhere. We can both sleep here, in my bed."
"Oh, no," Parr protested. "I couldn't do that."
"Don't worry, you'll be perfectly safe. I'll keep my hands to myself," I said.
He opened his mouth, then closed it again. Silently he helped me remake the bed, tucking the quilt in well. I got in; he turned out the light and got in after me, scrunching himself up as far away as possible—which wasn't very far. Our shoulders touched.
"Are you comfortable? Have I left you enough room?" he asked.
"I'm fine."
"You're still shivering. Are you warm enough?"
"Yes, I'm fine. I know it's cold in here, but I'm used to it. What about you—are you warm enough?" Not that there was anything I could do if he wasn't; hold him in my arms, maybe. I shivered and turned toward the wall, leaving his shoulder behind.
"Toasty. Embarrassed, but toasty. Good night, Julia. And thank you."
"Good night, Grandison."
For a long time we lay at our separate edges of the bed, back to back, the inch between us burning like lava. I felt the blankets move with his breathing. Was he asleep? He couldn't be. What was he thinking? I wanted to turn and put my arms around him and breathe in his smell. I wanted to curl myself into a trembling ball and shrink away to nothing, far, far away from him and everything else, never to emerge again. I wanted the night to last forever, the two of us side by side, with no end and no consequences.
A long time later I woke to find myself strangely warm in my cold room, with warm, steady breaths in my ear. After a moment I remembered who was there. Parr had turned over sometime in the night. He had his arm over my waist, his knees bent behind mine, like a pair of spoons. I felt his chest against my back, rising and falling with his sleeping breath. Blissful, I fell asleep again.
I next woke in the gray of dawn. It had stopped snowing. Parr was standing by my bed, dressed, wearing his boots and holding his coat. "Shh—I didn't mean to wake you again," he whispered.
"What time is it?"
"Six o'clock. The gate should be open by the time I get back."
"You found your boots."
"Yeah, you hid them pretty well, but I found them. Thank you, Julia. You're the best." He smiled that white-and-blue smile of his, bright with the turquoise of his eyes, upheld by his vertical dimple.
"Be careful going down the tree."
"I will." He gently lifted the end of my braid and kissed it, like a gentleman kissing a lady's hand. "Good-bye, Rapunzel."
I woke for the third time an hour later, dreaming I was kissing someone. Was it all a dream, then?
Apparently not. Pinned to my bulletin board, under the sonnet Ashleigh had found on the tree, was a note:
_Generous Julia,_
_Graceful and truly a_
_Port in a storm:_
_So calm, so warm._
The handwriting looked familiar. With good reason: it was the same as in the sonnet.
I was right, then. Parr was the mystery poet—Parr had written the sonnet.
But to whom? That was still a mystery. To me? To Ashleigh? If it was to me, I thought with a little laugh, how disappointed he must have been when he got upstairs. "Warm rooms would never lure me from this place," he had written. Well, that was for sure! No warm rooms in this house! And "Zero degrees down here: July above." July—ha! More like February.
Like February in my room, yes; but not in my bed—not in his arms.
Some minutes passed while I stared ahead of me, the hair-brush frozen in my hand, contemplating my bed and his arms.
"Julie! Julie, honey, are you up?" called my mother up the stairs, breaking my reverie. "It's almost eight o'clock."
"Coming, Mom!"
Whoever the sonnet was addressed to, it cast doubt on Parr's explanation of what he was doing under my window. Had he really found himself locked out of Forefield and come here for refuge? Possibly. But he had been downstairs at the foot of our tree at least once before, when he left the sonnet. Wasn't it possible that he had come again last night for the same reason, drawn by the presence of one of us—just as Ashleigh had dragged me to visit _his_ house and look up at _his_ window over Christmas vacation?
**_Chapter 19_**
_A song_ ~ _an Unspeakable Scandal_ ~ _my Mother takes a new Job_ ~ _the Talk_ ~ _a theatrical disaster._
_**I**_ just made the bus. "Jules, I have a surprise for you," said Ashleigh as we got on.
"What is it?"
"Has it occurred to you that once the play's over, it's no more Forefield for us?"
"Well, yes, it has," I said. It had indeed occurred to me, and today it was almost too painful to bear.
"So Ned and I have been thinking what to do about it, and we came up with a solution. Here!" She produced a piece of paper from her loose-leaf binder and handed it to me proudly.
"What's this?" I said.
"Look at it!"
I did. It was a sheet of music, a song apparently—and the lyrics were my poem from _Sailing_.
"Wow, Ashleigh—did you write this?"
"Yes. Well, mostly. Ned helped."
"Wow! That's amazing." I tried to hum it. I'm not so great at sight-reading; Ashleigh sang it to me. I had to admit, it was a good tune. "That's beautiful," I said. "I'm really impressed. But Ash, how does it solve the problem?"
"Ned wants me to collaborate with him on a song cycle. This is the first song. And Ms. Wilson agreed! I get to go to Forefield every week to work on it. And you can write the lyrics, so you can come too."
"A song cycle! What's a song cycle?"
"Oh, you know—a bunch of songs. If it weren't Forefield, we could just start a band, but this way it's all fancy and official and everything. They're calling it community outreach. It's supposed to improve Forefield's relations with the town if they include Byz High students in some of their programs. Anyway, the point is, we get to work with Ned on writing songs, and we get to go on seeing the guys. Isn't that crisp?"
I heard very little of what my teachers said that day. I lived in reverie. Mr. Klamp said, "Julie, snap out of it" twice, then gave up on me. Ms. Nettleton asked me to read aloud, and I did, but I have no idea what scene I read, even though it was from _Pride and Prejudice_ ; I didn't hear a word I was saying. Instead, I relived the night before. How much of it I had slept through! How much I had wasted! What did it mean?
On the way to Forefield, my heart beat harder than it had since the first day of rehearsals. My eyes found Parr as soon as I arrived. He stood in the back but faced the doors, as if he were waiting for me. He looked right at me and smiled. I met his eyes as long as I could, then looked down, blushing. The intensity was too much for me.
The theater was buzzing, as if in sympathy with my heart. A group of actors greeted us excitedly. "Have you guys heard?" said Emma. "Do you know about the disaster?"
"What disaster?" said Ashleigh.
"The sets. Mr. Hatchek got fired. The sets aren't even half done, and nobody can find his plans," said Ravi.
"Why'd he get fired?" I asked.
Chris was lounging a little apart, as if he considered himself above the conversation, but I noticed he still managed to hear what we were saying. "An unspeakable scandal," he said nonchalantly, with, however, a trace of satisfaction.
"What are you talking about? What does _that_ mean?"
"Nobody knows," he answered. "Except the administration, presumably. Everybody's speculating. The second formers think he embezzled the art supply fund, and that's why you can't ever find any charcoal."
"Whatever it is, there's no art for the fourth form until they find a new teacher," said Ravi.
"But what about the sets? Opening night's practically next week," cried Ashleigh.
"We'll have to go minimalist," said Ravi. "Empty stage, no curtain, create the sets with sheer acting and the imagination of the audience."
"Or the new teacher could wrestle them into shape, if the school can find one," said Parr, who was somehow standing at my elbow. How had he gotten there? My heart pounded at the sound of his voice. I leaned closer to him; I couldn't help it. So much for easy restraint. Our arms touched.
"Give it a rest, guys, okay?" said Dean Hanson, breaking into our circle. "There's no unspeakable scandal. But we _are_ looking for a new art teacher."
"So will we have new sets, or will we finish the old ones?" asked Emma.
"That's for the new teacher to decide, assuming we complete our search this week. Don't count on it, though. It's not easy to find someone qualified this late in the year."
"Jules—what about your mother?" said Ashleigh suddenly.
"Who—Mom?" I asked, like an idiot.
"Why not? She has an art degree, doesn't she?"
"That's true. She has an MFA. She did a lot of teaching before she married my father."
"Is that so, Julie? Well, tell her to send me her resume. As soon as possible," said the dean. "Now, shouldn't we all get practicing?"
That Saturday I went with Seth to a reading at the bookstore in town. I had agreed to go under the impression that other people from our magazine would be there, but the only one I saw was Ms. Nettleton. The author, a small, nervous person with a huge head and tiny hands, read a chapter from a novel in which the narrator's mother, dying of cancer, recalls in detail her passionate love affair with a wounded soldier in the French Resistance during World War II. Seth listened with rapt interest, leaning slightly toward me in his folding chair. Did he think the reading would put me in the mood?
As the story rambled from the narrator's mother's bedroom into a description of the French countryside, my mind began to wander to recent events in my own bedroom, and then to the stage at Forefield. I realized with a start that I'd left my copy of my _Insomnia_ script, with all my notes in it, on top of the piano where Ned had been using it to rehearse. I had promised to go over Ned's newest changes with Ashleigh—and unless Ned had remembered them and written them down on his script after rehearsal, I had the only copy.
Could I get Seth to drive me to Forefield and pick it up? But what if we ran into someone I knew? No, I would just have to apologize to Ashleigh and wait until next week.
After the reading, I made Seth drop me off at the Lius' instead of at my father's, so Dad and Amy wouldn't have a chance to invite him to dinner again.
"Hot date?" asked Samantha as he drove away.
I made a face. "No, thank God. A book reading, and Ms. Nettleton was there."
"You could let him know you don't like him, you know."
"I know. But he's a decent guy, and I don't want to hurt his feelings."
Sam rolled her eyes. "Well, you're going to be mad when you see who you just missed."
"What do you mean?"
"You had a visitor. Ask your father."
Dad looked up when I came in. "Was that Seth's car? Why didn't you invite him in?"
"He had to get back."
"Too bad, he could have stayed to dinner. Oh, before I forget, a friend of yours came by looking for you. Grant, or something like that? I told him you were out with your boyfriend, so he gave me this for you. He said you left it at school." Dad handed me my script.
My first impulse was to e-mail Parr and deny everything. But what would I say? "Dad's wrong, Seth isn't my boyfriend, it's _you_ I like, but so does Ashleigh and therefore my lips are forever sealed"? I had to content myself with kicking the fluffy pillows Amy had made for my new bed and tearing the flier from the reading into a thousand pieces.
My mother quit her job at the Nick-Nack Barn and started at Forefield two days later. She went whistling around the house, mostly songs from the play. I was glad to see her happy again.
Because there was so little time left before _Insomnia_ opened, she scrapped Mr. Hatchek's elaborate designs and replaced them with simple colored backdrops—slate gray and white for the lab, institutional yellow for the classroom, leaf green for the magical forest. She worked with the fourth-form painting squad as well as Mark, the lighting designer, and his team of techies to create an atmosphere of enchantment using colored scrims—screens that could look opaque or transparent, depending on how the light hit them. I may be biased, but I thought her designs were much more effective than Mr. Hatchek's fussy backdrops.
And I wasn't the only one who approved. Everyone in the production liked Mom, especially little Alcott Fish, who developed a crush on her that made him turn pink and squeak whenever she was nearby. Ashleigh and I laughed about it privately, but we were careful never to let him see that we'd noticed.
There was one disadvantage to having Mom around, though: no more hanging out with the guys while we waited to be picked up. Mom drove us home as soon as rehearsal ended. I hardly ever got a chance to talk to Parr, and never in private. Not that he seemed eager to talk to me now.
In a whirl of impersonal activity, I watched what I feared might be my last precious hours in his company drain away.
That Tuesday, my stepmother arrived a little earlier than usual to pick me up. She and my mother exchanged words of chilly politeness.
"What was Helen doing there?" asked Amy as we drove away. "Didn't she remember it was Tuesday?"
I explained that Mom had a job at Forefield.
"How nice. I was wondering when she was going to get around to getting a real job. I hope she's planning to tell your father soon. I think their settlement requires her to inform him within sixty days of any change in income," said Amy.
"Of course she is. This is only her third day working there. Has she ever tried to cheat you out of anything that's yours?"
"Hmp," said the Irresistible.
We drove the rest of the way in silence.
After dinner, my father cleared his throat. "Julie, now that you have a boyfriend, there's something Amy and I have to talk to you about," he said. "I know Seth is a trustworthy, reliable young man, and I hope that we've taught you some responsibility over the years. And of course, you're still very young; if we've done our jobs right, it will be a long time before you need to use this knowledge. However, I feel that it's my duty as a physician and a parent—that is, _our_ duty as _parents_ —" (here he gave Amy a saccharine smile, which she returned) "—to make sure you understand—," etc., etc., etc.
It was—can you believe it?—the Birth Control Talk.
The fourth one, chronologically speaking: Mom had given me the Talk a few years before, when I first got my period; and it had been repeated two consecutive years in Health and Hygiene, the second time with props, including a banana. Mom kept a you'd-better-not-need-these-but-just-in-case-you-do box of condoms in what I thought of as the Embarrassing Corner of the bathroom, updating them when they passed their expiration dates. (I checked.)
Hearing Seth's name coupled with the subject of the Talk made it doubly disgusting. I begged the floor to open and swallow me, as I had done so often during this distressful year. However, it had never yet obeyed. Why should it start now?
In no time at all, the day of the dress rehearsal arrived. I woke hours early and couldn't get back to sleep. Just one more day, and I would be singing in public.
I felt a horrible foreboding, but I dismissed it as stage fright. I slipped on my lucky thumb ring.
The first hint that something really was wrong came in homeroom. Yolanda sat in uncharacteristic silence, brushing tears away with her tapered fingers.
"Landa," I said hesitantly (since she seemed almost quiet enough to be Yvette), "what's the matter?"
She gave a little yelp and began to cry audibly.
I patted her back. "What is it? What's wrong?"
"Mom caught Yvette being me."
This was serious indeed. In the Gerard household, masquerading as one's sister was a grounding offense.
"Oh, no! How long are you down for this time?"
"Two whole weeks—both of us! We'll miss the play!"
Yvette confirmed the news at lunchtime. "I told you we should have switched the nail polish too," she said bitterly.
Yolanda started crying again. "Mom never noticed before," she gulped.
"That's 'cause you never wore green before."
What would Benjo do? I shuddered to think, but there was no warning him. Forefield boys were forbidden to use cell phones, except during certain evening and weekend hours.
Ashleigh broke the news as soon as we arrived. It took Benjo a while to understand, since he hadn't even known about Yvette's existence, much less her role in his production. As the news sank in, his face grew taut. I watched him pull himself together. He stood up straighter.
"Is there anything we can do to convince Yolanda's parents to change their minds?" he asked.
"Maybe if Ms. Wilson or the dean or somebody goes and talks to them?" said Ashleigh.
"Maybe my mom," I suggested. "She's friendly with Mrs. Gerard."
Benjo sent a second former to find the adults in question.
"Well, there's nothing else we can do about it today," he said. "Julie, you'll have to take over Tanya for now."
"What?" I gasped.
"You play Tanya. You know the part, don't you? You helped them rehearse. I thought you understood—you're the understudy."
"But my part—who'll play Headmistress Lytle?"
"Ned can do it."
"Uh, Benjo?" said Ashleigh. "Ned's a guy. He's a _bass_."
"Well, I know _that_. He'll have to be Head _master_ Lytle. One thing's for sure, he knows the part. He'd better—he wrote it. Okay, guys, help me get the cast together so I can make the announcement."
Hard as this may be to believe, it wasn't until Parr said, "So Julia's going to be Tanya?" that I realized what my new part meant.
**_Chapter 20_**
_My Fifth Kiss_ ~ _Mom to the rescue again_ ~ Midwinter Insomnia ~ _Conservatory flowers_ ~ _Ting is such Sweet Sorrow._
_**I**_ dreamed about kissing Parr. Asleep in my bed, awake in my bed, in that limbo between waking and sleeping that's known as tenth-grade European history, I dreamed about it. But I never dreamed that our first kiss would take place onstage, in front of the entire production of _Midwinter Insomnia_ , including my mother.
Although this was the dress rehearsal, there was clearly no way to make the twins' Tanya costume fit me. The clothes I had put on that morning—jeans and a sweater over a long-sleeved, scoop-necked T-shirt—would have to do.
I stood in Tanya's position, twirling my thumb ring on my upstage hand and looking out over a sea of furrowed brows. Concern shone from every eye in the audience, which included everyone in the production not actually onstage. I watched them worry: Would I remember my lines? Would my voice carry? Would I ruin the production they all had worked so hard on?
Gratifyingly, though, after a few minutes the brows began to clear. I was indeed going to remember the lines, my watchers decided one by one. My acting might not be as nuanced as the twins', my singing voice nowhere near as strong, but at least I wasn't going to totally flub it. My mother smiled encouragement at me. Part of me began to relax.
At the same time, though, the rest of me—the better part—began to clench up. For as I stormed at Parr, ordered Alcott Fish around, fell under the spell of the tainted drinking fountain, and fawned over Kevin Rodriguez in his Butthead costume, I knew that the moment I had so often dreamed of was about to arrive, in the most humiliating form imaginable. I would be kissing Parr—Parr, who had been avoiding talking to me, even looking at me—and I'd be doing it in front of an audience. My throat went dry. My voice dropped to a whisper, and Benjo had to say, "Speak up, Julie! Let's take it again from 'Do you admit you were a jerk?' "
Then there was no postponing it. As Owen, Parr admitted the error of his ways. As Tanya, I forgave him. He drew me close—and kissed me.
Was it like kissing Zach? Only the way the merry-go-round is like the Cyclone at Astroland. Only the way sliding down the hill behind the elementary school on your mother's roasting pan is like skiing down Mont Blanc.
I was glad I had kissed Zach. Because of that experience, I didn't flub the kiss onstage any more than I flubbed my lines. I met Parr's lips head-on, without slipping or crashing, and the outside world went dim.
When it was over—rather quickly, I think, because I didn't hear any hooting from the audience, and they _must_ have hooted if the kiss had really lasted as long as it seemed to me—I looked up at Parr. His eyes were opaque, abandoned. He looked as overthrown as I felt. Upstage, out of sight of the crowd, he crushed my left hand in his right. I heard a crack and felt my onyx ring snap in two and fall from my thumb.
We stood that way for only an instant; then Alcott Fish entered downstage right, Parr spoke his next line, and the rehearsal swept on to its finale. I spoke and sang mechanically, weak as a kitten.
Afterward, the entire cast and crew gathered around to congratulate me. I was their heroine. I had saved the day, and could now be counted on to save tomorrow too. I looked around for Parr, but it was too public to ask him anything or to tell him anything.
"Come on, girls, get your coats," said Mom. "We'd better get going if I'm going to have time to tackle Marie Gerard before bedtime."
And that was it. Parr and I parted without a word or a touch. Until tomorrow, that is—and tomorrow's kiss.
But it didn't work out that way. Mom's mission was successful. Mrs. Gerard agreed to extend the twins' sentence a week in exchange for their limited release over the next two days.
"How did you do it?" I asked.
"I explained the situation. Marie's a reasonable person," she said.
I gave her a doubtful look. Reasonable or not, Mrs. Gerard had never before reversed a punishment, to my knowledge anyway.
"Oh, all right. I threw myself on her charity. I told her that I was on trial for a job at the school, and that if I managed to get the girls back in the play, it would impress the dean and maybe land me the job."
"Very clever, Mom! That's worthy of Samantha Liu!"
"Yes, and it has the advantage of being true."
With no clear prospect of another kiss from Parr, then, I dwelled on today's. What did it mean? I had watched Parr kiss one twin or another dozens of times apiece, but this kiss seemed different. I had never before seen that look in his eyes—drowned, burning, transformed. Even though he'd hardly spoken to me since my father's horrible remark about Seth, he'd kissed me as if he meant it. I thought it must mean something.
I thought it must mean he liked me.
But Ashleigh! Ashleigh. Even if he _did_ like me, that didn't release me from my obligation to my best friend. As long as _she_ liked _him_ , my hands were tied.
Had Ashleigh noticed anything strange? Apparently not. "You were wow, Jules!" she cried, bursting through the front door after dinner. "I told you you could do it! Did your mom get Mrs. Gerard to relent? She did? Really? Too bad! I mean, crisp for Yv and Yo, of course, but too bad for you, you were so incredible as Tanya! And Ned was great too. I don't see why he didn't want a part in the first place, he has a loudly crisp voice. I loved you in your scenes with Kevin, you were both so, so funny, and you were great with Parr too. You're a natural. Next time you'll get a bigger part. No question! You just needed the practice. I bet you could even get into a Byz production now, if it wasn't such a popularity contest with Michelle Jeffries and all those people." Et cetera. Evidently the struggle going on within me had made no impression on my friend.
After the excitement of the dress rehearsal, opening night seemed almost tame. I relaxed into my old role of Headmistress Lytle with a calm and control that surprised me, and I handed over Tanya's part to Yvette with relief. Yolanda had agreed that after all her sister's hard work and risky pretending, it was only fair for Yvette to go first.
Our parents came to opening night—the Rossis sat in the front row, clapping wildly at pretty much everything—but mostly the audience was a sea of boys in blazers. Ravi missed the line he always missed; he smiled his beguiling smile, and the audience forgave him with a laugh. Ashleigh sang loud and clear, Alcott sweet and true. We all hit our high notes and our low notes. The ensemble numbers went smoothly, nobody tripping or crashing. Numb with adrenaline, I watched from the wings as Parr kissed Yvette. I even enjoyed my bow and the applause that came with it. How far I had come from the terror of the audition so many months ago!
The cast party afterward didn't last very long, since the performance was only a small part of the packed Founder's Day schedule. Chris had managed to smuggle in a fifth of vodka, but Mr. Barnaby found it in the prop room and confiscated it with grim warnings before Chris could use it to spike the hot chocolate, punch, and other virtuous beverages provided by the school. Mr. Barnaby, Ms. Wilson, Benjo, and Ned all made speeches. Everyone hugged or hit one another on the back.
I saw Parr across the room. He looked away quickly. Was he not going to say anything, even tonight? I felt I couldn't bear it. Everyone was happy, everyone was hugging. Even if he _was_ Ashleigh's crush, even if he didn't seem to want to talk to me anymore, at least this one night nobody would think it was strange if I . . . I walked across the room and put my arms around him.
"Congratulations, Grandison, you were great," I said, managing to keep my voice steady.
He hugged me back, hard. "Julia!" he said. "You too—last night, especially." He looked at me at last, his eyes close enough to burn me with their gas-blue flames, and I thought . . . But then the twins and Emma came over to deliver their own hugs, and he let me go. The party ended soon afterward.
The second and final performance the next day was much the same as the first, but with Yolanda's sunnier Tanya and an older audience, Old Boys (alumni) instead of current students.
After our curtain calls, Dean Hanson and the headmaster took over to make what amounted to a fund-raising pitch. Ned stayed onstage as the Live Performance Scholar, an example of the great things that resulted when Old Boys opened their checkbooks. But Parr slipped away and found me backstage where I was waiting for Ashleigh. "Here—these are for you," he said. He handed me a bunch of flowers wrapped in blank newsprint.
Ashleigh came up, carrying an armload of bundled costumes and props. "There you are," she said. "I don't think we can wait for Ned—he said it would take another hour. We better get going. Your mom'll be waiting."
"I'll walk you," said Parr.
On the way out of the theater, a woman in the audience stopped him. They had the same eyes. "Snip, that was wonderful," she said.
"Thanks, Mom. But not in public, remember?"
"Oh—right—sorry, Snip, I forgot."
" _Mother!_ Matricide!"
"Sorry, sorry, I mean Grandison."
"That's better. Mom, this is Ashleigh Rossi and Julia Lefkowitz. My mother, Susan Parr. I'll be right back, Mom, I'm just going to see Julia and Ashleigh out."
"It's nice to meet you, girls. Don't be too long, Sn—Grandison, your father's trapped in there with the headmaster."
" 'Snip'?" I asked as we walked down the drive.
"It's short for Parsnip, I'm sorry to say. She's not supposed to call me that in public. I wish she hadn't. I love her, and it'll pain me to kill her."
"Snip is better than Junior," said Ashleigh.
"It's better than Parsley or Parboiled. Or Sley or Boiled," I suggested.
"Don't," said Parr. "It would pain me even more to kill _you_."
"Tridge," I said. "Terre. Ticipation. Kinglot. Liament."
"Enough! Mercy!"
"All right, Typooper." I was giddy with relief that we seemed to be on speaking terms again. We approached the end of the drive.
"When will I see you again?" asked Parr. "You're coming to the Spring Frolic, aren't you? I'll send you tickets. But it's not until April."
"Didn't Ned tell you?" said Ashleigh. "We're collaborating on a song cycle. Ms. Wilson said we could—it counts as community outreach. We meet on Thursday afternoons, when the music studio is free."
"Oh, Ash! You didn't say it was Thursdays! I can't make it then," I said. "That's when _Sailing_ meets."
"I didn't know you sailed," said Parr. "So do I—my father's obsessed with sailing. Maybe I'll go out for it in the spring. We could meet on the river."
"Not sailing boats— _Sailing to Byzantium_ , our literary magazine," I explained.
Parr stiffened. "Oh, I see," he said.
Oh, no! He was clearly thinking of Seth. Had I ruined everything? Was there anything I could say? "I wish I could quit—I would, but Dad would kill me, especially now that _Insomnia_ 's over and I don't have any other extracurriculars," I said.
Parr relaxed slightly. "Well, I'm sure I'll think of something," he said. "Ski break isn't that far off, anyway."
"Ski break? What's that?" said Ashleigh.
"You know—mid-February vacation—Presidents' Day and all that. Don't you get off for it?" We shook our heads. "Well, we do," said Parr. "My parents like to go to Vermont, but I think this would be a good year to stay at our place in Steeplecliff instead."
We reached the gate and my mother's car and said our good-byes.
When Ashleigh deposited her armload of props and costumes in the backseat, I saw she was also carrying a bunch of flowers in newsprint. Hers were tulips; she looked at mine, which were something tall and lilylike. "Oh, Ned gave you flowers too!" she said.
"These are from Parr."
"Yeah, Ned told me he stole them from the Conservatory. Turkeyface almost caught him," she said proudly. "It's just like him to share them with Parr."
As we drove away, I saw Parr standing by the gate, looking after us until we turned the corner of the drive.
**_Chapter 21_**
_A Nonstatic Screen Wipe_ ~ _Ashleigh's new Craze._
_**A**_ nd that was it. No more _Midwinter Insomnia._ No more Parr. The weeks stretched out before me, blank and numb.
Ashleigh, lucky thing, began her musical collaboration at Forefield that Thursday, while I stayed at school for the _Sailing_ meeting.
"Slim pickings here," said our editor, Eleanor, waving a few pages, the only submissions so far. "Come on, guys, beat the bushes. Pound the pavement—pester the talent. Get your ear in gear. What's the matter, doesn't _anyone_ have a masterpiece in a drawer somewhere? Maggie? Andrew? Julie? What's wrong with you! Come on, Julie, I know you have something squirreled away. Of course you do, you always have ink up and down your arms."
"Don't be shy, Julie," said Seth. "What about that sonnet you said you were writing?"
I denied it. Any expression I might have given to my feelings was too private, too sacred for those eyes.
Since my mother was still at work, Seth drove me home after the meeting. "Don't you want to show me your sonnet?" he coaxed, parking in front of my house. He was clearly angling to be invited in. "I could help you make the rhyme and meter work before you submit it to the board, if that's what you're nervous about. I bet it won't be too hard to fix it."
"There is no sonnet! Leave it alone, okay?" I said irritably, getting out of the car.
"All right! Sorry. I didn't realize you'd be so touchy," said Seth. "You don't have to be, you know—you're really a pretty good writer."
"Yeah, thanks, see you tomorrow," I said, shutting the door hard and going into the house quickly.
I went upstairs and e-mailed Parr. I tried not to, but I couldn't help it.
Thanks again for the beautiful flowers. They've just finished opening. I have them on my desk, where I can see them whenever I look up. Did Ned really steal them from the conservatory, like Ashleigh says?
Parr wrote back at once:
Dear Julia,
Do you think I would let someone else commit my crimes for me? I stole every one of those amaryllises with my own hands.
I miss you.
CGP
He missed me! The words made my inky arms tingle, and I confess I kissed the screen where they appeared. Did he mean it? Did he miss me as much as I missed him? But what good would all the missing in the world do, when he was _there_ and I was _here_ and Ashleigh lay between us? Almost screaming with frustration, I got a nonstatic wipe out of my desk drawer and cleaned the mark of my lips off the screen.
When Seth drove me home again a week later, Ashleigh was waiting for me on her porch, wrapped in the big down throw from the Rossis' couch. "There you are, Julie," she said, hurrying down the steps. The corner of the throw trailed in the dry grass. "I need to talk to you."
Seth set his jaw sourly. By now he must hate Ashleigh as much as my stepmother did, but I was grateful to have a chaperone for the dangerous end of the drive, the most likely moment for a guy to lunge. I knew he wouldn't do anything with Ashleigh hovering over us. He let me out and drove off at once.
"Thanks, Ash," I said after he was gone. "I keep being afraid he's going to kiss me good-bye. What's up?"
She looked grave and uncomfortable. "It's cold out here. Let's go up to my room," she said.
I followed her upstairs and sat down on her bed. She sat on her desk chair, fidgeting, weirdly quiet.
"What's wrong?" I asked. "Why are you acting all weird?"
"Jules, I . . ." She stopped, took a deep breath, and started again. "Julie, is it really true . . ." She trailed off.
"What? Is what really true?"
"Is it true that—is it true what you're always saying about Ned?"
"What? Ash, tell me what's wrong. I don't understand what you're talking about. What am I always saying about Ned?" She opened and closed her mouth a few times, but didn't manage to answer. "What do you mean?" I said. "I'm not the one who talks about Ned all the time—you are. I don't know why you want to believe I like him, but I don't. I mean, he's a nice guy and everything, but I just don't _like_ him."
"That! That's what I mean," said Ashleigh. "Is it really true? You're not just saying that?"
"What, that I don't like him? _Yes_ , it's really true. Why would I be just saying that? I keep telling you it's true! I keep telling you over and over! Why don't you want to believe it?"
"You're sure?"
"YES, I'M SURE! Why are you going on about this?"
"Because—" Ashleigh took a deep breath. "I . . . He . . . We . . ."
My heart began to pound before I knew why. Then I knew why. "Ash! _You_ like him! Is _that_ it?"
She gave a strangled nod.
I had never before seen her speechless like this. I felt like whooping. I threw my arms around her. "Ash! You're perfect for each other!"
"You don't mind, then?"
" _Mind_? Why would I mind? That my best friend likes a really nice guy? And he likes you too, right? It's so obvious! The flowers! The music! Why didn't I see it? He does, doesn't he?"
She nodded. "I think so," she said. "At least—he kissed me."
"He _kissed_ you? What? When? Tell me!"
It had happened in the soundproofed rehearsal room. "When you spend a lot of time with someone, and you realize all the things you have in common, like music and liking to do fun things like playing little tricks on people and trying out different instruments and really talking about stuff, and there we were sitting on the same piano bench in complete privacy because nobody could hear us, and oh, Julie! He's so wonderful! He has the most beautiful voice! And his hands are so strong from playing the piano and his left hand has these wonderful calluses from the cello. Don't you love the cello? It has that soulful, sexy sound—just like Ned's voice. Kissing him is absolutely nothing like kissing Ravi. He was a little shy, so I kissed him first, but he said afterward that he was about to kiss me a split second later."
Once they realized how they felt, said Ashleigh, the only thing that stood in their way was Ashleigh's loyal determination not to destroy what she thought, generous girl, was my happiness. She still had trouble believing that I was telling the truth—she had trouble believing that anyone could know Ned and not love him as she did. I had to reassure her over and over.
I considered admitting that there was Someone Else for me too, but I held off. I knew how much trouble she would have turning her focus from the subject that engrossed her—but once she did, there would be no holding her back. My tender feelings weren't yet ready for the full force of Ashleigh. Besides, I didn't want to spoil her moment.
For the rest of the evening she poured out her joy. I soon realized that her new attachment represented not merely a change of love interest, but a full-out craze change. How had I missed it? The signs had all been there: her relenting about whether to expose the lower limbs, the intensity of her interest in _Midwinter Insomnia_. Her parents had noticed her new enthusiasm for Broadway long before I did—hence the tickets to _Fascination!_ And our visit to Parr's town house, I now realized, had been for Ned's sake, not Parr's.
"What about Parr?" I asked at length, my heart beating hard.
"What about him?"
"You said you liked him back in October—remember? You seemed pretty serious about it."
"Oh—yes—well, I thought I did, but that was before I really understood what Love was. You were right after all when you thought Ned was Darcy! Nothing against Parr, he's a really nice guy, but he's no Ned. He just doesn't have the same fire—the talent—the intensity—the inventive good humor—the _life_. You know what I mean?"
Smiling to myself, I said I could see how she would think so.
**_Chapter 22_**
_The B-word_ ~ _Seth vanquished_ ~ _a Ring_ ~ _my Sixth Kiss_ ~ _an Acrostic._
_**I**_ fell asleep that night in a dazzle of happiness. Honor no longer stood between me and my heart's desire.
I awoke the next day, however, to a gray, spitting drizzle and the realization that, although everything had changed, nothing had changed. True, I was free to love Parr. But I wasn't free to see him.
Also, I remembered, I had promised in a weak moment to hang out at the Java Jail with Seth that afternoon after school. When I made the promise, it had seemed like ages in the future, too far away to matter; but now the time had arrived. I saw myself sinking slowly into the swamp of Seth's expectations, while the golden sail of my love twinkled out of sight over the horizon.
_rescue me ash,_ I text-messaged my friend. _meeting seth @java j this aftnn. be there pls. pls pls pls. need you. jl_
_you shd dump him already. quit messing around. its too imptnt. dont worry tho ill be there. ash,_ she TM'ed back.
And she was. "Jules! Seth! Come sit over here," she shrieked from the back, patting two seats at her table. I headed stubbornly in her direction, with Seth dragging behind and trying to draw me off to other tables.
Once we had sat down, Ashleigh pounced on Seth. "As a literary person, what qualities would you say it's important to look for in poetry if you want to set it to music?" she asked him.
It was the perfect question, at once flattering and absorbing, and even useful (at least to Ashleigh). After a few increasingly feeble attempts to get away, Seth warmed to the subject. He almost seemed to forget his irritation at Ashleigh and his resentful yearning for me. He turned his face and shoulders toward her, leaving me behind at his elbow. I was never more grateful to Ash.
Their conversation left my mind free to wander. It headed off in the usual direction—toward Parr.
And then, as if I had summoned him, there he was. He was weaving his way through the crowded coffee bar in front of Zach Liu.
"Here you go, Stringbean, a late birthday present," said Zach with a smirk, pushing Parr forward.
"Zach! Parr! Hey, have a seat," cried Ashleigh, pushing out a chair. Zach sat down next to her. "Seth, you know Zach Liu, don't you?" said Ashleigh. She gave Parr a wink and a kick as she continued with the shocking words, "And have you met Grandison Parr, Julie's boyfriend?"
"Your boyfriend!" exclaimed Seth.
I felt the blood drain to my feet. I looked at Parr with terrified inquiry. He smiled back, a sweet, wicked smile, full of mischief and hope. I took a breath and decided to go with it. "Yes," I said, "my boyfriend, Grandison. I think you guys met before, right?"
"Hello, _sweetie_ ," said Parr, coming over to sit next to me.
"I didn't realize you were going out," said Seth stiffly.
"Oh, we weren't—then, I mean," I said. "That is, we . . ."
Zach looked as if he might burst out laughing at any moment.
"We were just talking about what makes a poem a true lyric," said Ashleigh quickly, drawing the attention to a safer corner of the table. "Seth says it's the meter and the quality of the assonance and alliteration, but what do you think, Parr? Parr wrote all the lyrics for _Insomnia_. He's amazing. That's what brought him and Julie together. She's really sensitive to poetry," she babbled.
I felt the old sensation, familiar from years of Ashleigh: mortal embarrassment. I turned my face away. Parr put his arm around my shoulder. "Are you all right, _sweetie_?" he said.
"She'll be fine, now that you're here," said Ashleigh. I straightened back up and kicked her under the table.
Seth cleared his throat. He looked pale. I felt bad for him. "Well, I'd better be going," he said, standing up. "Lots of homework this weekend."
"Oh, must you? Well, nice to meet you," said Zach.
"Bye, Seth, see you Monday," I said.
"Tuesday," said Ashleigh. "Long weekend."
"Right. Tuesday."
Seth made a pained little bowlike gesture and left.
"Ashleigh!" I said. "That was so embarrassing. And kind of mean."
"Why? You've been complaining for weeks about how you need help getting rid of him."
"Have you?" said Parr.
"Yes, she has," said Ashleigh. "You know it's meaner to let him keep hanging around when you don't actually like him. Now I won't have to chaperone you all the time."
Something in that sentence made Zach look at his watch. "Ope! Gotta go. Come on, Ashcan, I'll give you a ride home," he said.
"Thanks, Zach, that's okay, you don't have to," began Ashleigh.
"Don't be an idiot—come on—there's something my sister needs you to do," said Zach, taking her firmly by the shoulder.
"What? Oh. Oh! Right, that thing for Sam," said Ashleigh, grabbing her coat with one hand as Zach propelled her to the door. "Later, guys."
Then I was all by myself in the crowded coffee bar with Charles Grandison Parr. He grinned at me, took my hand, and said, " _Sweetie!_ Alone at last!"
Maybe it was all a big joke, but I noticed his hand was as cold as ice.
Did it tremble a little? Mine certainly did.
"Ashleigh can be so embarrassing," I said. "Sorry! Or, I guess, I mean, thanks."
"Don't mention it. I'm honored that I could be of use to you. Especially to get rid of guys who pester you. Anytime you find a guy troublesome, please feel free to tell him I'm your boyfriend."
"You mean it?" I said.
"You know I do. If you think it'll help, I'll even come by and threaten him with my epee—my dueling sword. Speaking of getting rid of guys," he added, "are you done with your coffee? Would you like to walk down to the river? I see some of the guys from the fencing team heading this way, and I don't particularly want to hang out with them. I get to see them all the time. I never get to see _you_."
"Sure," I said.
Parr left a tip on the table and helped me into my coat. Nobody had done that since I was a little girl; I fumbled around for the sleeves a bit before he found my arms and lifted the coat around my shoulders.
It was a warm afternoon for February, the earliest edge of spring. The rain had stopped, leaving a breath of moisture in the air. We walked the six blocks to the train tracks in silence, smelling the river just beyond, and crossed the tracks by the underpass, with its buzzing lights and loud echoes. The other side seemed quiet by contrast, hushed with the soft, deep slipping of the river.
"Let's see if anyone's in the band shell," said Parr.
No one was. Everyone else, apparently, remembered it was February.
We sat down on one of the wooden benches overlooking the river; the band shell kept the worst of the wind off.
"What are you doing out of school, anyway?" I asked.
"Ski break, remember? I'm staying with my folks in Steeplecliff. Actually, I was looking for you. I wanted to give you something." He took a little box out of his pocket and handed it to me.
"What is it?" I said. I had to take my gloves off to open the box. My hands trembled. I held it carefully, trying not to drop it. Inside was a ring: one side solid silver, the other side silver encasing something black.
"Does it still fit?" asked Parr. "Try it on. I was worried I might have made it too small—I had to add a strip of silver underneath the onyx. If it doesn't fit, give it back. I can make it bigger."
"You _made_ this?"
He nodded.
The ring was too big for my fingers, but it fit my left thumb perfectly. I looked at it more closely. "Wait," I said. "Don't tell me it's my onyx ring! The one that got broken?"
"I felt terrible about breaking it," he said. "I thought I should do something."
"You didn't have to . . . You _made_ this? But it's so beautiful. Is there _anything_ you can't do?"
He laughed. "Well, yes. Tons. Most of the important things. One thing I thought I probably couldn't—but I don't know, I'm starting to think maybe I can. Let's see."
And he kissed me.
How cold his lips were—and then how warm. My sixth kiss—but my first. The blue sky, the blue river, his blue drowned look, our breath steaming together into one cloud. My cold fingers—his cold neck, warm under his scarf. I touched his dimple. We kissed again.
"I take it back," he said, his voice rough. "You're right. There's nothing I can't do." He took off my hat and kissed my forehead, my cheekbones, the edges of my face next to my eyes. "I wanted to do that so badly," he said. "Especially that night."
"Why didn't you?" I said.
"Why didn't I? What do you take me for—barge into a girl's room in the middle of the night and start kissing her? And I wasn't sure you even liked me."
"But I was being so obvious—joining the play and hanging around you all the time helping you rehearse."
I was trembling. He hugged me to him and put his chin on top of my head. I heard his voice through my bones. My heart pounded and pounded.
"You call that obvious?" he said. "You never talked to me unless I said something first. And then there was that Seth person; everyone kept saying he was your boyfriend. I almost gave up."
"Oh, God, Seth—he was so awful, I wanted to die. But I thought you had a girlfriend too. Some friend of Samantha Liu's saw you dancing with a tall blonde at the Columbus dance. She said it was your girlfriend."
He drew back and looked at me. "What? You're kidding, right?"
"No, that's what Sam said."
He laughed. "Well, she was almost right. I _was_ dancing with an enchanting blonde, but she wasn't my girlfriend—not then, anyway. Do you really not know who it was?"
I felt as if I were standing inches from a sheer cliff, balanced over sharp rocks of jealousy. I hid my face against him again. "Sam's friend thought it was Kayla somebody?" I mumbled into his coat.
"No, silly! It was _you_! You really didn't know? There I was making a gigantic fool of myself, mooning around your house and writing you poetry, and I couldn't even tell if you had read it. That was the first thing that made me hope: seeing you had my sonnet up on your bulletin board. But you never said anything about it."
"But I wasn't sure you wrote the sonnet for _me_. I thought it was for Ashleigh."
"For Ashleigh!" He drew back again and looked at me. "But it had your name in it!"
"You mean _July_? 'Zero degrees down here, July above?' That's what Ashleigh said, but I didn't believe her."
"No, I mean your name! Well, July too—I put it in for the echo—but I'm talking about your actual name, Julia Lefkowitz. Going down the side, the first letters of the lines. It's an acrostic—fourteen letters, fourteen lines. You mean you didn't even notice? Wow, I feel silly."
Not as silly as I felt. My own name! Right there in the sonnet that the Person of My Heart wrote for me—and I didn't even see it.
"Okay, I'm a marshmallow brain," I said. "Do you hate me now?"
The answer took a while and was more absorbing than I could have thought possible. Afterward, I no longer knew how many times I'd been kissed.
**_Chapter 23_**
_Bliss_ ~ _Farewell._
_**T**_ hen followed ten days of unprecedented bliss. Parr found a way to come to Byzantium almost every day, and although the weather retreated into winter again, we barely noticed the cold. We held hands through the thriller and the romantic comedy at the Cinepalace, without noticing a single explosion or kiss (on-screen, at least). We spent hours talking about books in Andrezo's Diner, the Java Jail's unfashionable rival, where the coffee was hot, the patrons were scruffy, and the booths had high backs.
Ashleigh and Ned, who was staying at Forefield over the vacation, sometimes joined us. When they did, the noise level in the diner tripled.
After Mom's success with the Gerards, Dean Hanson persuaded the headmaster to offer her a contract for the following year. She informed my father immediately by registered mail. Flush with her settled new income, she turned the thermostat up to 68 degrees, and in her happiness she even converted the Treasures storeroom back into a painting studio, as it had been during the early years of my parents' marriage. Parr and I spent an afternoon helping her.
Parr brought me to lunch at his parents' on the second Saturday of the break. Their house in Steeplecliff had stone walls, low ceilings, and slanted floors; I could tell it was very old.
"So this is what was fascinating Snip in Byzantium all week! I was starting to wonder," said Ms. Parr—or Susan, as she told me to call her—with a familiar flashing smile.
To my great relief, I found that my table manners were not noticeably different from the Parrs'; Charles Grandison Sr.—Chip—even punctuated his points by gesturing with his chicken leg. And to my surprise, he took to me at once, insisting on giving me a tour of the barn out back where he was building a sail-boat. "See if you can get Snip to take an interest," he told me. "Half his ancestors were sea captains."
I found Parr's room upstairs delightfully revealing. Although he had clearly cleaned up in my honor, he was just as clearly a natural slob. Books, abandoned bird nests, and bits of fencing equipment lay in loosely squared stacks in the corners. He turned out to be an avid bird-watcher. It was the wrong season for the more exotic migrants, but he regaled me with stories of the loves and rivalries of the local crows.
He had the entire Patrick O'Brian series of naval novels in a heap behind the door and admitted to having read them all—"But don't tell my dad, it would please him too much," he said. He lent me the first one after making me promise to keep it away from Ashleigh for a few weeks. "Let her go on sharing Ned's interests for as long as possible."
I found it hard to drag myself to school during that heavenly period. I particularly resented the Thursday afternoon wasted at _Sailing_. Seth took care to talk to me just as much as ever, as if to prove that there had been nothing particular behind his attentions. But he soon took up with Margaret Barsky, a tall, pretty girl in Ms. Milburn's third-period bio, who had hair the same color as mine. They appeared regularly as a couple at the Cinepalace and the Java Jail. I often caught her looking at me with triumph tinged with dislike. From time to time, too, I caught a glance from Seth so full of some strangled emotion that I regretted ever having allowed him to think—whatever it was he _had_ thought.
My father and stepmother mourned Seth's loss as if he had been one of their own dreamed-of babies. Parr, they pointedly let me know, would never replace him in their affections.
When Forefield started up again, Parr and I had to sustain ourselves with e-mail for several weeks. Then we had the happy thought of volunteering at the Byzantium Senior Center at the same time on Tuesday evenings. It was a savvy move for college applications—or at least, that's how I presented it to Dad.
Yvette Gerard, after her _Insomnia_ experience, found she liked acting as much as her sister did. With some help behind the scenes from Samantha, the twins and Ashleigh seized control of Byz High's spring musical from the Michelle Jeffries clique. Ash volunteered to compose the music, and after some persuasion, I agreed to write the lyrics.
With all my new activities, time flew by. It's April already. The Forefield Spring Frolic is this Saturday. Parr and Ned gave us our tickets as soon as they were printed, and we look forward to producing them at the first sight of Turkeyface.
Ash has been giggling mysteriously all week, hiding sheets of music whenever I show up at her window. (The tree lost its ice weeks ago, but I have to be careful not to tear the tender young leaves.) I suspect she and Ned may be planning to surprise me with a waltz or a quadrille arrangement of the tune to which Ashleigh set the poem I wrote so long ago, when Parr seemed to me only a hopeless dream.
So far, Ashleigh's musical craze has held strong, and Parr and I have high hopes that even when it changes, as it inevitably will, her loyalty to those she loves will not allow her to leave Ned behind.
_The End_
ACKNOWLEDGMENTS
No writer could have a warmer friend or a more generous reader than Anna Christina Büchmann, or a keener, more loving husband than Andrew Nahem, who gave me my best joke and taught me everything I know about happy endings. For their insight, encouragement, and generosity I'm greatly indebted also to Nancy Paulsen, my editor; Irene Skolnick, my agent; and Michael Abrams, Mark Caldwell, Eunice Chan, Stacey D'Erasmo, Lisa Dierbeck, Carol Dweck, John Hart, Elizabeth Judd, Katherine Keenum, Eleanor Liu, Anne Malcolm, Shanti Menon, Christina Milburn, Laura Miller, Laurie Muchnick, James O'Shea, Lisa Randall, Jenna Reback, Maggie Robbins, Andrew Solomon, Cindy Spiegel, Jaime Wolf, and Shenglan Yuan. And for their love, support, intelligence, and humor I'm grateful to my family: my brother, Theodore Shulman; my mother and stepfather, Alix Kates Shulman and Scott York; my father and stepmother, Martin and Beverly Shulman; and all the Nahems, especially my niece Emily and my father-in-law, Sam, who was as beloved as he was bald.
**Enthusiasm**
**READER'S GUIDE**
# Little-known facts about _Enthusiasm_ and Jane Austen
• Jane Austen's niece, Anna, wrote a novel called _Enthusiasm_ , which she sent to her aunt. Jane had many encouraging things to say, including suggesting that Anna change the title to _Which is the Heroine?_
• The character of Charles Grandison Parr—or Parr for short—was named after Sir Charles Grandison, the hero of Samuel Richardson's 1753 novel of that name and one of Jane Austen's favorite literary characters.
• Jane Austen published her novels anonymously, as was the custom of female writers at the time.
• Like many of Austen's heroines, Jane herself turned down an offer of marriage that would have allowed her to live a more comfortable life and be less dependent on her family. In the end, she never married.
# The Life and Legacy of Jane Austen
Jane Austen (1775-1817) lived her entire life in the English countryside with her mother, father, sister, and two brothers. She never married nor ventured far from the confines of her family's home, yet she wrote some of the most enduring novels of her time, including _Pride and Prejudice_ , _Sense & Sensibility_, and _Emma_.
When Jane Austen penned her first novel in 1789, little did she know that the stories she acted out in her drawing room with her sister and brothers would affect popular culture hundreds of years later. Dozens of movie adaptations of her novels have been made and continue to be popular, starring actors such as Keira Knightley, Gwyneth Paltrow, Kate Winslet, Colin Firth, and Hugh Grant. Her writing has inspired other books as well, such as _Bridget Jones's Diary_ , _The Jane Austen Book Club,_ and _Jane Austen's Guide to Dating_.
# Discussion Questions
1. Why do you think Jane Austen and her books have endured as long as they have? Why do Jane Austen's stories translate so well into modern stories?
2. Have you ever read any of Jane Austen's books? If so, what similarities and differences do you see between Austen's works and _Enthusiasm_?
3. "There is little more likely to exasperate a person of sense than finding herself tied by affection and habit to an Enthusiast." Do you know/have known an enthusiast? Were you ever one yourself? Although Julie complains about her friend's enthusiasm, what admirable qualities can be found in Ashleigh's exuberance?
4. If you could produce a movie based on a Jane Austen story, which would you choose and from what angle would you approach it: Comedy or drama? Present day or historical setting?
5. Have you ever had a crush on the same person as your best friend? If so, what happened?
6. Throughout the story Julie is careful to point out what a good friend Ashleigh is to her. Unfortunately, Ashleigh's not always a very good listener. At the same time, Julie is keeping secrets from Ashleigh. Could you still say they are great friends? Why or why not?
7. Class was an important issue for people in Jane Austen's time. In what ways does the issue of class/money come up in _Enthusiasm_?
8. Do you think Julie handles her relationship with her stepmother well? What could Julie and her stepmother do to improve their relationship?
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\section{Introduction}
The Earth's liquid outer core consists mainly of iron, but its density
is about 10\% too low to be pure iron (Birch, 1952), so that it must
contain some light element. The nature of this element is still
uncertain, and during the last 45 years the main candidates have been
carbon (Birch, 1952; Clark, 1963; Urey, 1960; Wood, 1993), silicon
(Birch, 1952; MacDonald and Knopoff, 1958; Ringwood, 1959, 1961,
1966), magnesium (Alder, 1966), sulphur (Clark, 1963; Urey, 1960;
Birch, 1964; Mason, 1966; Murthy and Hall, 1970; Lewis, 1973), oxygen
(Dubrovskiy and Pan'kov, 1972; Bullen, 1973; Ringwood, 1977), and
hydrogen (Birch, 1952; Fukai and Akimoto, 1983; Suzuki et al., 1989).
For a given light element, it is also uncertain what concentration is
needed to explain the inferred density in the core. The arguments for
and against each of the candidate light elements have been reviewed by
Poirier (1994).
The aim of this paper is to use first-principles calculations to
investigate the possibility that oxygen is the light element.
First-principles calculations are well established as a reliable way
of predicting the thermodynamic, structural and dynamical properties
of solid and liquid materials, including liquid metals (\v{S}tich et
al., 1989; Kresse and Furthm\"uller, 1993). We have recently reported
calculations of this kind on pure liquid iron under core conditions
(Vo\v{c}adlo et al., 1997; de Wijs et al., 1998), which show that it
is a simple close-packed liquid with a viscosity not much greater than
that of many liquid metals at ambient pressure, contrary to some
earlier suggestions (Secco, 1995). We have also used first-principles
simulations to investigate a liquid iron-sulphur alloy under the same
conditions (Alf\`e and Gillan, 1998a). We showed that the properties of
the liquid are scarcely affected by the small sulphur concentration
needed to explain the observed density.
The proposal that oxygen is the light element has a long and
controversial history. Among the earliest proponents were Dubrovskiy
and Pan'kov (1972) and Bullen (1973), the latter of whom suggested an
outer core composition in the region of Fe$_2$O (equivalent to 12.5
wt$\%$). Ringwood (1977) argued that oxygen should be seriously
considered, and used seismic data to estimate the oxygen content as 28
mol percent (10 wt$\%$). However, it is not completely certain whether
the Fe/O liquid is thermodynamically stable against phase separation
under Earth's core conditions at the composition that would be
necessary to explain the density. It is known that the solubility of
FeO in liquid Fe is very low ($\approx 1$ mol percent) near the
melting temperature of pure iron (1811 K) at atmospheric pressure
(Distin et al., 1971). However, the solubility increases rapidly with
temperature, becoming $6.5 \%$ at 2350 K (Fischer and Schumacher,
1978) and rising to the region of $\approx 35 $ mol $\%$ at 2770 K
(Ohtani and Ringwood, 1984). According to Ohtani and Ringwood (1984),
an extrapolation of the available phase measurements would suggest
that the region of immiscibility disappears entirely above $\approx
3080$ K at atmospheric pressure. It is also well established that the
solubility of FeO in Fe increases with increasing pressure, and that
the partial molar volume of FeO in liquid Fe is lower than that of
pure liquid FeO itself. Ohtani et al. (1984) used their high pressure
measurements on the solubility of FeO to suggest that the Fe/O system
may show simple eutectic behaviour above a pressure of $\approx 20$
GPa, with no region of liquid immiscibility at any temperature.
Subsequently, Ringwood and Hibberson (1990) showed by direct
measurements that at 16 GPa addition of FeO to pure iron causes a
depression of melting point, leading to a eutectic point at oxygen
mole fraction of 28 $\%$ and a temperature of ca. 1940 K. Boehler's
(1992) measurements on the melting of Fe/O mixtures are consistent
with these ideas. Experiments of Knittle and Jeanloz (1991) and
Goarant et al. (1992) on the reaction between lower mantle material
and molten iron at pressures above 70 GPa revealed that the liquid
dissolves significantly amount of FeO. However, Sherman (1995) has
recently used first-principles calculations on crystalline Fe/O phases
to argue strongly against significant amounts of oxygen in the
core. His calculations gave values for the enthalpy of formation of
crystals of composition Fe$_3$O and Fe$_4$O starting from Fe in the
hexagonal close packed structure and FeO in the NiAs structure. The
Fe$_3$O and Fe$_4$O compositions were used to model substitutional and
interstitial oxygen respectively. The enthalpies of formation were
found to be so large that phase separation into FeO and Fe appears to
be inevitable. However, it is not clear that Sherman's results have
any relevance to the outer core, since in the liquid phase oxygen does
not have to be either substitutional or interstitial. Even for the
solid phase it is not obvious that Sherman's argument is robust, since
Fe/O crystal structures other than those he studied might well give
much lower enthalpies of formation.
We are mainly concerned in this paper with the Fe/O system
in the liquid state.
Our first-principles calculations, based on density functional
theory and the pseudopotential method, will be used to
address three questions: (a) Is the Fe/O liquid stable against phase
separation under Earth's core conditions? (b) If it is, what
oxygen concentration is needed to reproduce the observed
density? (c) At this concentration, do the structural
and dynamical properties of the liquid differ appreciably
from those of pure liquid iron at the same pressure and
temperature? Our first-principles simulations of the liquid
will provide strong evidence that it is thermodynamically
stable, and that the observed density requires an oxygen
concentration of $25-30$ mol-percent. In studying the properties of
the liquid, we shall be particularly concerned with the
viscosity, since this is one of the most poorly determined
properties of the outer core, with estimates from different
experimental and theoretical methods spanning many orders
of magnitude (Secco, 1995). We shall also investigate a number of other
properties, including the nature of the short-range order,
the atomic diffusion coefficients, and the electronic structure.
Although we are mainly interested in the liquid, we shall also
present some results for the energetics of various crystalline
forms of the Fe/O system. These crystal calculations
serve two purposes: first, they demonstrate that our techniques
are in complete agreement with those used by Sherman in
predicting large enthalpies of formation for
Fe$_3$O and Fe$_4$O in the structures he assumes;
second, they demonstrate that there are crystal structures
that give much lower formation enthalpies -- an important
fact in understanding how the Fe/O liquid can be stable
against phase separation. It is not our intention to come to
definite conclusions about the possible phase stability of
the Fe/O solid solutions themselves, but our calculations
suggest that their stability cannot be ruled out.
The paper is organised as follows. In section 2, we summarise the
first-principles techniques on which the work is based. Section 3
presents our calculations on the energetics of Fe/O crystals.
In section 4 we report our results on liquid Fe/O, including
its structural properties, the evidence for its stability against
phase separation, and its dynamical and electronic properties.
The final sections present discussion and conclusions.
\section{Methods}\label{method}
In first principles calculations, the solid or liquid is represented
as a collection of ions and electrons, and for any given set of ionic
positions the aim is to determine the total energy and the force on
every ion by solving the Schr\"odinger equation. This is a formidable
task if the number of atoms is large, but it was made feasible by the
introduction of density functional theory (DFT) many years ago
(Hohenberg and Kohn, 1964; Kohn and Sham, 1965; Jones and Gunnarsson,
1989; Parr and Yang, 1989). DFT treats electronic exchange and
correlation in a way that allows the electrons to be described by
single-particle wavefunctions, with the interaction between them
accounted for by an effective potential. DFT can be applied in two
ways: all-electron calculations, or pseudopotential calculations. The
first approach includes such standard techniques as full-potential
linearised augmented plane waves (FLAPW) and linearized muffin-tin
orbitals (LMTO). In the pseudopotential approach, only valence
electrons are explicitly treated, the effect of the core electrons
being included by an effective interaction between the valence
electrons and the cores. In both approaches, the accuracy with which
the real material is described is governed by the approximation used
for the electronic exchange-correlation energy. Until recently, the
local density approximation (LDA) was the standard method. But in
order to achieve the highest accuracy for transition metals it is
essential to use an improved method known as the generalized gradient
approximation (GGA) (Wang and Perdew, 1991). The present work is based
on the pseudopotential approach and the GGA. A non-technical review
of first-principles calculations based on the pseudopotential approach
has been given recently by one of the authors (Gillan 1997).
There have already been extensive first-principles calculations on
crystalline iron both at ambient pressure and at pressures going up to
Earth's core values (Stixrude et al., 1994; S\"oderlind et al., 1996).
The calculations have been performed using different all-electron
techniques and the pseudopotential technique, and a variety of
properties have been studied, including the equilibrium volume, the
elastic constants, the magnetic moment, the volume as a function of
pressure and lattice vibration frequencies. The agreement between
results obtained with different techniques is generally very close, and
the agreement with experimental data is also good. Particularly
relevant here is the recent comparison of the pseudopotential results
for the pressure-dependent volume of hexagonal-close-packed iron
up to core pressures with earlier all-electron results and with
experimental measurements (Vo\v{c}adlo et al., 1997).
Static first-principles calculations on crystals have been in routine
use for many years. But to study liquids we need to do dynamical
first-principles simulations, in which the calculated forces on the
atoms are used to generate time evolution of the system, with every
atom moving according to Newton's equation of motion. This kind of
first-principles molecular dynamics (FPMD) pioneered by Car and
Parrinello (1985) has been extensively used to study liquid metals
(\v{S}tich et al., 1989; Kresse and Furthm\"uller, 1993; Holender and
Gillan, 1996; Kirchhoff et al. 1996a, 1996b), and it is known to give
an accurate description of both structure and dynamics.
The present work was performed mainly with the VASP code (Vienna Ab
initio Simulation Package) (Kresse and Furthm\"uller, 1996a,
1996b). As usual in pseudopotential work, the electron orbitals are
represented using a plane-wave basis set, which includes all plane
waves up to a specific energy cut-off. The electron-ion interaction
is described by ultrasoft Vanderbilt pseudopotentials (Vanderbilt,
1990), which allow one to use a much smaller plane-wave cut-off while
maintaining high accuracy. When we perform FPMD with VASP, the
integration of the classical equation of motion is done using the
Verlet algorithm (1967), and the ground-state search is performed at
each time-step using an efficient iterative matrix diagonalisation
scheme and a Pulay mixer (1980). This method differs from the original
Car-Parrinello technique, which treated the electronic degrees of
freedom as fictitious dynamical variables. In order to improve the
efficiency of the dynamical simulation, the initial electronic charge
density at each time step is extrapolated from the density at previous
steps as described in our previous work (Alf\`e and Gillan, 1998a). In
FPMD on metals, the discontinuity of occupation numbers at the Fermi
level can cause technical difficulties. Since we are interested in
high temperatures in our liquid simulations, the electronic levels are
occupied according to Fermi statistics corresponding to the
temperature of the simulation. This prescription also avoids problems
with level crossing during the ground state search. The FPMD
simulations are performed at constant temperature (rather than at
constant energy), using the Nos\'e technique (1984).
The iron pseudopotential we use is the same as that used in our
earlier work (Vo\v{c}adlo et al., 1997; de Wijs et al., 1998; Alf\`e
and Gillan, 1998a,b), and was constructed using an Ar core and a
$4s^13d^7$ atomic reference configuration. The oxygen pseudopotential
was constructed using a He core and the $2s^22p^4$ reference
configuration. At Earth's core pressures the distance between the
atoms becomes so small that the Fe($3p$) orbitals respond
significantly. The net effect is a small repulsion, which we
determined from calculations on the h.c.p. crystal using a Ne core for
the iron pseudopotential instead of Ar. We represent this repulsion by
Fe-Fe and Fe-O pair potentials, as in our earlier work (Vo\v{c}adlo et
al., 1997; de Wijs et al., 1998). The resulting corrections to energy
and forces are generally small. Non-linear core corrections (Louie et al.,
1982) are included throughout the work.
For all the calculations to be reported, we use a plane-wave cut-off of
400 eV, which gives total energies converged within $\approx 10-20$
meV/atom. In the calculations on crystals Brillouin-zone sampling is
an important issue, and the sampling density we use is described in
the next section. But for the liquid we use $\Gamma$ point sampling,
which experience suggests should be satisfactory. (We have done
separate tests on pure liquid iron using 4 {\bf k}-points, and we have
found no detectable structural effects, while the average total energy
difference with respect to the $\Gamma$ point only calculations is of
the order of 10 meV/atom, which is completely negligible for the
purposes of the present work). The time step used in the dynamical
simulations was 1 fs and we generally used a self-consistency
threshold of $1.5 \times 10^{-7}$ eV/atom. With these prescriptions
the drift of the Nos\'e constant of motion was less than $\approx 60 $
K per ps.
\section{Fe/O solid solutions}\label{solid}
We have used the techniques described in the previous section to
calculate the equilibrium properties and the enthalpy of formation
$\Delta H$ of various members of the Fe/O system, including those
studied by Sherman (1995). Although Sherman's calculations and ours
are both based on DFT, the technical methods are completely different,
since Sherman used the FLAPW method, whereas ours are based on the
pseudopotential approach. One of our main aims is therefore to make
detailed comparisons with his results in order to ensure that the two
methods agree about the energetics of the systems. We shall also point
out that there are Fe$_3$O structures with enthalpies of formation
much lower than those reported previously.
We begin by presenting our results for the equilibrium density
$\rho_0$ (i.e. the density for which the pressure is zero), the bulk
modulus $K$ at this density, and the pressure derivative $K' \equiv
dK/dP$ for crystals of Fe, FeO, Fe$_3$O and Fe$_4$O. Pure Fe is in the
$\epsilon$ structure (hexagonal close packed); FeO is in the B8
structure (the NiAs structure); Fe$_3$O is in the structure obtained
from face-centred cubic Fe by replacing the atoms at the corners of
the conventional cube by O atoms; and Fe$_4$O is in the structure
obtained from f.c.c. Fe by inserting an O atom at the centre of the
conventional cube (see Fig. 1). We have done the
calculations both spin restricted (the occupation numbers of every
electronic orbital are equal for up and down spins, so that there are
no magnetic moments) and spin unrestricted (the occupation numbers for
up and down spins are allowed to vary independently). To sample the
Brillouin zone (BZ) we have used Monkhorst and Pack grids (1976)
using the sampling level that corresponds to 20 and 36 {\bf
k}-points in the irreducible wedge of the BZ respectively for the
cubic (Fe$_3$O and Fe$_4$O) and the hexagonal (Fe($\epsilon$) and
FeO(B8)) crystals. Using these values, the total energies are converged
within 5 meV/atom for the Fe$_3$O, Fe$_4$O and FeO(B8) structures, and
within $10-15$ meV/atom for the Fe($\epsilon$) structure. We find that,
except for Fe, all the structures are magnetic at low pressures, so
that the system is stabilised if spin restriction is removed. This
disagrees with the results of Sherman (1995), where a significant
magnetic moment was found only for the FeO(B8) structure (a weak
moment was found for Fe$_3$O).
Since we regarded the disagreement concerning magnetic properties as
disturbing, we repeated our calculations of the equilibrium magnetic
moment of Fe$_3$O and Fe$_4$O using a completely independent
electronic structure technique, namely the LMTO method (linearised
muffin-tin orbitals), using the LMTO-46 code due to Krier et
al. (1994). These calculations completely confirm the magnetic
ordering in Fe$_3$O and Fe$_4$O and give numerical values for the
magnetic moments that agree well with those given by our
pseudopotential calculations. This suggests that the minimisation of
the total energy with respect to magnetic moments may not have been
fully under control in Sherman's work.
For each system we
have calculated the static internal energy $E$ for a series of volumes, and
fitted the results to the Birch-Murnaghan equation of state:
\begin{eqnarray}\label{murna}
E = E_0 + \frac{3}{2}V_0K \left [ \frac{3}{4}(1+2\xi)\left
(\frac{V_0}{V}\right )^{4/3} - \frac{\xi}{2}
\left ( \frac{V_0}{V} \right )^{2}
-\frac{2}{3}(1+\xi) \left ( \frac{V_0}{V}
\right )^{2/3} +
\frac{1}{2} \left ( \xi + \frac{3}{2}\right ) \right ] \\
\xi = \frac{3}{4}(4 - K'), \hspace{12cm} \nonumber
\end{eqnarray}
where $K$ is the zero pressure bulk modulus, $K'= (dK/dP)_{P=0}$,
$E_0$ is the equilibrium energy and $V_0$ the equilibrium volume. Our
calculated values of $\rho_0$, $K$ and $K'$ are compared with
Sherman's results in Table 1, which also reports our
calculated magnetic moments. The overall conclusion from the
comparison is that the results agree well in the cases where magnetism
is absent: pure Fe and spin-restricted Fe$_3$O and Fe$_4$O (Sherman's
calculations are effectively spin-restricted for Fe$_3$O and Fe$_4$O,
since he found no moments). The equilibrium density agrees in those cases
to better than $2 \%$, and the values of $K$ and $K'$ to about $10
\%$. However, our results show that magnetism has a strong effect on
the equilibrium properties, so that it is important to treat it
correctly. The case of FeO(B8) is problematic. Our spin-unrestricted
calculations give a $\rho_0$ value in respectable agreement with that
of Sherman, but the agreement is poor for $K$ and $K'$. It is clear
that the accurate treatment of the volume dependence of magnetic
moment is important in obtaining reliable values for these parameters,
and our suspicion is that problems with moments may have affected
Sherman's values. However, we shall stress below that magnetic effects
become unimportant at core pressures, so the most important feature of
Table 1 is the good agreement between the two sets of
calculations for the non-magnetic cases.
We turn now to the enthalpies of formation $\Delta H$ of Fe$_3$O and
Fe$_4$O, defined by:
\begin{eqnarray}
\Delta H({\rm Fe_3O}) = H({\rm Fe_3O}) - H({\rm FeO(B8)}) - 2H({\rm
Fe}(\epsilon)), \nonumber \\
\Delta H({\rm Fe_4O}) = H({\rm Fe_4O}) - H({\rm FeO(B8)}) -
3H({\rm Fe}(\epsilon)),
\end{eqnarray}
where $H \equiv E + PV$ is the enthalpy per formula unit of each
material. The total energy is taken directly from our Birch-Murnaghan
fit, and the pressure $P = -dE/dV$ is calculated from the derivative
of the fitted form. We have done these calculations both spin
restricted and spin-unrestricted, and our results are reported in the
two panels of Fig. 2. In the spin-restricted panel we also
show Sherman's enthalpy results. The most important conclusion is that
$\Delta H$ for both Fe$_3$O and Fe$_4$O becomes very large at core
pressures. In the range from 135 GPa (core-mantle boundary) to 330 GPa
(inner-core boundary) $\Delta H$ is at least 3 eV, which corresponds
roughly to a temperature of $3.5 \times 10^4$ K, so that it is
exceedingly unlikely that Fe$_3$O and Fe$_4$O could be
thermodynamically stable in the assumed structures.
Our spin unrestricted results indicate that the true values of $\Delta
H$ are considerably lower than Sherman's results at low pressure, but
at high pressure the differences between the magnetic and non-magnetic
values of $\Delta H$ become very small, so the conclusion is
unaffected. The detailed agreement with Sherman's values of $\Delta H$
is only moderately good, but again this does not affect the
conclusions about the very large size of $\Delta H$.
We now want to ask whether the assumed crystal structures for Fe$_3$O
and Fe$_4$O are actually the most stable. A glance at Wyckoff's book
{\em Crystal Structures} (1964) shows that compounds having the
composition A$_3$B crystallise in a bewildering variety of
structures. We have picked some likely candidates and calculated their
formation enthalpy. Most turn out to be unfavourable, with $\Delta H$
values at least as great as those already reported in
Fig.~2. However, we have discovered one that has a much
lower value. This is the BiI$_3$ structure, which has a rhombohedral
unit cell containing two formula units. Putting Fe$_3$O into this
structure and relaxing both the atomic positions and the shape of the
cell, we end up with the triclinic structure shown in
Fig.~1. To characterise the structure briefly, we note
that at 300 GPa each oxygen is surrounded by 11 Fe neighbours at
distances of between 1.76 and 2.57 \AA, and two O atoms at distances
of 2.02 and 2.34 \AA~(by contrast, in the cubic Fe$_3$O structure at
the same pressure, each oxygen has 12 Fe neighbours at 2.05 \AA, and
the nearest oxygen neighbours are at 2.9 \AA). We have calculated the
fully relaxed total energy of this structure at several volumes, and
the resulting structural parameters are reported in Table
2. Spin unrestricted calculations show that the structure
is weakly magnetic, but the moment and the energy stabilisation are so
small that the effects can be ignored.
Fitting of the energies to the Birch-Murnaghan equation of state
yields the enthalpy of formation shown in
Fig. 2. Remarkably, $\Delta H$ is very much lower than for
the previous structures, and it decreases with increasing pressure. At
the pressure of the inner core boundary it is only just over 1
eV. Since we arrived at this distorted BiI$_3$ structure in a rather
haphazard way, it is quite likely that there are other Fe$_3$O
structures with even lower enthalpies. We cannot say at present
whether there are Fe$_3$O structures that are stable against phase
separation under Earth's core conditions, but it certainly does not
look impossible. The low $\Delta H$ for the BiI$_3$ structure will be
highly relevant to our study of phase stability in the liquid Fe/O
system.
\section{The liquid}\label{liquid}
\subsection{Thermodynamics}
Our aim in choosing the thermodynamic parameters for our liquid
simulations was to model liquid Fe/O near the thermodynamic state it
would need to have to reproduce the known outer-core density at the
inner core boundary (ICB). The temperature at this point is very
uncertain, with estimates ranging from 4000 to 8000 K (Poirier,
1991). We took the value of 6000 K, which is intended to be a
reasonable compromise. However, the density and the pressure are quite
accurately known to be $ \approx 12000$ kg~m$^{-3}$ and $\approx 330 $
GPa. This density is about $10 \%$ lower than it would be if the core
consisted of pure iron (Birch, 1952). The main problem in choosing
thermodynamic parameters is that we do not know in advance the
required oxygen concentration, so that a certain amount of trial and
error is needed.
We started from our previous 64-atom simulation for pure liquid iron
(Vo\v{c}adlo et al., 1997), which had a mass density of $13300$
kg~m$^{-3}$ and a calculated pressure of $358 \pm 6$ GPa. Our first
move was to hold the volume of the system fixed and to transmute the
appropriate number of iron atoms into oxygen atoms to produce the
density of $12000$ kg~m$^{-3}$. This resulted in a large reduction of
the pressure, and we therefore reduced the cell volume to restore the
original pressure. Naturally, this increased the density, and we
therefore converted more iron atoms into oxygen to regain the density
of $12000$ kg~m$^{-3}$. By repeating this cycle many times, one could
in principle achieve the required density and pressure. But the
calculations are very demanding, since at each state point one has to
equilibrate the system and run it for long enough to obtain adequate
statistics for the pressure, so that in practice a compromise between
accuracy and computational effort is needed. After several
iterations, we ended up with a simulation box containing 43 iron atoms
and 21 oxygen atoms, i.e. mole fractions of $x_{\rm Fe}\approx 0.67$
and $x_{\rm O}\approx 0.33$. The resulting mass density of $11600$
kg~m$^{-3}$ and pressure of $342 \pm 4$ GPa are close to the known
values at the ICB.
Since the mass density of $11600$ kg~m$^{-3}$ is slightly below the
known value at the ICB, it is likely that the concentration of $x_{\rm
O}\approx 0.33$ is an overestimate. We have therefore taken a second
thermodynamic state with a lower concentration. In order to facilitate
comparisons with our calculations on crystalline Fe$_3$O, we chose the
value $x_{\rm O} = 0.25$. This second simulation was performed on a
system of 48 iron atoms and 16 oxygen atoms at the mass density of
$12200$ kg~m$^{-3}$, and the resulting pressure was $366 \pm 8$
GPa. We shall refer to the two simulations in the following as the
`$33 \%$ simulation' and the `$25 \%$ simulation'.
From the thermodynamic results just mentioned, we can estimate the
oxygen concentration that would be needed to reproduce the known
density and pressure at the ICB. Interpolating between the calculated
density values and applying a small correction for the slightly
different pressures in the two simulations, we estimate that the mole
fraction $x_{\rm O} = 0.28$ would reproduce the density $12000$
kg~m$^{-3}$ at the ICB pressure.
In the next sections we describe the structural, dynamical and
electronic-structure properties of the Fe/O liquid alloys.
\subsection{Structure}\label{structure}
We have simulated the $33 \% $ system for 4.2 ps after 2 ps of
equilibration. The structural properties of the system have been
inspected by looking at the partial radial distribution functions
(rdf), $g_{\rm FeFe}(r)$, $g_{\rm FeO}(r)$, and $g_{\rm OO}(r)$. The
partial rdf's are defined so that, sitting on an atom of
the species $\alpha$, the probability of finding an atom of the
species $\beta$ in the spherical shell $(r,r+dr)$ is $ 4\pi
r^2n_{\beta} g_{\alpha \beta}(r) dr$, where $n_{\beta}$ is the number
density of the species $\beta$ (the mole fraction of species
$\beta$ times the total number of atoms per unit volume).
We have calculated averages of the rdf's over different small time
windows of the simulation and we find no meaningful differences
between the windows. This confirms that the system is well
equilibrated. In Fig. 3 we display the rdf's calculated
from the whole simulation. These show that the distance between
neighbouring iron and oxygen atoms is significantly smaller than the
iron-iron distance, the maximum of $g_{\rm FeO}(r)$ being at $\approx
1.7$ \AA, while the maximum of $g_{\rm FeFe}(r)$ is at $\approx 2.1$
\AA. It is interesting to notice that $g_{\rm OO}(r)$ has a first
maximum at $\approx 2.1$ \AA, which is much greater than the chemical
bond length expected for O-O single or double bonds (1.47 \AA~ and
$1.21$ \AA~ respectively). This is clear evidence that there is no
covalent bonding between oxygen atoms. The presence of the O-O peak
at $\approx 2.1$ \AA~ indicates that oxygen atoms repel each other
with an effective atomic diameter of $\approx 2.1$ \AA. This fact shows
that oxygen has two effective sizes in the liquid: a small one when it
interacts with iron and a large one when it interacts with itself.
It is interesting to compare the structural properties of the alloy
with those of pure liquid iron. In Fig. 4 we display
the rdf calculated earlier for pure liquid iron at ICB conditions
(Vo\v{c}adlo et al., 1997) and the $g_{\rm FeFe}$ calculated here.
The two are not very different, the only apparent effect being the
broadening of the peak in the liquid alloy, which is probably due to
the greater disorder in the alloy.
The integration of the first peak of the rdf's provides a definition
of the coordination number $N^c_{\alpha \beta}$ (the average number of
neighbours of species $\beta$ surrounding an atom of species $\alpha$):
\begin{equation}
N^c_{\alpha \beta} = 4 \pi n_\beta \int_0^{r^c_{\alpha \beta}}
r^2 g_{\alpha \beta}(r) dr,
\end{equation}
where $r^c_{\alpha \beta}$ is the position of the minimum after the
first peak of $g_{\alpha \beta}$. We find the values $N^c_{\rm
FeFe}=11.0$, $N^c_{\rm FeO}=4.5$, $N^c_{\rm OFe}=9.2$, and
$N^c_{\rm OO}=4.5$. For comparison, the average coordination number
found in our earlier simulation of pure liquid iron at ICB conditions
was $N^c_{\rm FeFe}=13.8$ (Vo\v{c}adlo et al., 1997). In interpreting these
numbers, it is helpful to consider the coordination numbers that would
be found if iron and oxygen atoms had exactly the same size and if
atoms were packed in the same way as in pure liquid iron. In that
case, the total number of neighbours of each iron atom, $N^c_{\rm
FeFe} + N^c_{\rm FeO}$, would be the same as in pure iron, whereas in
fact it is 15.5. This increase of coordination number is clearly due
to the smaller size of oxygen, which allows more atoms to be fitted
into the first shell of neighbours. On the other hand, the total
number of neighbours of each oxygen atom, $N^c_{\rm OFe} +
N^c_{\rm OO}$ is 13.7, which is almost the same as the coordination
number in pure iron. We interpret this as the result of two competing
effects. The smaller size of oxygen would lead to a smaller
coordination number if all atoms in its shell of neighbours were
iron. But since on average 4.5 of the neighbours are oxygen, which
have a smaller size when interacting with iron atoms in the shell, the
coordination number is increased again.
We note that the structure of the liquid is very different from that
of the cubic Fe$_3$O and Fe$_4$O crystals discussed in section
\ref{solid}. Oxygen atoms in these crystals have respectively 12 and 6
iron neighbours. The coordination number of 9.2 in the liquid is roughly half way between the two. In the
liquid, the radii from oxygen to iron and oxygen neighbours are equal
to $\approx 1.7$ and $\approx 2.1$ \AA~ respectively, whereas in the
cubic Fe$_3$O crystal at a similar pressure, the distances are 2.05
and 2.9 \AA. On the other hand, in the BiI$_3$-structure Fe$_3$O, the
O-Fe neighbour separation is spread over the range $1.76-2.57$ \AA,
and the O-O separation is in the range $2.02-2.34$ \AA.
We now want to ask whether our simulated system is really in a single
phase and whether we can detect any sign of phase separation. In
studying this it is very helpful to calculate the static structure
factors $S_{\alpha\beta}(k)$ defined by:
\begin{equation}
S_{\alpha\beta}(k) = \langle \rho_\alpha^*({\bf
k})\rho_\beta({\bf k}) \rangle,
\end{equation}
where $\langle \cdot \rangle$ denotes the thermal average (in practice
evaluated as a time average). Here, $\rho_\alpha({\bf k})$ is the
Fourier component of the number density of species $\alpha$ at
wavevector ${\bf k}$, given by:
\begin{equation}
\rho_\alpha({\bf k}) = N_\alpha^{-1/2}\sum_{i=1}^{N_\alpha} {\rm
exp}(i{\bf k} \cdot {\bf r}_{\alpha i}),
\end{equation}
where $N_\alpha$ is the number of
atoms of species $\alpha$ and ${{\bf r}_{\alpha i}}$ is the position of
the $i$th atom of this species. Phase separation is associated with
fluctuations of the concentrations of the two species, and the
structure factors give us quantitative information about the
intensities of these fluctuations.
The connection between phase separation and structure factors can be
made more precise. In the limit of zero wavevector, the structure
factors of a liquid alloy can be rigorously expressed in terms of
thermodynamic derivatives (Bhatia and Thornton, 1970):
\begin{equation}
\lim_{ k \rightarrow 0} S_{\alpha\beta}(k) = \frac{k_BT}{(N_\alpha
N_\beta)^{1/2}} \left ( \frac{\partial N_\alpha}{\partial \mu_\beta}
\right )_{V,T,\mu_\beta'}.
\end{equation}
where $\mu_\alpha$ are the chemical potentials, and the notation
indicates that the derivative is to be taken with the volume $V$, the
temperature $T$ and all chemical potential except $\mu_\beta$ held
fixed. But the condition for thermodynamic stability with respect to
phase separation is
\begin{equation}
(\partial \mu_\alpha/\partial x_\beta)_{P,T} > 0.
\end{equation}
At the consolute point (the point in the phase diagram at which phases
start to separate) the derivatives $(\partial \mu_\alpha/\partial
x_\beta)_{P,T}$ become zero. This implies that the matrix of
derivatives $(\partial \mu_\alpha/\partial N_\beta)_{V,T,N_\beta'}$
has vanishing eigenvalues, corresponding to variations of the numbers
$N_\alpha$ that maintain the pressure constant at fixed volume. But
the matrix $(\partial N_\alpha/\partial \mu_\beta)_{V,T,\mu_\beta'}$ is
the inverse of the matrix $(\partial \mu_\alpha/\partial
N_\beta)_{V,T,N_\beta'}$, so that when the latter becomes singular the
former must acquire infinite eigenvalues. The consequence is that the
values of the structure factors in the zero-wavevector limit must
diverge if the system is unstable with respect to phase
separation. This is also intuitively clear: as one passes from the
miscible to the immiscible region, concentration fluctuations become
ever larger, becoming of macroscopic size when the phases separate,
and the increase in the fluctuations is reflected in the divergence of
the quantities $S_{\alpha\beta}( k \rightarrow 0)$.
Our calculated structure factors for the $33 \%$ simulation are
reported in Fig. 5. They have the form usually found in
liquid alloys, with prominent peaks in $S_{\rm FeFe}(k)$ and
$S_{\rm OO}(k)$ in the region $k \approx 4$ \AA$^{-1}$ signalling the
approximate spatial periodicity associated with the packing of the
atoms. The significant feature for present purposes is the lack of any
anomalous behaviour at small wavevectors. We recognize, of course, that
because of the limited size of the repeating simulation cell, there is
a lower limit to the wavevector that we can examine, which in the
present case is 0.86 \AA$^{-1}$. But at least in the accessible region
of wavevectors there is no indication of any tendency towards phase
separation.
Before leaving the description of structure, we outline another
method we have used to search for signs of phase separation. To
explain this, let us imagine for a moment that the system had
separated into phases of pure Fe and pure FeO. Then the Fe atoms in
the Fe phase would have no oxygen neighbours, whereas the Fe atoms in
the FeO phase might be expected to have 6 oxygen neighbours (we assume
the FeO liquid to have a structure resembling that of crystalline FeO
in the NiAs structure). On the other hand, in the unseparated Fe/O
phase, the number of oxygen neighbours surrounding each Fe atom
fluctuates around the value $4-5$ (see above). We can therefore
distinguish between the two situations by studying the probability
distribution for the number of oxygen neighbours surrounding Fe atoms.
To do this, we use the cut-off distance $r^c_{\alpha \beta}$ defined
above to decide when an atom of species $\beta$ counts as a neighbour
of an atom of species $\alpha$, and we define the function $P_{\alpha
\beta}(n,r^c_{\alpha \beta})$ as the probability that an atom of
species $\alpha$ has $n$ neighbours of species $\beta$. If there is a
complete phase separation, we expect $P_{\rm Fe O}$ to have peaks in the
region of $n=0$ and $n=8$, but if there is no separation we expect a
single peak in the region of $n=4-5$. Note that the rdfs contain less
information than the $P_{\alpha \beta}$ functions, and cannot by
themselves deliver the discrimination we need.
We present in Fig. 6 the function $P_{\rm Fe O}(n,r^c_{\rm
Fe O})$ calculated from our $33 \%$ simulation with the cut-off
$r^c_{\rm Fe O}=2.5$ \AA. This shows a single peak at $n=4$, and no
sign of any structure at lower or higher values of $n$. This means
that there is no indication whatever of any tendency towards phase
separation.
\subsection{Confirmation of phase stability}
Our failure to find any evidence of phase separation strongly suggests
that the Fe/O liquid is in fact stable. But it might be objected that
a simulation lasting only $4-6$ ps does not give enough time for
separation to occur, and that it would occur if the simulation were
longer. To eliminate this possibility we have devised a method in
which phase separation is artificially induced by an external
force. We shall show that when the force is removed the phases
spontaneously re-mix very rapidly. We performed this procedure on the
$25 \%$ simulation (it is not significant that the $25 \%$ case was
chosen for this, and we believe that the $33\%$ system would have
behaved in the same way).
In our procedure, we notionally divide our cubic cell into two parts,
consisting of the left region $0 < x < 0.4$ and the right region $0.4
< x < 1.0$ ($x$ is the coordinate along one of the edge directions in
units of the cell length). The first step is to sweep all the oxygen
atoms into the left region with an external force, so that this region
contains something resembling FeO, while the right region contains
pure Fe. To achieve this, we apply a constant force along the $x$-axis
to all oxygen atoms lying in the right region. This force is in the
positive $x$ direction for $0.7 < x < 1.0$ and in the negative
direction for $0.4 < x < 0.7$. No force is applied to the iron atoms,
and these are left free to redistribute themselves. Initially, the
magnitude of the force was taken to be 1 eV/\AA, but since this proved
to be too weak it was increased to 3 eV/\AA. After $\approx 1$ ps a
complete phase separation was achieved, with all oxygen atoms in the
left region, and we let the system equilibrate with the external force
still present for a further 1 ps. We show in Fig. 7 a
snapshot of one configuration taken from this period, which clearly
shows the complete separation of phases. The external force was then
switched off and the system was allowed to evolve for a further 2 ps.
Remarkably, we found that after only 1 ps had elapsed the oxygen atoms
became completely randomized throughout the cell, as can be seen in
the snapshot shown in in the lower part of Fig. 7.
To characterise these events quantitatively, we use the probability
function $P_{\rm Fe O}(n,r^c_{\rm Fe O})$ described in section
\ref{structure}. Fig. 8 shows this function calculated by
averaging over three short windows of 0.1 ps each, starting 0 ps , 0.5
ps and 1.0 ps after the external force was switched off. The first
window shows a clear bimodal form, as would be expected for a two
phase system. (We note that because our system is rather small, many
atoms are near the boundary between the phases, so that the peaks in
$P_{\rm Fe O}$ are not as sharp as they would probably be in a larger
system.) After 0.5 ps, the distribution has already become unimodal,
and after 1 ps it is very similar to what we showed in
Fig. 6 for the equilibrium $33 \%$ simulation. The
conclusion is clear: the system is not stable in the separated state
and returns very quickly to the homogeneous state.
In section \ref{solid} we have used the calculations on Fe/O crystals
to show that phase stability in high-temperature Fe/O systems might
well be expected. In particular, we gave an example of an Fe$_3$O
solid structure whose enthalpy of formation is only $\approx 1$ eV. We
have used our $25 \%$ simulation to create another such
structure. This was done simply by quenching the liquid at the rate of
3000 K ps$^{-1}$ until the atoms came to mechanical equilibrium in an
amorphous structure (this is clearly a {\em local} minimum of the
total energy function). We can regard this as a crystal with the
Fe$_3$O composition having an unusually large supercell. The
calculated enthalpy of formation of this solid at the pressure of 290
GPa is reported in the left panel of Fig. 2, and we see
that its stability is even greater that that of the BiI$_3$ structure
proposed for Fe$_3$O in section \ref{solid}. This confirms the idea
that even more stable structures may yet be found.
\subsection{Dynamics}\label{dynamics}
In studying the dynamical properties of the Fe/O liquid, our main
concern is with the viscosity. However, we first give results for the
atomic diffusion coefficients $D_\alpha$, which give a simple way of
characterising the motion of the atoms.
These are straightforwardly calculated from the mean
square displacement of the atoms through the Einstein relation
(Allen and Tildesley, 1987):
\begin{equation}\label{einstein}
\frac{1}{N_\alpha}\langle \sum_{i=1}^{N_\alpha} |{\bf r}_{\alpha
i}(t_0+t) - {\bf r}_{\alpha i}(t_0)|^2 \rangle \rightarrow 6 D_\alpha
t, ~~ {\rm as} ~~ t \rightarrow \infty,
\end{equation}
where ${\bf r}_{i \alpha}(t)$ is the position of the $i$th atom of
species $\alpha$ at time $t$, $N_{\alpha}$ has its usual meaning, and
$\langle \cdot \rangle$ is the thermal average, in practice evaluated by
averaging over time origins $t_0$. In studying the long time
behaviour of the mean square displacement, it is convenient to define
a time dependent diffusion coefficient $D_\alpha(t)$:
\begin{equation}\label{tempo}
D_\alpha(t) = \frac{1}{6 t N_\alpha}\langle \sum_{i=1}^{N_\alpha} |{\bf
r}_{\alpha i}(t_0+t) - {\bf r}_{\alpha i}(t_0)|^2 \rangle,
\end{equation}
which has the property that
\begin{equation}
\lim_{t\rightarrow \infty} D_\alpha(t) = D_\alpha.
\end{equation}
In Fig. 9 we display the iron and the oxygen diffusion
coefficients calculated using Eq. (\ref{tempo}). From this data we
estimate $D_{\rm Fe} \approx 0.8 \times 10^{-8}$ m$^2$ s$^{-1}$ and
$D_{\rm O} \approx 1.0 \times 10^{-8}$ m$^2$ s$^{-1}$. These values
should be compared with those obtained for pure liquid iron, $D_{\rm
Fe} \approx 0.4-0.5 \times 10^{-8}$ m$^2$ s$^{-1}$ (Vo\v{c}adlo et
al., 1997; de Wijs, 1998), and those obtained for Fe and S in our Fe/S
simulation $D_{\rm Fe} \approx 0.4-0.6 \times 10^{-8}$ m$^2$ s$^{-1}$,
and $D_{\rm S} \approx 0.4-0.6 \times 10^{-8}$ m$^2$ s$^{-1}$ (Alf\`e
and Gillan, 1998a). This means that the two species of atoms in liquid
Fe/O diffuse somewhat more rapidly than the atoms in both liquid iron
and the liquid Fe/S alloy at the same pressure and temperature.
In the past, atomic diffusion coefficients have often been used to
estimate the viscosity of liquids via the Stokes-Einstein relation,
and this was the procedure used in our previous work on liquid Fe and
Fe/S. Since the diffusion coefficients are larger in the present case,
this procedure would lead us to expect a lower viscosity for liquid
Fe/O. We have recently demonstrated (Alf\`e and Gillan, 1998b) that the
viscosity can be more directly (and more rigorously) calculated in
first-principles simulations using the Green-Kubo relations, i.e. the
relations between transport coefficients and correlation functions
involving fluxes of conserved quantities (Allen and Tildesley, 1987).
The shear viscosity $\eta$ is given by:
\begin{equation}\label{shear}
\eta = \frac{V}{k_BT} \int_0^\infty dt \langle P_{xy}(t_0+t) P_{xy}(t_0)
\rangle,
\end{equation}
where $V$ is the volume of the system and $P_{xy}$ is the off-diagonal
component of the stress tensor $P_{\alpha\beta}$ ($\alpha$ and $\beta$
are Cartesian components). The stress tensor is straightforwardly
calculated, so that the stress autocorrelation function (SACF) $\langle
P_{xy}(t_0+t) P_{xy}(t_0) \rangle$ can also be obtained, but at first
sight it might appear that very long simulations would be needed to
gather adequate statistical sampling. However, we have recently shown
that perfectly adequate $\eta$ values can be obtained with
surprisingly short runs, and we have reported results for liquid
aluminium and liquid Fe/S (Alf\`e and Gillan, 1998b).
In the left panel of Fig. 10 we display the the average of the five
independent components of the traceless SACF divided by its value at
$t=0$ which we denote by $\phi(t)$. Since the traceless part of the
stress tensor has zero average, $\phi(t)$ goes to zero as $t
\rightarrow \infty$. The statistical error on $\phi(t)$ for all
values of $t$ is $\approx 5\%$ of the value at $t=0$, and after $
0.2-0.3$ ps the magnitude of $\phi(t)$ falls below that error. In the
right panel of Fig. 10 we display the integral $\int_0^t dt'\phi(t')$
of $\phi(t)$ as a function of time. The limiting value of the integral
for $t\rightarrow \infty$ is the shear viscosity. The error that one
makes in evaluating that integral grows with time, since one
integrates the noise together with $\phi(t)$. We estimated the error
in the integral as a function of time using the scatter of the
SACF's. Combining this estimate with an analytic expression for the
error, we obtain the error estimate displayed in Fig. 10. From the
point where $\phi(t)$ falls below the noise one integrates only the
latter, so one gains nothing by evaluating the integral beyond that
point. If we assume that $\phi(t)$ is zero above $t\approx 0.3$ ps,
we obtain the value $\eta = 4.5 \pm 1.0 $ mPa~s. This value is
roughly half the value reported earlier for liquid Fe (Vo\v{c}adlo et
al., 1997) and Fe/S (Alf\`e and Gillan, 1998a) at ICB conditions, and
is not very much greater than the viscosity of typical liquid metals
under ambient conditions; for example, the viscosity of liquid
aluminium at atmospheric pressure 100 K above its melting point is
1.25 mPa~s (Shimoji and Itami, 1986).
This result confirms our earlier conclusion (de Wijs et al., 1998) that the
viscosity of the outer core is towards the lower end of the wide range
of theoretical and experimental values reported in the literature.
\subsection{Electronic structure}\label{electronic}
We have studied the electronic structure of our simulated Fe/O liquid
in order to shed light on the nature of the bonding between the atoms
and to help to interpret the structure discussed in section
\ref{structure}. The main tools used in this analysis are the
electronic density of states (DOS) and the local density of states
(LDOS). The DOS represents the number of electronic states per unit
energy as a function of energy, while the LDOS is a projection of the
DOS onto states of chosen angular momentum on atoms of chosen
species. In performing this projection we took spherical regions of
radius $R$ on the atoms, and in practice we chose $R=0.6$ \AA~ for
both iron and oxygen. This $R$ value is considerably smaller than half
the interatomic distance in the liquid (see Fig. 3), so we
expect to distinguish clearly between the electronic structures on
different atoms. The results are not averaged over time but are
calculated from the electronic energies and wavefunctions for selected
time steps taken from the $25 \%$ simulation.
The calculated DOS is shown in the upper panel of Fig. 11,
and consists of four main features (energies are referred to the Fermi
energy): two fairly narrow peaks at $-24$ and $-11$ eV; a large
dominant peak spanning the range $-9$ eV to $3$ eV; and a broad
feature extending well above the Fermi energy. The LDOS shown in the
lower panel of the Figure allows us to interpret these features. The
peak at $-24$ eV consists entirely of O$(2s)$ states. The peak at
$-11$ eV is mainly O$(2p)$, but with an appreciable contribution of
Fe$(3d)$. The dominant peak from $-9$ eV to $3$ eV consists mainly of
Fe$(3d)$ states, but there is a significant peak associated with
O$(2p)$ states just above the Fermi energy. The broad feature above
the Fermi energy comes from both Fe$(3d)$ and O$(2p)$ states.
To see the implications of this structure for the interatomic bonding,
we note that if the O$(2p)$ states were much lower in energy than the
Fe$(3d)$ states, there would be very little hybridisation between the
two kind of states, the O$(2p)$ levels would be completely filled, and
the oxygen atoms would carry a net charge of $-2|e|$. The Fe-O bonding
would be ionic. On the other hand, if the O$(2p)$ states were at the
same energy as Fe$(3d)$, we should expect strong hybridisation and
little charge transfer, so that the bonding would be covalent. It is
clear from Fig. 11 that the Fe-O bonding is intermediate
between ionic and covalent. The O$(2p)$ states are below Fe$(3d)$, but
not low enough to suppress hybridisation. There is a clear splitting
of the O$(2p)$ levels into bonding and anti-bonding orbitals, though
the bonding orbital clearly has much larger weight (in pure covalent
bonding we should expect the weights to be equal). The peak in
Fe$(3d)$ at $-11$ eV is also clear evidence for hybridisation between
O$(2p)$ and Fe$(3d)$. The implication is that there is a partial
charge transfer from Fe to O, but not enough to give oxygen a charge
of $-2|e|$.
The bonding between Fe atoms is metallic. The partial filling of the
Fe$(3d)$ levels gives the well known bonding mechanism emphasised by
Friedel's (1969) analysis of the cohesive and elastic
properties of transition metal crystals.
Since the oxygen atoms carry partial charges and their $2p$ orbitals
are almost full, no covalent bond is expected between them, and this
is also clear from Fig. 3. To investigate the electronic
state of oxygen in more detail, we have calculated the LDOS for oxygen
atoms in different environments. We show in Fig. 12 the LDOS
for two oxygen atoms denoted O$_{\rm a}$ and O$_{\rm b}$, which have
been chosen from the $25 \%$ simulation so that O$_{\rm a}$ has 10 Fe
neighbours and one O neighbour while O$_{\rm b}$ has seven Fe and four
O neighbours. If there were any covalent bonding between O atoms, we
should expect a larger bonding-antibonding splitting of the $2p$
states for the O$_{\rm b}$ atoms than for O$_{\rm a}$, and we might
expect a similar splitting (or at least a broadening) of the O$(2s)$
states for O$_{\rm b}$. The LDOS curves show neither of these
effects. Instead, the main difference is the upward shift of the peaks
for O$_{\rm b}$ compared with O$_{\rm a}$. We believe this is direct
evidence for the partial charge transfer: the valence electrons of
O$_{\rm b}$ feel a repulsive electrostatic potential due to the
partial negative charges on the four oxygen neighbours, which raises
their energy.
The main bonding mechanisms in the liquid are therefore the ionic-covalent
Fe-O bond and the metallic Fe-Fe bond. Since the O$(2p)$ atomic
orbitals are more compact than Fe$(3d)$ orbitals we expect the Fe-O
distance to be shorter than the Fe-Fe distance, and this effect is
clear from the rdf's shown in Fig. 3. The partial charge
transfer presumably also contributes to the shortening of this
distance.
\section{Discussion}
Two of our main aims in this paper have been to probe the phase
stability of liquid Fe/O under Earth's core conditions, and to
determine the oxygen concentration that would be needed to reproduce
the known core density. In practice, these aims must be taken
together: we want to know whether the alloy is thermodynamically
stable at the appropriate concentration.
The results we have presented leave little doubt that {\em if} the
liquid is stable then the mole fraction of oxygen must be in the
region $25-30 \%$ (our best estimate is $28 \%$), because anything
much less than this would give a pressure that is too low at the known
density. This is essentially the same as the value proposed many
years ago by Ringwood (1977). In judging the robustness of this
conclusion we recall some important facts: First, DFT
electronic-structure methods of the type used here generally predict
the density of materials at a given pressure to within a few
percent. Particularly relevant here are recent DFT calculations
(Stixrude et al., 1994; S\"oderlind et al., 1996; Vo\v{c}adlo et al.,
1997) on h.c.p. iron over the pressure range $0-350$ GPa which are in
excellent agreement with each other and with the experimental
results. Similar comparisons for FeSi (Vo\v{c}adlo et al., 1997) are
also relevant. DFT calculations on oxides (including transition-metal
oxides) generally predict the density with similar accuracy. A second
important fact is that our earlier first principles simulations of
pure liquid iron, based on exactly the same techniques, gave a
prediction of the density at the pressure of the inner core boundary
which is correct to $\approx 2 \%$. (In fact the comparison was done
the other way round: at the density of 13300 kg~m$^{-3}$ and the
temperature of 6000 K, our simulations gave a pressure of 358 GPa,
compared with the value of 330 GPa estimated from experimental data.)
Third, our estimate of the oxygen concentration is based on {\em
changes} of pressure and density compared with our simulated pure
iron. The calculations should be even more reliable for these
differences then they are for the absolute values. We therefore
believe that our value for the oxygen concentration required should be
subject to an error of no more that $\approx 5 \%$.
It is interesting to compare the liquid composition with that for the
case of Fe/S. In our recent simulations (Alf\`e and Gillan, 1998a), we
showed that liquid Fe/S is not far from reproducing the known pressure
and density at the inner core boundary with a sulphur mole fraction of
$18 \%$. At this composition, the mass density of 12330 kg~m$^{-3}$
gave a calculated pressure of $349 \pm 6$ GPa. If we make a
correction to bring the density to 12000 kg~m$^{-3}$, we find a
sulphur mole fraction of $23 \%$. This means that to achieve the
required reduction in density we actually need a higher mole fraction
of oxygen than of sulphur, even though the atomic mass of oxygen is
only half that of sulphur. The reason is that the oxygen atom is
smaller, a point to which we return below.
The question of phase stability is more complex. What seems certain is
that earlier arguments against phase stability based on {\em ab
initio} calculations of the energetics of Fe$_3$O and Fe$_4$O crystals
(Sherman, 1995) do not really deliver the intended conclusion. This is
not because the calculations were wrong. On the contrary, our
calculations fully support their correctness. It is simply that the
crystal structures assumed were not the most stable. We have presented
an alternative structure for Fe$_3$O which gives a formation enthalpy
that is low enough ($\approx 1$ eV) to make phase stability under core
conditions quite plausible.
We have used our simulations to probe the phase stability of the Fe/O
liquid in the appropriate region of concentration, and all the
indications are that it is stable. We therefore fully support the
conclusion that has been drawn from high pressure experimental
measurements (Ohtani et al., 1984; Ringwood and Hibberson, 1990) that
liquid Fe and FeO are miscible under core conditions in the
concentration region of interest. However, a word of caution is in
order. The fact is that our simulated systems are rather small, and it
is quite conceivable that a system that would be unstable in the
thermodynamic limit could be stabilised by the artificial periodic
boundary conditions in small simulation cells. Nevertheless, our
calculations certainly provide strong support for the thermodynamic
stability of liquid Fe/O under core conditions at the relevant
concentration. In the end, we believe that the definitive theoretical
approach to this question will be first principles free energy
calculations on the appropriate solid and liquid phases. These would
be computationally very demanding, but should certainly be feasible in
the near future. (First principles free energy calculations have
recently been used with success to calculate the melting properties of
silicon and aluminium (Sugino and Car, 1995; de Wijs et al., 1998).)
The small size of oxygen is clear from our analysis of the liquid
structure: the Fe-O nearest neighbour distance (the position of the
first peak in the rdf) is only at $\approx 1.7$ \AA, compared with
$\approx 2.1$ \AA~ for the Fe-Fe distance. We recall that in the Fe/S
liquid, the Fe-S distance was $\approx 1.95$ \AA~ (Alf\`e and Gillan,
1998a). An important feature of the liquid is that each oxygen has on
average only 9 iron neighbours, whereas it has between 4 and 5 oxygen
neighbours. This atomic environment of oxygen is very different from
that produced by the cubic structure of Fe$_3$O. By contrast in the
BiI$_3$ structure of Fe$_3$O oxygen has 11 iron neighbours and 2
oxygen neighbours. The dilemma in making crystal structures for Fe/O
solid solutions at core pressures is that we want to achieve close
packing because of the high pressure, but the atoms that are being
packed have different sizes. It seems that the cubic structure of
Fe$_3$O is not a good solution. The BiI$_3$ structure is better, even
though it means putting oxygen atoms next each other. There may be
yet better ways.
Our analysis of the electronic structure of the liquid gives further
insight. Here, the important feature is that Fe-O bonding is only
partially ionic, with a substantial covalent contribution. The
implication that there is partial electron transfer from Fe to O is
relevant, because presumably if O carried a full ionic charge of
$-2|e|$ oxygen atoms would be more reluctant to become neighbours of
each other. The electronic structure shows that there is no
detectable covalent interaction between oxygens, so the fact that they
become neighbours cannot be attributed to covalency.
Finally, we have studied the diffusion coefficient of iron and oxygen
atoms and the viscosity of the liquid at $33 \%$ composition. The
finding is that both species diffuse more rapidly than in either pure
liquid iron or the liquid Fe/S alloy: the diffusion coefficients have
roughly twice the value that they have in pure Fe and Fe/S at the same
pressure and temperature. We also find that the viscosity of the Fe/O
liquid is about half what it is in those other systems. This means
that if the major light element in the outer core was oxygen, our
earlier conclusion (de Wijs et al., 1998) about the low viscosity of
the outer core would be confirmed, and indeed strengthened.
\section{Conclusions}
We are led to the following conclusions: if the light impurity in the
outer core is mainly oxygen, then its molar concentration would have
to be $\approx 28 \%$; in this region of concentration, we have
strong evidence that the liquid is stable against phase separation;
the proposed miscibility is not in conflict with the large formation
enthalpies predicted by earlier {\em ab initio} calculations, because
those calculations were based on assumed structures that are not the
most stable; the proposed Fe/O liquid alloy has an even lower
viscosity than that of pure Fe and the relevant Fe/S alloy under the
same conditions.
\section*{Acknowledgments}
The work of DA is supported by NERC grant GST/O2/1454 to G. D. Price
and M. J. Gillan. We thank the High Performance Computing Initiative
for allocations of time on the Cray T3D and T3E at Edinburgh Parallel
Computer Centre, these allocations being provided through the Minerals
Physics Consortium (GST/02/1002) and the U.K. Car-Parrinello
Consortium. We thank Dr. G. Kresse and Dr. G. de Wijs for valuable
technical assistance, and Dr. L. Vo\v{c}adlo
and Dr. D. Sherman for useful discussions. We thank Dr. W. Temmerman
for useful advice about the LMTO-46 code, and the Psi-k Network for
making the code available.
\newpage
\section*{References}
{\small
\medskip\noindent Alder, B. J., 1966. Is the mantle soluble in the
core? J. Geophys. Res., 71: 4973-4979.
\medskip\noindent Alf\`e, D., and Gillan, M. J., 1998a. First principles
simulations of liquid Fe-S under Earth's core
conditions. Phys. Rev. B, {\it in press}.
\medskip\noindent Alf\`e, D., and Gillan, M. J., 1998b. The first
principles calculation of transport coefficient. Phys. Rev. Lett.,
{\it submitted}.
\medskip\noindent Allen, M. P., Tildesley, D. J., 1987. Computer
Simulation of Liquids. Oxford University Press, Oxford.
\medskip\noindent Bhatia, A. B., Thornton, D. E., 1970. Structural
aspects of the electrical resistivity of binary alloys. Phys. Rev. B,
2: 3004-3012.
\medskip\noindent Birch, F., 1952. Elasticity and composition of the
Earth's interior. J. Geophys. Res., 57: 227-286.
\medskip\noindent Birch, F., 1964. Density and composition of mantle
and core. J. Geophys. Res., 69: 4377-4388.
\medskip\noindent Boehler, R., 1993. Temperature in the Earth's core
from melting-point measurements of iron at high static
pressures. Nature. 363: 534-536.
\medskip\noindent Bullen, K. E., 1973. Cores of terrestrial planets. Nature,
243: 68-70.
\medskip\noindent Car, R., and Parrinello, M., 1985. Unified approach
for molecular-dynamics and density-functional theory.
Phys. Rev. Lett., 55: 2471-2474.
\medskip\noindent Clark, S. P., 1963. Variation of density in the
Earth and the melting curve in the mantle. In: The Earth Sciences,
edited by T. W. Donnelly, University of Chicago Press, Chicago.
\medskip\noindent Distin, P. A., Whiteway S., and Masson, C., 1971.
Solubility of oxygen in liquid iron from 1785$^{\rm o}$ to 1960$^{\rm
o}$. A new technique for the study of slag-metal
equilibria. Canad. Metallurg. Quart., 10: 13-18.
\medskip\noindent Dubrovskiy, V. A. and Pan'kov, V. L., 1972. On the
composition of the Earth's core. Acad. Sci. USSR Phys. Solid Earth, 7:
452-455.
\medskip\noindent Fischer, W. A., and Schumacher, J. F., 1978. Die
S\"attigungsl\"oslichkeit von Reineisen an Saurstoff vom Schmelzpunkt
bis 2046$^{\rm o}$ C, ermittelt mit dem Schwebeschmelzverfahren.
Arch. Eisenh\"uttenwes., 49: 431-435.
\medskip\noindent Friedel, J., 1969. Fundamental aspects in the bonding of
transition metals. In: The Physics of Metals, ed. J. Ziman, Cambridge
University Press, Cambridge, 494 pp.
\medskip\noindent Fukai, Y., and Akimoto, S., 1983. Hydrogen in the
Earth's core. Proc. Jpn. Acad., Ser. B, 59: 158-162.
\medskip\noindent Gillan, M. J., 1997. The virtual matter
laboratory. Contemp. Phys., 38: 115-130.
\medskip\noindent Goarant, F., Guyot, F., Peyronneau, J., and Poirier,
J.-P., 1992. High-pressure and high-temperature reactions between
silicates and liquid iron alloys in the diamond anvil cell, studied
by analytical electron microscopy. J. Geophys. Res., 97: 4477-4487.
\medskip\noindent Hohenberg, P., and Kohn, W., 1964. Inhomogeneous
electron gas. Phys. Rev. 136: B864-B871.
\medskip\noindent Holender, J. M., and Gillan, M. J., 1996.
Composition dependence of the structure and electronic properties of
liquid Ga-Se alloys studied by ab initio molecular dynamics simulation.
Phys. Rev. B, 53: 4399.
\medskip\noindent Jones, R. O., and Gunnarsson, O., 1989. The density
functional formalism, its applications and prospects. Rev. Mod. Phys.,
61: 689-746.
\medskip\noindent Kirchhoff, F., Holender, J. M., and Gillan, M. J.,
1996a. The structure, dynamics and electronic structure of liquid
Ag-Se alloys investigated by ab initio simulation. Phys. Rev. B, 54: 190-202.
\medskip\noindent Kirchhoff, F., Gillan, M. J., Holender, J. M.,
Kresse, G., and Hafner, J., 1996b. Structure and bonding in liquid Se.
J. Phys. Condens. Matter, 8: 9353-9357.
\medskip\noindent Knittle, E., Jealnoz, R., 1991. Earth's core-mantle
boundary: results of experiments at high pressures and
temperatures. Science, 251: 1438-1443.
\medskip\noindent Kohn, W., and Sham, L., 1965. Self-consistent equations
including exchange and correlation effects. Phys. Rev. 140: A1133-A1138.
\medskip\noindent Kresse, G., and Hafner, J., 1993.
Ab-initio molecular-dynamics for open-shell transition-metals. Phys. Rev. B,
48: 13115-13118.
\medskip\noindent Kresse, G., and Furthm\"{u}ller, J., 1996a.
Efficiency of ab-initio total-energy calculations for metals and
semiconductors using a plane-wave basis-set.
Comput. Mater. Sci., 6: 15-50.
\medskip\noindent Kresse, G., and Furthm\"{u}ller, J., 1996b.
Efficient iterative schemes for ab-initio total-energy calculations
using a plane-wave basis-set. Phys. Rev. B, 54: 11169-11186.
\medskip\noindent Krier, G., Jepsen, O., Burkhardt, A., and Andersen,
O. K., 1994. LMTO-46 code, Max-Planck-Institut f\"ur
Festk\"orperforschung, Stuttgart.
\medskip\noindent Lewis, J. S., 1973. Chemistry of the planets.
Annu. Rev. Phys. Chem., 24: 339-351.
\medskip\noindent Louie, S. G., Froyen, S., and Cohen, M. L., 1982.
Non-linear ionic pseudopotentials in spin-density-functional calculations.
Phys. Rev. B, 26: 1738-1742.
\medskip\noindent MacDonald, G. J. F., and Knopoff, L., 1958.
The chemical composition of the outer core. J. Geophys., 1: 1751-1756.
\medskip\noindent Mao, H. K., Wu, Y., Chen, L. C., Shu, J. F., and
Jephcoat, A. P., 1990. Static compression of iron to 300 GPa and
Fe0.8Ni0.2 alloy to 260 GPa: implications for composition of the
core. J. Geophys. Res., 95: 21737-21742.
\medskip\noindent Mason, B., 1966. Composition of the Earth. Nature,
211: 616-618.
\medskip\noindent Monkhorst, H. J., and Pack, J. D., 1976. Special
points for Brillouin-zone integrations. Phys. Rev. B, 13: 5188-5192.
\medskip\noindent Murthy, V., and Hall, H. T., 1970. On the possible
presence of sulfur in the Earth's core. Phys. Earth Planet. Inter., 2:
276-282.
\medskip\noindent Nos\'e, S., 1984. A molecular dynamics method for
simulations in the canonical ensemble.
Molec. Phys., 52: 255-268.
\medskip\noindent Ohtani, E., and Ringwood, A. E., 1984 Composition
of the core, I. Solubility of oxygen in molten iron at high
temperatures. Earth Planet. Sci. Lett., 71: 85-93.
\medskip\noindent Ohtani, E., Ringwood, A. E. and Hibberson, W.,
1984. Composition of the core, II. Effect of high pressure on
solubility of FeO in molten iron. Earth Planet. Sci. Lett., 71:
94-103.
\medskip\noindent Parr, R. G., and Yang, W., 1989. Density-Functional
Theory of Atoms and Molecules, Oxford University Press, Oxford.
\medskip\noindent Poirier, J.-P., 1991. Introduction to the Physics
of the Earth's Interior. Cambridge University Press, Cambridge.
\medskip\noindent Poirier, J.-P., 1994. Light elements in the Earth's
outer core: A critical review. Phys. Earth Planet. Interiors. 85:
319-337.
\medskip\noindent Pulay, P., 1980. Convergence acceleration of
iterative sequences. The case of SCF iteration. Chem. Phys. Lett., 73:
393.
\medskip\noindent Ringwood, A. E., 1959. On the chemical evolution and
densities of the planets. Geochem. J., 15: 257-283.
\medskip\noindent Ringwood, A. E., 1961. Silicon in the metal phase of
enstatite chondrites and some geochemical
implications. Geochim. Cosmochim. Acta, 25: 1-13.
\medskip\noindent Ringwood, A. E., 1966. Chemical evolution of the
terrestrial planets. Geochim. Cosmochim. Acta, 30: 41-104.
\medskip\noindent Ringwood, A. E., 1977. On the composition of the
core and implications for the origin of the Earth.
Geochim. Cosmochim. Acta, 11: 111-135.
\medskip\noindent Ringwood, A. E., and Hibberson, W., 1990. The system
Fe-FeO revisited. Phys. Chem. Minerals, 17: 313-319.
\medskip\noindent Secco, R. A. 1995. Viscosity of the Outer Core.
In: Mineral Physics and
Crystallography: A Handbook of Physical Constants, ed. T. J. Ahrens,
Americal Geophysical Union, 218 pp.
\medskip\noindent Sherman, D. M., 1995. Stability of possible Fe-FeS
and Fe-FeO alloy phases at high pressure and the composition of the
Earth's core. Earth Planet. Sci. Lett., 132: 87-98.
\medskip\noindent Shimoji, M., and Itami, T., 1986. Atomic Transport in
Liquid Metals, Trans Tech Publications, Aedermannsdorf, p. 191.
\medskip\noindent S\"oderlind, P., Moriarty, J. A., and Wills, J. M.,
1996. First principles theory of iron up to the Earth's core
pressures: Structural, vibrational, and elastic properties.
Phys. Rev. B, 53: 14063-14072.
\medskip\noindent \v{S}tich, I., Car, R., and Parrinello, M., 1989.
Bonding and disorder in liquid silicon. Phys. Rev. Lett., 63:
2240-2243.
\medskip\noindent Stixrude, L., Cohen, R. E., and Singh, D. J., 1994.
Iron at high pressure: Linearized-augmented-plane-wave computation in
the generalized-gradient approximation. Phys. Rev. B, 50: 6442-6445.
\medskip\noindent Sugino, O., and Car, R., 1995. Ab-initio
molecular-dynamics study of first-order phase-transitions. Melting of
silicon. Phys. Rev. Lett., 74: 1823-1826.
\medskip\noindent Suzuki, T., Akimoto S., and Yagi, T.,
1989. Metal-silicate-water reaction under high pressure, I, Formation
of metal hydride and implications for composition of the core and
mantle. Phys. Earth Planet. Inter., 56: 377-388.
\medskip\noindent Urey, H. C., 1960. On the chemical evolution and
densities of the planets. Geochim. Cosmochim. Acta, 18: 151-153.
\medskip\noindent Vanderbilt, D., 1990. Soft self-consistent
pseudopotentials in a generalized eigenvalue formalism. Phys. Rev. B,
41: 7892-7895.
\medskip\noindent Verlet, L., 1967. Computer `experiments' on
classical fluids. I. Thermodynamical properties of Lennard-Jones
molecules. Phys. Rev., 159: 98-103.
\medskip\noindent Vo\v{c}adlo, L., de Wijs, G. A., Kresse, G., Gillan,
M. J., and Price, G. D., 1997. First principles calculations on
crystalline and liquid iron at earth's core conditions. Faraday
Discuss., 106: 205-217.
\medskip\noindent Wang, Y., and Perdew, J., 1991. Correlation hole of
the spin-polarized electron gas, with exact small-wave-vector and
high-density scaling. Phys. Rev. B, 44: 13298-13307.
\medskip\noindent de Wijs, G. A., Kresse, G., and Gillan, M. J., 1998.
First order phase transitions by first-principles free energy
calculations: The melting of Al. Phys. Rev. B, 57: 8223-8234.
\medskip\noindent de Wijs, G. A., Kresse, G., Vo\v{c}adlo, L., Dobson,
D., Alf\`e, D., Gillan, M. J., Price, G. D., 1998. The viscosity of
liquid iron at the physical conditions of the Earth's core. Nature,
392: 805-807.
\medskip\noindent Wood, B. J. 1993. Carbon in the core. Earth
Planet. Sci. Lett., 117: 593-607.
\medskip\noindent Wyckoff, R. W. G., 1964. Crystal Structures, 2nd
edition, Vol. 2, Interscience, New York. }
\newpage
\begin{tabular}{lcccc}
\hline \hline
& & PP(spin-unrestricted) & PP(spin-restricted) & FLAPW \\
\hline
Fe($\epsilon$) & $\rho_0$ & 8910 & 8910 & 8780 \\
& $K$ & 283 & 283 & 260 \\
& $K'$ & 4.39 & 4.39 & 4.53 \\
& & & & \\
FeO(B8) & $\rho_0$ & 5650 & 6980 & 5810 \\
& $K$ & 92 & 258 & 173 \\
& $K'$ & 4.96 & 4.4 & 2.93 \\
&$\mu$ & 2.0 & & not reported \\
& & & & \\
Fe$_3$O(cubic) & $\rho_0$ & 6500 & 7550 & 7420 \\
& $K$ & 123 & 226 & 223 \\
& $K'$ & 4.23 & 4.04 & 4.02 \\
&$\mu$ & 2.34 & & small \\
& & & & \\
Fe$_4$O & $\rho_0$ & 6880 & 7840 & 7830 \\
& $K$ & 135 & 273 & 310 \\
& $K'$ & 4.82 & 4.26 & 4.17 \\
&$\mu$ & 2.0 & & \\
& & & & \\
Fe$_3$O(BiI$_3$) & $\rho_0$ & & 7930 & \\
& $K$ & & 248 & \\
& $K'$ & & 4.29 & \\
\hline \hline
\end{tabular}
\bigskip
\noindent Table 1: Calculated equilibrium properties of crystals in the Fe/O
system: equilibrium mass density $\rho_0$ (kg~m$^{-3}$), bulk modulus
$K$ (GPa), the pressure derivative $K'=dK/dP$, and magnetic moment per
atom $\mu$ (units of Bohr magneton). Results are given for the present spin
unrestricted and restricted pseudopotential (PP) calculations and the
FLAPW calculations of Sherman (1995). The structures of
the cubic Fe$_3$O, Fe$_4$O and BiI$_3$-structure Fe$_3$O are shown in
Fig. 1.
\bigskip
\bigskip
\begin{tabular}{ccccccc}
\hline \hline
Volume (\AA$^3$) & Pressure (GPa) & $b/a$ & $c/a$ & $\alpha$ &
$\beta$ & $\gamma$ \\
\hline
22 & 467 & 1.01016 & 0.95443 & 123.76 & 90.70 & 79.04 \\
24 & 325 & 1.00988 & 0.95322 & 123.82 & 90.88 & 78.79 \\
26 & 226 & 1.01010 & 0.95164 & 123.85 & 91.03 & 78.55 \\
28 & 156 & 1.00772 & 0.94984 & 123.89 & 91.26 & 78.28 \\
30 & 105 & 1.00574 & 0.94874 & 123.85 & 91.39 & 78.07 \\
32 & 67 & 0.99711 & 0.94465 & 123.82 & 91.79 & 77.69 \\
34 & 40 & 0.99452 & 0.94303 & 123.76 & 91.97 & 77.37 \\
36 & 19 & 0.99193 & 0.94258 & 123.57 & 91.98 & 76.95 \\
\hline \hline
\end{tabular}
\bigskip
\noindent Table 2: Cell parameters and pressure $P$ of Fe$_3$O in the BiI$_3$
structure (see Fig. 1) calculated at a series of volumes
(per Fe$_3$O unit). The quantities $a$, $b$, $c$ are the magnitudes of
the primitive translation vectors, and $\alpha, \beta, \gamma$ are the
angles between the pairs $(a,b), (a,c)$ and $(b,c)$ respectively.
\newpage
\section*{Figure captions}
\medskip\noindent{\bf Fig. 1:} The crystal structures of cubic Fe$_3$O (left),
cubic Fe$_4$O (centre) and the BiI$_3$ form of Fe$_3$O (right) used to
calculate the formation enthalpies of Fe/O solid solutions. Large and
small spheres represent iron and oxygen respectively.
\medskip\noindent{\bf Fig. 2:} Calculated enthalpies of formation (per
formula unit) of solid solutions having compositions Fe$_3$O and
Fe$_4$O. Left panel shows spin-restricted results from present work
compared with FLAPW results of Sherman (1995); right panel shows
present spin-unrestricted results. Key to style of curves: present
cubic Fe$_3$O \protect\rule[.5mm]{5mm}{1mm}, cubic Fe$_3$O of Sherman
(1995) \protect\rule[1mm]{5mm}{0.3mm}, present cubic Fe$_4$O
\protect\rule[.5mm]{1.5mm}{1mm} \protect\rule[.5mm]{1.5mm}{1mm}
\protect\rule[.5mm]{1.5mm}{1mm}, cubic Fe$_4$O of Sherman (1995)
\protect\rule[1mm]{1.5mm}{0.3mm} \protect\rule[1mm]{1.5mm}{0.3mm}
\protect\rule[1mm]{1.5mm}{0.3mm}, present Fe$_3$O in the BiI$_3$
structure \protect\rule[1mm]{1.5mm}{.3mm}
\protect\rule[1mm]{.3mm}{.3mm} \protect\rule[1mm]{1.5mm}{.3mm}
\protect\rule[1mm]{.3mm}{.3mm} \protect\rule[1mm]{1.5mm}{.3mm}
\protect\rule[1mm]{.3mm}{.3mm}. The isolated point shows formation
enthalpy of the amorphous structure obtained by quenching the liquid
(see Sec. \ref{liquid}).
\medskip\noindent{\bf Fig. 3:} Radial distribution functions
$g_{\alpha\beta}(r)$ obtained from simulation of liquid Fe/O at oxygen
molar concentration of $33 \%$.
\medskip\noindent{\bf Fig. 4:} The iron-iron radial distribution function
$g_{\rm FeFe}(r)$ from the present simulation of liquid Fe/O (oxygen
molar concentration of $33 \%$) compared with $g_{\rm FeFe}(r)$ from
simulation of pure liquid iron at similar pressure and temperature
(Vo\v{c}adlo et al., 1997).
\medskip\noindent{\bf Fig. 5:} Partial structure factors $S_{\alpha\beta}(k)$
calculated from simulation of liquid Fe/O at oxygen molar
concentration of $33 \%$.
\medskip\noindent{\bf Fig. 6:} Probability distribution $P_{\rm
FeO}(n,r^c_{\rm FeO})$ for number $n$ of oxygen neighbours of an iron
atom calculated from simulation of liquid Fe/O at oxygen molar
concentration of $33 \%$.
\medskip\noindent{\bf Fig. 7:} Snapshots of the simulated liquid Fe/O system
along three principal Cartesian axes. Top three panels show a
configuration from the period when phase separation was artificially
induced by application of an external force. Bottom three panels show
a configuration from the later period after removal of the external
force. Large and small spheres represent iron and oxygen
respectively.
\medskip\noindent{\bf Fig. 8:} Probability distribution $P_{\rm
FeO}(n,r^c_{\rm FeO})$ for number $n$ of oxygen neighbours of an iron
atom calculated from simulation of liquid Fe/O at oxygen molar
concentration of $25 \%$. Results are average values for three
windows of length 0.1 ps at times of 0 ps, 0.5 ps and 1 ps after
removal of the external force used to induce phase
separation.
\medskip\noindent{\bf Fig. 9:} Time dependent diffusion coefficients
$D_\alpha(t)$ for iron and oxygen calculated from the simulation of
liquid Fe/O at oxygen molar concentration of $33 \%$.
\medskip\noindent{\bf Fig. 10:} Average over the five independent components of
the autocorrelation function of the traceless stress tensor $\phi(t)$
(left panel) and viscosity integral (solid curve) with its statistical
error (dashed curve) (right panel).
\medskip\noindent{\bf Fig. 11:} Electronic density of states (upper panel) and
local densities of states (lower panel) calculated for liquid Fe/O at
oxygen molar concentration of $25 \%$. Energy is referred to the Fermi
energy $E_f$.
\medskip\noindent{\bf Fig. 12:} Local densities of states for two selected
oxygen atoms taken from the simulation of liquid Fe/O at oxygen molar
concentration of $25 \%$. Atom O$_{\rm a}$ has 1 oxygen neighbour and
10 iron neighbours; atom O$_{\rm b}$ has 4 oxygen neighbours and 7
iron neighbours. Energy is referred to the Fermi energy $E_f$.
\newpage
\bigskip\centerline{FIGURE 1}
\bigskip\centerline{\psfig{figure=fe3o.ps,height=1.8in}\hskip 30pt
\psfig{figure=fe4o.ps,height=1.8in}\hskip 30 pt
\psfig{figure=b1.ps,height=1.8in}}
\newpage
\bigskip\centerline{FIGURE 2}
\bigskip\centerline{\psfig{figure=nonmag.ps,height=3in}\hskip 20pt
\psfig{figure=mag.ps,height=3in}}
\newpage
\bigskip\centerline{FIGURE 3}
\bigskip\centerline{\psfig{figure=g21.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 4}
\bigskip\centerline{\psfig{figure=compare.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 5}
\bigskip\centerline{\psfig{figure=s1.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 6}
\bigskip\centerline{\psfig{figure=gnum21.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 7}
\bigskip\centerline{\psfig{figure=front.ps,height=2.0in}\hskip 10pt
\psfig{figure=top.ps,height=2.0in}\hskip 10pt
\psfig{figure=side.ps,height=2.0in}}
\bigskip\centerline{\psfig{figure=front1.ps,height=2.0in}\hskip 10pt
\psfig{figure=top1.ps,height=2.0in}\hskip 10pt
\psfig{figure=side1.ps,height=2.0in}}
\newpage
\bigskip\centerline{FIGURE 8}
\bigskip\centerline{\psfig{figure=gnum16.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 9}
\bigskip\centerline{\psfig{figure=m1.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 10}
\bigskip\centerline{\psfig{figure=sacf.ps,height=2.8in}\hskip 20pt
\psfig{figure=visc.ps,height=2.8in}}
\newpage
\bigskip\centerline{FIGURE 11}
\bigskip\centerline{\psfig{figure=dos.ps,height=3.5in}}
\bigskip\centerline{\psfig{figure=ldos.ps,height=3.5in}}
\newpage
\bigskip\centerline{FIGURE 12}
\bigskip\centerline{\psfig{figure=ldos1.ps,height=3.5in}}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,208 |
Q: Testing whether a classification model is better than another predictor I hope this has not been asked before.
I have got data(my prediction variables) of 3 populations of individuals, A,B and C. These individuals can be described as:
*
*A is the part of the risk group showing symptoms,
*B is the part of the risk group not showing symptoms, and
*C is not a risk group and does not show symptoms.
I have now created two logistic regression models for the questions: "Does an individual belong to the risk group?", i.e. "classify (A and B) vs C" and "Is an individual showing symptoms", i.e. "classify A vs (B and C)".
Both models are better than chance level on the test set and therefore i want to answer the following question: can i predict the symptoms better than just classification to the risk group?
Is there a good statistical test for that? Both models are tested on the same data. I am thinking that i should just run both models on the task A vs (B and C) and use a one-sided McNemors test for that. But the models are trained on different class prior class probabilities, so this would not give the right result. Another alternative would be a DeLong test on the AUC.
Any other options?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 349 |
«Маленькі трагедії» — твір А. С. Пушкіна.
Інші значення
Маленькі трагедії (фільм, 1971) — екранізація спектаклю ленінградського театру драми імені О.С. Пушкіна.
Маленькі трагедії (фільм, 1979) — екранізація твору О.С. Пушкіна.
Маленькі трагедії (фільм, 2009) — екранізація твору О.С. Пушкіна. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 673 |
Q: NSButtonCell of Check type within NSTableView does not allow to have value changed I have window with 3 table views (10.7.2, Xcode 4.2).
They are all created in IB and NSButtonCells are connected with outlets.
I created controller class and I filled all three views with some sample data:
- (NSInteger)numberOfRowsInTableView:(NSTableView *)aTableView {
return 10;
}
- (id)tableView:(NSTableView *)aTableView objectValueForTableColumn:(NSTableColumn *)aTableColumn row:(NSInteger)rowIndex {
NSButtonCell *buttonCell;
if(aTableView == dimensionTable) {
[dimensionButtonCell setTitle:@"Dimension"];
buttonCell = dimensionButtonCell;
}
else if(aTableView == shopTable) {
[shopButtonCell setTitle:@"Shop"];
buttonCell = shopButtonCell;
}
else if(aTableView == countryTable) {
[countryButtonCell setTitle:@"Country"];
buttonCell = countryButtonCell;
}
return buttonCell;
}
I have 2 questions:
*
*I cannot change checkbox state through GUI. I can change it programatically, though. It blinks a bit, when you hold down mouse button, but doesn't allow change...
*I tried to fill data as with views, without outlets to cells. It didn't work. Are NSButtonCell cels within cell views somehow different as view based Table Views or "normal" cel based Table Views?
A: After long struggle I manage to find the solution for the problem. One part of the problem was simple bug at the data model side, but it wasn't crucial, something much more difficult was to be done with NSTableView delegate and datasource.
THere were mainly 3 difficulties that prevented good understanding and managing this problem:
*
*apple's documentation lacks of any reasonable explanation about differences and typical usage of - (id)tableView:(NSTableView *)aTableView objectValueForTableColumn:(NSTableColumn *)aTableColumn row:(NSInteger)rowIndexin table view's data source and - (NSCell *)tableView:(NSTableView *)tableView dataCellForTableColumn:(NSTableColumn *)tableColumn row:(NSInteger)row of its delegate. While it may seem that you would need latter method, because NSButtonCells are custom NSCells it turns out it is not necessary, but I left it at the end anyway.
*internal conversions in NSTableView methods
*problem is not documented almost anywhere on the net
Here are steps you should do:
- (id)tableView:(NSTableView *)aTableView objectValueForTableColumn:(NSTableColumn *)aTableColumn row:(NSInteger)rowIndex {
buttonCell = [aTableColumn dataCell];
NSString *columnKey = [aTableColumn identifier];
return buttonCell;
}
You can see this method has to be implemented whether you use it or not.
- (NSCell *)tableView:(NSTableView *)tableView dataCellForTableColumn:(NSTableColumn *)tableColumn row:(NSInteger)row {
buttonCell = [tableColumn dataCell];
NSString *columnKey = [tableColumn identifier];
if(tableView == dimensionTable) {
// returnObject = @"Dimension";
// [dimensionButtonCell setTitle:@"Dimension"];
// buttonCell = dimensionButtonCell;
}
else if(tableView == shopTable) {
[buttonCell setState:[[mySelectedShops objectAtIndex:row] integerValue]];
[buttonCell setTitle:[myShops objectAtIndex:row]];
}
else if(tableView == countryTable) {
[buttonCell setState:[[mySelectedCountries objectAtIndex:row] integerValue]];
[buttonCell setTitle:[myCountries objectAtIndex:row]];
}
return buttonCell;
}
you can see I used second method, however objectValueForTableColumn could be used solely.
You can also see, I have NSMutableArray mySelectedShops and mySelectedCountries to hold NSInteger (1 or 0) wrapped in NSNumber for each row in Table View.
If you set the state or integerValue of NSCell makes no difference. Both will check and uncheck NSButtonCell with values 1 or 0 of NSInteger type.
- (void)tableView:(NSTableView *)aTableView setObjectValue:(id)anObject forTableColumn:(NSTableColumn *)aTableColumn row:(NSInteger)rowIndex {
NSString *columnKey = [aTableColumn identifier];
if(aTableView == dimensionTable) {
// [dimensionButtonCell setTitle:@"Dimension"];
// buttonCell = dimensionButtonCell;
}
else if(aTableView == shopTable) {
[mySelectedShops replaceObjectAtIndex:rowIndex withObject:[NSNumber numberWithInteger:[(NSCell*)anObject integerValue]]];
}
else if(aTableView == countryTable) {
[mySelectedCountries replaceObjectAtIndex:rowIndex withObject:[NSNumber numberWithInteger:[(NSCell*)anObject integerValue]]];
}
}
Although I passed NSInteger value to NSCell object, anObject here is of __NSCFBoolean type, which means something doesn't work as expected. To be able to replace object value to arrays I have casted it to NSCell only to get integerValues. It actually works without cast as well, so it is another mystery to me, but I like it more that way.
It is clear Apple is moving to view based cells like in UITableView. Still, I hope this will help to somebody.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,737 |
EMMANUEL BOVE (1898–1945) was born Emmanuel Bobovnikoff to a Jewish émigré from Kiev and a Parisian chambermaid from Luxembourg. His childhood was spent in Paris, marked at times by extreme poverty in the company of his mother and younger brother, and wealth in the company of his father and stepmother. With his stepmother's patronage, Bove acquired an education in Paris, Geneva, and, during the First World War, England. Back in Paris, he began writing while supporting himself with a series of odd jobs. He had been publishing popular novels under the pseudonym Jean Vallois for several years when Colette helped him publish the novel _Mes amis_ ( _My Friends_ ) under his own name. He continued publishing successful novels until the Second World War, at which time he was forced into exile in Algeria. He died of heart failure soon after his return to Paris.
ALYSON WATERS's translations from the French include works by Louis Aragon, René Belletto, Eric Chevillard, and Albert Cossery. She is the 2012 winner of the French-American Translation Award for her translation of Chevillard's _Prehistoric Times_. Waters has received a National Endowment for the Arts Translation Fellowship, a PEN Translation Fund grant, and residency grants from the Centre National du Livre, the Villa Gillet, and the Banff International Literary Translation Centre. She teaches literary translation at New York University and Columbia University and is the managing editor of _Yale French Studies_. She lives in Brooklyn.
DONALD BRECKENRIDGE is the fiction editor of _The Brooklyn Rail_ , co-editor of _InTranslation_ , managing editor of _Red Dust_ , and the author of more than a dozen plays, a novella, and the novels _6/2/95_ , _You Are Here_ , _This Young Girl Passing_ , and _And Then_. He lives in Brooklyn.
HENRI DUCHEMIN AND HIS SHADOWS
EMMANUEL BOVE
Translated from the French by
ALYSON WATERS
Introduction by
DONALD BRECKENRIDGE
NEW YORK REVIEW BOOKS
New York
THIS IS A NEW YORK REVIEW BOOK
PUBLISHED BY THE NEW YORK REVIEW OF BOOKS
435 Hudson Street, New York, NY 10014
www.nyrb.com
Translation copyright © 2015 by Alyson Waters
Introduction copyright © 2015 by Donald Breckenridge
All rights reserved.
Cover image: Hans (Jean) Arp, _Torso-Navel_ , 1921; © 2015 Artists Rights Society (ARS), New York / VG Bild-Kunst, Bonn; photo: Yale University Art Gallery
Cover design: Katy Homans
Library of Congress Cataloging-in-Publication Data
Bove, Emmanuel, 1898-1945.
[Henri Duchemin et ses ombres. English]
Henri Duchemin and his shadows / by Emmanuel Bove ; Introduction by Donald Breckenridge ; Translation by Alyson Waters
1 online resource. — (New York Review Books Classics)
Originally published: Paris : Emile-Paul, 1928.
Description based on print version record and CIP data provided by publisher; resource not viewed.
ISBN 978-1-59017-833-1 () — ISBN 978-1-59017-832-4 (alk. paper)
I. Waters, Alyson, 1955- translator. II. Title.
PQ2603.O87
843'.912—dc23
2014046070
ISBN 978-1-59017-833-1
v1.0
For a complete list of books in the NYRB Classics series, visit www.nyrb.com or write to:
Catalog Requests, NYRB, 435 Hudson Street, New York, NY 10014
# CONTENTS
Biographical Notes
Title Page
Copyright and More Information
Introduction
Night Crime
Another Friend
Night Visit
What I Saw
The Story of a Madman
The Child's Return
Is It a Lie?
# INTRODUCTION
Emmanuel Bove was a master of hyper-objectivity. His characters, drawn from all classes, are often paralyzed by a failure of will, poisoned by envy, cursed with bad luck or betrayal. With relentless clarity, Bove imparts a deeply felt and lasting impression of the lives of these solitary and emotionally shattered young men whose fortunes and futures hinge on a stroke of luck, an immoral act, an accident. The author's own youth was a harsh one, characterized by instability and discord; and yet, like the lives of his characters, it was occasionally graced by wealth and privilege. Born in Paris in 1898, Bove was the son of a Belgian-born housemaid, Henriette Michels, and an immigrant Ukrainian Jew, Emmanuel Bobovnikoff. Bove's father was a largely absent womanizer whose financial contributions to the family were infrequent at best. Bove and his brother, Léon, lived in abject poverty with their mother who moved frequently within the slums of Paris to find work, always shadowed by bill collectors. However, Bove's childhood took a decisive turn when his father's affair with Emily Overweg, a wealthy painter and the daughter of the British consul in Shanghai, led to an unlikely marriage. Sent to live with his father and stepmother, Bove experienced the twilight of belle epoque opulence, while Léon, who would become a doctor, remained with his mother in an unforgiving cycle of grinding poverty. And like the fleeting encounters with fortune that Bove employed in his fiction, this unexpected stretch of good luck would not last.
At the age of eight Bove decided to become a writer and at fourteen, with the financial backing of his stepmother, he was sent to a boarding school in England. The outbreak of the First World War soon disrupted his studies and forced him to return to France. His father succumbed to tuberculosis and his stepmother, whose fortunes had all but evaporated as a result of the war, could offer little assistance. While thousands of young men were dying in the trenches of Somme, Arras, and Verdun, Bove was living in a transient hotel—a familiar setting for nearly all of his novels—and working menial jobs (busboy, waiter, Renault factory worker, and tram operator) while attempting to write. Before he could be called up, the armistice was declared—one wonders if he missed the opportunity to distinguish himself on the battlefield and pursue an officer's career. Instead he spent a month in jail on account of a vagrancy charge aggravated by an anti-Semitic gendarme's inability to pronounce his last name. Thus Bobovnikoff became Bove, and one of the last century's finest authors was agonizingly birthed from a seemingly endless series of unfortunate circumstances.
Barely in his twenties, Bove married Suzanne Vallois, a young teacher, and they migrated to a suburb of Vienna, where, under the name of Jean Vallois, he attempted to make money writing pulp fiction. Living in postwar Austria might have appeared to be an affordable alternative to France, but rising inflation and economic stagnation quickly devoured what little savings they had managed to relocate with. Vienna did provide Bove with enough distance to approach his craft, yet it offered no financial support for an aspiring writer, his wife, and their daughter. When the money ran out, they were forced to return to Paris. And then, with the help of Colette, who was taken by the wisp of a manuscript this deeply reserved unknown young writer pressed upon her, Bove was able to publish _Mes amis_ ( _My Friends_ ) under his own name. A thin yet dynamic book that borrowed heavily from the transient hotel years of his late teens, _Mes amis_ remains Bove's best-known novel. In it we meet Victor Baton—profoundly lonely, old for his relatively young age, and penniless—and witness him casting about among the destitute in a grim, postwar Paris. Victor is forever idle, emotionally paralyzed, unemployable, and never short on real and imagined slights.
When I wake up, my mouth is open. My teeth are furry: it would be better to brush them in the evening, but I am never brave enough. Tears have dried at the corners of my eyes. My shoulders do not hurt any more. Some stiff hair covers my forehead. I spread my fingers and push it back. It is no good: like the pages of a new book it springs up and tumbles over my eyes again.*
The thin novel _Armand_ followed _Mes amis_ and was very well received. In _Armand_ Bove further expands on the destructive roles of alienation, abject poverty, and disenfranchisement while disseminating a doomed relationship.
The language Bove employed in these early works _,_ and throughout his entire career, is precise and elegant. The urgency in his deceptively simple, pitch-perfect tone lends itself to an immediate intimacy. The precision in his writing evokes landscapes and interiors, emotional and otherwise, that read like high-resolution photographs. And while his craft was forever attuned to the complete story—exploring the motives and complexities that lead his characters to do what they must—he wrote in a simple, everyday language. Bove's work is a fine distillation of lived experience expressed with seemingly effortless artistry.
Throughout his brief yet productive career, Bove captured the experience of a lost generation of war veterans. He recorded the odious aftereffects of the November 1918 armistice with its widespread unemployment, and the growing disaffected and largely reactionary working class that was teetering on the brink of revolution. Bove portrayed the young widows burdened with illegitimate sons who grow up into pathetic needy men wrecking havoc on anyone unwise enough to show them a modicum of compassion. He captured bourgeois families grasping at the titles dangling from rapidly diminishing fortunes, and a well-trodden and forever alienating Paris.
At the time of Bove's debut, Colette, André Gide, Philippe Soupault, Max Jacob, and Rainer Maria Rilke all adored his writing. Samuel Beckett was another early admirer who claimed, "Emmanuel Bove, more than anyone else...has an instinct for the essential detail." Considering how obscure his work became after his death, it is hard to imagine Bove as a literary star, yet he was a successful and productive author in the 1920s and '30s. While he avoided the spotlight that came with fame, his writing had a profound influence on many authors during his lifetime, and later on those writing after the Second World War. Bove informed Albert Camus's humanistic preoccupations; Claude Simon's elegant details and the obsessive autobiographical accounting that drive his novels; Nathalie Sarraute's delicate and intimate precision; and the bleak humor and disparaging, otherworldly hopelessness of Beckett's contributions to the theater of the absurd.
Bove's German translator, the Austrian novelist Peter Handke, made known Bove's reputation beyond France at a time when his novels were forgotten. The poet and translator John Ashbery has also been a passionate advocate. After more than fifty years of obscurity following his untimely death in the summer of 1945, Bove's fiction has resurfaced in earnest and nurtured a younger generation of French novelists. His influence among writers in English has also begun to blossom—thanks in no small part to stellar translations by Dominic Di Bernardi ( _A Singular Man_ , _Quicksand_ ), Nathalie Favre-Gilly ( _The Stepson_ , _A Winter's Journal_ ), Janet Louth ( _My Friends_ , _Armand_ , _A Man Who Knows_ ), and now Alyson Waters, who has added another outstanding contribution to Bove's work in English. _Henri Duchemin and His Shadows_ collects many of his short stories and the novella _Night Crime_ in one volume for the first time in English. This collection was originally published by Gallimard in 1939—just months before the outbreak of the Second World War.
_Night Crime_ gloriously illustrates Bove's ability to comfortably nest in the realms of the macabre with a delicate Kawabata-meets-Poe touch. In the richly nuanced and evocative dreamscape of the novella, we are introduced to the lonely and destitute Henri Duchemin on what can only be described as a singularly miserable Christmas Eve.
Henri Duchemin dreamed of supplicants, of owning houses, of freedom. But once his imagination had calmed down, it seemed the disorder of his room had grown, in contrast as it was with his reveries.
A mirror in a bamboo frame reflected his face. He forgot everything and, talking to himself, gazed at his reflection to see what he looked like when he spoke.
The flame was becoming so weak that now it lit only the table. It flickered on its wick. Suddenly it went out.
Henri Duchemin, groping for matches, knocked over objects he did not recognize.
Weary from searching, he sat in the armchair and closed his eyes so as not to see the darkness.
The warmth from his body was slowly drying his clothes. He felt better. Soon it seemed to him that the floor was slipping away beneath his feet and that his legs were swinging in the void, like those of a child on a chair.
We follow a dreaming Henri though Paris as he contemplates suicide and befriends a criminal who convinces him that murdering a banker will guarantee him a lifetime of wealth. After the murder, Henri flees into the night with the banker's bulging wallet tearing apart the breast pocket of his threadbare overcoat and attempts to assuage his guilt by offering handfuls of francs to the strangers he encounters. When dawn finally breaks, Henri awakens to the realization of his innocence, and the story ends with a great relief that lasts only for as long as the final paragraph allows.
It was around this time that Bove left Suzanne and their two daughters without an explanation or even a farewell. As soon as his divorce was finalized, Bove married Louise Ottensooser, a young Jewish woman from a well-to-do family. Louise reintroduced him to the high society that he first had experienced while living with his stepmother, and he found himself completely alienated by it. At the same time his novels were growing longer and more nuanced, his writing having reached its full maturation, what was to be his peak, and he was able to live comfortably enough to write without having to worry about money.
Bove won the highly coveted Prix Figuière in 1928 for what many consider his finest work, _La Coalition_ ( _The Coalition_ ), which remains criminally untranslated. For the next few years, until the Great Depression nearly wiped out the Ottensooser family fortune, he was at his most content. Throughout the 1930s, Bove produced a steady output of writing that enabled him to provide for his mother, his second wife, and his ex-wife and daughters.
A staunch antifascist, Bove was revolted by Petain's Vichy, and he courageously decided not to publish during the German occupation. Bove and Louise went underground and were eventually able to flee to Algiers in 1942, where he wrote his last masterpiece, _Le Piege_ ( _Quicksand_ ), and the novels _Depart dans la nuit_ ( _Night Departure_ ) and _Non-lieu_ ( _No Place_ ) while living among the exiled. All three books depict the paranoia and hopelessness of those living compromised lives during the German occupation. These three books were not written from the luxury of perspective and are all the more remarkable for their artistry and focus, penned in real time. When the Germans were pushed out of France in 1944, Bove and Louise were forced to linger in North Africa until they could scrape up enough funds to return to Paris. Bove was suffering from cachexia, a body-wasting disease, when his heart failed. He died in Paris on July 13, 1945, at age forty-seven.
—DONALD BRECKENRIDGE
* _My Friends_ , translated by Janet Louth (Manchester: Carcanet Press, 1986).
# HENRI DUCHEMIN AND HIS SHADOWS
## NIGHT CRIME
It was Christmas Eve.
Henri Duchemin sat on a worn-out bench in a restaurant, waiting for the rain to stop. The holes in his trouser pockets and the long hair tickling his ears were constant reminders of his poverty.
Tired of sitting still, he was preparing to leave when he recalled the dark hallway of his house, the damp courtyard, the narrow stairway, and his unheated attic room.
He preferred the restaurant's mild warmth to all that.
A few regulars were reading the evening papers. A draft caused the gas mantle's slender chain to sway. The barmaid, elbows resting on the sideboard, wanted to go home.
Suddenly the customers raised their heads. A beggar had just walked in.
"He's a hunchback," one of them said.
The wind from the street nearly extinguished the gas lamp. Shadows fell from the ceiling along the walls.
"Close the door!"
The beggar obeyed and, hat in hand, moved forward, glancing furtively from right to left.
"What do you want?"
"A bit of charity."
This beggar was like an actor who appears at last on an empty stage. The barmaid, torn between the pleasure of being entertained and that of chasing the poor wretch away, hung back for only a moment.
"Go on now, get out. There's no begging in here."
The customers took advantage of the incident to get to know one another. Although they did not all share the barmaid's opinion, they had a vague sense they would end up approving of her action.
A sort of kinship developed from this, and they held forth for a long time on the subjects of begging, prostitution—on social problems, as they said drily.
The clock chimed four times, although the hands pointed to nine.
Henri Duchemin sensed that these strangers were harboring unkind thoughts. He checked to make sure the cotton plugging his ears had not fallen out and, shaking out his overcoat, walked to the door that, as he opened it, briefly caused the light from the restaurant to bathe the opposite side of the dark street.
The rain dripped from the painted cast-iron street lamps. The shimmering sidewalks seemed to be moving. The lamps of cars and taxis were dim.
He went into a café. The awning, battered by the wind, threw down sheets of water.
Condensation was everywhere, dulling the glasses, the counter, the electric light bulbs. Some customers had drawn on the mirrors.
Henri Duchemin ordered a coffee, a very hot coffee, which he swallowed in one gulp before the sugar had a chance to dissolve.
A woman in a damp fur coat was drinking milk that must have been sweetened by the red of her lips. Her heavily made-up eyes remained continually open, like a doll's.
"What a sad Christmas Eve!" she said.
Henri Duchemin knew that certain women spoke to men to ask for money, but he preferred not to think about it, remaining hopeful of some new experience.
"Yes, what a sad Christmas Eve indeed!"
He watched the door, afraid that his neighbor, Monsieur Leleu, would come in. If he did, he would sit down right there beside him and without a doubt take his place.
"You must be bored, Monsieur."
"Oh, I am, but don't be offended. If you knew how I'm suffering. I'd like so much to open my heart. I'm a stranger in your eyes. Be patient. I shall tell you the story of my life. It's a very sad story."
He was so happy to be speaking that he seemed younger. He was sure he would be liked and this gave him confidence. He was about to go on when the woman burst out laughing:
"Don't be ridiculous. If you're so unhappy, just kill yourself."
Henri Duchemin blushed. For a minute he tried to find a way to respond. When he could not, he got up and went out, his heart heavy with bitterness.
The rain whipped his face, reviving him. Two rows of gas lamps converged at the end of an avenue. The heads of the passersby touched the fabric of their umbrellas.
Kill myself! She's out of her mind, he thought. The world is so cruel.
His damp trousers clung to his thighs. His feet slid in his shoes that leaked even when the sidewalks were hosed down in summer. He saw nothing, not even the streams of rainwater swallowed up by the sewers with the gentle sounds of a small waterfall.
At last he recognized a small recessed lot cluttered with tarred pipes where he often would come to watch the men at work while he warmed himself over a brazier.
He had arrived home.
The wind was so strong as he opened the door that it felt as if someone wanted to prevent him from going in.
Henri Duchemin climbed the stairway slowly and then, once inside his room, gently closed the door so as not to wake Monsieur Leleu.
When he lit the lamp, it revealed a disorder that surprised him—he had forgotten the housework had not been done.
The items of furniture, with their shadow twins, seemed to touch one another. Icy air crept beneath the window, stirring the curtains. The damp blistered the ceiling plaster. The wallpaper flapped like old posters. The unmade bed was cold. When the wind rattled the door, the lock squeaked.
"Kill myself, come now, she's lost her mind!"
To drive the woman from his memory, Henri Duchemin paced the room, counting his steps, elated to find the same number going and coming. He then noticed that his intake of breath was sharper when he faced away from the lamp.
The shutters, unhinged by the wind, slammed so violently against the wall that he was afraid the neighbors would complain.
He opened the window wide: the flame of the lamp flickered, the curtains rose behind him like ghosts, a tram ticket flew around the room.
Across the street he saw a lit window and, through the blinds, a woman's shadow gesticulating.
Leaning out, his hair tangled in the wind, his hands blackened by the window sill, Henri Duchemin spied on this woman. He stood still and his eyes were so wide that his pupils seemed smaller in the middle of so much white.
But the light went out. Hoping she would turn on a light at another window, he waited. The night was black. The wind, burrowing in his sleeves, chilled his body. The rain shimmered around a street lamp.
He closed the window and, motionless in front of the only armchair, he saw women everywhere, in the depths of the walls, standing on his bed, languidly waving their arms. No, he would not kill himself. At forty a man is still young and can, if he perseveres, become rich.
Henri Duchemin dreamed of supplicants, of owning houses, of freedom. But once his imagination had calmed down, it seemed the disorder of his room had grown, in contrast as it was with his reveries.
A mirror in a bamboo frame reflected his face. He forgot everything and, talking to himself, gazed at his reflection to see what he looked like when he spoke.
The flame was becoming so weak that now it lit only the table. It flickered on its wick. Suddenly it went out.
Henri Duchemin, groping for matches, knocked over objects he did not recognize.
Weary from searching, he sat in the armchair and closed his eyes so as not to see the darkness.
The warmth from his body was slowly drying his clothes. He felt better. Soon it seemed to him that the floor was slipping away beneath his feet and that his legs were swinging in the void, like those of a child on a chair.
He had been sleeping for a long time when he felt the heat of a flame on his cheek, a little like someone's breath.
He opened his eyes.
Monsieur Leleu was beside him holding a lamp.
Monsieur Leleu was a calm fifty-year-old man who lived in poverty. He was interested in the lives of criminals and always sided with the police. He read the local crime news but never detective novels because he felt uncomfortable reading tales of things that did not exist.
"Are you asleep, Duchemin?"
"No."
Monsieur Leleu set his lamp on the fireplace mantel. It continued to light the floor.
"I need to speak with you, Henri."
Monsieur Leleu stroked his beard, honing it to a point.
"Do you remember the woman in the café?"
"Yes."
"You have to do what she told you."
"Kill myself?"
"Yes."
"You think I must?"
"Yes, because you are unhappy."
The rain, driven by the wind, relentlessly bombarded the windowpanes.
"But I wouldn't dare."
"Why not, Henri? I've brought you a rope. The slipknot has been made. You see, everything is ready. I'll come back once you're dead; that way, no one will suspect me."
Monsieur Leleu rose.
"You'll come back once I'm dead!"
"Yes. I'll wake the other tenants. _Adieu_. I'll leave you the lamp; I'll retrieve it later."
Monsieur Leleu went out without a sound.
Left alone, Henri Duchemin rubbed his eyes, looked at the lamp and, realizing he wasn't dreaming, wanted to write down his last thoughts. But he did not know what to say.
Suddenly, either because he was afraid of dying or because he feared Monsieur Leleu would return, he decided to flee.
He blew out the lamp, checking that the flame was really extinguished, and left.
* * *
Although Monsieur Leleu's door was closed, Henri Duchemin walked on tiptoe.
Outside, the cold air gnawed at a nerve in one of his teeth. The slope of the street made him want to run. The bubbles floating on the puddles did not burst because they did not move.
Henri Duchemin walked through the _faubourg_. There were words written in chalk on the walls. A fence concealed an empty lot. Curtainless windows glinted like mica in the light from a lantern.
A cabaret, painted in red, flooded a cul-de-sac with light. Shadows shifted on the panes still splashed with rain.
Any passerby would have hesitated to enter this dive.
Henri Duchemin, who on this night feared nothing, went in and sat down in the back like a regular.
A few other customers were standing around, chatting with the female owner. She was washing glasses, her apron damp around her waist, her feet secure and dry on a duckboard.
"What may I serve Monsieur?"
"A glass of rum."
Henri Duchemin downed it like medicine.
Then he drank beer, wine, liqueurs, and, since this was not his habit, he was drunk in an hour. Alcohol made him overemotional, and he grew worried at the idea that he could not pay for his drinks.
Soon his thoughts became muddled. He blinked his eyes as if blinded by the sun. He no longer perceived the glistening of the counter or even the clinking of the bottles.
Just then, despite his state, he noticed a man dozing before him, his head on the table, his arms between his legs.
Henri Duchemin could not believe his eyes. Thinking he was dreaming, he reached out and with a fingertip touched the sleeping man's hair.
The latter woke with a start. His eyelashes were sticky. He must still have been half-asleep because he searched for his handkerchief in all his pockets. Although he was unshaven and his hat had no hatband, he was wearing a detachable collar. He had enormous veins on his hands at the spot where one would kiss them.
"A drink!"
No doubt, like many people, he favored a drink when he woke up.
As soon as the proprietress had brought him a bottle of wine, he swigged two glasses in a row.
He smiled, trying to strike up a conversation.
"What awful weather!"
Henri Duchemin did not respond. He liked to chat, but distrusted strangers.
The customers, realizing their conversation was not changing the world, left the establishment.
The proprietress arranged her hair with her damp fingers. The two men observed each other.
"Listen," said the stranger.
No word in reply encouraged him to continue.
"Listen, I said."
"Yes?"
"Tell me your name."
Henri Duchemin did not know how to answer.
He thought he would be weaker, exposed, if he placed himself at the mercy of this stranger by telling him his name but, taken by surprise, he did not have the presence of mind to invent a false one.
Very quietly, as if he did not want to be heard, he said:
"Henri Duchemin."
"Do you want to be my friend? Like you, I wouldn't mind having a lot of money."
Indeed, Henri Duchemin did want to have a lot of money. Because he thought that this yearning could come only from a bold man, he was flattered that his tablemate had noticed. And so, even though this alliance seemed risky to him, he accepted.
"But what is your name?"
"I have no name."
"You have no name?"
"I have one, but you don't need to know it."
"And what do you do?"
"Nothing. But from now on, we must act. Do you want to get rich, old pal?"
"Yes, if possible."
When the proprietress came to serve them again, the man without a name took her by the waist.
"Do as I do, then, Duchemin."
He would have been happy to do so if his strength had not been sapped by his timidity.
"You mustn't blush, young man," said the proprietress as she pulled away from the man without a name.
"Duchemin, I have important things to talk to you about. Pay attention."
"I'm listening, pal," responded Henri Duchemin, determined to echo the familiarity of his interlocutor.
"Would you like to be rich?"
"Yes."
"Don't just say 'yes.' Say 'I'd love to.'"
"I'd love to."
A customer, dozing off near the stove, gave a start. The moisture evaporating from his overcoat and shoes enveloped him in a transparent cloud. The proprietress, reading a novel, had trouble turning the pages.
"Are you listening to me, Duchemin?"
"I'm listening."
"Between the life you're leading and riches, which do you choose?"
"Riches."
From a leaky faucet drops of water fell into a tub.
"You choose riches."
"Yes."
"Congratulations! You are saved!"
The man without a name drew close and took Henri Duchemin's hand.
"Are you brave?"
"Yes."
Everything was motionless in the brightly lit room.
"Good. In a little while, we'll go into a house. A banker is to spend the night there."
"A banker?"
"Yes. When he falls asleep, you..."
The man without a name removed his hat so that the sweat on his forehead would not dampen the leather.
"When he falls asleep, you..."
"I..."
"You'll kill him."
"I'll kill him?"
"Yes..."
Henri Duchemin felt dizzy, as if he had not eaten. His vision became blurry. The ceiling lamp and the bottles fell behind the counter then moved through the room.
"You'll enter his bedroom, the moon will light your way. You'll just need to strike, and you'll be rich."
"Help! Help!" cried Duchemin.
The proprietress did not even raise her eyes.
As for the other customer, he swayed on his chair, waking and falling back to sleep by turns.
"You'll buy clothes, Duchemin, new clothes."
Henri Duchemin took a deep breath. The warm air dried his teeth.
"Shall we have a toast?"
"Yes."
"Two cognacs, please!"
The woman poured their drinks with small, careful gestures so that the glasses would not overflow.
A minute later the two men headed for the exit. The trapdoor to the cellar trembled beneath their footsteps. The man without a name drew his mustache to his lips to suck up the last drops of cognac.
"Good evening."
"Good evening, gentlemen."
We did not pay for our drinks, and she didn't ask us for anything, thought Henri Duchemin.
He wanted to share this thought with his companion, but he was afraid of appearing ridiculous.
* * *
It was raining again. Without exchanging a word, the two men, slipping wherever the sidewalk sloped, set out for the house about which the man without a name had spoken.
Henri Duchemin was ambivalent. It seemed to him, in this street that belonged to everyone, that the murder would be more difficult to commit. In the end he realized he should not have accepted and, because it was too late now to get out of the deal, he was determined to flee. But either because he was waiting for the right time, or because he was afraid of the man without a name, he kept postponing the moment.
Finally, at the sight of an empty lot, he ran away as fast as his legs would carry him. In order not to trip over a clod of earth or a stone, he raised his knees high, like a horse on parade. His tie floated behind him. Hollows and mounds followed one after the other beneath his feet, reminding him of the time when as a child he would jump from the top of a hillock the better to climb the next one.
A stitch in his side forced him to stop running. Henri Duchemin was sluggish by nature, prone to stitches.
Intoxicated by his freedom, his neck stiff, he wandered down a muddy path. Hedges with dead branches scratched his hands. The wind cut his breath short.
A tin can he knocked over with his foot splashed his ankles as it toppled. Despite this, he felt like whistling, but the air came out of his lips as if out of a tube. He did not know how to whistle. So he sang the only song he knew by heart.
"Duchemin!" cried a distant voice, one of those lone voices that can be heard in the woods on Sundays.
He listened without breathing. He was afraid. He wanted to run. But his legs were shaking like they did during the war when he was a stretcher-bearer and had to carry a fellow soldier.
"Don't be afraid. It's me."
It was the man without a name. So as not to frighten Henri Duchemin, he did not scold him. On the contrary, he told him he would have done the same thing in his place.
The two men left the path and, on the sidewalk, treaded as if they had clubfeet, trying to unstick the mud from their shoes.
Henri Duchemin, who had been too warm, was now trembling, which made him fear he was coming down with bronchitis. He no longer thought about running away; all he wanted now was a bed to sleep in.
The two men wandered the streets for a full hour. Sometimes they stepped in a puddle and were splashed to the knees.
These events had no importance in relation to what was about to happen.
At last the man without a name stopped in front of a new house.
"It's here."
He rang. A window lit the street. Grumbling and the clattering of old slippers could be heard even outside.
"Who is it?"
"Me!"
The lock clicked and the door opened. A light bulb fixed on the ceiling made the upper part of the foyer brighter. The man who had just opened the door was in shirtsleeves. You could tell from his hair and the blotches on one cheek that he had been sleeping.
"Come in, follow me," he said.
He showed his guests into the dining room where, winter or summer, a basket of artificial flowers always sat on the sideboard. A white porcelain lampshade covered an electric lamp hanging motionless at the end of a wire.
Henri Duchemin took off the overcoat that was numbing his shoulders and, more comfortable, his arms longer, he inspected his jacket for stains. They had disappeared.
The man without a name lay down on a sofa with his feet hanging off so as not to dirty the red velvet. He shut his eyes and fell asleep.
Henri Duchemin sat in a wicker armchair that creaked loudly even when he did not move, and blew on his hands. Eyes closed, he imagined his whole body bathed in warm breath. He felt his feet were cold and wet, but this did not bother him. His feet were so far from his body. Every now and then a car drove down the street, almost grazing the shutters.
Suddenly there was a knock at the door.
The man without a name got up like a passenger on a train who had been occupying two seats. Henri Duchemin, trying to find his bearings, did not understand what was happening.
"Duchemin, he's here."
"Who?"
"The banker."
Yes, it was the banker. He was wearing an overcoat with a silk lining and holding a top hat in his hand. He came in, bowed in greeting, sat down in a chair, unfolded a newspaper, and studied the stock prices.
The silence was marred only by the rustle of the large sheet of paper.
Then the banker stood, motioned good-bye, and left the room.
The two men who remained alone wore the scheming expression of servants who had just won the sympathy of their masters.
"Follow me, Duchemin."
On tiptoe, one hand against the wall, they walked down the dimly lit hall.
"In here."
They entered a room with walls covered in flowered fabric.
"Sit down, Duchemin."
"Fine."
"Take off your shoes."
Henri Duchemin obeyed. It seemed to him that it was not his own shoes he was removing.
"Listen to me, Duchemin."
"I'm listening."
"The bed is on the right, the window is open, the moon will light your way."
"But there is no moon."
"I'm telling you, the moon will light your way. You'll strike as if you wanted to split a tree trunk, and then you'll be rich."
Tiny sounds came through the wall.
"Take this hammer. The banker is in bed."
"What if he's not sleeping?"
"Go. It's for your own happiness."
Henri Duchemin rose. His damp socks left the imprint of his feet on the wood floor.
He stopped a few feet from the door.
"I'm frightened."
"Go. Afterwards, you'll be rich."
"I'll be rich?"
"Yes."
Still, he hesitated.
"Go on, I tell you. You'll be rich."
Henri Duchemin entered the banker's bedroom. He had held the doorknob tightly in his hands for so long that his fingers smelled of copper.
Exactly as the man without a name had said, moonlight illuminated the room. It was the light of insomnia, a light for sick eyes.
The banker's body was hidden by blankets and his head, resting on the pillow, seemed to lack a torso. There was also something ridiculous about this older man's head perched on its exposed neck.
Henri Duchemin knew that if he did not want his courage to flag, he must not think at all. And, understanding that what he was doing was not right, he headed straight for the bed so that he would not be able to stop himself.
His knees knocked against the bed.
He raised the hammer as high as he could. He closed his eyes. When he opened them, he saw blood on the sheets and the hammer in the eiderdown.
A wallet lay on the night table. He took it without thinking he would not have needed to kill the man to do so.
Then he went back to the room where the man without a name had led him.
It was empty. The lamp's forlorn light lit only motionless objects.
Henri Duchemin called out, opened the wardrobes, touched the furniture without taking his eyes off the switch for fear someone would shut off the light.
There was no one. It was impossible. He was going crazy. He fell to the ground. For a long time he remained crouching, his forehead pressed against the wood floor, for he thought no one could find fault with him in that position.
When he stood up, he felt better. He put on his shoes, looked around to make sure he wasn't forgetting anything, walked through the dining room, slipped on his overcoat, and went out.
* * *
The rain had stopped. A few clouds floated among the stars. Henri Duchemin wanted to run but in order not to attract attention, he walked rapidly instead. He held his hand in his inside pocket, which the fat wallet had unstitched.
He drew himself up. To look at him, who would have guessed he was carrying a fortune over his heart? Who would have believed that this poorly dressed man was now a person of independent means?
The gas lamps drew two dotted lines at the level of a second story. They appeared brighter in the crisp air.
Lulled by the rhythm of his footsteps, Henri Duchemin imagined women sitting on bank notes, and all the while he took detour after detour so that the police would lose all trace of him.
As he passed in front of a café, he heard the exquisite music of a player piano, half tin, half crystal. Women were laughing, probably over nothing. He attempted to look above the curtain at what was going on inside, but he was too short.
So he went in, sat down quickly, and waited until the attention he had attracted died down.
Three women were sitting on a velvet bench.
Henri Duchemin gazed at them lustfully, wondering which one of them attracted him the most. And although he was determined to be a different man now, he still did not dare invite them to his table.
Nevertheless, without his having to make a single gesture, one of them came to sit next to him. Her necklace of small pearls was too tight. She had the white, unattractive skin of women who never blush.
Henri Duchemin rested a hand on the young woman's lap and felt her garter button beneath his fingers.
He wanted to sing, laugh, shout, but did not dare.
Little by little, however, he began to feel at ease. No one was making fun of him. The customers went so far as to get on well with him for, one by one, glasses in hand, they came to his table.
"Music! Music!" he cried.
Although he realized he had raised his voice, it did not bother him.
The barmaid slipped two coins into the piano slot.
"How about a game of poker?" asked a young man who was entertaining himself shuffling a deck of cards.
"Good idea!" cried Henri Duchemin.
The young man spread out a small red carpet. The center of a slate was wiped clean. The deck was cut and the game began.
It did not last long.
Even though Henri Duchemin did not know how to play, he kept on winning. The other players, at the ends of their resources, had to give up. They were not happy and conversed softly.
Their bad mood annoyed Henri Duchemin. He could not explain to himself how he had won; he never had any luck. And so, for fear of alienating his friends, he suddenly gave them back all the money he had won.
Stunned, they stopped speaking. Then, having recovered, they thanked him with exaggerated warmth. In their entire existence, they had never known such a generous man. He was a true friend, no doubt about it! And might the whole world follow his example!
Henri Duchemin rejoiced at the thought of having so many friends.
"Let us be brothers," he said with eyes raised.
He was not crying, but tears were streaming down his cheeks. He glanced at the woman next to him.
"I am so happy! Life is so wonderful! What is your name, my child?"
Not receiving a response, he continued:
"Allow me to kiss you. Oh! If you were to accept, we'd get married. I have money. I'd buy you everything you wanted. I'd rescue you from this dive. You are too pure to live here. We would love one another."
He stopped talking when he realized he could no longer be heard over the laughter.
"Please! Be quiet, let him speak," said a customer, winking in case anyone took his words seriously.
"My friends, if you'd like, we'll never leave each other. Love will unite us until death. I have money. Why should I have any and not you? Let us share, share."
This time, all hell broke loose. Everyone cheered him, except the woman next to him who pinched him under the table.
"Why should we despise one another? Let us love each other, let us show the way, we who are brothers."
He rose amid the cheering. For a second he thought about throwing his wallet to his admirers, but something held him back. He simply threw down a handful of bills.
"Take it, my friends. My true friends. It's for you. Are we not brothers? And you, my darling, be happy like the others. I love you, life is beautiful."
"Let's go," she said.
"Where?"
"To my place."
The room filled with boos.
"No!"
"Leave him with us."
"He amuses us!"
"She wants the money."
Everyone was speaking at the same time, and Henri Duchemin began to sense that they did not like him. The ugliness of life appeared to him. Until then, as long as they had been listening to him, he had lived in a dream.
Now, everything was finished.
Head in hands, he walked to the door. They begged him to stay, but it was useless.
Standing on the sidewalk he tried to hear through the door what was being said about him, but only a murmur reached his ears.
He wiped his lips so the cold would not chap them. He now knew that all men were ingrates. And let them stay that way! Henri Duchemin had no need to worry about them. He could do without the entire world because he was rich.
He had been walking for an hour when the idea came to him to return to the new house where he'd killed the banker. No matter how he tried to convince himself that there was no point in doing so, the temptation was too great.
Hoping to lose his way, he wandered aimlessly, his hands scraped by the walls; yet in spite of himself, each step brought him closer to the house.
Suddenly he saw shapes moving behind the lit windows of a building. He drew closer. He recognized the new house. Two police officers, whose shadows stretched to the middle of the road, were chatting in the entranceway.
The crime had been discovered.
Henri Duchemin thought about turning himself in. But, changing his mind, he fled. His unbuttoned overcoat floated behind him. A gust of wind carried off his hat. He was getting ready to chase after it when he had the feeling time was running out.
Bareheaded, he took off at a run. Arc lamps lit a boulevard from above. The stores' metal shutters were drawn down to the sidewalk. Against the darkened café storefronts, cane chairs were stacked one atop the other.
Upset by the loss of his hat, Henri Duchemin did not dare look at the few pedestrians he encountered.
For the second time, he thought of turning himself in, but the law terrified him. He was familiar with it because he had already ventured into criminal court with Monsieur Leleu. With a flushed face, he had pushed open the heavy padded doors. They had seen lawyers whose feet seemed huge beneath their robes. He had found not peaceful city policemen, but municipal guards dressed in the same sky blue as soldiers.
No, he would not turn himself in. It was better to remain free, for these heartless people would never understand the reasons behind his crime. Indeed, no one would understand them. He would have been happier among madmen in whose company he would have skipped, laughed, and sung.
Henri Duchemin heard the rumbling of a carriage. In the silence of the night, the noise terrified him. He imagined a prison van was following him and that the little slanted shutters were hiding policemen.
But the noise faded and he relaxed.
Not daring to return home or to take a room in a hotel because they might have his description, he went into a train station.
There was no one in the main hall, which was cheerless like all places abandoned by the crowd. In the distance locomotives sat idle. A lantern swayed to the rhythm of footsteps.
Henri Duchemin entered a waiting room and walked over to a sheet-metal stove that blew little puffs of warm air through the openwork toward his face. From time to time his gaze would meet the staring eyes of a traveler who was awake.
Sleepiness was making Henri Duchemin's eyelids heavy and, like a horse, he dozed standing. His head fell forward.
Suddenly shouts rang out.
His teeth chattered. He shivered. He looked at the room. Newspaper photographs shading a lamp formed dark squares. People were getting up.
"Passengers for Dijon, Mâcon, Lyon, and prison, all aboard," shouted an employee.
He had been found out.
Terrified, he stepped over packages and, running, opened a door that slammed when he was already far away.
Soon he stopped. The street was deserted.
I'm so foolish! he thought.
He wanted to retrace his steps, but though he was sure he had been tricked by his imagination, he did not dare.
* * *
Henri Duchemin was overcome by such an enormous desire to sleep that he closed his eyes as he walked—but not for long, because he was afraid of veering off course.
A lantern, like a common star, was twinkling in the distance. He had no reason to be anxious; people had every right to light lanterns. Still, he did not take his eyes off it because it seemed to him that on this night everything that was lit was lit because of him.
As he approached this lantern, he could read, etched in its blue paint: "Police Station." So, without turning around or paying attention to the streets he ran down, he fled.
When he was out of breath he stopped and began to think. Wasn't it ridiculous to be afraid like this when he possessed a fortune? In the morning, everything would be better.
He was wandering aimlessly in the streets when exhaustion forced him to sit on a bench. The air was bitterly cold. He shoved his hands in his pockets and did not move. He knew that cold could kill. And so he tried hard to stay awake. To help, he thought of every joy his fortune could bring him.
His legs grew heavy. He stood up.
The streets were becoming narrower and narrower. Not a single light shone in the windows. From time to time he would cross a street, then cross back to the sidewalk he had just left. Sometimes he would stop, turn around as if someone had called to him, then take off again.
As he walked along the barred windows of a night shelter, he read: "Post No Bills." And, to show that they weren't fooling, "Law of 29 July 1881."
The shelter seemed abandoned. He went in, making sure to leave the door open so he could flee if necessary. The silence was bottomless. A disagreeable odor floated in the air. The black pipe from a stove led straight up to the ceiling. The bunks, in rows along the whitewashed walls, were all occupied. The beggars must have been tormented by bad dreams for their clothes hung down to the floor or lay scattered among the beds. In a glass booth, the watchman, partly lit by a lamp with a shade, was reading a book whose pages curled at the corners.
Henri Duchemin lay down on the floor. He felt safe. For a few minutes the rays from the lamp shone between his eyelashes. Then everything grew dim. Despite the hard stone bruising his hips and elbows, despite the cold tugging at his face, he had fallen asleep.
Who was it who was stubbornly striking him on the shoulder? One of his enemies, no doubt. Or a policeman. Henri Duchemin did not move a muscle. He knew that there was nothing easier to do than to pretend to be asleep. But what he did not know was that one never tires of trying to wake somebody.
And, indeed, the irksome person did not tire.
So Henri Duchemin imagined that a prison guard, who naturally held a lamp in his hand, was offering him a final cigarette. In order to know what was going to happen, he took it while he was asleep and, for the first time in his life, swallowed smoke. Then he got up and followed the guard. A guillotine appeared on a square. He saw its steel blade.
He was about to die when he was bullied awake.
"What are you doing here?"
"Sleeping."
"You have to leave. No one's allowed in after 9:30 at night."
Henri Duchemin obeyed. As he left, he saw the watchman's empty booth, the book resting on the table, and the lamp lighting the chair.
Henri Duchemin tried to forget everything that had just happened by walking hurriedly, which also warmed him. As he was crossing a street, the fact of not having to watch out for cars seemed odd to him. His shoes struck dry asphalt. Sometimes, he searched the sky in the hope of seeing the dawn, but the stars, still in the same place, remained clear and bright.
He saw a small park where mothers strolled with their children during the day. The hope of finding a bench and the fact that the fence was not high prompted him to enter. The guard was asleep, so he climbed over the roll bars and paced the frost-covered lawn with a pleasure that was all the greater since he knew only the gardeners had the right to step there. Then he looked through the panes of the guard's kiosk. He imagined a multitude of objects filling the booth, but all he saw were a few chestnuts on a table of black wood.
Disappointed, he sat down on a bench. Across the way, between the bare trees, he saw a building, pale in the moonlight, whose shutterless windows and balcony railings reminded him of a city hall in a toy construction set. Not a breath of wind. The motionless cold of an icebox.
Eyes wide, his eyelids not once closing over them, even for a moment, Henri Duchemin was thinking. He was thinking that now he would be respected. And this respect would have been even greater had he not given half his fortune to those people who, rather than being grateful, had made fun of him. But since Henri Duchemin did not like regrets, he filed that memory away.
The loss of his hat annoyed him as well, especially since he would have had the time to pick it up. But what's done is done, one's thoughts must not linger on the past. What good would it do him to go back in time? Tomorrow, he would buy a brand new hat and a vest. He liked vests. Aren't they a bit like the face of one's body and don't they wear a satisfied expression when the jacket is unbuttoned?
And at dawn, he'd go abroad. He pictured himself on a train. He even felt slight bumps as he passed imaginary switching points. He saw the countryside and a very red sun rising over the frozen plowed fields. A peasant opened a barn door. He was just starting his day's work whereas he, Henri Duchemin, was escaping into the unknown.
Henri Duchemin got up and began to walk quickly to give himself the feeling he was traveling.
He soon found himself on a crowded street where, despite the late hour, people were enjoying themselves. The crowd, the illuminated shops, the rosy poultry gave the impression of a celebration. The copperware glistened in the light, so much so that it looked like liquid. The scent of tangerines was in the air. Everyone was laughing, having a good time. The pavement was dry. Along the sidewalk frozen puddles riddled with trapped bubbles gleamed in the gold of the lights.
"I want to be happy," Henri Duchemin whispered as he stared at the women passing by.
One of them took him by the arm.
"I love you," she said.
She was tottering slightly, but you could hardly tell because the unsteadiness of women's legs is hidden by their dresses.
"Let's go eat."
"All right."
They went into a local restaurant. The heat coming from the food, the lights, and people's breath warmed the room. It was disagreeable, like any heat that doesn't come from a fire.
Henri Duchemin removed his overcoat, smoothed his hair, and furtively threw the cotton from his ears under a chair.
As he was wiping off his cutlery, he gazed around him. People envied him. Surely they thought that the woman with him was his mistress.
"Do you love me?"
"Yes."
"Do you swear it?"
"Yes."
Customers came and went. The electric light bulbs were reflected high up in the mirrors. Outside, groups walked by, singing. The squeaking of a balloon could be heard now and again in the room.
The young woman opened and closed her mouth, as if she were tasting something.
Henri Duchemin was thinking about the future. Yes, his heart would no longer race whenever someone knocked on his door. He would take care of his health. It's wonderful to do so when you feel well. He would go to the dentist; he'd had a toothache for a long time. Gone was that awful sense that each day the pain, which could be cured if only one had the money, would continue to grow more acute.
"Listen. Let's go away, away."
"Where?"
"Abroad."
The meal finished, Henri Duchemin felt better. He lit a cigar. The young woman's eyes were closed. He looked at her more easily. Only the air passing between her lips proved this face was alive.
"Let's go."
She gave a start, then let her dull gaze flit from table to table.
"Your hat, Monsieur?" asked the waiter.
"No, no, I don't have one."
This incident upset Henri Duchemin. To hide his distress, he opened his overcoat, which he had just closed.
"Let's go, let's go, let's go."
A group of people passing by forced him off the sidewalk. He turned back and, in a voice he thought sounded like that of every man, he swore at them. He was sure of himself. No one could have managed to intimidate him, not even a policeman.
Despite the crowd, they arrived quickly at the young woman's hotel. Shoulder against the wall, she went in first, opened the glass door of an office half way, and took her key.
A maid was making up her room. When the couple arrived, she withdrew.
Expressing his surprise that people were compelled to work at night, Henri Duchemin went in. The curtain around the dressing table was drawn back. He saw a blue pitcher and basin. There were photographs on the mirror. Pollen from a branch of mimosa mixed with ashes from the fireplace.
"Are you tired?" he asked her.
"I don't feel comfortable."
"Do you need a bit of air?"
"Yes, open the window."
Henri Duchemin opened the window. A house so close that you could reach out and touch it got lost in the dark night.
"How are you feeling now?"
"I'm cold."
"Do you love me?"
"I don't know."
"A little while ago you did."
"Too bad."
She took off her skirt, stepped over it, and began to wash. Half undressed as she was, her torso seemed too long.
"You're beautiful."
He went over to her and tried to take her by the waist.
"Leave me alone."
She splashed him. Taken by surprise, he let her go. His lips were dry. A drop of water rolled down his nose.
"You don't love me?"
"Leave me alone or I'll scream."
"No, don't scream, don't scream. I'll go."
"Go then."
He opened the door. The sound of his footsteps filled the corridor as if he were a giant. He raced down the stairs, imagining he was falling with each step, for he did not have the time or the strength to move his legs.
* * *
When he got to the street, he walked away with long strides. The lights from the stores bothered him. He passed in front of a cinema and saw a poster. It was of the heroine of a film. She was crying. The candor on this face awakened a need for love in Henri Duchemin that made him cry along with her.
The farther he got from this neighborhood, the more numerous the streetlamps seemed, the wider the sidewalks, the bigger the windows.
Henri Duchemin was walking along the slatted wall of a cemetery when he noticed someone in front of him. He picked up his pace. Soon he was next to an old man. The sleeves of his too-long overcoat hid his hands.
"It's bitter cold," said Henri Duchemin.
The stranger's white beard inspired trust. Henri Duchemin was afraid of being alone with himself. Talking with this old man until morning would make the time pass.
"Indeed it is."
"You're on your way home, I assume?"
"Yes."
There was a moment of silence. The two men walked side by side. Henri Duchemin would have wanted to walk faster, but he did not.
"And you, young man, where are you going?"
"I'll be leaving at dawn."
"What's your job?"
"I'm an office worker."
A few black crosses rose above the wall. Farther on, behind the cemetery, were new houses.
"Perhaps you don't have a place to sleep?" said the old man.
"I don't."
"Come home with me. It will be warmer. I don't live far from here."
The two men ventured down a dark street. From time to time they passed beneath an archway. It began to grow lighter. The moon was gone. It had not waited for the sun in order to disappear.
At last they entered a detached house whose sides had been battered by the wind.
There was no light to guide their steps; they groped their way up the stairs. At each landing, afraid of bumping into each other, they raised their feet one too many times. Above their heads, the woodwork presented a reverse image of the stairs. Drafts blew the doors shut noisily.
"Wait a moment. I have to find my key."
A few seconds later, the two men entered a hovel. The old man lit a candle. A newspaper covered the table. Henri Duchemin sat down in an armchair no sturdier than the one in his room.
When the old man took off his overcoat, he appeared in a worn morning coat with a pocket in each of its distinct tails. Now, with an old man's clipped movements, he paced back and forth, he bent down. Before lighting the fire, he had to pull the grate on the stove several times. The cloud of ash that rose settled on his shoes, turning them white.
Old clothes hung on nails fanned out near the floor. There was very little air in the garret. A doily lined a shelf. On the shelf, a fork, salt, a tin. Everywhere, broken, ravaged furniture, the kind found in handcarts.
The fire blazed. It could be seen through the stove's bands. The old man was straightening things up. From time to time he stopped to ask Henri Duchemin if he were cold. Or else he would bring his hand close to the dormer window to make sure no air was seeping in.
At last he sat down. His face was lit by the candle flame. He sat straight on his stool, legs next to each other, hands clasped.
The circle of smoke the candle made on the ceiling moved ceaselessly. The only sound was the crackling of the wood. A gentle warmth pervaded the garret. Drops fell from the ceiling like diluted ink.
The old man poured some ashes on the fire. It seemed to go out. Thick smoke came out of the ill-fitted pipe. Then, all of a sudden, the fire blazed again.
Henri Duchemin noticed with joy a pale dawn through the dormer window. He had a feeling that all was for the best. More than anything else, he must not think, because it might make him sad, which would be ridiculous just when day was dawning.
He really had deserved an easier life. He had suffered his share. Now, he was able to see that the world was well designed. Aren't both happy and unhappy people necessary?
He looked at the pained face of the old man.
"You are unhappy!" he said.
"Yes."
"You haven't been lucky!"
"No, I haven't."
"Now, you know, it's too late. I don't know what I'd do if I were you."
"What can I say? A person can get used to anything. I'm not as unhappy as I seem," the old man answered.
"You're not unhappy?"
"No, nor happy."
"Well I, I am happy. I can do anything I want. I won't be made fun of any longer. I'm going abroad in a little while. And I have a lot of money on me. One would never know it."
"No."
"You see. One can be wrong. I have a lot more money that you realize."
"Yes, but you murdered someone."
Henri Duchemin grew pale. It seemed that all the blood in his body was draining out through a hole. He looked at his hands. They were open. He had never looked at them when he was suffering.
The old man spoke. He said: "I obey the voice of the heavens. It tells me to stay poor. It tells me of the joy that comes from the love of God."
A pale light was falling from the dormer window. The stains on the wall circled the entire garret.
The old man was praying. He swayed as if his stool were resting on a cloud.
Henri Duchemin stammered:
"What will become of me? What will become of me? I am lost, I've killed, I've killed."
The old man raised his eyes.
"In order to redeem yourself, you must suffer."
The sky was still growing lighter. The stars were disappearing one by one. Suddenly an infinite elation entered Henri Duchemin's soul. A beatific vision replaced the sordid walls that surrounded him. Slowly, in the light of day, the old man, standing with one hand raised, began to move away. A myriad of stars flashed like diamonds. Dazzled, Henri Duchemin was walking along the paths of paradise. Everywhere were baskets of flowers, gilded vases, and angels flying upside down.
"Yes, I have killed, but I shall suffer, suffer my entire life. I shall redeem myself. I shall be forgiven. I will do everything. I'll endure anything to be forgiven. Oh, to be forgiven! I shall be so happy. I shall suffer, suffer, my entire life."
But like a flock of birds, the angels flew off together toward a corner of the sky.
Henri Duchemin followed them with his eyes. He saw them growing ever smaller. Then, he turned his gaze toward the vases: they were no longer gilded.
He opened his eyes wide to see better. He awoke.
Henri Duchemin got up. The cold had chilled his body to the bone. Now he recognized the wallpaper and the sideboard to which he did not have the key. The light of dawn was coming through the curtains. The marble fireplace, the two chairs, the bed had never seemed so still.
Henri Duchemin picked up his hat and went out. For the first time, he saw flowerpots in the concierge's window.
The street was empty. A frightening calm fell from the starless sky. With a few flaps of its wings, a bird slowly crossed the empty space.
Henri Duchemin walked straight ahead. On the horizon, wisps of smoke stood motionless against the gray sky. It was Christmas Day.
He vaguely remembered his dream. He recalled an old man who had said that in order to redeem oneself, one must suffer. But that did not concern him, for he had never done anyone any harm.
## ANOTHER FRIEND
I prefer English gardens to French gardens. It's not that order and harmony are distasteful to me, nor that the imitation of nature delights me. I simply like not knowing exactly where I am. English gardens are mysterious with their waterfalls and secret alleyways. Though you quickly end up where you began, for a few moments you have the wonderful illusion of being lost. Most of all, you don't have to walk across vast open spaces where so many people look at you.
One hot August day I was strolling in the Parc Montsouris. Although it was noon, the sun was not in the middle of the sky. I could see it without moving my head, simply by raising my eyes.
The morning hours are the finest in the whole day. All those evening thoughts—too ambitious or too modest—have vanished. Night has made me a new being.
For me, the joys of the day never last beyond noon. That day, however, I was happy. I listened to the singing of the birds. I did not understand how some people could find it so appealing. Nothing in this chirping brought me any solace.
I was walking very slowly down a shaded alleyway looking for an isolated bench as close to the center of the park as possible, so that all around me an equal expanse of trees and lawn would separate me from the city.
The sky was blue. The air shimmered in the sunlight. A few insects that did not need to fear other, stronger insects hopped about on the grass. The intense, buzzing life of the fields and woods did not burst forth from this sheltered environment. The ground on which I walked reverberated. It did not absorb my footsteps the way country soil does.
I like giving bread to the birds. I do it because it's a sign of a charitable soul. I'm even more commendable in that nothing attracts me to them. Like most people, I am fond of their grace and independence, but not to the extent that I find contentment throwing them crumbs.
As soon as I had located the bench I was looking for, I removed from my pocket the piece of bread I'd brought with me.
There were already a dozen or so birds around me when I noticed, a few yards away, a man watching me. I will not say, as some people would, that I felt him looking at me. That would be a lie. Yet I am sure that a woman in my position, seeing this stranger as I saw him then, that is, out of the corner of my eye without turning my head, would certainly have sworn she felt this gaze weighing on her.
Still, I continued tossing crumbs. I tossed them as close to me as possible. It's always very satisfying to see birds come close. The trust they show in us enchants us and, although we know they would trust anyone, we want to believe they have gleaned our good intentions.
The stranger was still looking at me, and so I spoke to the birds. I even gave them nicknames. I wanted one of them to come take a crumb from my fingertips, which would have made it seem that those birds knew me, and that I often came to the park. Sadly, none of them did.
And as interested as I appeared to be in what I was doing, I didn't stop thinking about the man watching me. He must have been saying to himself: "Some people are odd. Here is a poor wretch sharing the little he has with the birds. If nothing else, he must have a big heart. I've never seen a poor man do this."
Surely he was telling himself this. I was conscious of my generosity. Since I had only a tiny piece of bread left, I divided it into crumbs. The stranger took a few steps toward me. The birds flew away. I turned to him humbly, my expression reproachful.
"Don't be angry with me, monsieur," he said gently. "The birds will return."
Only then did I dare observe the stranger closely. He was an elderly man of average height, well-dressed. He had on pince-nez and those rubber boots that can be worn on either foot. He was looking at me with so much kindness that for a moment his pince-nez seemed to mist over.
"Do you come here often?"
"Yes, monsieur."
For the first time in my life I was not embarrassed to meet someone. I was in such a perfect position to be liked that I could speak to anyone without being afraid.
"You must be fond of animals?"
"Very."
I stood up, and without really thinking, simply to give myself something to do, I threw the bread in the grass where the birds had been.
"You're a good soul," he said after a moment of silence.
I did not answer. And yet these were not words that should have remained between two silences. No one ever complimented me before. No one ever said to me what other people hear so frequently. These fine words filled me with joy. I even felt I could have cried had I wanted to.
I continued to throw smaller and smaller crumbs. This stranger surely was very sensitive. He was embarrassed. When I looked at him, I had just enough time to see his eyes, for he lowered his head at almost the same moment.
"You know," he said, pointing to the birds so I would not look at him, "they'll come back."
"But I don't have any more bread."
Now I have to confess something. When I said "But I don't have any more bread," there was a spiteful tone to my voice. We all have our weaknesses. No one is perfect. I said "But I don't have any more bread" as if I were criticizing him for not having any, as if he should have foreseen I would run out, as if I wanted him to buy me some so I could go on giving it to the birds.
Fortunately, I am intelligent. Right away I understood what was petty in my attitude and I made up for it by saying in a natural voice:
"The birds have had enough for today."
"Do you think so?"
The stranger was so kind he had not even noticed my little outburst.
We moved away. He was walking slowly, at his own pace. I matched my step to his. From time to time, he stopped to look at the sky.
"What a day!"
An immense joy filled me. I could tell that this stranger had a great love of simple things. He took interest in a thousand little trifles. He was, then, a man like me. Someone who does not know me well could think at first that I am hard to please and that this is why I am unhappy. No, all I ask for is a little friendship. I know the sign of great wisdom is not asking men to give what they cannot. One must take men as they are. I know this. I am wise. I ask only to take them as they are. But even this is denied me.
I walked next to the stranger with cautious steps, prepared to speed up or slow down, like those girls who proposition passersby.
I could hear each and every noise. The garden was almost deserted. Sometimes, across a lawn, we could see someone going by.
The stranger walked with his head bowed. I watched him. We didn't know where we were going.
On a bench, a poor man was eating a bit of bread with a slice of meat. One always wonders, where do people who eat outside sleep? The stranger looked at him with pity. Oh! Don't think I was jealous. Not at all; it was a great joy for me to see that, in spite of everything, there were men on earth who felt compassion for the wretchedness of others. No, I was not jealous. I am not jealous of actual beggars, of people whose poverty does not surprise them, who desire nothing and don't notice when someone feels sorry for them. The man eating on his bench was not a schemer. He did not even exchange a look of complicity with the stranger. He was truly a poor man, the sort of poor man I like.
We were still walking without saying a word. It's so pleasant to walk next to someone who is well dressed, whose thoughts you don't know, who will perhaps change your life—someone you sense to be powerful.
This stranger was almost a father to me. I felt a protective strength in his gait, in his silence. Even as a child when I went out with my father, I never had this same sense of security.
From time to time, the stranger turned to me and stared, shaking his head. And, imbecile that I am, I did not know how to look at him. To look meekly would have been ridiculous because he was the stronger man; coldly, impolite; submissively, undignified. So I carefully avoided his gaze, which I sensed skimming over my worn-out clothes, my shoes too big for my feet and, what was particularly painful, over my collar.
We were nearing the exit. In a few seconds, it would be necessary to speak. How I longed for us still to be in the center of the park.
We stopped. Near the gate was a park keeper's hut painted the same yellow as the iron chairs.
So it was over already! We were going to part.
I shivered. Luckily the stranger was not looking at me just then. It was hot. When I lowered my eyes, I could feel that my eyelids were damp.
Though his face was covered with sweat, the stranger did not wipe it off. This inattention pleased me. I attributed it to his extreme shyness and his immense fondness for me.
For the first time in years, I had the impression that at long last I had a friend.
The stranger pulled a handkerchief from his pocket, one that had not yet been unfolded and, before wiping his face, he asked:
"Where are you having lunch?"
"I don't know, monsieur."
I sensed that there was probably an answer that would have been more advantageous to me, but I am not quick and I did not have time to come up with it.
"Would you like to eat with me?"
A lunch is such a small thing; it's over so soon. Still, if you only knew how this invitation filled me with joy.
Unfortunately, I never have the courage to accept what is offered to me. I'm always afraid of accepting too quickly.
"No, thanks, I would only be a bother to you," I stammered.
"Come now. I invited you, didn't I? Let's go."
I thought neither of the heat nor of my poverty. I forgot my life. I saw the blue sky above me, the park to my right, the street to my left. It was all so immense.
"Oh, monsieur, all right. Yes."
Yes—I had said yes. If only you knew how difficult it is for me to say yes. I have never said yes. I don't know how to say yes. It seems to me that yes means freedom, happiness.
* * *
The stranger lived on a mezzanine floor. Maybe it's because I've always lived on the attic floor or perhaps for some other obscure reason, but I know even if I were rich I could never live on a mezzanine floor.
When we arrived on the landing, even though he was returning to his own home, the stranger did not look in his pocket for the key. He rang. A maid, young and innocent-looking, but probably stubborn too, opened the door.
"Come in, my friend," said the stranger, motioning toward the foyer.
I obeyed, but without wiping my feet because the loose sole of one of my shoes might have caught on the carpet. I was about to remove my hat when the stranger said:
"Don't trouble yourself. Leave it on. You're at home here."
I could say at this point that these words humiliated me because they were no doubt addressed only to people like me, but what good would it do? There are so many things that hurt me, it's best not to draw attention to them all.
I removed my hat anyway. I took two steps forward, looked at a stuffed animal mounted on the wall, and waited.
The stranger had left me in the foyer. He returned a few moments later.
"Come. Let's go into the dining room. I've asked for a place to be set for you."
I followed him.
"Sit down. Make yourself at home."
The stranger looked at my hands, then said: "You must be wondering, my dear friend, who I am. I shall tell you. My name is Boudier-Martel. I am fond of those whom life has treated harshly. I could see that behind your timid appearance you have a pure soul. That's why I wanted to get to know you, to be of help to you, to encourage you. Don't let your pride suffer from this. I could be your father. You have a friend in me. Every time I am able to make someone's life a little less painful, I do so. You are someone who is worthy of being looked after."
I listened to these words as if they were spoken by the perfect being about whom I had thought so often. I listened without trying to understand them because I was afraid some of them might displease me. I focused my attention on the words I love: dear friend, help to you, pride. I could not believe the friend I had been seeking for so long was there, in front of me. And yet, there he was, and I felt how ill prepared I was to speak to him.
"You mustn't think, my friend, that I have a cold heart. I do everything in my power to make life a little less difficult for the needy. I know nothing greater than turning one's attention to the misfortunes of the meek."
These words soothed me. It seemed as if the chair on which I was sitting had no legs, that my heels were no longer resting on the parquet floor, that I was living in a dream. A new life was about to begin for me. I had a friend. With all his gifts, with his heart, he was coming to me.
"Oh, monsieur, how happy everything you are saying makes me!"
"Yes, yes, I thought as much. Come now, let's eat. And then, on Sunday, I'll come see you in your little room. It is a little room on the top floor, isn't it?"
"Yes, monsieur."
"If only you could understand how well I know you. I can picture your whole life. You wake up and get out of bed, then go for a little stroll. You are very fond of animals. You eat lunch; you stroll about some more; you eat supper; you go to bed. Alone. You are alone, completely alone. No one bothers you. By the way, what do you live on?"
"My annuity."
"Of course! You have a small annuity. You are happy. You are wise. I admire you."
I will remember that lunch my whole life. There was so much trust between Monsieur Boudier-Martel and me, so much solicitude, that I can hardly believe nothing remains of it today.
* * *
Sunday came at last. Monsieur Boudier-Martel was to arrive at four in the afternoon, after the heat of the day had died down.
I spent the entire morning getting ready. I bought wine, a tin of biscuits, some lemon soda. My room, tidied up, seemed larger than usual. I sat down on my bed, at the spot where there is a big hole in the quilt, and I waited. The window was open. The blinds are broken, so the harsh light from outside flooded the room.
I was in that contented state in which you find yourself when you've just finished a thousand little chores that are so easily forgotten. There were just the two glasses that I had not yet washed. I knew it. I was saving that task so I would have something to do when Monsieur Boudier-Martel arrived.
Suddenly I heard footsteps on the stairway. It had to be him. I stood and picked up the glasses so I would be busy rinsing them when he knocked.
I heard him on the landing. Although I had explained to him which door was mine, he was looking down the other end of the landing, where Lecoin lives. How I wished my neighbor could see Monsieur Boudier-Martel coming into my place!
There was a knock. I went to open the door.
There he was. Despite the fact that it was Sunday, he had put on old clothes to come see me. No doubt he had done so out of tact. He walked in, removing his hat at the door.
"As you can see, I'm just rinsing the glasses. Please sit down," I said, and offered him my best chair.
"Oh, don't trouble about me. I can sit anywhere."
He sat on the bed, at the same spot I had been sitting because the sagging mattress forms a hollow there.
"Why, this is a very nice room. It's clean, it's airy. It's a bit high up, but it's airy."
"You think so?"
"Rooms like this are rare."
His admiration for my quarters displeased me. I had hoped that after he'd seen my place he would offer me a big, comfortable room in his apartment. Now I realized it was pointless to count on that.
"Do you do your own cooking?"
"Oh, no, monsieur!"
"You don't?"
"No, I eat out."
"You eat out?"
"Yes, monsieur."
"But restaurants are very expensive."
"I have a little arrangement."
"Oh, that's different! When one is in a situation like yours, you have to know how to make little arrangements."
"Indeed, monsieur."
There was a moment of silence. While looking out the window, Monsieur Boudier-Martel was testing my bed with the back of his fist. Every now and again he raised his heel and struck the floor. He also turned around, looking everywhere.
While I was trying to find a dishrag, he said: "No, don't dry the glasses. You mustn't trouble yourself. I like drinking from glasses that have been freshly rinsed. You know, it's not as bad as all that here. You probably have running water not too far."
"Yes, on the landing."
"Excellent. The other day, I couldn't talk to you as I would have liked. I barely knew you. Now I want to tell you how noble I find your self-denial, your simplicity."
These words, which I found full of truth, moved me. I looked tenderly at Monsieur Boudier-Martel. I felt that whatever was still separating us was about to vanish.
"Would you like a little wine, monsieur?"
"As you please, my child."
My child. He said my child. This time, all my sorrow vanished. I was trembling as I poured the wine. He was about to get up to take his glass, so I said:
"No, don't bother." And I brought it to him, not without spilling a little.
He drank leaning forward, the way one drinks at a bar.
I found this tactless. I don't think he should have noticed that I had filled the glass too much because if I did, it was because his kind words had disconcerted me. Even if he were to spill a little on himself, he should have drunk as if he were at home.
"You are very sensitive, my friend."
For a second I thought he was reading my mind.
"I am fond of people like you. Human misery moves me. Tell me about your life. If something is troubling you, confide in me."
Tell him about my life! Can one tell the story of one's life to a friend? Can one tell the story of one's life without making it more beautiful, or uglier, without lying? As for confiding, is it possible to do it just like that, on demand? To talk about my life, about myself, to a man who had just walked in—no, it could not be done.
Monsieur Boudier-Martel was waiting for me to speak, pretending to be very attentive. Yes, I said pretending: though his gaze was fixed on me, from time to time his eyes would momentarily turn away toward some object in my room.
"Do you wash in that basin?"
"Yes, monsieur."
"That must not be easy. Come now, tell me about your life, confide in me. You have a friend in me, a brother."
"A brother?"
"Yes. I too have suffered from poverty."
"You've suffered from poverty?"
"Yes."
I sensed he wanted me to rejoice at this news. Yet deep down I respected him less.
"Would you like a little more wine, monsieur?" I asked, expecting a polite refusal.
I was mistaken. Monsieur Boudier-Martel accepted.
Have you noticed how often we are wrong about people? We are sure they will say one thing and they say the opposite. But this mustn't change our opinion of them. For some infinitesimal reason that was unknown to me, Monsieur Boudier-Martel had not said no, yet his whole being was refusing the wine I offered.
This time I poured the wine slowly so that Monsieur Boudier-Martel would spare me from seeing him lean forward to drink. Though the glass was only half full, he still leaned forward.
"Well, then, when are you going to tell me about this life of yours?" he asked, looking for a place to set down his glass.
If only you could have seen how he searched! If he had truly cared about me, if he had truly been drawn to me by some feeling, he would not have had that self-conscious air about him; he would have set the glass on the floor.
"So, what about that life of yours?"
"Oh, monsieur! It's not all that interesting."
He stood, came over to me, and stroked my hair.
I lit up with joy, even though I was torn between the desire for him to stop and for him to go on—for him to stop, because there is something grotesque about emotional outpourings between men; for him to go on, because it was a sign of such deep friendship.
"Oh, child, child," he said, pulling away from me. "I'm leaving now, my friend."
"You're going to leave?"
And I had thought we would stay together until nightfall!
"Come have lunch with me whenever you like. I'm not insisting. You are a free man. I'm not setting a date. I respect other people's freedom too much."
If only Monsieur Boudier-Martel knew how little one values one's freedom when one is alone.
He took his hat and did not wait to go out to put it on. I realized he had made an effort to be tactful at first and now he was tired and couldn't be bothered.
I caught a glimpse of the vast solitude awaiting me.
I stood up also.
"You're leaving?"
"Yes, I must get back."
I lost my head.
"Monsieur, monsieur, don't go."
Disconcerted, Monsieur Boudier-Martel drew back a step. As a precaution, pretending to be surprised, he opened the door as if he weren't thinking.
"Don't go; I'll be so alone without you, if only you knew how I suffer when I'm alone. Stay. You'll talk to me. You have been so kind."
Reassured, Monsieur Boudier-Martel released the doorknob.
"Come, come, my child, calm down. You know you can count on me."
I realized it was impossible to keep him. I don't know anything more agonizing than the feeling that, no matter what you do, you cannot prevent someone from leaving.
With a final burst of energy, I approached him and, kneeling awkwardly, the way people who don't go to church do, I stammered:
"Don't be angry with me, I acted without thinking. You understand me, please forgive me. You can count on me for everything. I'll go to any lengths, please stay, monsieur."
I got up. Monsieur Boudier-Martel, who had stepped back even farther, was on the landing.
"Come now, my friend, take heart. I shall not forget you. I am very fond of you. Good-bye. Come see me."
And he went out without even having heard that I had said I would go to any lengths for him.
Alone again, I sat down on the bed. It was still quite light out. Someone was playing a guitar in a house nearby. At times he played the same tune twice in a row. Birds flew across the blue sky so quickly they seemed to be following a straight line. They were black, the way all birds are in the late afternoon.
I got up. I picked up my hat. I waited a bit so I would not catch up to Monsieur Boudier-Martel. I opened the door; the landing was deserted. I went out and strolled about until night fell.
* * *
I will always remember that radiant day that was one of the saddest of my life.
The previous night I had fallen asleep late because in my bed I had been thinking about Monsieur Boudier-Martel. I am such a good person that whenever I am far from people I no longer see their faults. I had foolishly imagined that Monsieur Boudier-Martel, in his bed, was also thinking about me. So I looked at my watch. I decided right then to go to his house the next day to tell him that at 11:10 p.m. our thoughts had crossed.
In the morning the idea seemed ridiculous to me. But since it had already been three days since we'd seen each other, I did not go back on my decision. He had been so insistent for me to come to lunch at his house that I was not afraid of abusing his kindness.
I put on my best clothes. When I'm in my room, I always find I look fine, but as soon as I go out, as soon as I am on the street mixing with the crowd, I realize how poorly I'm dressed. It's not a question of contrast. No one notices me. It's because I think everyone can see the life I lead and that they are all saying: "He's only got what he deserves." So I am fearful, and I lack confidence. I'm wrong. No one pays me any mind.
I left my room at eleven thirty. Usually I leave earlier, but that day I wanted to arrive at Monsieur Boudier-Martel's fresh and clean, without a trace of dust.
The heat was overwhelming. A vehicle washing down the streets wet my feet. I walked slowly because even though the visit I had decided to pay was entirely justified, I was nervous.
The noon bells were ringing everywhere when I arrived in front of Monsieur Boudier-Martel's. I went straight in. The corridor was not as cool as it had been the previous time; all the doors were closed. In summer, doors seem as though they are never supposed to open.
The elevator wasn't there. I climbed the stairs. The railing was too wide to be held. When I got to the door, I removed my hat, then put it back on. I was panting, but with emotion—I couldn't blame it on having climbed six flights of stairs.
I had to ring. Without bothering to switch on the hall light, I pressed the bell. I waited a few seconds.
"Is Monsieur in?" I asked the maid, one hand on the wall, the other in my pocket.
I struck this pose as soon as I saw her because I cannot bear servants. I wanted to show her that, although I was poorly dressed, I was above her. No doubt she felt it and, either out of unkindness or to get even, she asked:
"Monsieur who?"
I almost lost all the confidence I had worked so hard to gain.
"Your master," I answered insolently.
I immediately regretted this outburst. I realized I was, after all, at the mercy of this woman. What could I have done if she had answered, "My master! He's not in!" So right away I added:
"You must recognize me. I came for lunch the other day."
And although I stammered, humiliated and frightened now, I was nonetheless reserving the pleasure of speaking ill of her later to Monsieur Boudier-Martel.
"Yes, he's here. Come in."
I removed my hat, despite the fact that I was loath to do so in front of this maid. She was capable of thinking I'd done it on her account.
"And whom should I announce?"
I hesitated a moment.
"Announce the gentleman who came to lunch the other day."
"But which one? Gentlemen come every day."
This time, I had to say my name. She was going to make fun of me; she would laugh. Oh, well, after all, my name is my name. I don't have to be afraid of saying it.
"Monsieur Bâton."
"Bâton."
"Yes."
"Please wait here."
I sat in one of those foyer chairs where one sets down packages and hats, but where only people like me actually sit.
A door opened. Monsieur Boudier-Martel appeared without a collar, in his dressing gown. I leaped up.
Without moving, he held his hands out to me.
"Ah, it's you. I am so pleased to see you. Come in. Let me introduce you to a friend, a man like you. Come in, come in."
"A man like me?"
"Yes, come in."
I did not have time to think. I went in stunned, happy, as in dreams we remember.
Suddenly I stopped cold. Instead of circulating through my whole body my blood all rushed to my head. Monsieur Boudier-Martel was pointing at me. The maid was somewhere behind me. Someone was speaking. I heard some words. The door, gently and by itself, was closing.
I had just seen, right there, in the same chair where I had sat, a poor man, a poor man like me. I needn't look at them for long. I recognize them immediately. It was obvious that, right there, in the armchair, was a poor man.
"Come in, come in, my friend."
I said nothing. Now I understood everything. Monsieur Boudier-Martel did not love me. He loved poor people.
"Well, come in, Bâton. What's wrong?"
"No, no, I'm leaving. I don't feel well."
I was backing away. Monsieur Boudier-Martel followed slowly. I could tell he did not want to come any closer to me. You never draw close to people who abruptly change their attitude.
"Stay, dear man, stay. You're at home here; you are my friend."
I was still retreating, and then I opened the door.
"I'll come back shortly, Monsieur. I don't feel well. I'm ill. I have to go."
I went out, leaving the door open. I could have closed it, but I did not have the will. As long as it stayed open, something was still possible between Monsieur Boudier-Martel and me. He could follow me, beg me to come back. I don't know what I would have done in that case.
If I left the door open, it was so he would be the one to close it, so that he would be the one to break off our friendship forever, so that in my loneliness I would at least have reason to suffer because of other people's lack of understanding.
Monsieur Boudier-Martel remained in front of his door as I descended the stairs. It seemed as if the landing was as far as he could go, and that the stairway was an abyss. He leaned forward, calling to me, not daring to place a foot on the first step.
"Come back, Bâton. What's the matter?"
I, for my part, walked away very slowly. When I got to the hallway, I stopped. Was it because my suffering was not as great as I thought that I caught myself on guard, listening to what was happening on the mezzanine?
The door slammed shut. It was over.
In the blinding light of the street, it seemed that everything that had just happened in the shade of the house was already lost to the past. I did not weep. One never weeps right away. I was so on edge that, although I was not laughing, my face was contracted as if I were.
Days passed.
I would have forgotten this sad story a long time ago had I not retained the impression that Monsieur Boudier-Martel knew why I had left. He was certainly aware that a base sense of jealousy had forced me to flee, that I would have stayed if there had been a rich man instead of a poor man in the dining room. I'm sure he knew all the petty thoughts going through my mind at the time. Yes, without a doubt, he knew them all because, had I been in his shoes, I too would have guessed them.
## NIGHT VISIT
What was making me sad? My books—all my books—were sleeping on the shelves. No one had spoken badly of me. My family and friends had no particular worries. I found myself in the midst of all things. So I did not need to fear that events, in my absence, would take a turn I would be unable to change. I was not unhappy with myself. And, even had I been, this intensity of feeling was different.
It was eleven o'clock at night. A lamp without a shade lit my desk. I had not gone out all day. Whenever fresh air has not put color in my cheeks, I don't feel at ease. My wrists are smoother and I notice, with some displeasure, that the down covering them is silkier, and when I go to bed, my unexpended energy makes me uncomfortable.
I was dozing in an armchair. At the seam where the red velvet meets the wood, golden tacks form a border. One of them was missing and, there, the edge sagged a bit. I sat motionless. My hand tugged at this seam without my being aware of it, as it sought unconsciously to pull out the next tack.
It was only once I had managed to pull it out that I became aware of what I was doing. I felt a small joy at this discovery, as I feel each time I catch myself doing something without realizing it, or when I bring to light a sensation in me of which I was unaware. It makes me as happy as a ray of sunshine or a kind word. Anyone who would criticize me for this tiny joy will never understand me. I think that seeking knowledge of oneself is a pure deed. To criticize me for digging too deeply into myself would be to criticize me for being happy.
I have to say, though, that this joy is very fragile. It really is not equal to the joy a ray of sunshine brings. Quickly it disappears, and I have to look for something else inside me to bring it back to life. Then, in the intervals, it seems that everything is hostile to me and that the people around me, with their simple joy, are in reality happier than I am.
* * *
I was reading when there was a knock at the door. It was my friend Paul. He rushed in and the door, which he had yanked behind him so it would close, stopped half- way.
"What's the matter, Paul?"
"Nothing."
His face was pale, and his eyes were darker than usual. He dropped onto the sofa, which he knew was soft.
"But what is it?"
He stood, walked around the room as I put my book down, and lit a cigarette, then sat again. He was smoking the way nervous people do, his cigarette drooping from his mouth. From time to time, he would spit out bits of tobacco.
"Please, Paul, tell me what's happened to you."
I looked at him. I tried to find a gesture, an expression, something in his bearing that would reassure me. But there was nothing. If he had been holding some object, his fingers would have trembled. He must have realized this because he avoided touching anything whatsoever.
"Paul, I'm your friend. Tell me everything. You know if there's anything I can do for you, I'll do it. It hurts me to see you like this, without being able to help you."
He was so upset that I alone heard my words. I saw them drift over his head without ever reaching his ears. It seemed as if the words were balls that I was tossing haphazardly. I grew tired of his lack of concentration and stopped paying attention to what I was saying, and just then he appeared to listen to me.
Cautiously he drew near me. It was as if he were afraid that the slightest sound would close my mouth. He looked at it, blinking; his eyelids were missing a few lashes. The light from the lamp as it glided across the roundness of his eyes made their color fade. He burst out laughing. Yes, he burst out laughing. His fingers trembled one after the other, their thin fingernails molded to the flesh rather than sitting atop it. A few teeth I had never seen appeared at the back of his mouth, similar to the others but unfamiliar to me. They revealed physical mysteries. I was aware that I no longer had a friend in front of me, but a man like myself. And this did more to make me feel sorry for him than his desperate behavior.
"Why are you laughing?"
"Hmm! I don't know, you're right, I shouldn't be."
And he went on laughing. His nose seemed longer as the muscles of his face contracted. His mouth, which had lost the rhythm of his breath, was trying to recover. In spite of everything, Paul had to breathe through this confusion, and so his breath vibrated on his palate before escaping from his mouth.
At last he sat down, calmer, which made me wonder if his pain was as great as it had seemed.
A streetcar passed. Was it because the rain continued to fall that I thought, from the noise the streetcar made, that the electrical current was stronger?
There was a moment of silence. When the rumbling of a taxi interrupted it, I listened to the sound until it became imperceptible. And my concentration was so intense that I still heard it, even though it no longer existed. A pinkish light, projected from outside, lit up a spot on the wall where it would have been difficult to hang a painting.
My friend was not moving. Having parted his lips once and for all, he was breathing feebly, his tongue folded over so as to be out of the way. And his hair, which he had not combed with his hand, was untidy.
"Say something, Paul!"
His eyes left the comforting flame of the lamp, seemed to follow the flight of a bird, then landed on me. They were shrouded by a dark glaze, fringed by the shadow of his lashes. Perhaps because each eye was so intensely alive, I realized very clearly that there were two of them. As for my gaze, even though it was full of compassion, I felt it was not honest. As hard as I tried to open my eyes wide to see more clearly, it was hopeless. Paul took my hand. I questioned him.
"Tell me, what has happened to you?"
He leaned forward to take my other hand, which was quite far from him. He did it gently. Then he began to speak.
I did not hear the first sentences, as I was busy trying to find the mark of pain on my friend's features. I placed more importance on that than on whatever he was about to tell me. For when a man is suffering, what can he tell us that we don't already know?
"Jean, a great misfortune has just happened to me."
Now he was calm. Beneath his thick clothes, one could see he wasn't trembling. The circle of lamplight extended beyond us. We were sitting up in the middle of it, on the tangled shadows of the chair rungs.
"You know, dear Jean, how strong my attachment to you is. We met during the difficult days of war and from the start—and it had nothing to do with danger—we were attracted to each other. You would read me your letters. I read you mine. And we trusted each other enough not to hide anything. Sometimes we would get angry with each other and even though we both easily bear grudges, it never took us long to patch things up. We were true friends. And do you remember the demobilization? Do you remember our joy at finally being free? At the time, everyone's happiness was so great that all our friends no longer even thought about friendship. But we weren't like that. We had tears in our eyes when we parted. Do you remember? You came to Paris while I went to the South to join my fiancée. After only a few months, we met again by chance. How we celebrated that wonderful encounter! What a night! Well, my friend, in the name of this unblemished friendship I am asking you to listen to me. I want to believe that my presence in your home at this hour is not unpleasant for you. We have spent too many sleepless nights side by side for you not to want to spend one last night with me. It will be less dangerous than all the others, but much sadder. Before, we were waiting, hoping for something when we couldn't sleep. Today, everything has changed."
This preamble might seem affected. Obviously a man who is suffering, when he confides in a friend, does not go back to the beginning of the friendship that binds them. But ours is a very particular case. Paul and I, in truth, are no longer friends. We were friends only during the war. So it was natural for him to speak that night about what we had meant to each other in order to confer on the tenuous relationship we have today the significance it once had.
My friend, who had stopped talking, pulled out his handkerchief and mopped his brow oddly, insistently, with the care he would have taken for his entire face. I did not look straight at him so as not to embarrass him.
He rose, removed his overcoat, and sat back down in the larger chair. Without the fullness of his clothing, he seemed even more depressed. His hands were uneasy in the small pockets of his jacket. Everything he was wearing seemed no longer to belong to him.
Suddenly he burst into sobs. He hid his face in his hands and I could only see the lower half. His chin tensed so much that dimples appeared in unsuspected places. Have you ever seen dimples like that? They are made of tiny, trembling wrinkles that disappear then reappear somewhere else.
He was crying. How sad are the tears one hides! Why wasn't he crying freely, with his face uncovered? I could have consoled him. But like this, withdrawn into himself, he was completely alone with his pain.
"Paul! Paul!" I said, distressed.
My God, how firm my voice seemed! True, in order to ease someone's pain, you need not suffer as well. You must use a familiar, cheerful tone, the one everyone uses because everyone has always realized that no other tone of voice can offer consolation.
"Please, Paul. Be serious. You come here to tell me your troubles and you start crying like a child. A little courage, Paul! You know I'm your friend and I only want to help you. Be reasonable. We are men. It's ridiculous to cry like this before you've examined the problem. Afterwards, if there is no solution, there will be plenty of time to cry."
My friend must have been waiting—not for these words, which he did not even listen to—but for that consoling tone of voice because he raised his head. He had not cried long enough for his eyes to be red. Just wiping his handkerchief across his face was enough to erase all traces of his tears.
Exactly as I would have done, he began by apologizing for having cried. He did so in words I wish to note because they are shared by all men.
"Forgive me, Jean. I couldn't help it. But it's nothing, just a moment of weakness. If you knew what happened to me, you'd understand."
His handkerchief, damp from his tears, took up little space in his hand. One tear, to which he paid no mind, still glistened on his cheek.
"Come now, Paul. Tell me everything and then I can tell you what needs to be done."
"Yes, Jean, I'll tell you everything. Don't be angry with me if I'm emotional. You know Fernande. You know how much I love her."
I did not know Fernande. And when I nodded, I swear it was not because I was uncaring, or because I was afraid my friend was trying to prove to me that I'd already met his wife, but simply so as not to contradict him. I was too aware that a contradiction would upset him.
Paul spoke without a single gesture, like a sick person. From time to time, he would glance at the door and that was enough to make him lose his train of thought. It was as if he were reading and his eyes, distracted for an instant, could no longer find the line he had just read. And when he stopped speaking, I listened just as attentively so that he would begin again as quickly as possible.
"You know, Jean, I was living peacefully. I'm a simple man with modest tastes. Unfortunately, I am too good. My wife, my dear wife, often criticized me for this. Not because she would have wanted me to be unkind, but because she finds it disagreeable to know that I am good to others as well as to her. I never raised my voice. Even though I would have had reasons to do so, I've always understood that the volume of a voice adds nothing to the meaning of words. Tonight, as I speak to you, I will stick to this principle. I will tell you everything as simply as possible."
He stopped speaking for a second but went on acting as if he were swallowing something, which lifted his double chin slightly.
"My life was calm. At times I'd have the impression that one day some misfortune was bound to come and spoil my happiness. But the feeling would not last. I would have had to have a pathological tendency to imagine vileness everywhere not to be happy. Often my wife and I would go to the countryside. Nature would fill us with wonder. We would go into ecstasies over the perfection of the plants and insects and, as unlikely as it may seem, I felt that in Fernande's eyes, I held the key to mysteries I did not understand. There was, nonetheless, a small snag in our happiness: we had no children. I'm ashamed to admit it, but even that safeguarded our happiness. When we would go to visit friends, everything about us made it seem as though we regretted not having a child. And so our tranquility was preserved because, deep down, our friends pitied us. Life flowed on in this way, smoothly, without quarrels or conflict. And I have to say that sometimes, at the thought that I was happy with so little effort, I wondered if I truly was. But I would dismiss this misgiving as quickly as possible for I knew that if someone were to ask me to describe the happiness about which I dreamt, I would have had no choice but to paint a picture of the one I already possessed. To love, be loved, to do as we pleased, to have faithful friends, to never argue, never be ill, what more could one ask for when one is a simple, trusting soul? In any event, there are not that many ways to be happy! It is indeed happiness to have no worries, and to love. I don't think the vagabond on the road who has no idea where he will sleep at night is happy, poor man, although some people claim this to be so."
Paul broke off again, not as one does at the end of a period, but as if he were casting about for his words. His lips were moving. He gestured and finally managed to go on.
"It was eight in the evening when I got home. I took off my hat and went to Fernande in our bedroom. Lying on a divan, my wife seemed to me more beautiful than ever. Her eyes were closed, but I knew she wasn't sleeping. She was holding a book with the grace of someone who has dozed off while reading. I went to her and kissed her softly on the forehead. She gave an elegant little start, not right when I kissed her, but a few seconds afterwards. 'You, Paul!' You know how hard it is to give a gentle intonation to two words. Yet if you had heard the tone with which she murmured 'You, Paul!' it would have delighted you. Then she closed her eyes again, without hiding her face from me, with that trust of women who love. Fernande often closes her eyes. In the theater, while we're eating, everywhere. Not a day goes by when I don't see her before me, eyes closed, even though she isn't sleeping. It seems she has trouble tolerating the spectacle of life, that everything appears so trite to her that by closing her eyes she doesn't think she is missing anything. We've been married for four years, but I've never been able to tell if there is any kind of deceit in all that. I did not do anything to wake her from this feigned sleep. I sat close to her and waited. I stayed like that for a long time, without even daring to read the newspaper. I love Fernande and it seems natural for me to watch over her. If she found it amusing to pretend to sleep, why would I stop her? To look at her without her seeing me was a joy for me. She had let her book slip, no doubt so that her slumber would seem more natural. It slid slowly. I let it fall. She opened her eyes.
"'Paul, why didn't you catch the book?'
"'I was looking at you, my darling.'
"I understood then, in an instant, that she had realized that I knew her sleep was feigned. And instead of being slightly embarrassed, she said drily, almost vengefully,
"'Were you really looking at me?'
"And then we said no more. I like silence when I am near my wife. I don't believe in a meeting of souls, but in the evening, in silence, near the woman one loves, something happens."
Paul stood up abruptly. He released my hands, shoved his chair with his foot. He was overcome by so many different emotions that I couldn't have said if he was angry, upset, afraid, or filled with hatred. I felt lost. I can be quite a good psychologist, but only if people are calm.
"Listen to me, Jean. Everything on earth follows the same laws. Starting at one point, a man, an animal, a tree grows and grows, then slowly deteriorates. Whereas in Fernande's mind, a feeling, instead of being born humble and fragile like everything else, sprang forth powerfully, in a monstrous way. Given what I just told you, you can imagine that we were happy, that nothing disturbed our tranquility. Then how to explain what happened? Do you want to know what happened? Well, it's very simple. Just like that, all of a sudden for no reason, when I asked Fernande to come have dinner, she got up suddenly like I did just now, shoved her chair with her foot, and announced: 'I don't love you anymore. Tomorrow, I'll be leaving.' Did you hear what I just said? Tomorrow, she'll be leaving. She doesn't love me anymore. Why? I haven't the slightest idea. Am I mad? I've begun to wonder if my wife's calm hasn't always been concealing some outburst. At first I thought she was playing, imitating an actress. She often does this, she exaggerates. But no, it wasn't that. She doesn't love me anymore. Tomorrow, she'll be leaving."
I looked at Paul. He was now gesticulating wildly. He raised his arms to the sky, then wrung his hands so hard that he almost broke a finger. He paced the room, turned around abruptly, started off again, hovered far from me and then suddenly came at me with great strides, as if he were walking down a road.
"Paul, calm down. All is not lost. Perhaps she said that without thinking."
He dropped a book and did not pick it up.
"Don't get so worked up."
Then, either because a sudden rage swept over him or because he wanted to prove his strength, he struck the floor with his heel several times.
"I swear that's what she said. She's leaving tomorrow. She doesn't love me anymore."
"Of course she does!"
"So you think I'm crazy? You haven't grasped what I just told you. I'm not making anything up. She said she doesn't love me anymore. She said she would be leaving tomorrow. Don't you understand? It seems very clear to me."
I did not like his insolent tone of voice. I was only trying to console him and this is how he answered me! We were not, after all, such good friends for him to allow himself to treat me this way. When someone comes to tell you his troubles, he should at least be polite. I assure you if Paul's pain had not seemed so genuine to me, I would have answered him coldly.
I raised my eyes. Paul was sitting on the sofa. All his anger had vanished. He appeared so unhappy, so weighed down, that my displeasure evaporated.
Poor Paul! How you were suffering. You who, during the war, spoke to me about peacetime with so much ardor, you who were expecting so many joys from it, how disappointed you must be! And to think that for a moment I was angry with you for being on edge.
My friend, although he had barely cried, was in that state of semi-consciousness that follows sobbing. He was looking at a corner of the room without even having chosen it. His hands were far apart, whereas when one is suffering they are like two close friends, never wanting to leave each other. His shoulders hunched, his head to the side, he was daydreaming.
"Paul, be brave."
"I will be."
We did not hear a sound, not even the sound the last buses of the night should have made. We stayed like this for several minutes, without moving.
Suddenly, so abruptly that I was startled, Paul got up, took a few steps, and kneeled down too close to me.
"Jean, Jean, I'm begging you, do something for me. Perhaps you could fix everything. You are my friend. You are almost my brother. We spent unhappy times together."
I lowered my head and met Paul's tearful gaze. That gaze! I will remember it the rest of my life! Humble, despairing, looking up at me, it struck me as the gaze of an animal at my mercy.
"Paul, stand up. I'll do everything I possibly can."
"Jean, if you wanted to, you could go see Fernande right now, you could tell her how much I love her, how I'm suffering. You could speak for me and perhaps she would be sorry."
"Yes, Paul, I'll take care of everything."
My friend, holding on to a chair, rose with difficulty, stumbled a bit, then sat down. His face had brightened. His eyes had grown wide and looked at me without humility. He was breathing evenly. And for the first time, he did something normal: he looked at his watch.
* * *
It was still raining. Now, however, the drizzle was so fine that when I ran my hand across my overcoat, I wiped it away. It barely moistened anything, like a fountain on a windy day. It created a misty halo around the streetlamps.
We walked briskly, without speaking. Since my friend lived close to my place, we soon arrived in front of his house.
"Jean, let's go into this little bar. I have more to say to you."
We went in. It was a small, very clean café. There were mirrors everywhere, yet we were reflected in none of them. The nickel, the glasses, the tin counter gave the light the coolness of water. A bit of sand crunched beneath our feet, as though we had come to sit near a spring.
A waiter approached us. His left hand, folded in, seemed to be hiding a cigarette. We ordered coffee. So that our coffee would not taste like metal, we took the spoons out of the cups.
"Jean, listen to me. Since you are kind enough to go see Fernande, let me thank you with all my heart. My happiness is in your hands. I don't have the strength to go with you. I'll wait for you here. In fact, it will be much better if you go alone. You see the state I'm in. Tell her I cannot live without her. Tell her I love her so much that I would give my life for her. Perhaps I have not always acted as I should have but tell her that now I will obey her, that I will be her slave. I will do anything for her to stay, for her to want to continue the life we were living. I love her so much! You can make me the happiest or the unhappiest of men."
The little café was completely hushed. The owner was already counting his money. The waiter, leaning against a column, looked at us now and again. As for Paul, no doubt from a habit he had with his wife, he was holding my hand.
"Go now, Jean. I'll be waiting for you here. Oh! How I'd like to know already! My God, if you were to succeed in making her understand how much I love her, I think I would dance, jump for joy, and cry out with all my strength."
I stood up. As if he were at home, Paul walked me to the café door. Never have I seen a man so moved. I felt he was looking for one last word to say to me, one word that would sum up his pain, his hope, and he could not find it.
* * *
I shall not attempt to recount my visit to Fernande. All I can say is that she was not welcoming. Whenever I asked her a question, she would answer with these same words: "I am free to do as I please." Although I described her husband's suffering and his love, her attitude did not change. After hearing what my friend had to say about her, I had thought that she was, if not beautiful, at least pretty. Not at all. She was a rather corpulent, rather common woman whom I had difficulty imagining in the languid poses Paul depicted. She spoke in a disagreeable, aggressive voice. I wouldn't say she had the behavior of a shrew, but almost. In addition, she seemed very insolent. I knew that my visit, so late in the evening, was not likely to be met with good humor. Nonetheless, she should have behaved better with a stranger and not let her annoyance at my presence show so plainly on her face. Perhaps she thought I was defending her husband for want of anything better to do! Yet she must have been aware that all of this was as disturbing to me as it was to her, and she should have been grateful to me for defending so ardently a man who, after all, was her husband and whom she must have loved, whatever she said. The more I think about this visit, the more it seems that nothing but what happened could have happened. I assure you had I known it would end the way it did, I would not have troubled myself. Paul, naturally, is not to blame, poor fellow. He thought he was doing the right thing. But I will never understand how one can be so attached to such a woman. She must have influenced everything he did. And no doubt the anger she felt on seeing me stemmed from that fact that Paul had taken the liberty of sending me to her. She could not bear the idea that her husband had done something on his own. She took it out on me. Truly, you had to be someone as good as my friend never to get angry. But in the end none of that concerns me. What I especially disliked was the offhand, overbearing way she received me when in fact I was acting in her interest just as much as in her husband's. It was pointless of her to try to appear to be the victim of two men. Perhaps Paul had done some things of which I was unaware. But I? I simply came to try to make Fernande see what she had misjudged in her husband. That's all. I took no one's side. And had she treated me properly, had she answered me clearly, I would have had no reason to be angry with her.
In the end, all this only confirms what I think about the world. Let her do as she pleases, it's all the same to me. As for Paul, I pity him with all my heart, for it seems to me that, however this story turns out, he will not be happy.
* * *
When I left my friend's wife, it had stopped raining. I took a few steps before I was entirely sure. Then I looked up. The sky, deep and black like marble not yet dry, was filled with stars. In the distance, the long, furrowless cloud that always floats along after it rains was low in the sky. The stars twinkled in the translucent air as if threatened by a celestial breeze. The street was still damp, but there was no mud as there is after a storm. And the white, ethereal moon rose unexpectedly high on the horizon.
I returned to Paul in the little café. He was watching for me through a curtain, sitting the way children do, sideways on a bench.
As I approached, he turned around and, hands on the table, looked straight at me. He was trying to guess what had happened before I spoke. He did not dare ask me. There was such distress in his eyes that they appeared about to close. It seemed his eyelids would droop at the slightest puff of air, that they were folded open only because his eyes were so round, and that if he were to look off to the side they would slip down.
"Paul!"
"So?"
I was incapable of pronouncing a single word. The despair into which my friend was about to plunge frightened me. I was expecting so much pain, so much shrieking when I told him his wife's decision that I could not bring myself to give an account of my visit. I was waiting for him to infer Fernande's attitude from my silence.
"Jean, what did she say?"
"She wants to leave."
"She wants to leave?"
"Yes."
My friend seemed not to comprehend. He was trembling, but his face remained impassive. It was as if only his body had understood. I was overcome with pity. I sat next to him and, my arm around his shoulders, attempted to console him.
"You're young, Paul. You have your whole life in front of you. Be strong. You'll see there will be more moments of happiness for you. That woman did not know how to appreciate and love you as she should have. I could see from her behavior that she was too fickle for you. Believe me, later she'll regret what she did, she'll never find another man with all your fine qualities. Let her go, and if one day you should meet again, be distant. Nothing can hold her, so at least have the strength to pretend not to care about her. That's all it will take to humiliate her deeply. Without you in her life she is a lost woman. You were not only a husband to her, but also a father. One day she'll understand that, you can be sure. Unfortunately, it will be too late. She needed a man like you to be happy. She did not understand that. It's a shame. As for you, you loved her too much not to suffer from her behavior; you loved her too much not to miss her. I know. But you have to do something! Slowly, you'll forget her. And then, who knows, one day you'll meet another woman, more beautiful, more intelligent, who will love you with all her heart."
As I spoke, Paul was gazing at me with an astonishment I could not explain. His half-open mouth and his furrowed brow made him appear stunned. From time to time, he would turn his head away sharply, then stare at me again with an even more surprised look in his eyes. Despite this odd attitude, I continued speaking.
"I have suffered, too, Paul. Two years ago I was with a woman who, like Fernande, left me for no apparent reason. Well, I got over it. Not without long months of suffering. But one has to live and most of all not become discouraged. Fernande wanted to remain ignorant of your generosity. She imagined you wanted to bully her when all you wanted was to make her happy."
Suddenly Paul shoved the table away so that he could get out.
"Come, let's go, I can't stay here anymore."
We started down a deserted street, half white from the moonlight, half dark with damp stone, and we did not cross to the sidewalk that was bathed in light, as we would have done during the day to move from shade to sun. We walked past the houses. The streetlamps lining the sidewalk were all that lit our way. An echo made it seem as though two other passersby were in front of us and, bizarrely, they seemed to have more energy.
"What's to become of me?" Paul whispered.
My friend's voice was so plaintive when he said these few words that I feared he might resort to the most drastic measures. He was so depressed that if the idea of a crime were to enter his head, he would not have pushed it aside. Still, I wanted to try to comfort him.
"Paul, be strong. That woman is not worthy of your suffering. Don't think about your unhappiness any longer. Think of the future. Think that you have your whole life in front of you. Come on, make an effort. Let's go. I'll walk you to your door. You'll go home, go to bed, and tomorrow you'll come back and see me."
"Go home?"
"Of course, you must go home. It's late. You need to rest. You need to recuperate."
We were on a wide avenue. The moon, which had risen higher now, seemed even colder because the sun gives off more warmth when it is in the same spot. Trees cast shadows on the sidewalks. We were stepping on a thousand drawings of intertwined branches. I had a vague, childish desire to place my feet only on blank spaces, but I would have found no pleasure in it.
"Paul, we absolutely must go home."
My friend took me by the arm, leaned over to see me better and, hardly opening his mouth, whispered:
"You're leaving me?"
"I must. It's late."
"You're going to leave me alone?"
"We can't stay outside all night!"
His lower lip trembled then. The sweat already beading on his forehead flowed out of the wrinkles and dripped below his eyebrows. He released my arm and leaned against a wall, either so he would not fall, or else in order to feel something solid.
I realized how difficult it would be for me to leave him. Although my friendship for him was strong right then, it seemed ridiculous to spend a night consoling him. If it could have eased his pain, I would have done it. But, with me or without me, he would be just as miserable. And if he wanted me with him, it was not because he hoped I would be able to comfort him. He knew that all my words could not change his wife's decision in the least.
"Come on, Paul, we have to leave each other."
"You want to leave me?"
"Yes, what do you expect!"
"No, Jean, please, don't do that. Alone, I don't know what will become of me. I'll kill myself. Oh, I don't know."
He seemed completely distraught. He was not moving at all. It was as if he were no longer suffering, as if he had stopped fighting his pain, as if he were letting himself slip into unconsciousness.
Seeing him like that, I wondered if he was really determined to kill himself or if some sort of resentment was making him think I was the sole cause of his suffering; or perhaps he was trying to make me feel remorseful.
"Yes, I'm going to end my life," he murmured.
I, too, have suffered. I too have thought about killing myself, yet I never did anything about it. Why should I have taken his threat seriously? In a few days, he would cheer up. In a few days, we would both laugh about this episode.
"See you tomorrow, Paul. Be brave."
These few words that, in my opinion, should have left us in the same situation in regard to each other, brought him out of his dejection.
"So you're not my friend?"
"Of course I am, but what can I do for you right now? Show some fortitude. Only you can overcome your pain."
"I know that, Jean. But please take pity on me. Don't abandon me. Do you want to make me really happy? Let's stay together until tomorrow. I don't want to be alone. I don't think I have the strength. I'll go home with you. I'll sleep in an armchair. That's all I ask. You can't refuse."
"You're being ridiculous. How will that resolve anything?"
Suddenly Paul's attitude changed from imploring to remote.
"So you want to leave me, Jean?"
Although I sensed my friend had made a decision, my position remained the same.
"I do. It's late. We must part."
"Very well. Adieu."
He walked away without even offering me his hand. I had a foreboding of some misfortune. I am sure I'm no different from anyone else, yet I was afraid he would do what he said, that he would kill himself. I shouted:
"Where are you going?"
He did not answer, walking away with great strides.
"Paul!"
Already another streetlamp was lighting him.
For a moment I glimpsed the consequences of my refusal. He was going to kill himself. For the rest of my life I would be aware of being responsible for his death. And everything going on in my head became more and more confused as he walked away. I ran behind him.
"Paul, where are you going?"
"Leave me alone."
"Answer me! Be reasonable. Why are you running away like this?"
"I don't know, I don't know. Leave me be, I'm going to end it all."
He kept on walking, staring ahead.
"You didn't understand what I was saying before, Paul. Come on, let's go to my place. Tomorrow, everything will be sorted out."
He stopped and as he looked at me, he gradually realized what I had just said. He did not smile. Yet his face brightened. I took his arm and without a word we started off toward my place.
An automobile on its way to Les Halles passed very close to us. In the pure, freezing air, it left such a circumscribed scent of vegetables that when we took one step to the side, we could not smell it anymore. In the middle of the sleeping city, beneath the sky, we were alone. The moon had disappeared. And without it, as if they lacked a leader, the stars seemed to be in disarray.
## WHAT I SAW
I don't often write on an impulse like this. Something very serious has to have happened to me for me to decide to do so. So I shall ask, dear sir, for your indulgence. It is not an author you find before you. It is a man who is suffering and who is seeking the one word that will explain everything.
Slowly I had recovered from the great shock I'd had. Everything was going well. I felt strong again and then, suddenly, once more I began to doubt.
It would be impossible to explain why I am overwhelmed by anxiety. It returned, all-powerful, without my having any say in the matter. I was at home reading a book when, for no apparent reason, I realized I had not been mistaken. I tried not to think about it anymore, but you know that the harder you attempt to forget an ordeal, the more it clings to you.
Yes, I was reading a book that was as interesting to me as any book can be. I was so deep into this novel that I forgot where I was when, all of a sudden while turning a page, during that brief moment of distraction that interrupts the story with each new page, I had the clear realization that I had not been wrong.
I had seen the thing with my own eyes and as a result, it was true. My girlfriend could deny it all she wanted, but because I had seen it, it was true. The proof that I was wrong is all around me. My friends, to whom I made the mistake of telling this story, disagreed with me. My girlfriend's parents hinted that I had taken leave of my senses. Even my Henriette, after having heatedly defended herself, in the end simply shrugged whenever I mentioned this scene.
And so I managed, by the strength of my will, to doubt my own eyes. Gradually I forgot what I had seen. I forced myself to think I had been wrong. Life became bearable again. My girlfriend was ever more loving.
And now, in some idiotic way, I have begun thinking about this episode again. And so, all my efforts have been in vain! That painstaking and salutary process I suffered through in order to find peace was for naught!
Ridiculously, I again find myself anxious and desperate, like on the first day.
Yet I believe I was wrong, that my girlfriend is innocent, and that I was the victim of a hallucination. I want to believe this, even though my eyes will not let me. But despite all my efforts, I feel I will always have before me that ludicrous vision that pains me so.
This is why I am writing to you, so as not to be alone with my doubts. And perhaps for you to give me some advice. I must confess that I feel the need to ask you to forgive me for writing. When a man suffers as I am suffering, writing should not be a consolation. Forgive me, dear sir, for speaking to you like this. You are not used to such confessions. They seem to you some artifice meant to hold your attention, whereas in reality they are the proof of deep despair. It's true. I feel some embarrassment in writing. I know I shouldn't tell you this. One never admits that the person who is writing to us is doing so reluctantly—and with good reason. If, at the theater, an actor were to say he did not want to play his part, that it annoyed him to do so, I admit that I, like any theatergoer, would boo him off the stage.
But this, I must say, is a different situation. I am suffering as much as a man can suffer. And I am not writing to entertain or interest you, but simply to ask you what I should do.
I will expose the facts one by one, very clearly. I will tell you everything I know about my girlfriend and will ask you, afterwards, to tell me if I was mistaken or not. I am addressing myself to you because as an outsider you will be able to be an impartial judge. It is in the interest of my parents and friends for everything to work out. They know me. They know I am impressionable. And they will believe me less than you, who do not know me.
Because you have agreed to hear me out, I must first tell you what happened. You can see from the tone of what I just wrote that I am a sincere man, that I do not lie. I therefore beg you, while I am telling you this story, not to think you need to know my girlfriend's version before you form an opinion. Only the spineless want to know the pros and cons in order not to take sides. So I am asking you to judge this story simply through what I tell you, otherwise you will cause me great pain.
I shall recount the story you are about to read as if I were not the main character. I shall have no bias. On the contrary, I shall not mention anything that casts me in a good light. I shall lay out clearly everything that does credit to my Henriette. You can see that all I long for is to be wrong.
So I will begin. Pay close attention. Do not skip over anything, because my happiness is at stake. Some other time, I will write a long letter to amuse you, a letter full of youthful imagination. And if it annoys you, do not finish it. It will not matter. But today, I beg you, pay attention. At the risk of repeating myself, let me say yet again that my happiness is in your hands.
* * *
My girlfriend is as sweet as an angel. I must tell you that, although she was pure when she gave herself to me, she did not wait until we were married to abandon herself and I am open-minded enough not to reproach her for this. It would be human enough for me to use this fact to degrade her in your eyes. Believe me, I see nothing in this proof of love that could allow what my dear Henriette did to be predicted. If she gave herself to me without our being married, it is my fault.
A thousand signs prove to me that my girlfriend adores me. She has forgiven me what many women would never have forgiven. Even though she is beautiful, she recognizes that a man's lapse is not as great as a woman's. Naturally, she did not say this to me, but I felt, deep down inside, she knew it. When in the past I did what I should not have done, she was not angry with me, but rather, with man's very nature. And this fact alone demonstrates my girlfriend's immense goodness.
There are additional signs that make her pure in my eyes. Other men do not exist for her. I believe I can discern, from certain details, from certain attitudes, that they repulse her just as they do me. She often says exactly what I would say about a man were I a woman. She could not invent these feelings if she did not have them. And this is another reason why I love her so much.
A few times I asked her what she would do if I lost a leg. And she always responded ardently that she would love me just as much.
Please forgive me for providing such details but, when you want to prove a woman loves you, they are necessary.
There is something else that proves her love, and that is the way she admires me. She takes all my opinions for her own. Sometimes, when I have not finished voicing my opinion on a subject, embarrassed by the difficulty I have expressing myself, she will finish my thought differently from how I would have. As soon as she realizes this, she stops herself and is even ready to contradict herself until we agree. Is this not the mark of great love, to show such self-abnegation? Do you believe that if my adored Henriette did not love me she would follow my line of thinking in this way, step by step? No, of course not.
That's not all. So many things at every moment of the day and night demonstrate her love. When we are lying next to each other, I am always the first to turn away. Candy, cake, fruit—she always goes without in order to offer them to me and, if I don't take them, because I know how fond she is of them, she insists with so much love that I would be hurting her if I continued to refuse them. Nothing exists for her. She sees all of life through me. And when she arrives late for one of our dates, do not think it is because she is trying to be coy. She wants to imitate other women. She forces herself to be late because she is a woman and sometimes she is afraid she will lose me if she is not enough of one.
No, my Henriette, you did not do that, and yet...
One day she asked me if I ever had the feeling when I was away from her that I had not been as kind to her as I could have been. Without thinking, I said no. How can you detect in a question asked in an ordinary voice everything that someone expects from your answer? She became a bit sad. She did not say anything right then, but later in the evening she told me I was not kind, that I did not love her as much as she loved me. And she added that whenever she was away from me, she had the impression she had not pleased me enough.
Often she reminds me of things I said that I had forgotten and that she had thought about for a long time without my suspecting. Her sweet little brain works tirelessly to make me happy.
With her, as with little children, I never mention death. But I have the feeling if I asked her to die with me, she would. She led me to understand as much without pronouncing the word "death," out of modesty.
Now that you are familiar with my girlfriend from what I have said about her, I ask you to believe the portrait I have painted of her. Everyone will only have good things to tell you about her. Love has not deformed my judgment. This is how she is. And although it may be difficult to believe the portrait that one person paints of another, it is less difficult than believing in true love.
* * *
You know her well now, or at least you know how much she loves me, and that is what is important. So I am going to tell you what happened.
This is what happened. Two months ago, I was not feeling well. It was a Friday. The day was a cold one, but the sun was shining in the blue sky. We'd had lunch at home. We were just finishing up when Henriette came over to me and kissed me.
"Darling, will you let me take a little walk?"
"Of course."
"I'd like to buy a few things."
"Shall I come along?"
"Why not, my darling?"
Then she changed the subject, busied herself with this and that and, picking up one of my books, sat down in an armchair. Jokingly, I said to her:
"You're going to know that one by heart!"
Indeed, she only reads the books I have written, and since there aren't many, she reads them over and over.
"That's what I want, my love. I am jealous of your thoughts."
I did not really understand what she meant, but I felt she was trying to make me understand that my work represented a rival to her.
I know that, even though she loved me very much, what she said was not completely sincere. She said it because women are supposed to be jealous of their husbands' work. But I am indulgent. What is the use of taking offense at that? One shouldn't ask too much of a woman. And then again, this lack of sincerity is also a kind of love.
She sat back down and continued reading. Although she admired my writing, she closed the book before the end of a chapter, stood up, and said to me:
"You are really amazing! You notice everything. Well, I'm going out, darling."
"Don't you want me to come with you?"
"Yes, of course. But wait, there is still something at the end of your book that I want to reread. You know, the story about the unfaithful wife. It's amazing. Don't tell me you haven't been acquainted with a woman like her."
"You're mad, darling! You know how trustworthy I am."
"Still, there's something of you in this story. You are a bit like the husband."
So she reread the story of the unfaithful wife. Then, getting up, she went to dress without saying anything. She came back a few minutes later.
"Good-bye, darling. I'll be back around six o'clock. Be good and work well."
"You don't want me to come along?"
"How foolish you are! You are feeling poorly. You told me yourself you have a headache. It's cold outside. You have a slight fever. Give me your hand. See, your hand is burning!"
"Yes, but if I dress warmly?"
"I don't think it would be wise. A man like you must be well cared for."
"You know I don't like to stay home alone, darling."
"But I'll be back before six!"
And she left. It is obvious she did not want me to accompany her. But I pay this no mind. I understand that a woman can feel the need to be alone from time to time. It could even be that she truly did not want me to go out because I was feeling poorly. Perhaps she was thinking about my health, perhaps not. She wanted to be alone for no reason; she wanted to be alone for a variety of reasons. I know that just because one is hiding something, it does not mean that one is guilty. She could easily have been hiding the fact that she was going to see someone, meeting a girlfriend, without necessarily being unfaithful to me.
And so, I soon stopped thinking about her. Have you noticed that it takes several hours of absence before you think about the woman you love when she has gone out freely, happily, with her errands to run?
I sat down at my desk with the intention of writing. Do not think some vague suspicion was keeping me from working. I assure you I was not thinking about her. If I was incapable of doing anything at all it was more because I felt lazy than because I was worried.
To my great sadness it was then that, bored with my indolence, I decided to go out.
* * *
I shall always remember that radiant winter afternoon. No wind. A blue sky that grows dark before the evening papers come out. A pure sky where the sun seems to be an intruder. A white dust that surprises you because the previous day it was raining.
I was strolling calmly. It was pleasant to feel that my fever gave me permission not to hurry. Like a convalescent, I walked down a boulevard, taking interest in small things. Whenever our life is peaceful, whenever everything smiles on us, how agreeable it is to take interest in small things! You stop, you look. No one pays attention to you. These small things don't really interest us. It's our soul that is content with simple things, our soul that wants to find its youth again because it is happy contemplating small things, for no reason, simply not to think.
Such joy in being alive! And to imagine that we struggle to push away everything that could prevent us from moving along like this, gently, slowly, toward some intangible goal, almost unconsciously, happy to listen to the sounds, smell the aromas, see the light, touch a few objects.
A clock chimed. I did not count the chimes, but I sensed from the duration of the ringing that it was four o'clock.
And at that moment, dear sir, something horrible occurred. Have the kindness to read what follows attentively. I must tell you yet again that my happiness is in your hands. You know what a great responsibility it is for someone's happiness to depend on you. Think of a person in your life who made you suffer. Think that, in his shoes, you would not have acted as he did. I am not asking you to do for a stranger what you would have done for a loved one. I'm simply asking you to attempt to understand and advise me.
I looked at a storefront. I looked at it distractedly, like people do when they have no one with whom to share what they notice. Then I turned around. Now you will find out what happened.
For merely a second, I saw a taxi pass close to me and in the taxi, my girlfriend was kissing a man.
You've grasped what I just said. I saw a taxi and, in this taxi, my girlfriend was kissing a man. She was blocking my view of him, but not entirely; I could see that he was hatless.
I swear on everything that is most sacred in the world that I saw my girlfriend kissing a man in a taxi. I swear. I saw them. He was letting himself be kissed. It was she who was leaning toward him. The taxi passed a few feet in front of me. I saw them. I'm sure of it, absolutely sure. Why would I say that if I had not seen them? I even remember today, two months later, all the details with extraordinary precision. She was to the left of this man. And her left knee was higher than the right and hid the man's legs. I did not have the time to see her hands. I don't know where they were. But, on reflection, I really have the impression that her right hand was behind her companion's back, while her left hand must have been holding him around the neck. There is no possible doubt. She was kissing a man. I saw the bright color of the hat she had put on before going out. I saw her, my girlfriend in this taxi and I also saw a bit of the man she was kissing. Yes, it was her. But, then, I just don't understand. If she doesn't love me, why doesn't she leave me? It was her. I saw her. I was not thinking about her when I saw her. Otherwise it would be easy to imagine that, since I was thinking of her so vividly, I gave her features to the first woman I saw.
And now, since I have offered you all my certainty, let me tell you again that it is true: I saw her in this taxi; it was her.
I went back home, completely demoralized. Before my eyes I continually saw the inside of this taxi that in my mind—a bit dark, lit from the front, with its cushions—resembled a small bedroom. I even imagined flowers in this taxi, flowers I had not seen. It is impossible to describe what I was feeling. I would have to choose among a thousand fleeting thoughts. I need to present you, dear sir, with a few of these inconsistent thoughts that, in my head, followed one another with dizzying speed. And if I could manage to sort some of them out, to see them separately from the others, it would seem, by their insignificance, that my pain was not as great as I claim. So I shall not describe my pain. Can one really portray suffering with words? In this account, I don't think so. I am too removed from what happened. Any perfect description of pain presupposes an effort I can no longer make. I can only write as I am writing, just clearly enough for you to understand me.
I went home and lay down on a bed. Remaining motionless seemed odious to me, but by forcing myself to lie down, I wanted to prove I was still in control of myself.
Until my girlfriend returned, I never ceased thinking about her. No, I had not been mistaken. If I'd had even the slightest doubt, I would have done everything I could to fuel it until it became a certainty. But there was not even the shadow of a doubt. It is dreadful to find yourself confronting reality in this way. No matter what line of reasoning you come up with to forget it, it reappears quickly, more real than ever.
I spent two interminable hours like this, thinking, all the while waiting for the one I love.
Suddenly the door opened. She was there.
"I've been looking for you everywhere, darling. Why are you here? Do you feel ill? You should have undressed and gone under the blankets."
I did not answer. I was torn between the desire to tell her everything I had seen, immediately, and the desire to remain silent in order to hold onto a reason to be sad, in order to take an incomprehensible pleasure in hearing my girlfriend lie. I hid my confusion behind an imaginary headache.
"Get under the blankets, my love. If I had known you were so sick, I would not have gone out. I would have taken care of you. Lie down now. I'll make you something warm. And I'll sit next to you and read you the papers. I have never seen you so ill. What's wrong? Do you want me to go get a doctor?"
"He wouldn't be able to cure me."
My voice was filled with sadness as I pronounced this sentence. In my mind, it had a different meaning than the one my girlfriend heard and, like all such sentences, it required a disabused tone. I like those sentences, and you will see what I did with them.
"Go lie down now."
"What's the point?"
"Come now, don't be so discouraged. The minute you are a tiny bit ill you think you're going to die."
"I wish it were true."
My girlfriend, despite all my efforts, did not notice that my pain was entirely emotional. I would have wanted her to see it for herself so I could deny it feebly, and in the end explain to her why, so that she could reassure me. But she did not notice.
She undressed me by force, as badly as I would have undressed her.
"Now lie down. Close your eyes, my darling. I'll bring you something warm."
I obeyed. I felt I would not be able to keep what I had seen to myself. In spite of how sure I was of my girlfriend's unfaithfulness, I still wanted to believe that a word from her would dispel my certainty.
"Henriette, what did you do this afternoon?"
"Errands of no interest to you, my little lamb. I brought you the papers. You see, I thought of you."
"But what kind of errands?"
"How jealous you are, Jean."
"I'm not jealous, my darling. I'm interested in everything you do."
"Well, in that case: I went to the milliner's. Then to Anne's. We went for a walk together. Then she felt ill. We went into a café. And you know, when I was at the milliner's, there was some old man who waited for me for at least an hour at the door. If you had seen him! He was looking at hats, but how he looked at me! In the café, it was the same thing. Two young men wanted to sit down at our table. You can imagine how Anne, who wasn't feeling well, sent them packing!"
"And if she had been feeling well?"
"Oh! You are mad! You see everything in a distorted way. You know I would never talk to a man I didn't know."
"And you didn't go down rue Saint-Lazare?"
As I said these words, I stared at my girlfriend, as much as one can when one is lying down. She answered me without hesitating at all:
"No, why?"
"Because I saw you."
"You saw me?'
"Yes."
"So you went out? That was not wise, sick as you are. You should have told me you wanted to go out. We would have gone out together."
"I saw you."
"You're mistaken. What would I have been doing on rue Saint-Lazare? I was not even in that neighborhood. You dreamed it, and it doesn't surprise me, my adorable little lamb."
"I saw you in a taxi."
"Well, that takes the cake! In a taxi, now! I swear I wasn't. If I had taken a taxi, I wouldn't hide it. And why would I have taken a taxi?"
"To kiss a man."
My girlfriend, who was stirring a cup of tea, stopped. She looked at me with large, surprised eyes in which there was that hint of calm that precedes indignation.
"To kiss a man?"
"Yes."
"My poor Jean, what is the matter with you? You are going mad, mad, mad. How can you think such a thing of me? Me, kiss another man? So you take me for a streetwalker? You are mad, completely mad."
"I saw you."
"Listen, Jean. You don't know what you're saying. You have a fever. You are so jealous that you're losing your mind."
"I saw you. Do you understand what that means? I saw you, you who are in front of me. I saw you kissing a man."
"You're lying. I swear on my life that I didn't take a taxi, and that I have never kissed a man besides you."
"But I saw you."
"That's impossible. What would keep me here if not love? We are not married. If I loved someone else, I would not be able to put on such a loathsome act, I would be incapable of concealing it. You know how frank I am. If I loved someone else, I would tell you. Even if it would make you suffer, I would tell you. You could not have seen me. It's impossible. I belong to you alone."
"I saw you."
"Perhaps you saw someone who looks like me."
I had been waiting for these words for several minutes and yet I did not know how to answer them immediately. I was afraid of them. I knew that they alone were capable of making me doubt my eyes without providing proof of my girlfriend's innocence.
You, dear sir, will perhaps think what my friends thought, that I fell victim to a resemblance. When one is trying to console someone, one always manages to make statements that one would not believe oneself. To claim I fell victim to a resemblance is such a statement.
Let me tell you, sir, that I recognized my girlfriend, not just her clothing, but her neck, the color of her hair.
"You are the one I saw in the taxi."
"It was not me. I told you exactly what I did and when. You can ask Anne if I spent the afternoon with her. You can come with me to the milliner's, and to the café where we went when she felt ill. You can ask the waiter, if it's the same one, what we had to drink. I cannot do any more, my Jean, to prove to you that I am faithful."
I listened to these words without believing them. I know I would have got lost trying to figure out my girlfriend's schedule. Yes, I could have seen a waiter who would have told me "It seems to me that I waited on those two women," or who would not have remembered. My girlfriend would have shown me the table where she sat. But what would that have proved? The fact remained that I had seen her, in this taxi, kissing a man.
Or if she had said to me, "Yes, I took a taxi with Anne and I kissed her," I would have believed that. I did not have enough time to see the person who was with my girlfriend and to be sure it was a man. But the fact that she denied so obstinately that she had taken a taxi proves to me she was unfaithful.
"I saw you."
"Listen to me, Jean. I swear to you on our love, on your life, on my parents' life, that I did not take a taxi, that I was not unfaithful to you, that I love you more than anything in the world, more than my family, and I am ready to do anything you tell me, I am your slave and your wife. I swear to you, my love, that if you were to die tomorrow, I could not survive. You are my sole joy in this world. I only live for you and through you. Look me in the eyes. You see, I don't lower my lids. Do you believe if I had done what you say I would not die of shame beneath your gaze?"
Dear sir, I wound up believing my girlfriend. I wound up believing her but, in spite of everything, some doubt has remained in me. It is this doubt I am asking you to drive out. I repeated word for word what the woman whom I still love said. I also told you that, while I wound up believing my girlfriend, I am still sure I saw her in the taxi. It seems that nothing can shake this certainty. And yet, Henriette loves me so, she is so honest! Let me tell you as well that, if you had been in my shoes, you certainly would have recognized my girlfriend. You would have recognized her as I did. So it is useless to tell me that perhaps I did not see clearly.
Before you can reach your decision, you probably think you will need to know my girlfriend better. It is not worth the trouble. You know her. She is unable to do anything behind my back. She loves me. You were able to see that. Do not think I am blinded by love. She is exactly as I have presented her to you. And as sure as I am of having seen her kiss someone else, I am just as sure of her total love.
I am waiting, dear sir, for a letter from you that will allow me to know the truth. If you are not sensitive to my pain, perhaps you will respond with indifference. Know that I shall read your words with the same attention, for I am hoping nonetheless to find in what you say the word that will bring me peace.
## THE STORY OF A MADMAN
To put the reader at ease, I need to state from the start that I am not crazy. And if anything could be proven by words, the fact I am affirming my lucidity should be enough to show I am in full possession of my faculties.
I know I may seem crazy at times. It's true, it doesn't take much for that to happen. But let's be clear. To be sure, I may often seem crazy, but not so much so that two people would bother mentioning it to each other. I seem just crazy enough for one person to think so without his neighbor thinking so as well. And if I always provoke this feeling by some ridiculous action or question, I must say that I manage to stop myself when I sense that this inner misgiving might be externalized. If I do this it's not to amuse myself, nor is it to make fun of the people for whom I am putting on an act; nor is it to ennoble myself in my own eyes by inflicting some kind of humiliation on someone. I do it simply, perhaps precisely because I am crazy.
No, I am not crazy. I just wrote what I did because I was driven by the need to explain illogical acts. And when one has such a need for clarity, I guarantee you, one is not crazy.
Still. None of that is important. It has no relation to the story you are about to read.
But what is oddest of all is that I have no willpower. I have always done just as I please. Fortunately, I am a good person at heart. I have no inclination to do harm. Otherwise I would surely have come to a bad end. I would have gone to prison. I would have killed people who had not done anything to me.
It's quite funny. It's funny because you will see with what willpower I have acted. They cried, they begged me, and I did not bend. It's funnier and funnier. Honestly, I am both an odd and a likable guy. I am a man who will no doubt succeed in life, who will do great things.
But wait. Let us proceed in an orderly fashion. A person who does not put his mind in order is lost. Without order, nothing is possible. I who, according to what some people claim, am half mad, will show you how reasonable I am.
Above all, you need to understand who I am. I never knew my mother. I was raised by my father, which made me precociously mature. You cannot imagine how good I am. My goodness is so immense that everyone has always made fun of me, and the most incredible things have happened to me. I would not say I am incapable of killing a fly, for does any man exist who has no fits of bad temper? What makes me sad is that I have never been able to provide examples of my goodness. It seems to me that goodness held up as an example is not goodness anymore. But that's something else entirely. Don't be afraid, I am not losing my train of thought. I will recount the story you are about to read without going off track. The only thing I can tell you is that I am truly good. I assure you. I swear it. And what I swear is true. I am not like some other people who swear to anything on their parents' lives.
I cannot bear to see someone suffer, and as a result it would seem I am incapable of doing any harm. But this is not so! You will be surprised at what I did. Suddenly, I discovered in myself unsuspected strength, which is the mark of great youth, and I plunged everyone around me into the deepest distress, including my parents and the woman I love. Yes, that is what I did. And now that I am free, now that everything is over, I wonder if I won't regret it.
How could I have done that, I who am so good and whose heart melts into tears at the slightest pain felt by someone else? It's incomprehensible, and that is why I am writing this story. Perhaps there will be someone to explain it, to feel sorry for me, because there is no question that strangers can be closer than friends.
As I said earlier, we are going to proceed in an orderly fashion so that everyone can understand what happened. But here's the thing. At bottom, nothing happened. My poor head aches. I wish everyone could understand immediately what is seething in my brain without my having to write about it. That would simplify everything. What can I do? I like what is simple. As soon as I attempt to explain complicated feelings, I become confused and begin to lose my concentration. It's very funny. I can see the thoughts that are in the foreground of my mind very clearly, but as soon as I try to go behind them, I find myself in a haze.
So, in order to be able to finish this story, I am going to recount it as simply as can be. It would be ridiculous to begin like this, very clearly, and then become confused at the end without the reader having been able to understand what I meant. And not only would it be ridiculous, but there would be no hope for me in the reader's eyes. It would be better for me to write calmly, taking my time, and then you would understand me and say I am right. You'll see that the people who think badly of me are wrong and that, despite what I did, I am a reasonable man.
If you doubt this, you should tell me. I won't get angry. I am intelligent enough to grasp that everyone may be right. No, I won't get angry. I will then find out that something in me is not normal. I will look after myself. I will take trips, I will get my mind off things, and later, when I feel better, when no one will be able to speak badly of me anymore, I will tell this story again. And this time, people will be forced to understand what I meant.
Now I will begin. Pay attention. Let's be serious. I am about to begin. The first thing I have to say, which to my mind is very important, is that everything you are about to read is true. I am inventing nothing. This whole thing happened to me yesterday. I plunged everyone who knows me into sadness. And for the first time in my life, I am not suffering. I was right to act as I did. If I had hurt only one person, I would be unhappy today. But since everyone is crying because of me, I am smiling. I am alone. I am not suffering. Everything was perfectly calculated. We shall see what will happen now. I am gathering all my strength to remain in this state. I am fine. Everything happened the way I thought it would. I did not have to face any unforeseen circumstances, which no doubt would have disconcerted me. Now, my life's goal has been attained. My happiness will not be compromised by some blunder. What will happen tomorrow? I have no idea. But for the moment, all is well. So let's not talk about all that anymore. Let's examine the facts.
* * *
Yesterday morning I woke up as usual at about eight o'clock. My eyes were not shocked by some unexpected position of the hands on my watch. So everything was starting out very well.
If you had seen me in my bed, you would have laughed. I did not move when I awoke. I did not move a muscle. I gazed at the ceiling. I closed my eyes again for no reason. I opened them again. I closed them again. It was funny.
None of that is serious. There is something much more serious. I made my family suffer, and my friends.
Now I am writing. You can see that I am writing because you are reading what I write. Well, they are suffering, they are suffering because of me. But I must not feel sorry or else I won't finish this story, and that would be a shame.
Yes, had you seen me in bed, you would have laughed aloud. Had you seen how serious I was when I woke up! It was as if a scientist had awoken. Well, I don't want to dwell on that. It's only of secondary interest. What follows is much more serious. Now I must write seriously. You can imagine that if I am writing now, with my head aching, it is because I have something important to say, otherwise I would go out for a stroll.
So, it was eight in the morning. I waited a few minutes before getting up. That's natural. Everyone does that. I don't have to hide it. And I got dressed. Yes, I got dressed. So you see! If I were what some people think, I would have gone out naked. I dressed slowly, but I did dress. I was in no hurry. I had the whole day before me. I ate breakfast as usual. I put on my hat and went out.
Now pay attention! We are nearing the serious, very serious thing. No more smiling. Don't think that it bothers me if someone smiles while I am speaking seriously. I don't care. But now I am asking you, humbly, not to smile. It is too serious. Pay attention, reader! Read these lines by yourself. No one should be around you. I too am alone. We are both alone. Pay attention. Lock the door. Mine is already closed. You will see what happened. Shhh! Shhh! Listen. Don't get up, don't move. So. I was at my father's. I was allowed into his bedroom because I am his son. So I saw my father in his shirtsleeves. He's quite ugly like that. I can't look at him. My father must be completely dressed, or else I feel ill at ease.
"Father!"
He turned around. He has a thick black beard and almost no hair on his head. He was surprised. He did not know why I had come. He suspected nothing. Ha! He did not know what I was about to tell him. But I knew. Everything was prepared in my mind. Nothing could make me change, not even his surprise. Nothing. I was determined. Ha! My poor father. He was about to know and he did not know yet.
"Father!"
"What is it?"
Don't think that I started to laugh. No, I am not like that. I came because of something serious. So I had to act appropriately in his presence.
"Father, I have something to say to you."
"Go ahead, go ahead."
Watch out! Watch out! You're going to see what I said. Ha! But it's true. I said it in a firm voice. I could have weakened at the last moment. But I did not! I am finished with weakness. I am fed up with always changing my mind in front of someone. I want to be myself. After all, I'm a man. I fought in the war. I have seen dead people. It's over now. Ah, yes. It's good to obey when one is a child. Now everything has changed. My life is before me. You'll see right away that I'm not lying, that I am also capable of great deeds. Ha! Now no one can say that I have no willpower.
"Tell me what you need to tell me!"
"I don't want to see you anymore, Father."
"You don't want to see me anymore?"
"No. That's what I've decided."
"You're mad."
"No, I am not mad. If I were mad, I would know it, and I would not have made a decision like this."
"What's wrong, son? You must be ill."
"No, I don't want to see you anymore."
"But what is it? You haven't slept. You're losing your mind."
"It's settled."
"I'm going to have you locked up."
"I am twenty years old. I am free to do as I please."
"Explain yourself."
"I won't."
"Why, why? I don't understand what you mean. Not see me anymore?"
"Yes."
"There's no reason. You are free. You come here whenever you want."
Ha! I left very quietly, without answering my father's questions. He was still speaking and I was already outside. I could hear him through the door. Hmm, I was wrong to laugh. My father must be suffering! Poor Father, I loved you so! You alone had affection for me. Poor Father. Now you no longer have a son. You are suffering. You must be crying. And I, I am here in this locked room. Ha ha! You no longer have a child, my dearest father, you who thought I would always be beside you to brighten your old age! My smile will not warm your aged heart. The name you bear and of which you are so proud will die with you. You won't know the joy of welcoming your child in your solitary retirement. Just when you thought you would at last live out peaceful days, when it seemed that the rigors of life would leave you alone so that you could regain the childlike soul that a man must have in order to die without suffering, once again I plunged you into pain. Farewell, poor Father. Don't be angry with your son. He did what he had to do. Soon death will come and set you free.
* * *
But this story is not finished. I have a girlfriend whom I love more than anything in the world. She is beautiful, more beautiful than every other woman because I love her. Not a minute goes by when I don't think about her. I love her so much that when I am far from her it always seems to me that I have not been as kind as I should have been. Now, like my father, she is suffering. I am sure my beloved Monique is suffering.
I went to her place. Monique is the only woman I have ever loved passionately. Monique is blond, her eyes are blue, and her skin so delicate that the slightest stroke leaves a mark.
Monique! I'll never see you again. Poor Monique, how you will suffer, you who should never have known the ugly side of life! You are twenty and already you have lost all your illusions. You were made to go through life in a glow of happiness. And I, coward that I am, no longer wanted to see you. I've shattered your youth. Monique, do not forgive me. Suffer in silence until all is forgotten. Yes, suffer, my beloved. You will still know moments of happiness when you remember me. You will recall my silly, childish words. They won't make you laugh as they did before, but they will remind you of so many delightful moments.
I went to Monique's. My heart was pounding as I was about to knock on her door. She was asleep when I entered her bedroom. I saw her, half-naked, in her bed. I looked at her for a long time before waking her. You don't know why I looked at her? Ha! It was so she would be embarrassed when I told her. Yes, I looked at her for a long time. She is so young and healthy that her sleep is restful. I was not simply looking at someone who had slept well. I was looking at a young woman completely prepared to live life again until the following evening.
I went up to the bed and woke her with the tenderness of someone who has been close to the sleeping person for a long time. She opened her eyes and immediately the blue of her pupils dazzled me. She stretched a rosy arm above her curls, an adorable arm, an arm that extended from shoulder to hand without a bend at the elbow. She raised a knee beneath the sheets with no trouble because her legs are not long. Then she turned her head toward me. A loving smile appeared on her face, still warm from trouble-free dreams.
She had no idea, poor Monique, of what I had come to tell her. She thought I was going to lie down next to her, kiss her, and that she would ask me to make silly faces to amuse her.
No, I am not like that. When I've made up my mind about something I do not back down. You have to have absolutely no willpower to assign yourself a task and not carry it out. You have to be a man with no backbone. I am not like that. You'll see exactly what I said to my beloved Monique, what I had decided to tell her. I did not weaken my resolve at any moment. Nonetheless, I could have felt sorry for her. I am not heartless. When someone cries in my presence, well, it moves me. No, I resisted. I wanted to show once and for all who I am.
"Monique!"
"Fernand."
Watch out here. I am going to bring you deep inside me. You will understand. Listen carefully. I love Monique. She loves me.
"Listen, Monique, I have to leave you."
"Where are you going?"
"I'm leaving you for good."
Monique sat up in bed. For the first time, she looked me in the eyes. But she could not read anything there because my eyes do not betray me. Ha! Eyes are like a nose. They are always the same.
"Are you crazy, Fernand?"
"I am not crazy."
"But where are you going?"
"I'm leaving you for good."
Then, getting out of bed, she wrapped her arms around me.
"Fernand, Fernand, you're mad. What's wrong? You know I love you and only you. Do you have anything to reproach me for? Tell me so I can defend myself."
This is when you should have seen me. I was absolutely calm faced with the immense pain of the woman I love. I did not move. Although she begged me to speak, I said not a word. Ah! I was exactly as I always was. This, especially, is how great men are recognized. Major events have no hold over them. They are always the same. I was insanely in control of myself. Perhaps I was pale. But it takes so little to grow pale.
"Fernand, Fernand, you know I adore you."
"I know," I said with a deliberate smile.
"How could you, my Fernand, suspect me of anything? I love only you. You are the most beautiful man to me."
Poor Monique, how you suffered!
But I did not weaken for a second. I know all too well that if I had simply consoled her, I would have lost my confidence. I absolutely had to remain pitiless.
"Fernand, I cannot believe it. You're joking."
"I am not joking."
"Well, I just don't understand. Don't you love me a lot?"
"Of course I love you."
"I don't understand, I don't understand. Is it because of André? You know I don't love him. I love you."
Poor Monique! What did I do to you? From the purest happiness, I sent you into the greatest despair. You cried leaning against me. I saw the tears fall from your eyes. And they were the saddest tears of all because they fell from your eyes that remained open. Poor Monique, you're alone now. You no longer have any reason to live because we are no longer together.
"Farewell, Monique. Be strong! One must know how to be brave in the face of unhappiness."
"So it's true, Fernand?"
"Yes. Farewell."
"But I didn't do anything to you."
"Farewell, Monique."
And I left. I shut the door very gently so that my darling would not think I was angry. I left. Yes, I left. You see, it's not difficult. And to think I believed it would be impossible to leave someone. It is not. It's very easy. You need simply feel no pity. I know this kind of coldness cannot be acquired from one day to the next. But after a time, one manages. You can see that deep down it was not so terrible.
* * *
I spared no one. When I left my beloved Monique's, I went to see my friend Léon. He was not home. I sat down at his desk, took a sheet of paper, and wrote:
My only friend,
I would have liked to see you one last time. It was not to be. This is sad, because I am leaving you forever. Farewell, Léon. From now on you will follow a path different from mine. It was what fate desired. But know that in difficult moments I will think of you with all my heart. I will always remember our perfect friendship. It will be the invisible companion of my life. I will turn to it whenever I am in pain. I will ask advice of it. My only friend, you are my only friend! Your feelings for me were not overly tender. They were what they should be between men. But they were sweeter to me than all the complicated feelings of love.
You are my only friend and so you will remain in my memory. I am sure you will suffer from my abandoning you, but know that it was necessary. I wanted, I do not yet know why, to be alone in the world. This cannot happen without making those who love me suffer. I beg you to forget me. It will be hard for you. One cannot forget people one loves. They remain as alive as ever in our memory and those who follow do not chase them away. Love does not die. The years have no hold on love.
Above all, Léon, do not be angry with me for destroying your life. It would have been better had you never known me. You would have been happy. The future smiled on you and now, because of me, you are losing everything. My poor friend, how I feel sorry for you! You are deprived of all joy. You are alone and helpless. And if a final word can console you, let me tell you that I, like you, am alone.
Farewell, my best friend. Forgive me for all the pain I am causing you. Forgive me, because forgiveness is the only thing that can revive a man who is losing his life.
Farewell, Léon. Be strong.
Fernand Blumenstein
Ha! That's what I wrote. Never would I have been able to say all that to him directly. It was better he was not there. I left the letter on his table. When he comes home, he will find it. Before opening it, he will wonder who could have written to him. Then he will read it.
* * *
I also went to see my sister who just married a very commendable man who, it's true, does not like me all that much, but who in spite of this has always remained very courteous to me. My sister even suffered a bit from this state of things. It was clear she would have liked her husband and me to be good friends.
Yesterday afternoon I went to their home. And in fact they welcomed me amiably. They went out of their way for me.
You poor young couple! If you had known what I had come to say, surely you would not have been so cheerful!
I went into their place with my usual casualness and sat down. They asked me several questions to which I responded calmly. Monsieur Laloz, my sister's husband, came up to me and, placing his hand on my shoulder, talked to me of my future. He told me that if I was serious, hardworking, and honest, I would find an excellent position. He advised me to watch out for some of my friends and, with a great deal of tact, he let on that Monique was not entirely the woman who could understand and appreciate me. Then we talked about my father. I must say right away that Monsieur Laloz demonstrated a certain bias, rather excusable when you know my father was opposed to his marrying my sister.
Nonetheless, he lauded my father's good qualities, which proves that Monsieur Laloz is open-minded, generous, and does not hold grudges. My sister, who is older than I am, listened to our conversation without seeming embarrassed in front of me about the authority her husband had over her. I took note of everything they said to me and when on occasion I tried to express my opinion, if Monsieur Laloz continued to speak, I did not insist that he hear me out.
As he was coming to the end of discussing my father's intelligence with me—for in an hour of conversation one can touch on a variety of subjects—I said to him:
"You know, I don't want to see the two of you anymore."
My sister did not even raise her head. No doubt she had not heard, and Monsieur Laloz looked at me without any bewilderment whatsoever.
"You know, I don't want to see the two of you anymore," I said again, trying to give this sentence the tone of the conversation that had preceded it so that it would not seem too incomprehensible.
But it was precisely this tone, I believe, that caused Monsieur Laloz not to grasp what I was trying to say.
"You know, Monsieur Laloz, I don't want to see either you or my sister anymore."
My brother-in-law, who is in the habit of looking me in the eye, turned his head ever so slightly without taking his gaze off me to show me that he was truly looking at me.
"It's true. I don't want to see you anymore."
My sister, who had surely heard what I said, stood up and went over to her husband, asking him:
"What did Fernand say?"
Married sisters have an odd habit of wanting to learn everything from their husbands' mouths.
"I don't know. I don't understand. It appears your brother is saying he doesn't want to see us anymore."
"Did you say that, Fernand?"
"Yes."
"But why?'
"Just an idea I had."
"So you're not thinking about the pain you will cause us?"
It's true, you poor young couple! You were happy together, side by side in life, and I came to spoil your happiness. What will become of you now? Your life, which promised to be full of future joys, is destroyed. Without me, you would have had children, you would have watched them grow up, you would have loved them!
Later, when you had grown old, these children would have brought joy into your home, whereas now, because of me, you will live in sadness. Poor young couple! Be brave.
"Fernand, you wouldn't do that," said Monsieur Laloz, with a sad, sad smile.
I like my brother-in-law's sad smiles. They bring to his face an expression of pain mixed with bitterness. Poor sister, poor brother-in-law, I won't see you anymore. I pity you with all my heart.
"Speak to us, Fernand," said my sister, holding her husband by the arm.
"Yes, I'll speak to you. I came to see you one last time, to say farewell."
Monsieur Laloz leaned toward my sister and whispered:
"Leave him be. At that age, a person knows what he is doing."
Poor Monsieur Laloz! He did not realize how final my decision was. He thought I would come back tomorrow. He did not want to believe I was leaving the two of them forever, that it was the last time we would see each other.
And I walked out. My sister called to me in the stairwell. I did not go back.
* * *
I went to a shaded park. I don't know anything more delightful than the walk I took there. The sun was pouring its gentle rays on the plants, directly into the dust raised by the children's hoops. It was barely distorted behind its own heat. It held itself up in the sky without any system. Everything was calm. The guard made a comment to a child who was throwing white stones. He was wearing a uniform. Men in uniform are so charming in parks! They have all that space around them. And the statues. Why are they on pedestals? Why aren't their bare feet resting on the grass?
I sat down on a chair. I was happy. I had finally arrived at what I wanted, though I might have weakened.
You can see I am not as weak as they say. I do have willpower. There are people who let themselves go, who are spineless, whom events can influence. They are at everyone else's mercy. Not I. I have a lot of willpower when I want to. All I need is to will something, whereas others, even by willing, have no willpower. To do what I do takes courage. It is not just anyone who can plunge people dear to him into pain and find the strength to go on living without friends, as I decided to do.
And the funniest thing about it is that I am right. I am definitely right. What I did makes sense, otherwise I would not have done it. They are all suffering now. I've wanted to do that for a long time. Until now, I did not dare.
And you, reader, perhaps you are thinking that all this is not very logical. Right? Is that what you think? No, you find it all very clear. You understand what I meant. I left my parents, the woman I love, my friends. It's understandable.
If something does not seem clear to you, I can do better. No, it's not worth it. Everyone has understood.
Oh, yes! I just realized something. I know what you are thinking. You are thinking that what I did is not extraordinary, that it was not difficult. Yes, I am sure that's what you are thinking. But you do not know me. I am intelligent enough to grasp that lots of people could have done what I did. I sense that you do not admire me, am I right? Am I mistaken?
I have an idea. Tomorrow morning I will carry it out. And then you will be forced to understand. Above all, do not say a word.
It has to remain a secret so that no one will bother me. I will walk along the Seine, and then, well, never mind, I would rather not tell you anything now. I have my idea. No doubt everything will happen as I imagine it. And tomorrow I'll tell you what I did.
## THE CHILD'S RETURN
On the train taking me back to the village I had left five years earlier, all was silent. The heat was stifling on this July afternoon. The sun's rays must have been keeping pace with us, for otherwise—based on my memories of the long minutes I had spent motionless, magnifying glass in hand, browning sheets of paper—I would have been surprised that the sun was as strong shining through the windows as it would have been had the train come to a standstill.
I drew the curtains, which were the same color blue as the flagpoles. It was only three o'clock. I was unsettled by the idea that time was not passing more quickly even though we were traveling through the countryside faster than a man on foot.
Occasionally, in those spots where the rails were smoother, it seemed as if we weren't moving anymore. Only the metal ring of the alarm trembled, as if the jolt that had set it in motion had been strong enough to make it sway for several hours.
My natural apprehension prevented me from placing my feet on the foot warmers: perhaps they had become boiling hot since a few moments earlier when I had leaned over to feel them with the outer edge of my fingers, the most sensitive part. I imagined it would not have taken more than the flick of a switch by some conductor, in the head or rear car, for the heat to have been turned on again in the interval. This flick, I feared, could have occurred while the conductor was distracted, smoking a cigarette or reading the paper.
Through a window, held half open by a strip of cloth (cloth because of an administrative decree from 1917, when the soldiers had been cutting off the leather straps to use as uniform belts), the artificial wind of speed penetrated the compartment, coating our hair with coal dust. Field insects sometimes fell on the seats, slipping between the cushions as if into a chasm, which made me feel sorry for them like I did for the little worms in lettuce carried off by water from the faucet toward some dark sewer.
The tunnels came in series, like everything in life, like good luck and bad. Not knowing how long they were, I would hesitate to close the window. The smoke would surround us and, long afterwards, the taste of it remained on my tongue.
My neighbor was sleeping. At times he moved like a man awake, even though he kept his eyes closed. He would take a handkerchief out of his pocket, nodding his head, blowing his nose, all while he slept. Some remote force was commanding his limbs to move in order to find a comfortable position, but faltered when it came to tidiness and propriety. In a small railway station, the train stopped for three long minutes, which, even though they were scheduled, seemed to delay my arrival by three minutes.
When the passengers who disembarked had reached the road, we pulled out again. Bags in hand, they watched us go by.
From high up on a railway bridge, closer to the only cloud, the front tip of which was pointed so it could cleave through the air and follow us, we glimpsed small valleys, hillsides. In the distance steeples rose, their weathervanes crooked. The cars on the road seemed to be following a longer path.
We were traveling through an inhabited countryside where fields followed villages, a countryside furrowed by small streams crossed by wooden footbridges from which men were fishing.
This was the countryside of my youth, of illustrations to help you learn German, in which every object the small farmer uses is in its place, and nothing is lacking.
There were grindstones, wheelbarrows, thatched cottages, pitchforks, two horses pulling a plow, herds of cattle.
Cows grazed, chasing flies with their tails. Foals stopped in front of hedges, rearing up without danger to anyone in the vast pastures. An old woman carried a bundle of firewood so that the word "bundle" would stick in the children's memories more easily.
The landscapes we passed through—each of us thinking of arriving or departing, each of us filled with expectation or farewell—this countryside where we would never go except on train tracks unless by some impossible chance, but which cheerfully went on with its life nonetheless, filled me with melancholy.
A newspaper, which no one had bothered to fold back up and which had been tossed a number of times because it would not land properly, lay on the floor.
I tended to look for the absurd in everything, so I picked up this paper, bought in Paris in the morning and now doubly aged by time and distance, and attempted to find some stories from this countryside in it, stories whose journey had been pointless because I was bringing them back whence they came.
The people working in the fields had not yet read this newspaper. Being better informed than they were, carried along at uniform speed through landscapes where I would have stopped had I been alone, witnessing a rural life as private as the one inside houses—all this prompted me to dream, eyes half-closed, legs crossed so that no fatigue would bring me back to reality.
Passengers moved through the corridor in the direction opposite the train's, fearful as they walked between cars, gaining a few steps on the ground as we lost a thousand in the countryside. We did not have time to see the clocks in the stations. A train passing us in the other direction caused me a dreadful fright.
I was leaving behind friends whom I had not forewarned, as well as my landlady, because I am unable to part company forever with even the most insignificant people.
I was leaving behind a young woman, Julienne, whom I loved and whom, only nine hours after my departure, I already missed, and who seemed to me to have so many fine qualities, as if we had been separated a long time.
I was leaving behind a dark office where only one window looked out onto rue Drouot, and five others onto an inner courtyard. The window facing rue Drouot was not for me. I would have had to wait for seven employees to leave in order to become the most senior and sit anywhere near it.
I was leaving behind habits, odd little ways born of poverty, sudden awakenings in the half-light, the fear that people would not return what I had loaned them, bad restaurants, and indigence at month's end, prolonged by a day every month, that inevitably would have led me to total destitution.
I was leaving behind an ordered life, so well ordered that I was surprised, now, that it no longer existed. The person who would take over my room would not get up at the same time as I had. I would no longer buy my rolls at the neighborhood bakery. No one would see me anymore. In a month, after the customary time of an illness or a vacation, perhaps people would think of me for a moment, and then it would be over.
I was also leaving behind things, old things I had believed useful, and that I had never wanted to get rid of, even when I changed rooms: a pair of trousers, a metal can, a perfume bottle that was empty but made of heavy glass, photographs, worn-out shirts, letters and envelopes—I cannot tear up a letter, and I always keep the envelopes as if, without them, the letters would not be letters.
I had left everything. I was carrying only new things in my new suitcase. My brushes were wrapped in the day's newspaper. I did not want anything to remind me of my life in Paris and wanted to focus my attention solely on driving out bad memories.
I was returning to my family. I, their son, their brother, was going to see them again. In an hour, I would be among them. At first they would not recognize me. Then, with tears of joy, my mother would hold me in her arms—but not for long, because she would go fetch my father to tell him the good news.
In my mind, until it occurred to me that perhaps it would be my father I would first lay eyes on, it was without a doubt my mother who would be the first to forgive me.
But what is the difference? My father would kiss me, and he too would hold me in his arms, against his firmer chest, the feel of which was more difficult for me to imagine.
There were lots of animals—dogs and cats—at home the day I left. But, afraid of becoming sad, I was not thinking about them. Animals don't live as long as people. I did not know how old they were, so perhaps they had died. In fact, no one knew their ages. They were strays adopted by my mother, who was very kind.
Again I saw the small yard where as a child I would have liked to dig a hole as deep as I was tall.
Again I saw the rabbits. They would not be the same ones, but it did not matter because even if they had not died, I would not have recognized them.
Again I saw the well next to which I used to wash because I did not want to carry water anywhere or get my bedroom floor wet, and this would anger my father because he was afraid the soapy water that was absorbed by the earth would seep back into the well.
Again I saw the artichokes, so firm when they are raw; the low branches that allowed me to climb the trees; my collection of stamps, one of which was triangular; the little stream that ran right next to the house among the nettles; a sandy path for my bare feet; the donkey—bigger now; my kite, which was store-bought and made with whitewood rather than with young twigs; my bicycle that I took care of the same way I took care of myself, disregarding the dust that covered it, but very preoccupied with the ball bearings because of the predisposition of my mind to care more about fundamentals than form.
I heard the cock's crowing, the one at eight in the morning, more coppery because of the sunlight and the shimmering of the streams, the mallets of the washerwomen, the barking of the dogs. All these sounds became muted in space. They died slowly, mixing with the vibrations of the warm air without an unexpected echo, without reverberating on some zinc roof.
* * *
I recognized a farm, a barn. I had arrived. Because I was in the head car, I had the feeling the train was not going to stop.
The platforms were deserted. The clock, which I finally located, said 5:10. A warm, dry breeze caressed my face.
Although I had not told my family I was coming, arriving like this on a station platform where no one was waiting for me was vaguely disappointing.
The ticket windows were open. The account books, the jar of liquid glue, the telephones, the scales jammed full of bags, the announcements that were tacked but not glued out of respect for the ones underneath—all those things that are damp and sad in cities were cheerful here.
I found myself at the exit with a few other passengers. A railway worker held our tickets in his hand. It was as if we all had agreed among ourselves to turn in our tickets to the same man.
I had an inexplicable fear of coming home with a suitcase after an absence of five years, so I left mine at the baggage check.
I still had a little under half a mile to go, without a car, without a train, on foot; that is, I was sure to arrive in ten minutes, if there were no accidents or delays. My heart was pounding. I was truly happy that it was hot outside: my emotion would not show on my already flushed face.
Yet I did not dare leave the public square, where there were still people and where I could go unnoticed, to walk along the straight, open road where I could be seen from a distance.
The train left the station. So I had arrived. I had not lost any time. After a long journey like the one I had just made, I had a right to freshen up at the inn. Seeing the travelers there cheered me up.
Suddenly, I froze with fear at the thought that my father might walk in by chance and see me. The magnitude, the surprise of my return would be spoiled. I would not even dare to kiss him in front of the customers. He would bring me back to the house thinking I had not been in a hurry to see him. My mother would not welcome me in the same way. Foolishly, everything would be ruined.
I quickly downed an alcoholic drink. Flies were buzzing in the middle of the room. They were bigger than the ones in Paris. An open door gave onto a garden, onto the sky, onto my life to come.
* * *
The road that leads to my parents' house is lined with apple trees from which now hung wisps of hay that had been left by carts returning from harvest. Winter crows, black, slow, and sad, flew above a tree in the shadowless air.
I walked very quickly. Grasshoppers, trusting their weightlessness to exempt them from any pain when they fell, were jumping from the supple support of the grass to the middle of the road where their long legs, used to stubble, were ill at ease.
A dogcart drove by on its high, iron-rimmed wheels, between which a cloud of dust was rising at the spot where the dogs are attached. It had no license plate. It was as free as the air, as the fields, as the life I was about to lead.
A man, alone in an immense field, was reaping wheat. To my right, I could see the houses in the village from which, despite the hot weather, blue wisps of smoke were rising. They were fragile in a sky where they had too much room. You could sense that the slightest puff of wind would disperse them, not immediately, only after having caused them to fold in on themselves. A milestone reminded me that somewhere someone knew the road existed.
I now dreaded the moment I had been longing for a moment ago. I was afraid of catching sight of a member of my family in the distance. I was so hot that each time I passed beneath the shade of an apple tree I did not even realize it.
A butterfly flitted in front of me. It waited for me, not only on the flowers, but on the stones, and when I drew near it flew off—so delicate—without having seen or heard me, to set itself down farther on.
The scents of wood, grasses, ponds everywhere blended above the odorless road.
On a hill, between two trees of the same size, I suddenly saw my parents' house. The windows were open. I waited to feel some emotion, the joy I had been expecting. Nothing. This emotion, this joy, came up against a thousand thoughts, a thousand memories evoked by the host of insects, the blades of grass, the specks of earth that surrounded me, scattered me, and made it so that for a moment I did not know where I was.
I stopped. I stared hard at the house. It resembled the ones I had seen from the train. Everything was normal. Nothing about it attracted my attention more than any of the others.
Something moved in a window, something like a light-colored cloth with swirls.
I opened my mouth as if to call out. I stretched my hands toward this sign of life. It was neither an animal nor laundry hung out to dry. It was still moving on the second floor, at my parents' bedroom window. It was my father, my mother.
A shimmering spread across my field of vision. I was suddenly aware that what I had imagined was collapsing, that the words I had prepared would not come out, that traps would be set for me, that I would contradict myself, that I was as alone as I had been in the Paris train station this morning when I left. The distance I traveled had not brought me closer. No one was thinking of me. The one who, after five years of absence, found himself a few steps from his parents was still so far away, so forgotten that everything was going on normally: a piece of cloth was moving in a window, the house was white, the windows open.
Dust was falling on me. I could tell how thick it was from the blades of grass and the branches it covered.
I took a path that skirted the house so as not to arrive by the main door. The clearings, which only partly concealed me, did not allow me to be recognized from the way I walked. The long shadows from the trees accompanied me. When I passed near a swarm of mosquitoes, I looked beneath it to the ground, without thinking.
Birds sang and flapped their wings, ready to fly off again should the branches sag too much beneath their weight. The sun was setting. A buzzing life was being reborn with the first chill of evening. Other insects, those that like the rain, emerged from the cracked earth.
I continued to gaze at the house as I walked. Certain details began to make it familiar to me. A fence separated the yard from the vegetable garden. I recognized the curtains, the bench in front of the ground-floor window, a shovel whose smooth handle was worn thin from use. I saw a brand-new zinc pail that did not seem out of place because it had not been used without me. The trees had not grown taller.
I had only about fifty yards to go. My family would soon be in the dining room. My mother would be preparing supper. My father would be reading in his study. My sisters would be sewing.
I advanced slowly. My pulse, my temples, all my veins were throbbing in unison. Again I pictured the scene I had imagined so many times: the hugs, the tears, my parents' happiness. It would happen as I had foreseen. There was no reason for me to have been mistaken because, until now, everything was just as usual.
Tears came to my eyes, mixing with the perspiration on my face. Yet they were cooler as they fell.
I would be forgiven for having taken the money, for having made a scene before I left, for having gone five years without writing.
Then, smelling the sweet scent of the grass, it suddenly seemed to me that what I had done was much more serious than I had thought, that I would have to beg them to let me stay, beg them to forget.
Everything I had imagined was fading in the buzzing life all around me that would continue until evening, impervious to my motives and the complexities of my thoughts.
I was now very close to the house. I did not dare enter yet. I had placed my hand on the fence that surrounds our property. A bush from our yard concealed me. Because it belonged to us, and because it seemed to become my accomplice, for an instant I regained my confidence.
I did not have the strength to take another step. I, who had believed everyone would laugh together and feel sorry for me, sensed that I would be unable to utter a single word. I had a dizzy spell. Now, with all my heart, I hoped someone would come out and see me. I would have fainted then. They would have carried me. I would have woken up in a bed with my family at my side, attentive to all my gestures.
But no one came. I heard my sister singing, my mother speaking, but did not see anyone even though the windows were open.
For a second I let go of the fence to test myself, to wipe my forehead from which new perspiration was dripping. I almost fell. I stumbled. With both hands I again grabbed the fence that the rain had turned a greenish color. I wanted to call for help. The memory of what I had done, along with the hope that I would recover, stopped me.
Suddenly my father came out of the house in his shirtsleeves. I saw him clearly. I bent down and spied on him through the leaves where a world of insects was teeming in the coolness. He did not see me. I was no longer his son. I was hiding, watching him intently without his being aware of it, like a burglar. He went into the garden. He was carrying a light, empty basket. He had aged. I was so shaken by this that I was not sad.
Five years ago, even when it was very hot out, he would never remove his jacket, he would hold himself erect and never go into the garden. I was the one who fetched the vegetables.
I wanted to run to him then, throw myself at his feet, beg him to forgive me. But I did not budge.
He passed in front of me again, slowly, turning around at the sound of a rooster crowing. Soon I saw him from behind, stooped over, sadder it seemed because he was going back into the house.
It was too late to follow him, to stop him. My father was leaving me outside.
I could no longer remain like this, hidden. I had to go in.
I forgot everything and, letting go of the fence, I took a step, then two.
I was about to go in. The great moment had arrived. My father, my mother would see me, look at me before they recognized me.
I raised the latch on one of the gates. I was in the yard. I stopped short and stood straight, motionless, my hands needing no support because the ground was level.
No one was in the courtyard. In my blurred vision, the line of the horizon seemed to spin as I looked at it. No tree, no bush concealed me. I was facing the walls of the house, the window, and the slope of the roof at which I had tossed balls as a child.
A few yards lay between me and the door. All I had to do was walk straight ahead on the cleared ground, uncluttered now by any pail, wheelbarrow, or basket.
Suddenly my gaze fell on the walls whose thickness I could see in the embrasures, on the objects made of wood and iron, on the bench, the shovel, the stones of the well and, for a second, on the chickens moving about at my feet. A clear voice rose from all that. I did not understand the meaning of what it was saying. My ears buzzed. I stiffened. One by one my calm, my strength, my will abandoned me. A groan escaped my throat. Because of the stables nearby, and a dog sleeping on the warm sand, it drew no attention.
I took another step. I waited, my body bathed in sweat, my chest heavy. As in a dream, I couldn't catch my breath. I felt as though I had collapsed, that I was lying on the ground, that my feet were as close to the sky as my head.
I could not bear anymore. Even if I had continued moving forward, I would not have seen the door, I would have crashed into a wall. I was incapable of going any farther. I took a step backwards, without taking my eyes off the house. My breath returned to normal. Relief flooded my body. The chickens, more active now, pecked. The ground, in the shade, was slowly cooling off. A bird tried to carry off a wisp of straw. A sudden calm fell over the garden, the courtyard, the house, as if all I had to do was move away for the calm to return.
Slowly I reached the road. My dusty shoes had been wiped clean by the grass. I was coming back to life. Without turning around, I headed toward the village. The sun was setting behind me. It stayed with my parents' house. My long shadow preceded me. I spared it from colliding with the trees, the piles of stones. I was calm. I tried not to think.
On the hill, when I reached the spot where the house had appeared to me when I arrived, I turned around.
The house rose between the two trees already dark against the blue sky. One window was closed. A single pane flashed. The day was coming to an end in the same peacefulness as the previous day. I felt guilty for having almost disturbed it.
A puff of warm air that the insects followed enveloped me. I looked one last time at the countryside that had not changed, that surrounded the house I was leaving forever, and I went on my way.
## IS IT A LIE?
When the clock struck ten, Monsieur Marjanne began to worry. For the third time, he called the maid.
"Irene, madame didn't tell you anything?"
"Honestly, no, monsieur!"
"Did you see her go out? How was she dressed? Did she have her suede purse?"
Monsieur Marjanne had noticed that whenever she went to a friend's house or to the theater, his wife preferred her suede purse to any of the others.
"I didn't see madame go out."
Once alone, Robert Marjanne paced in the living room for a few minutes, then went into his wife's bedroom. Everything was locked. He had never managed to get her to leave the drawers unlocked when she went out. "I don't need the servants reading the letters you used to write to me. That's nobody's business," she would respond whenever he said "What an odd habit you have of locking everything up like this!" It was useless for him to point out that Irene did not know how to read. The drawers he could open contained only insignificant things. He remained for a moment in the room, looking for something abnormal, then, returning to the living room, sat down in an armchair.
Robert Marjanne was a short, very well-proportioned man. Had he been stooped or deformed, he would have been ugly and his intelligence would perhaps have seemed surprising, whereas short and well built, he was oversensitive, and his misanthropy was such that it verged on neurasthenia. When asked his age, he answered like those people who pretend not to know where they are in their lives: "I was born in '64. You do the math!" But he did not leave them time to obey this injunction, turning the conversation to other subjects. The only child of rich storekeepers, he had grown up surrounded by a great deal of care and attention so that once he had come of age, he had only a vague idea of life in general; until late middle age he had dreamed—and he kept this dream as hidden as a teenager might keep his knowledge of lovemaking—of a woman who would be an artist of some kind, of traveling, of the high life.
When he turned fifty, he had an abrupt change of heart. His parents had died. His commercial enterprises were running on their own steam. He wanted to live. As if audacity had come with age, he decided to move toward—but in his own way, that is, very slowly—what seemed to be his ideal. He had countless moments of leisure. It was during one such moment that he met a young woman, Claire Paoli, the daughter of an engineer. She was so beautiful that soon he confused her in his mind with the woman he had dreamed of marrying his whole life.
Some years earlier, Claire had left her family to be with a young man who had just finished medical school, but who did not have enough money to open his own practice. They had lived together for three years on rue Gay-Lussac using the money from a few private lessons he gave to young boys who always resided on the opposite side of Paris. Then they separated, and Monsieur Paoli had taken his daughter back solely, it seemed, to heap criticism on her day after day. So when Monsieur Marjanne offered to marry her, she accepted immediately.
From that moment on, Robert Marjanne lived as if in a dream. He couldn't do enough to make his wife happy. He was attentive to her every need. Not like a man in his twilight years who uses all his past experience in order to continue to please, but like a man who had wasted his youth and had never been attentive to anyone before.
And Claire became attached to him. Every day, she cheered him up with little jokes, gave him serious advice that she did not believe and that would change a few hours later. Due to her poor treatment by her parents and the medical student, she found her husband's thoughtfulness charming and no amount of pride could lead her to reject it. Nonetheless, she intended to retain a degree of independence. She had demanded to have her own bedroom. She always refused to give the slightest details about her schedule for the day. Once lunch was finished, she would go out and not reappear until dinnertime.
* * *
That evening, however, for the first time, Monsieur Marjanne waited for her in vain. He continually went to the window in the hope of seeing her step out of a taxi in front of the house. A few times he had even gone out to be at her side more quickly when she arrived. Then, suddenly, fearing she had come in without his seeing her, he climbed back up hurriedly and found himself in the empty apartment in front of the table, covered in a frosty white tablecloth, that had been set for dinner. Slowly time passed. The clock chimed ten, and Claire had still not come home.
After letting his mind wander for several minutes, Robert Marjanne got up again. For an instant he remained motionless. What could he do? Pace back and forth, sit down again, go out walking in front of the house once more, lean out the window? But how would any of that make his wife appear more quickly? He was in the most painful state of anxiety, the state in which, because the anxiety has lasted for so long, the weary mind seeks explanations, begins to try to reason, and, because there is nothing to be done, ends up becoming annoyed with itself.
"I'm just too tense," he thought. "It's ridiculous to get in such a state. She has been delayed. Why always look on the dark side? Everything seems complicated, but I'm sure it's very simple. Naturally, because I am by myself, I have come up with all the conjectures one can come up with. By myself, the truth escapes me. Right now, I have no more grounds for thinking the worst than thinking the best. She was delayed. That's undeniable. Everything else is just my imagination. Still, she could have found a minute to phone me."
Thinking that perhaps his telephone was not functioning, Monsieur Marjanne was overjoyed to have to verify this, while confusedly imagining he would gain a few minutes in doing so. Very slowly he walked to his study, then, so as to still have this task before him and because he secretly hoped to find another one, several times he lit the room then made it dark again, trying to persuade himself that the light switch was not working properly. But whichever way he flipped it, it obeyed him.
At last Robert Marjanne sat down in front of the telephone. "Let's see if it's working," he thought. "Whom can I call? The Bertrins? Perhaps Claire is at their house, after all!" But at the thought that, if this were true, she would make a scene when she got home or, in the opposite instance, it would be impolite to disturb friends at this late hour simply to ask them if his telephone were working, he refrained. "The best thing would be to telephone the operator." He picked up the receiver and requested that she call him back in a few minutes to make sure the line was functioning properly.
A moment later, the telephone rang. Even though he was absolutely certain that it was a telephone employee, he was filled with emotion.
Because he had now returned to the dining room, he suddenly noticed the clock on the mantel. It was a few minutes before eleven. All at once he realized Claire should have been back four long hours ago. The uneasiness he had been experiencing abruptly became a sharp pain. It was the middle of the night. Everything such an absence could imply filled his mind. "She must have a lover," he thought. "She is with him right now. He doesn't want her to go. She doesn't have the strength to leave him. If there had been an accident, someone would have called me. It doesn't take much common sense to guess the truth. She is at his place. They are not asleep yet. They're talking, laughing..."
He could not shake the idea that Claire was cheating on him while he was waiting for her. Yet he wanted to go on hoping she would come home from one minute to the next. But midnight chimed, then one in the morning. Robert Marjanne was hardly recognizable.
After going to get some blankets, he lit all the lights in the living room, sank down in the lowest armchair, and covered his legs. From time to time he heard the little clocks in the apartment. The same thoughts continued to come to him, one after the other.
Daylight arrived and Robert Marjanne got up from his armchair. He had dozed off, as one does on a journey, haunted by nightmares in which Claire had turned into an insolent, drunken woman, then into a repentant wife begging her husband to forgive her. He opened the window. The sky was gray like silty water. The bare trees of boulevard Raspail, which were not yet twenty years old, did not even reach the height of a third story. An east wind, heavy with rain, was shaking them and there was something infinitely sad in seeing them sway like this on the deserted boulevard. Monsieur Marjanne closed the window. The light of dawn and the light from the streetlamps blended together, forming a single pale glow that filled the room with a strange brightness. It was seven o'clock in the morning.
* * *
As he was about to enter the dining room, he suddenly found himself face to face with his wife, who had just come home and, before going to join him, had rushed to her bedroom, no doubt to reacquaint herself with her surroundings before seeing her husband again. She had already removed her hat and coat. She smiled and said:
"I see you didn't go to bed. You should have gone to lie down. Really! You shouldn't worry so much just because I was held up. You know that if something serious had happened, I would not have left you like this. Naturally I would have telephoned. And since I didn't, everything was fine."
Claire was talking volubly. Robert Marjanne did not take his eyes off her. He was filled with immense joy at seeing his wife again, so similar to the way she was every other morning, and at the same time he was overcome with anger. But he controlled his emotions. He was aware that if he reproached her, she would immediately withdraw into silence and what he wanted to know more than anything was what she had done during this interminable night. Still, he couldn't help asking:
"So what were you doing then?"
Still smiling, she answered:
"I'll tell you, but wait a bit. I need to get back to my routine. It's no joy, you know, to have to spend the night at friends'. I don't know if it ever happened to you, but as hard as they try, it's still uncomfortable. You just don't feel at home. But has Irene prepared breakfast? I slept so badly and I'm starved. Come into the dining room. I'll tell you all about it. You'll see how odd it is, and sad at the same time."
Claire had never spoken with such candor before and so Robert Marjanne's suspicions only grew stronger. It seemed strange that his wife who ordinarily worried so little about what he would think of her was trying so hard to seem sincere.
"Just tell me in a few words what you did last night. Afterwards, we won't mention it again. It will be over, buried..."
"It's impossible in a few words. I have to tell you everything. How do you expect me to explain everything that happened in a few words?"
"Well, simply tell me where you slept. That's all I want to know."
"Give me time. I'll tell you. I'll tell you. You'll know everything."
"What's the harm in telling me right away? It can't be that complicated. Just tell me where you slept. I don't need to know the rest."
"You should hear yourself! As if I had done who-knows-what!"
"All I'm asking is for you to tell me where you spent the night."
"I'll tell you everything or nothing," Claire said drily.
Robert Marjanne then got the feeling that the story his wife was about to tell had been invented out of whole cloth and that, just like when she had told it to herself, she needed not to break off, not to be interrupted.
"Well then, tell me everything."
"Now I'll tell you everything because you are being reasonable. I have nothing to hide. Obviously if I had wanted to do something that was not right I would go about it differently. I'm not a child. Isn't that so, Robert, that I'd go about it differently?"
She smiled again and continued:
"I told you yesterday, I think, when I left you, that I was going to spend the afternoon at Madeleine's. So I went to her place, as you know. She was alone. Another girl came a little later, Maud. Have I already mentioned her to you?"
It seemed to Robert Marjanne that his wife was stretching out the true parts so that the lie would blend into the stream of words that preceded it.
"No, you never mentioned her to me. Who is she?"
"Let me finish first. I'll explain later who she is."
Claire said these words more gently. The fact that her husband showed interest in a minor point of her story seemed to reassure her. No doubt for this reason she brought the conversation back to Maud.
"All right, I'll tell you right away. Then you'll understand better what happened next. Maud is the daughter of an Englishman who has lived in Paris for, I think, twenty years. He's a real character. He adores France."
Robert Marjanne surmised that his wife really knew this Englishman and his daughter and that it was because their eccentricity had struck her that she had put them into her story: their exceptional nature would make it easier to accept the exceptional nature of the story she was preparing to tell.
"But why didn't you come home?"
"Wait, I said. Let me explain everything or I won't say anything."
He was suffering so much at the thought that she had spent the night with a man that all he wanted was to be convinced of the contrary. He coaxed her to go on.
"Of course I'll go on. So the three of us chatted. You know what it's like when girls get together. We have no idea of the time. We talk about clothes and all kinds of things, and the time flies by. All of a sudden I realized that it was six o'clock."
Just then Robert Marjanne had such a clear sense that Claire was going to lie that he thought she could feel it. And in fact, she seemed not to dare to go any further into her tale.
"I realize that it's six o'clock," she repeated. "My friends are surprised. I leave them at last and then, remembering I had something to buy, I catch a cab to the Printemps department store."
"It must have been closed!"
"How annoying you are! Will you let me finish already? Don't you know that the department stores close at six-thirty?"
"What was it that you wanted to buy so urgently?"
"What I wanted to buy? You want to know?"
Claire went to her room and returned a moment later with a small package that she opened in front of her husband. It contained a pair of gloves.
"You see. Nothing to worry about."
Then, showing him the paper in which the gloves had been wrapped and on which was printed in a modern font "Le Printemps," she added:
"Here's the proof."
"None of that tells me where you spent the night. You had to sleep somewhere, after all!"
"If you interrupt me one more time, I'm warning you I won't tell you another thing. You think it's amusing to recount everything in such detail? Listen to me now. So I leave Le Printemps. It was exactly six-thirty and I say to myself, 'Robert must be waiting for me, I've got to hurry.' But instead of taking a cab in front of the store—you know how crowded it is there, I would have waited for an hour—I go on foot to boulevard Malesherbes. And right then, when I am on the corner of rue du Havre, I run into—you'll never guess who. Who do you think?"
"I don't know."
"Come on, guess."
"Maud!"
"No, no. I told you a moment ago that I had left her at Madeleine's."
In Monsieur Marjanne's mind Claire was only trying to give the illusion of truth. To be less alone with her lie, she wanted to make her husband participate in it. But he was determined not to let himself be dragged into it and simply answered: "I don't know" and "What can I say?"
"Well! I'm going to tell you. I ran into Olga and her mother."
"What _are_ you talking about?"
"Don't you remember? You have no memory! In Nice, at the Hotel Beauséjour."
"Of course I remember, but you are not going to make me believe that you ran into them just like that, out of the blue."
Right after their wedding, Claire and Robert Marjanne had gone on a long trip to Italy. When they came back to France, they had stopped in Nice for two weeks. Mme. Kalinina and her daughter Olga were staying in the same hotel as the Marjannes. They had become acquainted easily, even more so because at first Mme. Kalinina had thought that Monsieur Marjanne was Claire's father. Claire had sensed, to the point of teasing him about it, how flattered her husband had been—coming as he did from a family where everything was always a question of self-interest and from which he had dreamed of escaping all his life—to make the acquaintance of Mme. Kalinina, who had been admitted to the court of the Czar, who belonged to one of the greatest families in Russia and who now, driven out of her country, had taken refuge in Nice.
Thus the idea immediately occurred to Robert Marjanne that his wife, having remembered his high regard for the two foreigners, had thought that simply mentioning Mme. Kalinina would cause his suspicions to fade.
"So they are in Paris?" he asked.
"And they asked about you first thing."
Monsieur Marjanne then remembered the two delightful weeks he had spent in Nice where, the only man among these three women, he had accompanied them to the casino, to the theater, to tea, and how considerate everyone had been to him, and how proud he had been, and also the thought he had so often had: "If only my family could see me now!"
But Claire had begun speaking again.
"You can imagine, Robert, how surprised we were to meet again. All they could talk about was you. They asked me a thousand things, how you were, if you were happy, if you still loved me, if you spoiled me, if you hadn't changed, absolutely everything!"
Robert Marjanne still remembered the look of admiration for Claire's beauty he had seen in Mme. Kalinina's eyes.
"They were on their way home," Claire continued. "You can easily imagine that I felt I should accompany them. And as we walked they spoke to me for a long time, very frankly, as if I were part of the family. And I learned a lot of things about which you have no idea. They made me swear not to tell you anything, but now that they have gone, and I may never see them again, I will tell you everything."
"They're gone?" Robert Marjanne asked anxiously because he'd had a glimmer of hope at the thought that everything might be true, that he would go to see Mme. Kalinina that very day, and from her very mouth he would get confirmation of his wife's words.
"Yes," said Claire, looking surprised. "Didn't I tell you? If they hadn't had to leave for London this morning, you know very well that I would have had the time to see them again and I would not have stayed with them until now."
With these words Monsieur Marjanne had the feeling that everything was collapsing around him. All this was nothing but lies. He was sure of it. However, he forced himself not to show his distress.
"They've left?" he asked again without even realizing he was speaking.
"This morning, on the train to Boulogne. I don't know what time exactly, but you can check the schedule."
Monsieur Marjanne ran a hand across his forehead.
"So I won't see them?"
"Did you want to see them so badly?" Claire asked, feigning jealousy to tease him.
"But what did you do all night?"
"I was about to tell you a moment ago, but you interrupted me. I wanted to tell you everything. It's not very nice of me, though. If only you'd heard how they especially asked me not to say anything to you."
"But why didn't you invite them to come back here?"
"Why, why? Well, what I was about to tell you would have answered your question. It would have been perfect. You thought you were laying a trap for me, but you're the one who got caught in it. I don't know if you recall in Nice when we met them they had jewelry, furs, real furs from Siberia. Well, already back then they were selling everything to live on. I knew it. They had told me but begged me not to say anything. I kept my word and always hid it from you. Now it's different. When I met them yesterday, you cannot imagine how painful it was for me. They were not the same women. Mme. Kalinina was wearing a thin little black coat with a rabbit-fur collar dyed black to match. She wasn't wearing any jewelry, not even a wedding ring, and neither was her daughter. Naturally I pretended not to notice, but I was quite upset. They were too. When they talked about you, one could feel that their laughter, their cheerful tone was not natural, that they were trying to be as carefree as when we'd met them. As soon as we had exchanged a few words, they wanted to be on their way. It was then that I realized they felt abandoned and that they wished I would stay with them and say to them: 'Don't go, let's talk, let's spend the evening together.' And that's what I said. The thought of inviting them home came to me of course, but I sensed how much their pride would suffer seeing you again dressed as they were. So we walked and then quite naturally I said you were going to be with friends that evening and I invited them out to dinner. Because they felt my affection for them, little by little they became more trusting and told me everything they had endured. I took a taxi with them to accompany them back to their hotel and on the way Olga began to cry. She had been terribly upset by what her mother had recounted. Once we got there, I went up to their room with them, I consoled them. We ordered tea, we talked some more. And that's how it got to be midnight. And then I wanted to come home to you. I don't know if you know how it is, but when you've had a lot of sorrows and have given an account of them to someone else, you want that person to stay with you. If she leaves, you feel more depressed than before. When I said I was going home to you, you should have seen them! Olga stopped crying and grew very pale. Mme. Kalinina took my hand. They didn't say a word, but from their faces and everything about them I sensed that their situation would seem even more distressing to them once I had gone."
"So you slept in their room?"
"I didn't even lie down. We talked until very, very late. Then I sat down in an armchair and slept for maybe three hours at most. Early this morning we went to the Gare du Nord. And I left them barely an hour ago."
"You spent the night in their room? In the hotel?"
"Naturally. I think they spent two days in all in Paris."
"Which hotel?"
"A hotel. You know the street that goes up next to the Église de la Trinité?
"Rue Clichy?"
"No, the other one."
"Rue Blanche?"
"Yes, that's it. We took a taxi. The driver went up that street, then turned right and took another cross street. Two or three hundred meters farther along, he stopped. If you'd like we can go there one day together when we take a walk. You'll see, I'll be able to locate the hotel very quickly."
This time, Robert Marjanne had the clear impression that Claire's tale was nothing but one long lie.
"So, my Robert, were you bored without me? Did you sleep well? You see it was nothing as terrible as you supposed. Come on, admit that you thought, I don't know, that I had a lover, that I'd gone to the theater, then home with him, and that we spent an extraordinary night together. I'm sure that's what you thought. You were wrong. You never know what can happen in life. There are so many odd things, unforeseen events."
Robert Marjanne did not respond. Suddenly it dawned on him that, after all, what Claire had told him might be true. Claire came to him and took his hands.
"Are you angry?" she asked.
"I don't know. You told me a long story, but did you make it all up?"
"You're mad, darling. How could I invent such a story? Really, put yourself in my place. What if I had lied and, for example, we ran into Mme. Kalinina and her daughter tomorrow! If I had invented such a tale, I couldn't go on living. Every time I went out with you, I would say to myself, 'Maybe we'll run into them and Robert will find out that I lied to him!' Life would be unbearable. You always have such bizarre ideas. This isn't the first time."
Monsieur Marjanne looked at his wife sadly and, in a steady voice, asked:
"Is the story you just told me true?"
"I swear it is, darling. If I'm lying, may I die this instant."
"All right, I believe you."
He clasped his wife to him. He did not believe her. He was profoundly convinced she had lied. But suddenly it occurred to him that he was nearing old age and, rather than losing everything, it would be better to suffer in silence in order to have the joy of living with the woman he loved and who had enough respect and fondness for him to go to the trouble of lying.
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,334 |
\section{Introduction}
Let $[n]=\{1, \ldots, n\}$.
Only finite, simple graphs are considered.
Given a graph $G$, let $V(G)$ and $E(G)$ denote the vertex set and edge set of $G$, respectively.
The {\it vertex arboricity} of a graph $G$, denoted $a(G)$, is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ where $G[V_i]$ is a forest for each $i\in[k]$.
This can be viewed as a vertex coloring $f$ with $k$ colors where each color class $V_i$ induces a forest; namely, $G[f^{-1}(i)]$ is an acyclic graph for each $i\in [k]$.
The {\it girth} of a graph $G$ is the length of the smallest cycle in $G$.
Note that a graph with no cycles is a forest, and it has vertex arboricity 1.
Vertex arboricity, also known as point arboricity, was first introduced by Chartrand, Kronk, and Wall \cite{1968ChKrWa} in 1968.
Among other things, they proved Theorem~\ref{allplanar}.
Shortly after, Chartrand and Kronk \cite{1969ChKr} showed that Theorem~\ref{allplanar} is sharp by constructing a planar graph with vertex arboricity $3$, and they also proved Theorem~\ref{outerplanar}.
\begin{theorem}\label{allplanar}\cite{1968ChKrWa}
If $G$ is a planar graph, then $a(G)\leq 3$.
\end{theorem}
\begin{theorem}\label{outerplanar}\cite{1969ChKr}
If $G$ is an outerplanar graph, then $a(G)\leq 2$.
\end{theorem}
We direct the readers to the work of Stein \cite{1971St} and Hakimi and Schmeichel \cite{1989HaSc} for a complete characterization of maximal plane graphs with vertex arboricity $2$.
In 2008, Raspaud and Wang \cite{2008RaWa} not only determined the order of the smallest planar graph $G$ with $a(G)=3$, but also found several sufficient conditions for a planar graph to have vertex arboricity at most $2$ in terms of forbidden small structures; namely, they proved that a planar graph with either no triangles at distance less than $2$ or no $k$-cycles for some fixed $k\in\{3, 4, 5, 6\}$ has vertex arboricity at most $2$.
Chen, Raspaud, and Wang \cite{2012ChRaWa}
showed that forbidding intersecting triangles is also sufficient for planar graphs.
In \cite{2008RaWa}, Raspaud and Wang asked the following question:
\begin{question}\cite{2008RaWa}
What is the maximum integer $\mu$ where for all $k\in\{3, \ldots, \mu\}$, a planar graph $G$ with no $k$-cycles has $a(G)\leq 2$?
\end{question}
Raspaud and Wang's results imply $6\leq \mu\leq 21$.
The lower bound was increased to $7$ by Huang, Shiu, and Wang \cite{2012HuShWa} since they proved planar graphs without $7$-cycles have vertex arboricity at most $2$.
We completely answer the question for toroidal graphs, which are graphs that are embeddable on a torus with no crossings.
Kronk \cite{1969Kr} and Cook \cite{1974Co} investigated vertex arboricity on higher surfaces in 1969 and 1974, respectively.
\begin{theorem}\label{higherSurfaces}\cite{1969Kr}
If $G$ is a graph embeddable on a surface of positive genus $g$, then $a(G)\leq \lfloor{9+\sqrt{1+48g}\over 4}\rfloor$.
\end{theorem}
\begin{theorem}\label{higherSurfacesTri}\cite{1974Co}
If $G$ is a graph embeddable on a surface of genus $g$ with no $3$-cycles, then $a(G)\leq 2+\sqrt{g}$.
\end{theorem}
\begin{theorem}\label{higherSurfacesGirth}\cite{1974Co}
If $G$ is a graph embeddable on a surface of positive genus $g$ with girth at least $5+4\log_3 g$, then $a(G)\leq 2$.
\end{theorem}
Theorem~\ref{higherSurfaces} says every toroidal graph $G$ has $a(G)\leq 4$.
Theorem~\ref{higherSurfacesTri} says a toroidal graph with no $3$-cycles has vertex arboricity at most $3$, and Theorem~\ref{higherSurfacesGirth} only guarantees that toroidal graphs with girth at least $5$ have vertex arboricity at most $2$.
Both of these cases were improved by Kronk and Mitchem \cite{1974KrMi} who showed Theorem~\ref{toroNo3}.
Recently, Zhang \cite{00Zh} showed Theorem~\ref{toroNo5}, which says that forbidding $5$-cycles in toroidal graphs is sufficient to guarantee vertex arboricity at most $2$.
\begin{theorem}\label{toroNo3}\cite{1974KrMi}.
If $G$ is a toroidal graph with no $3$-cycles, then $a(G)\leq 2$.
\end{theorem}
\begin{theorem}\label{toroNo5}\cite{00Zh}.
If $G$ is a toroidal graph with no $5$-cycles, then $a(G)\leq 2$.
\end{theorem}
Since the complete graph on $5$ vertices is a toroidal graph with no cycles of length at least $6$ and has vertex arboricity $3$, the only remaining case is when $4$-cycles are forbidden in toroidal graphs; this is our main result.
\begin{theorem}\label{result}
If $G$ is a toroidal graph with no $4$-cycles, then $a(G)\leq 2$.
\end{theorem}
In section $2$, we will prove some structural lemmas needed in Section $3$, where we prove Theorem~\ref{result} using (simple) discharging rules.
Note that Theorem~\ref{result} implies that every planar graph without $4$-cycles have vertex arboricity at most $2$, which is a result in \cite{2008RaWa}.
\section{Lemmas}
From now on, let $G$ be a counterexample to Theorem~\ref{result} with the fewest number of vertices.
It is easy to see that $G$ must be $2$-connected and the minimum degree of a vertex of $G$ is at least $4$.
A graph is $k$-regular if every vertex in the graph has degree $k$.
A set $S\subseteq V(G)$ of vertices is $k$-regular if every vertex in $S$ has degree $k$ in $G$.
A {\it triangular cycle} is a cycle adjacent to a triangle.
A (partial) 2-coloring $f$ of $G$ is {\it good} if each color class induces a forest.
\begin{lemma}\label{partial}
If $V(G)$ contains a 4-regular set $S$ where $G[S]$ is a cycle $C$, then every good coloring $f$ of $G[V(G)\setminus S]$ that does not extend to all of $G$ has either
\begin{enumerate}[\mbox{Case} 1:]
\item $f(v)$ the same for every vertex $v\not\in S$ that has a neighbor in $S$, or
\item $f(x)\neq f(y)$ for all $v\in S$ such that $N(v)\setminus S=\{x, y\}$ and $C$ is an odd cycle.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $S=\{v_1, \ldots, v_s\}$ where $v_1, \ldots, v_s$ are the vertices of $C$ in this order.
For each $i\in [s]$, let $\{x_i, y_i\}=N(v_i)\setminus S$.
Obtain a good coloring $f$ of $G[V(G)\setminus S]$ by the minimality of $G$.
We will show that if $f$ does not satisfy one of the two conditions in the statement, then $f$ can be extended to all of $G$.
If $s$ is even and $\{f(x_i), f(y_i)\}=\{1, 2\}$ for each $i\in[s]$, then let $f(v_i)=
\begin{cases}
1 & \mbox{if $i$ is odd}\\
2 & \mbox{if $i$ is even}
\end{cases}$
to extend $f$ to all of $G$.
We know that there exists at least one index $j\in[s]$ where $f(x_j)= f(y_j)$ since we are not in Case 2.
For each $i\in [s]$ where $f(x_i)=f(y_i)$, let $f(v_i)=
\begin{cases}
1 & \mbox{ if $f(x_i)=f(y_i)=2$ }\\
2 & \mbox{ if $f(x_i)=f(y_i)=1$}
\end{cases}$.
Now, consider the vertices of $C$ in cyclic order starting with $i=j$, and for $f(v_i)$ that is not defined yet, let $f(v_i)=
\begin{cases}
1 & \mbox{ if $f(v_{i-1})=2$ }\\
2 & \mbox{ if $f(v_{i-1})=1$}
\end{cases}$ for all $i$.
We claim that this coloring $f$ is now a good coloring of all of $G$, which is a contradiction.
Note that $f$ cannot have a monochromatic cycle that only uses vertices of $V(G)\setminus S$.
Also, $f$ cannot have a monochromatic cycle where $x_i, v_i, y_i$ are consecutive vertices on this cycle since $f(x_i)=f(v_i)=f(y_i)$ never happens.
Moreover, $f$ cannot have a monochromatic cycle where $v_i, v_{i+1}, x_i$ are consecutive vertices on this cycle since $f(v_i)=f(v_{i+1})$ implies that $f(x_{i+1})=f(y_{i+1})\neq f(v_{i+1})$.
Thus, a monochromatic cycle in $f$ must be $C$ itself, which is possible only in Case 1.
\end{proof}
\begin{lemma}\label{reducible}
$V(G)$ does not contain a 4-regular set $S$ where $G[S]$ is a triangular cycle.
\end{lemma}
\begin{proof}
Let $S=\{v_1, \ldots, v_s, u\}$, so that $u, v_1, v_2$ are the vertices of a triangle and let $C=S\setminus\{u\}$.
Let $v_1, \ldots, v_s$ be the vertices of $C$ in this order.
For $i\in[2]$, let $v'_i$ be the neighbor of $v_i$ that is not in $S$.
We will obtain a good coloring of all of $G$ to show that $S$ does not exist.
Obtain a good coloring $f$ of $G[V(G)\setminus C]$ by the minimality of $G$.
Assume that the first case of Lemma~\ref{partial} happens and without loss of generality, assume $f(v)=1$ for every vertex $v\not\in C$ that has a neighbor in $C$.
For $i\in[s]\setminus\{1\}$, let $f(v_i)=2$ and let $f(v_1)=1$.
If $f$ is not a good coloring, then in the graph induced by $f^{-1}(1)$, there must exist a cycle where $v'_1, v_1, u, z$ are consecutive vertices on the cycle for some $z\in N(u)\setminus\{v_1, v_2\}$.
Now, alter $f$ by letting $f(u)=2$ to obtain a good coloring of all of $G$.
Assume that the second case of Lemma~\ref{partial} happens and without loss of generality, assume $f(v'_1)=f(v'_2)=1$ and $f(u)=2$.
Note that $s$ must be odd.
For $i\in [s]\setminus\{1\}$, let $f(v_i)=
\begin{cases}
1 & \mbox{if $i$ is odd}\\
2 & \mbox{if $i$ is even}
\end{cases}$, let $f(v_1)=2$, and change $f(u)$ from $2$ to $1$.
If $f$ is not a good coloring, then in the subgraph induced by $f^{-1}(1)$, there must exist a cycle where $u$ and two of its neighbors that are not $v_1, v_2$ are consecutive vertices on the cycle.
Now for $i\in[s]\setminus\{1\}$, alter $f$ by letting $f(v_i)=
\begin{cases}
2 & \mbox{if $i$ is odd }\\
1 & \mbox{if $i$ is even}
\end{cases}$ (but keep $f(u)=2$) to obtain a good coloring of all of $G$.
\end{proof}
\begin{figure}[h]
\begin{center}
\input{reducible}
\end{center}
\caption{Forbidden Configuration.
The white vertices do not have incident edges besides the ones drawn.}
\label{fig:tikz:reducible}
\end{figure}
Let a vertex $v$ be {\it bad} if $d(v)=4$ and $v$ is incident to two triangles;
a vertex is {\it good} if it is not bad.
Let $H=H(G)$ be the graph where $V(H)$ is the set of triangles of $G$ incident to at least one bad vertex and let $uv\in E(H)$ if and only if there is a bad vertex of $G$ that is incident to both triangles that correspond to $u$, $v$.
\begin{claim}\label{components}
Each component of $H$ is either a cycle or a tree.
\end{claim}
\begin{proof}
Assume for the sake of contradiction that $H$ has a component $D$ with a cycle $C$ where $C$ is not the entire component.
Let $v\in V(D)\setminus V(C)$ be a vertex that has a neighbor in $V(C)$.
The graph in $G$ that corresponds to this structure is forbidden by Lemma~\ref{reducible}, which is a contradiction.
See Figure~\ref{fig:tikz:helper}.
\end{proof}
\begin{figure}[h]
\begin{center}
\input{helper}
\end{center}
\caption{The cycle $C$ in the proof of Claim~\ref{components} (left) and the corresponding graph in $G$ (right).
The white vertices do not have incident edges besides the ones drawn.
The black vertices may have other incident edges. }
\label{fig:tikz:helper}
\end{figure}
Here is a lemma that will help later on.
\begin{lemma}\label{tree-leaf}
Every $n$-vertex tree where with maximum degree $3$ has exactly $2$ more vertices of degree $1$ than vertices of degree $3$.
\end{lemma}
\begin{proof}
Let $z_i$ be the number of vertices of degree $i$.
An $n$-vertex tree has $n-1$ edges and the sum of the degrees is twice the number of edges.
Thus we have $n=z_1+z_2+z_3$ and $2(n-1)=z_1+2z_2+3z_3$.
By eliminating $z_2$, we get $z_1=z_3+2$.
\end{proof}
\section{Discharging}
In this section, we will prove that $G$ cannot exist.
Fix an embedding of $G$ and let $F(G)$ be the set of faces.
We assign an \emph{initial charge} $\mu(z)$ to each $z\in V(G) \cup F(G)$, and then we will apply a discharging procedure to end up with {\it final charge} $\mu^*(z)$ at $z$.
We prove that the final charge has positive total sum, whereas the initial charge sum is at most zero.
The discharging process will preserve the total charge sum, and hence we find a contradiction to conclude that $G$ does not exist.
For every vertex $v\in V(G)$, let $\mu(v)=d(v)-6$, and for every face $f\in F(G)$, let $\mu(f)=2d(f)-6$.
The total initial charge is zero since
\begin{align*}
\sum_{z\in V(G)\cup F(G)} \mu(z)
=\sum_{v\in V(G)} (d(v)-6)+\sum_{f\in V(F)} (2d(f)-6)
=6|E(G)|-6|V(G)|-6|F(G)|
\leq 0.
\end{align*}
The final equality holds by Euler's formula.
For the discharging procedure we introduce the notion of a {\it bank}, which serves as a placeholder for charges.
For each component $D$ of the auxiliary graph $H(G)$, we will define a separate bank; let $b(D)$ denote the bank.
We give each bank an initial charge of zero and we will show that either some vertex or some bank has positive final charge.
The rest of this section will prove that the sum of the final charge after the discharging phase is positive.
Recall that a vertex $v$ is {\it bad} if $d(v)=4$ and $v$ belongs to two triangles and a vertex is {\it good} if it is not bad.
A good vertex $v$ is incident to a bank $b(D)$ if there is a vertex $u$ of $D$ where $v$ is incident to the triangle in $G$ that corresponds to $u$.
Note that each bad vertex of $G$ is an edge of $H(G)$.
Here are the discharging rules:
\begin{enumerate}[(R1)]
\item Each face distributes its initial charge uniformly to each incident vertex.
\item Each good vertex $v$ sends charge ${2\over 5}$ to each bank $b(D)$ each time $v$ is incident to $b(D)$.
\item For each component $D$ of $H(G)$, the bank $b(D)$ sends charge $2\over 5$ to each bad vertex in $G$ that corresponds to an edge in $D$.
\end{enumerate}
It is trivial that each face has nonnegative final charge.
Moreover, each face $f$ with $d(f)\geq 5$ sends charge ${\mu(f)\over d(f)}={2d(f)-6\over d(f)}\geq {4\over 5}$ to each incident vertex.
We will first show that each vertex has nonnegative final charge.
Then we will show that either some bank or some vertex has positive final charge.
Note that since $G$ has no $4$-cycles, each vertex $v$ is incident to at most $\lfloor {d(v)\over 2}\rfloor$ triangles, and therefore at most $\lfloor{d(v)\over 2}\rfloor$ banks.
\begin{claim}\label{vertex}
Each vertex has nonnegative final charge.
Moreover, each vertex of degree at least $5$ has positive final charge.
\end{claim}
\begin{proof}
A vertex $v$ with $d(v)\geq 6$ has nonnegative initial charge and receives at least ${4\over 5}\cdot{d(v)\over 2}$ after (R1).
Since $v$ is incident to at most $\lfloor{ d(v)\over 2}\rfloor$ incident banks, $\mu^*(v)\geq{4\over 5}\cdot{d(v)\over 2}-{2\over 5}\cdot\lfloor{d(v)\over 2}\rfloor>0$.
A vertex $v$ with $d(v)=5$ will receive charge from at least 3 incident faces and will give charge to at most 2 incident banks.
Therefore, $\mu^*(v)\geq -1+3\cdot{4\over 5}-2\cdot{2\over 5}> 0$.
A good vertex $v$ with $d(v)=4$ will receive charge from at least 3 faces and will give charge to at most 1 incident bank.
Therefore, $\mu^*(v)\geq -2+3\cdot{4\over 5}-{2\over 5}= 0$.
A bad vertex $v$ will receive charge at least ${4\over 5}$ from two faces and $2\over 5$ from exactly one bank.
Therefore, $\mu^*(v)\geq -2+ 2\cdot{4\over 5}+{2\over 5}= 0$.
\end{proof}
Given a component $D$ of $H(G)$, since an edge of $D$ corresponds to a bad vertex of $G$, we need to check that $b(D)$ has enough charge for each edge of $D$.
\begin{claim}\label{cycle-bank}
Each bank $b(D)$ where $D$ is a cycle has nonnegative final charge.
\end{claim}
\begin{proof}
Assume $D$ is a cycle $C$ with $n$ vertices.
Since $D$ is a cycle, each triangle in $G$ that corresponds to a vertex in $D$ must be incident to one good vertex; each good vertex will send charge $2\over 5$ to $b(D)$.
Thus, $b(D)$ receives charge ${2\over 5}n$ and there are $n$ edges in $D$ so $b(D)$ has nonnegative final charge.
\end{proof}
\begin{claim}\label{tree-bank}
Each bank $b(D)$ where $D$ is a tree has positive final charge.
\end{claim}
\begin{proof}
Assume $T$ has $n$ vertices.
$T$ has maximum degree at most 3 since a triangle in $G$ cannot be incident to more than 3 bad vertices.
For $i\in[3]$, let $z_i$ be the number of vertices of degree $i$ in $T$.
Each triangle in $G$ that corresponds to a degree 1 vertex in $T$ is incident to 2 good vertices, and each triangle in $G$ that corresponds to a degree 2 vertex in $T$ is incident to 1 good vertex.
Thus $b(T)$ gets charge ${4\over 5}z_1+{2\over 5}z_2$, and must spend ${2\over 5}|E(T)|={2\over 5}(n-1)$.
Since $z_1=z_3+2$ by Lemma~\ref{tree-leaf}, it follows that ${4\over 5}z_1+{2\over 5}z_2={2\over 5}n+{4\over 5}>{2\over 5}n-{2\over 5}$.
Thus, $b(T)$ has positive final charge.
\end{proof}
If $H(G)$ has a component that is a tree $T$, then $b(T)$ has positive final charge.
If $H(G)$ has a component that is a cycle, then there exists a vertex of degree at least 5 in $G$, and by Claim~\ref{vertex}, this vertex has positive final charge.
If $H(G)$ has no components, then there are no bad vertices, and we are done since either some bank or some vertex will have positive final charge.
\section{Acknowledgment}
The authors thank Alexandr V. Kostochka and Bernard Lidick\'y for improving the readability of the paper.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,564 |
\section{Introduction}
Electrons move and can gain energy in response to the electromagnetic fields of a laser pulse; the coupling of the laser pulse energy to the electrons regulates the entire relativistic intensity laser-plasma interaction. Many other secondary phenomena of interest arise from this electron heating, including ion acceleration \cite{ion_accn_2000,Willingale_PRL_2009,Silva_PRL_2004,Haberberger_NP_2012, Palmer_PRL_2012, Fiuza_PRL_2012, Albright_PoP_2007, Bin_PRL_2018}, high-harmonic generation \cite{HHG}, x-ray beam generation \cite{Rousse_PRL_2004,Kneip_PRL_2008,Stark_PRL_2016}, and positron production \cite{Chen_PRL_2009,Jansen_PPCF_2018}. Electron acceleration and heating in a plasma is surprisingly complex due to the collective plasma effects that affect both the laser pulse propagation and the electron motion itself.
Several parameters determine the dominant electron heating mechanism at relativistic intensities, the foremost factors being the plasma density ($n_{e}$), and the laser pulse duration and intensity. The classical critical plasma density is defined to be $n_{c} = m_{e} \epsilon_{0} \omega_{L}^{2} / e^{2}$, where $\omega_{L}$ is the laser frequency. The two extremes for target plasma densities have been studied extensively. For a very overdense ($n_{e} \gg n_{c}$), short scale-length plasma, the dominant heating mechanisms become vacuum heating \cite{Brunel_PRL_1987} and $\mathbf{j} \times \mathbf{B}$ heating \cite{Kruer_PoF_1985}, with the expected hot electron temperature scaling as the ponderomotive potential, $U_{p} \approx (a_{0} /2)^2 m_{e} c^{2}$ \cite{Wilks_PRL_1992}, where $a_{0}$ is the normalized laser amplitude. A significant scale-length underdense plasma ($n_{e} < n_{c}$) could be present ahead of an overdense target due to heating and expansion either during a laser pre-pulse or on the timescale of the laser pulse interaction~\cite{Kemp_PRL_2012, Sorokovikova_PRL_2016}. Such a pre-plasma is known to reduce the $\mathbf{j} \times \mathbf{B}$ heating and the overall energy conversion efficiency \cite{Wilks_IEEE_1997, Cai_PoP_2010,Yabuuchi_PoP_2010}. However, a characteristic enhancement in the high energy tail of escaping electrons is a typical observation from experiments~\cite{Jarrott_PoP_2014} and has been attributed to other acceleration mechanisms occurring in the underdense region~\cite{Cai_PoP_2010, Krygier_PoP_2014,Sorokovikova_PRL_2016}.
There is significant interest in using near-critical density plasma to enhance ion acceleration mechanisms \cite{Willingale_PRL_2009,Willingale_PoP_2011,Haberberger_NP_2012,Fiuza_PRL_2012,Bin_PRL_2015,Chen_SR_2017,Fedeli_SR_2018, Bin_PRL_2018}, or to generate bright x-ray \cite{Wang_PoP_2015} or electron-positron plasmas \cite{Zhu_NC_2016} by taking advantage of the high laser energy conversion to hot electrons and the high electron temperatures.
A significantly underdense plasma offers favorable conditions for electron acceleration well beyond the ponderomotive potential, as it allows the laser pulse to propagate with a phase velocity ($v_{ph}$) that remains close to the speed of light. The laser pulse could excite a co-propagating plasma wave in the underdense plasma leading to laser wake-field acceleration~\cite{Tajima_PRL_1979}. For a higher intensity laser pulse with a duration longer than a plasma wave period, the plasma wave development is inhibited due to the large and sustained ponderomotive force.
Instead, electrons are expelled from regions of highest intensity and, if the ponderomotive force persists to balance the electric field acting to return the electrons, a cavitated channel can form~\cite{Mora_PRB_1996, Tzeng_PRL_1998, Nilson_NJP_2010}. In this regime, direct laser acceleration (DLA) assisted by quasi-static transverse and longitudinal electric fields of the channel may become the dominant mechanism generating an electron population with characteristic energies many times greater than $U_{p}$~ \cite{Pukhov_PoP_1998, Pukhov_PoP_1999, MeyerterVehn_PoP_1999, Arefiev_PRL_2012, Robinson_PRL_2013, Arefiev_PoP_2014, Arefiev_PoP_2016, Khudik_PoP_2016, Jiang_PRL_2016}.
In this paper, we consider the energy transfer mechanisms in the intermediate range of near-critical densities ($n_{e} \sim n_{c}$), a regime that has received little attention.
One compelling reason to consider near-critical density targets is that they can become transparent at relativistic laser intensities, when $a_{0} >1$. Accelerating electrons to relativistic energies, the laser pulse effectively enhances the electron mass, thus reducing the effective critical density that determines the cutoff for an electromagnetic wave. As a result, the relativistically induced transparency allows the laser pulse to propagate in plasmas with electron densities up to $n_{\gamma c} \equiv \bar{\gamma} n_{c}$~\cite{Guerin_PoP_1996, Fuchs_PRL_1998, Willingale_PRL_2009}, where $\bar{\gamma}$ is the characteristic Lorentz factor.
The expected drawback of this regime is the enhancement of $v_{ph}$ of the pulse. This superluminosity leads to poor phase matching between the wave and the electron during DLA, severely limiting the electron energy gain~\cite{Robinson_PoP_2015}.
However, the presented experimental measurements from relativistically near-critical plasma does observe an enhanced super-ponderomotive electron tail formation.
It has previously been noted that even relatively weak oscillating longitudinal electric fields found in a focussing or defocussing laser pulse can play a significant role in understanding DLA \cite{Robinson_PoP_2018}.
Here, the two-dimensional particle-in-cell simulations show one of the dominant energy transfer mechanisms into the high-energy tail is mediated by the evolving longitudinal electric fields within the main plasma volume causing the electrons to experience huge, rapid acceleration via this mechanism.
This is in stark contrast to previously identified DLA mechanisms that have either occurred in the very underdense region or essentially in vacuum with the overdense region serving as a source of electrons.
\section{Experimental set-up}
\label{exp_section}
The experiments were performed using the Titan laser system at the Jupiter Laser Facility \cite{Stuart_OSA_2006}. A pulse energy of $\epsilon_{L} = 127 \pm 25 \; \rm{J}$ was delivered on target in a full-width-half-maximum (FWHM) pulse length of $\tau_{L} = 1 \pm 0.2 \; \rm{ps}$. It was focused with an $f/3$ off-axis parabolic mirror to a $w_{0} = 10 \pm 2 \; \mu \rm{m}$ FWHM focal spot diameter containing up to $50 \%$ of the laser pulse energy to produce a mean peak vacuum intensity of $(5.3 \pm 1.8) \times 10^{19} \; \rm{Wcm}^{-2}$, corresponding to an $a_{0} \approx 6.5 \pm 2.2$. The prepulse energy was measured using a fast photodiode behind a water-cell to be $16 \pm 5 \; \rm{mJ}$ (measurements available for about $20 \%$ of the shots), giving a nanosecond energy contrast ratio of $\sim 10^4$. The laser pulse was linearly polarized and had a wavelength of $\lambda_{L} = 1.053 \; \mu \rm{m}$, so therefore $n_{c} = 10^{21} \; \rm{cm}^{-3}$.
Very low-density foams were used, with mass densities of $3$--$100 \; \rm{mg/cc} \pm 5\%$ that fully ionize to produce plasma with electron number density range $(0.9$--$30) \times 10^{21} \; \rm{cm}^{-3}$ (previously used for the experiments in Refs.\ \cite{Willingale_PRL_2009, Willingale_PoP_2011}) to produce well-controlled near-critical density targets. The low density foam targets were fabricated using the \textit{in situ} polymerization technique and had a composition of $71 \%$ C, $27 \%$ O and $2 \%$ H by mass. The pore and thread structures were sub-micron, so a relatively homogenous plasma was expected on the $\lambda_{L}$ scale. The delicate foams were supported within $250 \; \mu \rm{m}$ thick washers, with the aperture filled with foam to produce $(250 \pm 20) \; \mu \rm{m}$ thick foam targets. The angle of incidence of the laser pulse onto the front surface of the foam at $s$-polarization was $16^{\circ}$. For comparison, some shots were taken onto Mylar foils (fully ionized plasma density of $433 n_{c}$, i.e.\ $\gg n_{c}$), with thicknesses of $23 \; \mu \rm{m}$, $67.5 \; \mu \rm{m}$ or $250 \; \mu \rm{m}$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{Figure_1.pdf}
\caption{Measured spectra, averaged over each density (upper plot) and simulated (lower plot) electron spectra from different density targets. The spectra were measured along the laser-axis direction. The simulated spectra are snapshots for the entire plasma volume.}
\label{Figure_1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{Figure_2.pdf}
\caption{Electron beam divergence and pointing from two different shots onto $1.5 n_{c}$ plasma. The electron spectrometer acceptance angle and position is shown as the orange dot.}
\label{Figure_2}
\end{figure}
\section{Particle-in-cell simulation parameters}
\label{PIC_section}
To gain insight into electron heating in near-critical plasmas, two-dimensional simulations were performed using a fully-relativistic particle-in-cell code EPOCH~\cite{EPOCH} for the same range of near-critical target densities.
The laser propagates along the $x$-axis, and is linearly polarized with the electric field in the $y$-direction.
The laser pulse was approximated by a Gaussian beam focused to a 14 $\mu$m spot (FWHM of intensity) with $\lambda = 1.053 \; \mu \rm{m}$ and 0.7 ps in duration.
The peak vacuum normalized vector potential was $a_0 = 6.5$.
This laser pulse duration was chosen to mimic the experimental setup while keeping the simulation box in the case of lower density targets manageable.
At lower densities, the laser pulse easily propagates through the plasma.
In order to prevent the laser pulse from burning though the target during the simulation, the plasma thickness would have to be increased by roughly $\delta l \approx c \delta t$ if the pulse duration is increased by $\delta t$.
Additionally, the plasma width would have to be increased as well, because instabilities cause unpredictable and sometimes significant changes of direction for the propagating laser pulse.
Again, the lower density runs are much more impacted by this than the runs with intermediate densities.
Initially, the plasma is uniform, with a sharp boundary at $x = 0$.
The cell size in all the runs was 0.02 $\mu$m by 0.04 $\mu$m to resolve the dynamics of the accelerated electrons~\cite{Arefiev_PoP_2015}.
There were 100 macro-particles per cell at $n_e = 30 n_c$ and $n_e = 13.5 n_c$, and 50 macro-particles per cell in the other runs.
The ratio of macro-particles in each cell representing electrons, protons, carbon ions, and oxygen ions was set at 10:2:7:1.
No ionization took place during the simulation, with the ionization states for carbon and oxygen ions set at $Z_C = 6$ and $Z_O = 8$.
To ensure that the plasma is initially quasineutral, the ion densities are initially set at $n_p = 0.04 n_e$ for protons, $n_C = 0.116 n_e$ for carbon ions, and $n_O = 0.033 n_e$ for oxygen ions, so that $n_p + Z_C n_C + Z_O n_O = n_e$.
The target thickness in each case was sufficient to prevent the laser pulse from burning through the target during the runs that lasted 2 ps for $n_e = 30 n_c$ and $n_e = 13.5 n_c$ and 2.5 ps for $n_e = 0.9 n_c$, $n_e = 1.5 n_c$, $n_e = 3 n_c$, and $n_e = 6 n_c$.
Specifically, the target thickness was 140 $\mu$m for $n_e = 0.9 n_c$, 110 $\mu$m for $n_e = 1.5 n_c$, 60 $\mu$m for $n_e = 3 n_c$ and $n_e = 6 n_c$, and 25 $\mu$m for $n_e = 13.5 n_c$ and $n_e = 30 n_c$.
Using shorter targets made these computationally demanding runs more manageable, particularly in the case of high density targets where the number of macro-particles per cell had to be doubled.
\section{Results}
\label{results_section}
\subsection*{Experimental results}
The experimental electron spectra were measured using magnetic electron spectrometers \cite{Chen_RSI_2008} with image plate detectors. The upper plot in Fig.~\ref{Figure_1} shows typical electron spectra measured along the laser axis for each target density.
The lower plot in Fig.~\ref{Figure_1} shows snapshots of the simulated electron spectra at the peak of the laser intensity.
The maximum vacuum transverse and longitudinal electron $\gamma$ associated with $a_0 = 6.5$ are $p_{y}/m_e c = a_0 = 6.5$ and $p_{x}/m_e c = a_0^2 / 2 = 22$ respectively, so $n_{\gamma c}$ is likely in the range $6.5n_c$--$22n_c$.
Both plots show higher maximum electron energies for near-critical target densities when compared with relativistically opaque densities, i.e.\ $30 n_{c}$.
The experimental data shows significant fluctuations at the lowest electron densities.
The likely explanation for this is a variable electron beam pointing, as illustrated in Fig.~\ref{Figure_2}.
The electron beam divergence, $\theta_{e}$, and pointing were measured using a stack of aluminum and image plate layers.
Figure~\ref{Figure_2} shows electrons beams from two different $n_{e} = 1.5 n_{c}$ shots with $\theta_{e} \leqslant 10^{\circ}$ (half angle). The beams have asymmetric distributions and shot B has hints of more than one beam.
These measurements also indicate that the electron beam pointing was unstable.
The center of the beam was offset by $> 10^{\circ}$ from the original laser-axis with apparently arbitrary and random direction.
These observations are consistent with the numerical modeling where for $n_e = 0.9 n_c$ and $1.5 n_{c}$ the simulations showed unstable beam propagation accompanied by significant off axis deviations.
The total electron spectra from the simulation should be unaffected by this instability, but it could lead to a seeming decrease of the measured electron spectrum at $n_e = 0.9 n_c$.
The feature of primary interest to us here is that the spectra from the relativistically near-critical target range of $3 n_c$ to 13.5$n_c$ exhibit a similar looking energetic electron tail.
The spectrum drops only as the density is increased to $n_e = 30 n_c$ and the target becomes relativistically opaque and hence overdense.
These trends are in agreement with the simulation study presented in reference \cite{Willingale_PoP_2011}, where the simulated electron spectra from different near-critical density targets are considered, but the electron acceleration mechanisms were not investigated.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{Figure_3.pdf}
\caption{The experimental $T_{e}$, (a), and average electron energy, (b), extracted from the spectra along the laser-axis direction. The crosses show the individual shot data, whereas the circles give the averaged data for each density with the corresponding error-bars showing the $95\%$ confidence interval using Student's t-distribution. The black line shows the average solid target values and shaded region the error and the blue line shows the average with the standard error (shaded region) over the foam target shots and to guide the eye, the dashed line shows the maximum values at each density. The error on the individual shots is not shown for clarity, but typical errors are $\sim 10 \%$. The green triangles show the simulation temperature and average energy.}
\label{Figure_3}
\end{figure}
The experimental spectra were generally reasonably exponential so a fit was made to the data to determine a Maxwellian-like temperature, $T_{e}$, along the laser axis and are plotted versus plasma density in Fig.~\ref{Figure_3} (a), albeit with significant error in some cases.
Individual shot data is plotted as crosses and the mean for each density is plotted by circles with the error-bars showing the $95\%$ confidence interval using Student's t-distribution.
The likely reasons for the fairly large variation in $T_{e}$ are the variable electron beam pointing, as already discussed, and uncertainties when fitting to data with non-Maxwellian features.
The average electron energy measured between $2 \; \rm{MeV}$ and the detection threshold is a different way to present the data (Fig.~\ref{Figure_3} (b)).
There was smaller shot-to-shot variation for the average electron energies making the trend clearer and the mean values (squares) have a reduced standard deviation.
For the highest density, the solid Mylar foil targets ($n_{e} = 433 n_{c}$), and $n_e = 30n_c$ foam, the $T_{e}$ is in reasonable agreement with $U_{p} \approx 5.4 \; \rm{MeV}$ for $a_{0} = 6.5$.
For the lower densities, the high-energy tail enhances the $T_{e}$ and average electron energy to significantly above $U_{p}$.
For both $T_e$ and the average energy, the solid target mean values (gray lines) are significantly lower that the mean values over all of the foam target shots (blue lines).
Also shown in Fig.~\ref{Figure_3} as green triangles are the $T_e$ extracted from the simulation spectra.
The trends in both $T_e$ and average electron energy are similar, albeit with slightly lower values.
This shift is likely due to the difference between two- and three-dimensional effects, as well as the larger effective collection angle for calculating the simulation spectra.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{Fig_4a.png}
\includegraphics[width=0.6\columnwidth]{Fig_4b.pdf}
\caption{Data from the $n_e = 3 n_c$ simulation at $\Delta t \approx 18 \; \rm{fs}$ after the peak of the laser pulse has arrived at $x =0$ $\mu$m (elapsed time since the beginning of the simulation is $t = 1.15 \; \rm{ps}$). (a): $n_{e}$ on a logarithmic scale and the $n_{e} = n_{c}$ contour is indicated. The total electric fields normalized to the peak electric field in the absence of the target, $E_{0}$, is overlaid to highlight the relativistically transparent channel. (b): the same $n_{e}$ on a linear scale with quasi-static magnetic field contours shown. Overlaid is an example electron trajectory that is color-coded to indicate the $\gamma$ at each position.}
\label{Figure_4}
\end{figure}
\subsection*{Simulated electron energy gain}
The key features of the laser-plasma interaction in the near-critical regime ($n_c < n_e < n_{\gamma c}$) observed in the PIC simulations are illustrated in Fig.~\ref{Figure_4}.
The electron density prior to the interaction with the laser pulse is uniform, with $n_e = 3 n_c$.
The intense laser pulse induces relativistic transparency, which allows it to propagate through the plasma beyond the $n_e = n_c$ surface shown with a red curve in Fig.~\ref{Figure_4}a.
The electric field amplitude $E$ in Fig.~\ref{Figure_4}a has distinct spatial modulations associated with the oscillating field of the laser pulse more than 20 $\mu$m beyond the $n_e = n_c$ surface.
The density and the field snapshots are taken at $\Delta t \approx 18$ fs after the peak intensity would have arrived at $x = 0$ $\mu$m in the absence of the plasma.
The elapsed time since the beginning of the simulation is $t = 1.15$ ps.
The laser pulse produces a narrowing, funnel-like channel in the plasma with a laser-driven longitudinal electron current that generates and sustains a relatively strong slowly evolving magnetic field $B_z$.
$B_z$ is averaged over ten laser periods to find the quasi-static component that denoted as $\langle B \rangle$.
Two contours, $\langle B \rangle = \pm B_0$, are shown in Fig.~\ref{Figure_4}b, where $B_0$ is the peak amplitude of the laser magnetic field in the absence of the plasma.
Evidently, the quasi-static magnetic field is not negligible compared to the magnetic field of the laser and should be expected to impact the electron dynamics inside the funnel-like channel~\cite{Stark_PRL_2016, Jansen_PPCF_2018}.
\begin{figure}
\centering
\includegraphics[height=8cm]{Fig_5ab.pdf}\includegraphics[height=8cm]{Fig_5cd.png}
\caption{(a-b): The example electron $\gamma$ (blue dashed) as a function of time. The longitudinal (green line) and transverse (pink dots) components of the electron momentum (a) and contributions to the energy gain due to the each electric field component (b) are shown. (c-d): The longitudinal (c) and transverse (d) electric fields in a window moving along the $x$-axis with $c$. The location of the center of the window is shown above panels as a function of the elapsed time $t$ since the beginning of the simulation. The example relative electron position is color-coded according to the energy gain ($\gamma$) from the corresponding electric field component.}
\label{Figure_5}
\end{figure}
The energetic electrons are tracked during their energy gain process and the majority of the electrons from the energetic tail are found to originate inside this relativistically transparent channel.
Figure~\ref{Figure_4}b shows a representative electron trajectory to be discussed in detail.
As evident from the color-coded $\gamma$-factor in Fig.~\ref{Figure_4}b, the energy gain for this electron takes place well inside the plasma where $n_e > n_c$.
Figure~\ref{Figure_5}a shows the time evolution of the electron momentum components and the $\gamma$-factor for the same electron, illustrating that the electron is accelerated primarily in the laser propagation ($x$) direction.
To determine the underlying mechanism, the contributions to the $\gamma$-factor from the work done by the transverse, $E_y$, and longitudinal, $E_x$, components of the electric field are calculated and shown in Fig.~\ref{Figure_5}b as functions of time.
Here we use the following definitions:
\begin{eqnarray}
&& W_{\parallel} \equiv - \frac{1}{m_e c^2} \int |e| E_{x} v_{x} dt, \label{Wx}\\
&& W_{\perp} \equiv - \frac{1}{m_e c^2} \int |e| E_{y} v_{y} dt, \label{Wy}
\end{eqnarray}
so that $W_{\parallel} + W_{\perp} = \gamma -1$.
Remarkably, half of the energy gained by this tracked electron is contributed by $E_x$.
The significant role of the longitudinal field is unexpected, since the longitudinal component is negligible in the considered incoming beam due to the large beam width.
In the incoming beam, it can be estimated from the condition $\nabla \cdot {\bf{E}} = 0$, which yields $|E_x| \approx |E_y| \lambda / R$, where $R$ is the characteristic transverse scale of $E_y$.
Taking into account that $R \approx \sqrt{2} w_0$, we find that $|E_x| \approx 0.05 |E_y| \leq 0.05 E_0$, where $E_0$ is the amplitude of the transverse electric field in the focal plane of the incoming laser pulse.
In order to determine the actual fields experienced by the considered electron as it travels into the target, we use a window that is moving with the speed of light along the beam axis ($x$-axis).
The tracked electron is in the center of the window when it begins its longitudinal motion at $t = 1153$ fs (from the beginning of the simulation) and $x = 6.55$ $\mu$m.
Figures~\ref{Figure_5}c and \ref{Figure_5}d show $E_x$, $E_y$, and the longitudinal electron displacement in the moving window.
In contrast with the transverse field, a strong longitudinal electric field with $|E_x| \sim0.5 E_0$ emerges well inside the near-critical plasma ($x > 6.55$ $\mu$m).
This is the field that contributes to the electron energy gain, rather than the weak longitudinal field that we estimated for the incoming beam before it enters the target.
The mechanism responsible for generating this field is explained towards the end of this section, but here we simply point out that it is critical for the electron acceleration: the simulations observe a 30 fold increase in the longitudinal field compared with the vacuum case.
\begin{figure}
\centering
\includegraphics[width=1.1\columnwidth]{Figure_6.pdf}
\caption{Electron heating in the $n_e = 3 n_c$ simulation during the time interval of 1.05 ps $ \leq t \leq$ 1.3 ps. The electrons are tracked inside a box with $|y| < 8$ $\mu$m and $x < 30$ $\mu$m during 1050 fs $< t < $ 1300 fs. The panels show the electron data for the electrons that leave the box with $\gamma > 40$ moving to the right through the boundary located at $x = 30$ $\mu$m during 1050 fs $< t < $ 1300 fs. (a) shows a relative contribution, $\Delta W_{\parallel}/\gamma$, of the work done by the longitudinal field towards the total energy of each tracked electron. (b) and (c) show a statistical analysis of the components of the work done for $t < 1.17 ps$ ($\Delta t < 38 \; \rm{fs}$) and $t \geqslant 1.17 ps$ ($\Delta t \geqslant 38 \; \rm{fs}$) respectively. The inset shows the count of macro-particles representing electrons in panels (b) and (c).}
\label{Figure_6}
\end{figure}
The electron momentum is primarily longitudinal and, in agreement with Eq.~(\ref{Wx}), this enables a rapid transfer of energy from $E_x$ to the electron, shown with the color-coded circles in Fig.~\ref{Figure_5}c.
The electron gained the remainder of its energy from the transverse field where the self-generated magnetic field plays an important role in enabling this energy transfer.
The initial contribution right after the electron reaches the axis of the beam and begins its longitudinal motion (see Fig.~\ref{Figure_4}) is made via the conventional direct laser acceleration mechanism.
However, the presence of the near-critical plasma considerably limits the resulting energy gain by increasing the wave phase velocity $v_{ph}$ and thus deteriorating the phase matching.
As shown in Fig.~\ref{Figure_5}, the phase velocity of the transverse electric field in side the channel is $v_{ph} \approx 1.075 c$.
According to Ref.~\cite{Robinson_PoP_2015}, we should expect an energy gain corresponding to $\gamma \approx a_0 \left[ 2 (v_{ph} - c)/c \right]^{-1/2} \approx 16$.
This matches well the $E_y$-contribution at about 1160 fs shown in Fig.~\ref{Figure_5}b.
The second significant increase in $W_{\perp}$ occurs after the electron encounters a region with a strong magnetic field at 1183 fs and becomes deflected (see Fig.~\ref{Figure_4}).
The transverse momentum increases as a result of the deflection, which is typically detrimental for the direct-laser-acceleration.
The magnetic field however also breaks the synchronism between $p_y$ and $E_y$ that otherwise prevents further energy gain.
Following the deflection, the electron enters a region of negative $E_y$ (see Fig.~\ref{Figure_5}d) with a substantial positive transverse momentum $p_y$ (see Fig.~\ref{Figure_5}a).
This then allows for a rapid transfer of energy shown in Fig.~\ref{Figure_5}d with the color-coded circles, similar to what was observed in the case of $E_x$.
Detailed electron tracking has also enabled us to determine average relative contributions by $E_x$ and $E_y$ over a wide range of electron energies, shown in Fig.~\ref{Figure_6}.
We have tracked electrons in a box enclosing the funnel-like channel, $|y| < 8$ $\mu$m and $x < 30$ $\mu$m, recording $W_{\parallel}$ and $W_{\perp}$ over 250 fs (1050 fs $< t < $ 1300 fs).
We show the results for electrons with $\gamma > 40$ that leave the box moving to the right through the boundary located at $x = 30$ $\mu$m during 1050 fs $< t < $ 1300 fs.
Figure~\ref{Figure_6}a shows a relative contribution, $\Delta W_{\parallel}/\gamma$, of the work done by the longitudinal field towards the total energy of each tracked electron.
As the funnel structure becomes more pronounced with time, the effect of the longitudinal electric field becomes more pronounced.
After $t \approx 1.17$ ps, there are electrons, shown with yellow markers, that have gained more than 60\% of their total energy from $E_x$.
The energy exchange with $E_x$ is positive only for some electrons, while others lose an appreciable amount of energy to $E_x$.
Figures~\ref{Figure_6}b and \ref{Figure_6}c provide a statistical analysis of the electron heating in order to determine the effect of $E_x$ for each energy range.
We split the electrons into those that leave the box before and after $t \approx 1.17$ ps.
For the electrons that leave at $t <1.17$ ps, most of the energy had been accumulated outside of the spatial region of interest or before we started tracking them.
For the electrons that leave after $t \approx1.17$ ps, most of the energy is accumulated inside the region with the funnel-like channel.
The inset in Fig.~\ref{Figure_6}b shows the count of the macro-particles representing electrons in the histograms of Figs.~\ref{Figure_6}b and \ref{Figure_6}c.
The curves are essentially the electron spectra.
They confirm that the heating for the first group is ineffective, so its contribution compared to that of the second group is relatively insignificant.
The most important trend for the second group is that the longitudinal electric field contributes a considerable amount of energy of the energetic electrons, with $\Delta W_{\parallel} / \Delta W_{\perp} \approx 0.3$ for $\gamma > 80$.
Contrary to what one might expect, the work by the transverse electric field inside the region of interest never exceeds 70\% of the total energy for the energetic electron tail with $\gamma > 60$.
\subsection*{Accelerating field structure}
We have determined that the longitudinal electric field that arises inside the narrowing plasma channel makes an appreciable contribution towards the electron energy gain.
Here, we show that it is caused by reflections of the incoming laser beam off the walls of the funnel-like channel rather than by beam focusing or space-charge effects.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Figure_7.pdf}
\caption{The transverse and longitudinal electric fields shown at the same time as the images in Fig.~\ref{Figure_4} ($t = 1.15$ fs and $\Delta t \approx 18$ fs). Both components are normalized to $E_0 \approx 2 \times 10^{13}$ volt/m, the peak amplitude of the electric field in the incoming laser beam in the absence of the plasma. The maximum and minimum values of these field components are: $\max \left( E_y/E_0 \right) \approx 1.8$, $\min \left( E_y/E_0 \right) \approx -1.7$, $\max \left( E_x/E_0 \right) \approx 0.9$, and $\min \left( E_y/E_0 \right) \approx -1.1$.}
\label{Figure_7}
\end{figure}
Snapshots of $E_y$ and $E_x$ shown in Fig.~\ref{Figure_7} have seemingly uncorrelated patterns.
The transverse component $E_y$ has almost flat wave-fronts as deep as 10 $\mu$m into the plasma.
In contrast to that, $E_x$ has what appears as tilted wave-fronts, such as in the region with $y>0$ $\mu$m and 0 $\mu$m $< x < 10$ $\mu$m where the wave-fronts of $E_y$ are still flat.
In the case of beam focusing, the wave-fronts of $E_x$ and $E_y$ are aligned (for example, see Ref.~\cite{Gong_arXiv_2018} where a narrow channel is used to amplify $E_x$).
However, this pattern is not visible in the incoming beam because the corresponding field, $|E_x| \approx 0.05 E_0$, is too weak.
The focusing in the narrowing channel is also insufficient to explain the observed increase of the longitudinal field.
The beam width would have to decrease at least by a factor of ten for $E_x$ to be visible in Fig.~\ref{Figure_7}b, but the beam width decreases by not more than a factor of two when $E_x$ becomes strong.
Figure~\ref{Figure_8} shows magnified snapshots of $E_x$ and $B_z$ in the region with tilted wave-fronts of the longitudinal electric field.
A comparison of Figs.~\ref{Figure_7}a and \ref{Figure_7}b reveals that the transverse periodic modulations of $B_z$ coincide with the wave-fronts of $E_x$ that are shown with contours in both panels to guide the eye.
The fact that there is a correlation between $E_x$ and $B_z$ indicates that space-charge effects are unlikely to be the cause of the strong longitudinal electric field.
The modulations are consistent with reflections.
In order to demonstrate the role of beam reflections in creating the observed field structure, we consider a simple model where three plane waves overlap, producing an interference pattern.
The electric and magnetic fields in each of the waves are given by
\begin{eqnarray}
&& E_x = - E_* \sin \theta \cos \left[ 2 \pi x' / \lambda + \psi(t) \right], \\
&& E_y = E_* \cos \theta \cos \left[ 2 \pi x' / \lambda + \psi(t) \right], \\
&& B_z = E_* \cos \left[ 2 \pi x' / \lambda + \psi(t) \right],
\end{eqnarray}
where $E_*$ is the wave amplitude, $\theta$ is the angle between the $x$-axis and the direction of the wave propagation, $\psi(t)$ is the time-dependent phase and
\begin{equation}
x' \equiv x \cos(\theta) + y \sin(\theta)
\end{equation}
is the distance along the direction of the wave propagation.
We mimic the case observed in the simulation by assuming that the main wave propagates forward along the $x$-axis, such that $E_* = E_0$, $\theta = 0$.
Without any loss of generality, consider two lower amplitude waves that come in at an angle, where $E_* = 0.25 E_0$, $\theta = - \pi/3$ and $E_* = 0.1 E_0$, $\theta = \pi / 8$.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{Figure_8.pdf}
\caption{Magnified region of the longitudinal electric field $E_x$ (a) and transverse magnetic field $B_z$ (b) at $t = 1.15$ fs. The magnetic field is normalized to $B_0 \approx 66.2$ kT, which is the peak amplitude of the magnetic field in the incoming laser beam in the absence of the plasma. The black curves in both panels indicate the contours of constant $E_x$, with $E_x / E_0 = 0.1$. Over the entire simulation domain, $\max \left( B_z/B_0 \right) \approx 2.1$ and $\min \left( B_z/B_0 \right) \approx -1.8$.}
\label{Figure_8}
\end{figure}
The interference patterns at $\psi = 0$ for the electric and magnetic fields are shown in Fig.~\ref{Figure_9}.
Similarly to what is seen in Fig.~\ref{Figure_7}, the wave-fronts of $E_y$ are vertical, but the wave-fronts of $E_x$ are clearly tilted without any correlation between the two patterns.
This pattern has a clear origin: the wave-fronts of $E_x$ are created exclusively by the lower-amplitude waves.
The explanation is further corroborated by the difference in the longitudinal phase velocities of $E_x$ and $E_y$ in Figs.~\ref{Figure_5}c and \ref{Figure_5}d.
These results were obtained from the PIC simulation and they show that the wave-fronts of $E_x$ are moving faster.
Since the lower-amplitude waves that are responsible for $E_x$ are moving at an angle with respect to the $x$-axis, their phase velocity along the $x$-axis is indeed increased.
The last point to emphasize is the correlation between the modulation of $B_z$ and the tilted wave-fronts of $E_x$ in Figs.~\ref{Figure_9}b and \ref{Figure_9}c. This pattern is again similar to what is seen in the PIC simulations and shown in Fig.~\ref{Figure_7}. The incoming beam has only one component of the magnetic field, which is $B_z$. Reflections do not alter the polarization of the magnetic field, as opposed to what happens to the electric field. As a consequence, the tilted wave-fronts contribute more to the magnetic field of the main wave than to $E_y$ and that is why the modulations in the magnetic field are much more pronounced than those in the transverse electric field.
This simple model elucidates the mechanism responsible for the observed 30 fold increase in the longitudinal field compared with the vacuum case.
The increase takes place without any significant laser beam focusing.
The field is particularly beneficial for energizing electrons that are accelerated in the forward direction by the pulse, i.e. the main component of the wave.
Since the electron momentum is primarily longitudinal, a rapid transfer of energy from $E_x$ to the electron takes place, shown with the color-coded circles in Fig.~\ref{Figure_5}c.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{Figure_9.pdf}
\caption{A wave pattern produced by three overlapping plane waves of different amplitude. The dotted curves in the right panel indicate the contours of constant $E_x$, with $E_x / E_0 = 0$.}
\label{Figure_9}
\end{figure}
\section{Summary}
\label{summary_section}
In conclusion, we have shown that laser beam propagation in near-critical plasmas, where $n_c < n_{e} < n_{\gamma c}$, can create conditions favorable for electron heating to energies well beyond what is achievable using transverse electric field DLA in such plasmas. Oscillating longitudinal electric and quasi-static magnetic fields generated by the narrowing plasma channel play a profound role in electron heating, enabling rapid and significant energy transfer to electrons from the laser pulse despite the appreciable super-luminal phase velocity. On average, the longitudinal electric field contributes roughly one third of the energy transferred by transverse electric field of the laser pulse to electrons of the super-ponderomotive tail.
Situations where this mechanism may be particularly important are in thin foil targets that decompress to near-critical densities on the timescale of the laser pulse \cite{Palaniyappan_NP_2008, Howell_NJP_2015}, for neutron beam generation \cite{Pomeratz_PRL_2014}, for hole-boring fast ignition \cite{Tabak_PoP_1994}, or for the next generation of laser systems, currently under construction, that will reach intensities accessing a ``QED-plasma'' regime -- where non-linear synchrotron $\gamma$-ray production and multi-photon Breit-Wheeler pair production become important -- and even solid aluminum targets will be in the $n_{\gamma c}$ regime \cite{Ridgers_PRL_2012}.
\ack
$^{\ddag}$Corresponding author.
$^{\dag}$Present address: Department of Physics and Astronomy, University of California, Irvine.
$^{\ast}$Present address: General Atomics, CA, USA.
$^{\ast \ast}$Present address: Naval Research Laboratory.
The authors gratefully acknowledge technical assistance from the staff of the Jupiter Laser Facility.
Work done by Lawrence Livermore National Laboratory was supported by the U.S. Department of Energy under Contract No. DE-AC52-07NA27344.
LW acknowledges support from the Department of Energy National Nuclear Security Administration under Award Number DE-NA0002028 and DE-NA0002723. AVA was supported by the U.S. DoE through agreements No. DE-NA0002008 and by the National Science Foundation (Grant No. 1632777). Simulations were performed using EPOCH code (developed under UK EPSRC Grants No. EP/G054940/1, No. EP/G055165/1, and No. EP/G056803/1) using HPC resources provided by the TACC at the University of Texas at Austin.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,808 |
Scott Weisenberger offers his customers much more than just freshly baked goodies at Sisters McMullen Cupcake Corner at 1 Pack Square in downtown Asheville. He's more like an Ambassador to the city for the countless visitors who can't resist stopping in after peering in the window and seeing the colorful display of pastries.
A young couple open the door and Scott gives them his biggest smile and a warm greeting. Their eyes light up at all the goodies. " Take your time" he says. "There's no rush. If you have any questions, feel free to ask." It doesn't take them long to make their decision, pay for their cookies and start munching!
When Scott sees me coming through the door, he knows to start packing up a few black and white cookies. For years, I searched all over Asheville for black and white cookies that resembled the ones from my Father's restaurant (Wolfies) in South Florida. It's like having a little bit of "home" each time I take a delicious bite!
Thanks to Scott Weisenberger, stopping in for my cookie "fix" is (as always) one of the nicest experiences of my week! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,860 |
All crime is unacceptable but offences that are driven by hostility or hatred based on personal characteristics set a particular challenge to a civilised society. For the Crown Prosecution Service (CPS) therefore, effectively addressing all forms of hate crime and crimes targeting older people remain a core commitment.
The CPS Core Quality Standards1 were introduced in 2010. The standards describe the quality of service that can be expected by victims, witnesses and defendants and set out the essential considerations for the prosecution of all crime. We regularly monitor the number and quality of our hate crime prosecutions and continually review the guidance provided to our prosecutors to ensure that our performance and the service that we provide to the victims of hate crime improves. This report sets out our performance and highlights any emerging trends.
outcome this is 11% more than the previous year. I am also particularly pleased that 85.5% of our successful cases involved a guilty plea as this demonstrates that our prosecutors are building stronger cases. This increase in guilty pleas benefits the victims of these crimes and the efficiency of the criminal justice system.
We continue to improve our understanding of all aspects of hate crime through our joint work with organisations such as MIND and the scrutiny and feedback on our handling of hate crime cases provided by our local scrutiny panels and our network of Hate Crime Coordinators. Therefore, although, I am pleased by the improvements in our performance during this year, there is still room for improvement so I am in no way complacent about the task ahead; particularly in relation to disability and transphobic hate crime where I believe we are very much at the beginning of our journey.
stakeholders locally and nationally. In 2011/12, our Equality and Diversity objectives will reflect our commitment to improving hate crime performance and we look forward to working with other government agencies on the development of the new Hate crime action plan and to working with the Equality and Human Rights Commission to take forward the recommendations from their Inquiry into disability related harassment with a view to building on our progress next year. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,912 |
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\begin{document}
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\begin{titlepage}
\nummer{MZ-TH/94-26}
\netnum{gr-qc/9409193}
\monat{\today}
\titel{Hamilton Formalism in Non-Commutative Geometry\footnote{
{\rm Work supported in part by the PROCOPE agreement between the
University of Aix-Marseille and the Johannes Gutenberg-Universit\"at of Mainz.
}}}
\autor{Wolfgang Kalau\footnote{
e-mail: kalau{\char'100}dipmza.physik.uni-mainz.de}}
\adresse{Johannes Gutenberg Universit\"at\\
Institut f\"ur Physik\\
55099 Mainz}
\begin{abstract}
We study the Hamilton formalism for Connes-Lott models, i.e., for
Yang-Mills theory in non-commutative geometry. The starting point is an
associative $*$-algebra $\cal A$ which is of the form $\cal A=C(I,\cal A_s )$ where
$\cal A_s $ is itself a associative $*$-algebra. With an appropriate choice of a
k-cycle over $\cal A$ it is possible to identify the time-like part of the
generalized differential algebra constructed out of $\cal A$. We define the
non-commutative analogue of integration on space-like surfaces via the
Dixmier trace restricted to the representation of the space-like part
$\cal A_s $ of the algebra. Due to this restriction it possible to define the
Lagrange function resp.~Hamilton function also for Minkowskian space-time.
We identify the phase-space and give a definition of the Poisson bracket
for Yang-Mills theory in non-commutative geometry. This general formalism
is applied to a model on a two-point space and to a model on Minkowski
space-time $\times$ two-point space.
\end{abstract}
\end{titlepage}
\section{Introduction}
In the last few years it has turned out that A.~Connes' non-commutative
geometry provides a framework which allows for new qualitative insights in the
spontaneous symmetry breaking mechanism of Yang Mills theories. The
cornerstone of this approach is a generalization of the algebra of
differential forms and its corresponding differential. This has been used to
construct models for the electroweak interaction \cite{colo,cobuch} and
Grand Unification \cite{chams1,chams2}. Since the generalization of
the differential algebra and its differential is not unique there are
alternative models for the electroweak interaction, like the one developed by
the Marseille and Mainz groups \cite{cev,hps1,hps2}. However, all models have
in common that the Higgs field is interpreted as a part of the generalized
connection form, although the precise form of the Higgs potential depends on
the model chosen.
Another feature, which is common to all models so far, is that they are purely
classical models, i.e. non-commutative geometry has been used to derive
classical actions. In this approach some coupling constants, like
the Higgs mass and the top mass in the
Connes-Lott model, appear naturally restricted. However, such relations at the
classical level cannot be translated to relations at the quantum level in an
obvoius way \cite{AGBM}. The reason for this is that it is not known so far
how to quantize a theory in the framework of non-commutative geometry and for
the usual quantization procdedure it does not matter if some coupling constants
of the classical action are fixed by hand or by some general principles of
non-commutative geometry. Therefore it seems desirable to have a translation
of the usual quantization procedure into the language of non-commutative
geometry in order to get new insight in quantized Yang-Mills theory.
The generalization of geometry to non-commutative geometry is achieved by
translating geometrical concepts into an algebraic language where
conventional geometry corresponds to commutative algebras. The generalization
is then obtained by extending those concepts to non-commutative algebras.
The quantization procedure which is closely related to algebra is the
canonical quantization method. This approach to quantum theory is based on
the Hamilton formalism. The purpose of this article is to develope an Hamilton
formalism for (generalized) Yang-Mills theories in non-commutative geometry
as they were introduced in \cite{colo,cobuch}.
This article is organized as follows. In sect.~2 we give a motivation for the
structure\linebreak
$\cal A=C(I,\cal A_s )$ of the associative $*$-algebra $\cal A$ which is the
starting point for the derivation of Yang-Mills theory in non-commutative
geometry. The universal differential enveloping algebra and the concept of
finitely summable k-cycles are briefly reviewed in sect.~3 where we construct
a k-cycle which is appropriate for our purpose. In sect.~4 the generalized
differential algebra $\Omega_D \cal A$ of A.~Connes\cite{cobuch} is discussed
where we use the structure on $\cal A$ and the k-cycle, introduced in the previous
sections, to show that there is a split
of $\Omega_D \cal A$ into a ``space-like'' and a ``time-like'' part. The trace theorem
of A.~Connes \cite{co} is used in sect.~5 to define an inner product on
$\Omega_D \cal A$. This definition differs from the usual definition in the sense that
it corresponds to an integration on a ``space-like'' surface. As a consequence
it is possible to define it also on space-time geometries with Minkowski
metric. After a brief review of Yang-Mills theory in non-commutative geometry
as it was introduced by A.~Connes and J.~Lott \cite{colo,cobuch}, the
Lagrange function and the Hamiltonian for Yang-Mills theory are defined in
sect.~6. The formal construction ends with the definition of the Poisson
bracket and time evolution in sect.~7. In sect.~8 the formalism is applied
to two examples, namely to a discrete space and to Yang-Mills theory
with symmetry breaking. The article ends with some conclusions in
sect.~9.
\section{The Algebra $\cal A$}
Hamilton formalism is related to Cauchy surfaces in space-time and
the separation of time which implies that the space-time manifold
$M$ has the topology
\begin{equation}
M = {I\kern -0.22em R\kern 0.30em} \times \Sigma
\end{equation}
where ${I\kern -0.22em R\kern 0.30em}$ corresponds to time and $\Sigma$ to a (compact) space-like
manifold. As consequence the corresponding $C^*$-algebra of continous
functions (vanishing at infinity) $C_0(M)$ is of the form
\begin{equation}
C_0(M) = C_0({I\kern -0.22em R\kern 0.30em}) \otimes C(\Sigma) = C_0({I\kern -0.22em R\kern 0.30em}, C(\Sigma)) \label{prod}
\end{equation}
where $C_0({I\kern -0.22em R\kern 0.30em}) \otimes C(\Sigma)$ denotes the completition of the
algebraic product of $C({I\kern -0.22em R\kern 0.30em})$ and $C(\Sigma)$ and $C_0({I\kern -0.22em R\kern 0.30em}, C(\Sigma))$,
or more generally $C_0({I\kern -0.22em R\kern 0.30em} , \cal A)$, is the algebra of continous functions
over ${I\kern -0.22em R\kern 0.30em}$ with values in $C(\Sigma)$ resp.~with values in some normed
algebra $\cal A$.
The starting point of A.~Connes' generalization of differential forms is
an associative $*$-algebra $\cal A$ (a subalgebra of a $C^*$-algebra).
Equation (\ref{prod}) motivates us to
require that $\cal A$ has some additional structure which allows to
introduce ``time'' to the formalism of generalized differential forms.
Thus we postulate that
\begin{equation}
\cal A = C(I,\cal A_s )\;\; ,\label{gprod}
\end{equation}
where $I$ is either ${I\kern -0.22em R\kern 0.30em}$ or $S^1$ and $\cal A_s $ is a normed associative
$*$-algebra with unit, possessing a finitely summable k-cycle.
If $\cal A_s $ is a $C^*$ algebra we have
\begin{equation}
\cal A = C(I,\cal A_s ) = C(I)\otimes\cal A_s
\end{equation}
where $C(I)\otimes\cal A_s $ again denotes the completition of the algebraic
product of $C(I)$ and $\cal A_s $. Since $\cal A_s $ has a unit we can identify $C(I)$,
the algebra of continous functions on $I$, as a subalgebra of $\cal A$ by
\begin{equation}
\begin{array}{rclr}
i_t & : & C(I) \longrightarrow \cal A & \\
& & & \\
i_t(f)& = & f\otimes 1_s & f\in C(I)
\end{array}
\end{equation}
where $1_s$ denotes the unit element in $\cal A_s $.
We shall assume that $\cal A$ has a unit element. If $I$ is compact
(i.e. $I=S^1$) then $C(I)$ and therefore also $\cal A$ has a unit element.
If $I={I\kern -0.22em R\kern 0.30em}$ then $C_0(I)$ does not have a unit. However we can always
formally add a unit element to $C_0(I)$ which induces a unit element in $\cal A$.
Furhtermore we can use the unit element $1_t$ of $C(I)$ to identify $\cal A_s $
as a subalgebra of $\cal A$:
\begin{equation}
\begin{array}{rclr}
i_s & : & \cal A_s \longrightarrow \cal A & \\
& & & \\
i_s(a)& = & 1_t\otimes a & f\in \cal A_s \;\; .
\end{array}
\end{equation}
\section{The Universal Differential Envelope and the k-cycle over $\cal A$}
In this and the subsequent section we follow A.~Connes construction of
generalized differential forms \cite{cobuch}. However, we will focus on the
structure of $\cal A= C(I,\cal A_s )$ which will lead to a
``time-split'' in the generalized differential algebra. For details of the
general construction we refer to \cite{cobuch,Kbuch,GBV}.
The first step is to construct a bigger algebra
$\Omega\cal A$ by associating to each element $A\in\cal A$ a symbol
$\delta A$. $\Omega\cal A$ is the free algebra generated by the symbols $A$,
$\delta A$, $A\in\cal A$ modulo the relation
\begin{equation}
\delta (AB)=\delta A\, B + A\delta B\;\; .\label{gl-4}
\end{equation}
With the definition
\begin{equation}
\begin{array}{rcl}
\delta(A_0\delta A_1\cdots\delta A_k) & \;:= &
\;\delta A_0\,\delta A_1\cdots\delta A_k \\
& & \\
\delta(\delta A_1\cdots\delta A_k) & \;:= & 0
\end{array}
\end{equation}
$\Omega\cal A$ becomes a ${I\kern -0.22em N\kern 0.30em}$-graded differential algebra with the odd
differential $\delta$, $\delta^2=0$. $\Omega \cA $ is called the universal
differential envelope of \cal A.
By defining
\begin{equation}
{\delta(A)}^*=-\delta(A^*)
\end{equation}
the $*$-operation is extended uniquely to $\Omega \cA $.
The next element in the construction is the k-cycle $(\cal H , D)$ over $\cal A$.
It consists of a Hilbert space $\cal H $ with a faithful $*$-representation
$\pi$
\begin{equation}
\pi : \cal A \longrightarrow {\cal B}(\cH)
\end{equation}
where ${\cal B}(\cH) $ denotes the algebra of bounded operators on $\cal H $. The second
part of the k-cycle is an unbounded self-adfoint operator $D$ on $\cal H $. Since
the k-cycle should also reflect the structure given by eq.(\ref{gprod}) let us
first discuss the representation $\pi$ a little bit further before we come
to structure of $D$. However, the main strategy will be to construct the
k-cycle $(\cal H , D)$ over $\cal A$ out of k-cycles $(\cal H_s , D_s)$ over $\cal A_s $.
Suppose $\cal H_s $ is a (seperable) Hilbert space with an inner product
${(\cdot ,\cdot)}_s$ and a faithful $*$-representation
${\tilde{\pi}}_s$
\begin{equation}
{\tilde{\pi}}_s : \cal A_s \longrightarrow {\cal B}(\cHs) \;\; .
\end{equation}
$\cal H _s$ can be extended to a bigger Hilbert space
\begin{equation}
\cal H = L_2(I,\cal H_s )
\end{equation}
with the inner product
\begin{equation}
(\Psi,\Phi) = \int_I \! dt {(\Psi(t),\Phi(t))}_s\;\; .
\end{equation}
The representation ${\tilde{\pi}}_s$ on $\cal H_s $ induces a representation $\pi_s$
on $\cal H $ of $\cal A_s $
\begin{equation}
\pi_s : \cal A_s \longrightarrow C(I,{\cal B}(\cHs) ) \subset {\cal B}(\cH)
\end{equation}
by indentifying ${\cal B}(\cHs) $ with the subalgebra of operators in $C(I,{\cal B}(\cHs) $ which
are constant in $t\in I$. There is also a representation $\pi_t$ of $C(I)$
\begin{equation}
\begin{array}{rcl}
\pi_t & : & C(I) \longrightarrow C(I,{\cal B}(\cHs) ) \\
& & \\
\pi_t(f)& = & f\, {id}_s \;\; ,\;\; f\in C(I)
\end{array}
\end{equation}
where ${id}_s$ denotes the unit element in ${\cal B}(\cHs) $. Because of eq.(\ref{gprod})
these two representations induce a faithful $*$-representation $\pi$ of $\cal A$
\begin{equation}
\pi : \cal A \longrightarrow C(I,{\cal B}(\cHs) )
\end{equation}
with
\begin{equation}
\pi(f\otimes a)=\pi_t(f)\pi_s(a)=\pi_s(a)\pi_t(f)\;\; ,\;\; f\otimes a\in \cal A
\end{equation}
Strictly speaking, $\pi_s$ and $\pi_t$ define a representation of a dense
subalgebra of $\cal A$, which can be extended to a representation of $\cal A$.
Let us now turn to the second element of the k-cycle, the operator $D$
on $\cal H $. The general conditions to be fulfilled by this operator are
\cite{cobuch}
\begin{itemize}
\item[{\bf i.}] $D$ is self-adjoint;
\item[{\bf ii.}] $[D,\pi(A)]$ is a bounded operator;
\item[{\bf iii.}] $D$ is unbounded with a compact inverse (modulo finite rank
operators) such that $|D|^{-1}$ is
$d^+$ summable for some $d\in{I\kern -0.22em N\kern 0.30em}$;
\end{itemize}
If $\cal A$ is a $C^*$ algebra condition {\bf ii.} holds only on a dense
subalgebra of $\cal A$ in general. Therefore we denote in the following by $\cal A$
a dense subalgebra of a $C^*$ algebra such that {\bf iii.} holds for any
element of $\cal A$, i.e.~$\cal A=C^\infty(I,\cal A_s )$, where $\cal A_s $ is also a
suitable dense subalgebra of a $C^*$ algebra.
However, since $D$ is closely related to the metric structure of the
underlying manifold, which is also the case for
non-commutative geometries\cite{cobuch}\footnote{In fact, if $D$ is a Dirac
operator it is possible to construct a gravity-action by taking the Wodzicki
residue of an appropriate inverse power of $D$\cite{coLS,kagr,KW}.},
we have to impose further conditions on $D$. They should reflect the topology
which is encoded in the structure (\ref{gprod}) of $\cal A$. This motivates the
additional requirement that $D$ is the sum of two operators
\begin{equation}
D= D_t + D_s
\end{equation}
with
\begin{itemize}
\item[{\bf iv.}]
$
[D_t,\pi_s(a)]=0\;\;\; \mbox{and} \;\;\; [D_s, \pi_t(f)]=0\;\; ,\;\;
\forall f\in C(I)\;\; ,\;\; \forall a\in \cal A_s \;\; ;
$
\item[{\bf v.}]
$
[D_t,\pi_t(f)][D_s,\pi_s(a)]+[D_s,\pi_s(a)][D_t,\pi_t(f)]=0\;\; ,\;\;
\forall f\in C(I)\;\; ,\;\; \forall a\in \cal A_s \;\; ;$
\item[{\bf vi.}] $D_s \in C^\infty(I, {\cal O}_s)$\footnote{Note, this implies
the second part of {\bf iv.}.}, (where ${\cal O}_s$ denotes the algebra of
operators on $\cal H_s $) thus $D_s$ is as a smooth 1 parameter
family of operators on $\cal H_s $. $D_s(t)$ fulfills conditions {\bf i.}-{\bf iii.}
with $\cal A$ replaced by $\cal A_s $, $\cal H $ replaced by $\cal H_s $. In other words
$(\cal H_s , D_s(t))$ is a smooth 1 parameter family of k-cycles over $\cal A_s $.
\end{itemize}
We now show how a k-cycle $(\cal H , D)$ over $\cal A$ can be constructed out of a
1-parameter family of k-cycles $(\tilde{\cal H_s }, \tilde{D_s}(t))$, $t\in I$, over
$\cal A_s $. Having the $3+1$-dimensional case in mind, we do not assume that there
is a grading on $\cal H_s $, i.e.~an automorphism $\gamma$ with $\gamma^2=1$ and
$[\gamma,\tilde{D_s}]_+ =0$\footnote{$[ \cdot, \cdot]_+$ denotes the
anti-commutator} and $[\gamma, \pi_s] = 0$. However, a substitute for this
automorphism can always be constructed by extending $\tilde{\cal H_s }$, which also
allows to drop the condition that $\tilde{D}_s$ is self-adjoint. We extend
the Hilbert space by $\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{\!2}$:
\begin{equation}
\begin{array}{rcl}
\cal H_s & = & \tilde{\cal H_s }\otimes \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{\!2}\\
& & \\
\pi_s&= & \tilde{\pi_s}\otimes 1_{\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{\!2}}
\end{array}\label{c2-ext}
\end{equation}
and
\begin{equation}
D_s =
\left( \begin{array}{cc}
0 & \tilde{D_s} \\
& \\
\tilde{D_s}^\dagger & 0
\end{array} \right) \;\; .\label{dsdef}
\end{equation}
The Hilbert space $\cal H_s $ and the representation $\pi_s$ can be extended
to a Hilbert space $\cal H $ and a representation $\pi$ of $\cal A$ in the above
mentioned manner. What is still missing is the operator $D$. More precisely
the part $D_t$ of $D$ has to be specified. A natural choice is
$D_t\sim \partial_t$. However, condition {\bf v.} has to be
taken into account. Therefore we introduce an element $\gamma^0 \in {\cal B}(\cH) $
(a substitute for the grading) with the following properties:
\begin{equation}
\gamma^0=\left(\begin{array}{cc}
\tilde{\gamma} & 0 \\
& \\
0 & -\tilde{\gamma}
\end{array}\right)\;\; ; \label{blockdia}
\end{equation}
\begin{equation}
{\gamma^0}^2 = \pi(N)\; , \; N\in\cal A \;\; ;\;\;
{[\gamma^0, \pi(A)]} =0 \; , \;\forall A\in \cal A \;\; ;\;\;
{\gamma^0}^{-1}\;\; \hbox{exists}\label{etacon}
\end{equation}
and
\begin{equation}
{\gamma^0}^\dagger = -\gamma^0\;\; ,\label{eukl}
\end{equation}
where the same block structure as in eq.(\ref{c2-ext}) is used.
Such an element always extists since $\cal A$ has an unit element.
We now define $D_t$ by
\begin{equation}
D_t = \gamma^0\partial_t \;\; .\label{dtdef}
\end{equation}
The anti self-adjointness of $\gamma^0$ ( eq.(\ref{eukl})) ensures the
self-adjointness of $D=D_t+D_s$.
It is straightforward to check that
for this choice of $D=D_t+D_s$, with $D_t$ resp.~$D_s$ defined as in
eq.(\ref{dtdef}) resp.~(\ref{dsdef}), and $\cal H $ $\; (\cal H , D)$ is a k-cycle
over $\cal A=C^\infty(I,\cal A_s )$ which fulfills conditions {\bf ii.} and
{\bf iv.}-{\bf vi.}. Condition {\bf iii.} is not needed for the definition
of the generalized differential algebra. It is crucial for the definition
of the operator theoretic substitute for integration. However, since we are
only interested in a substitute for integration on ``space-like'' surfaces
we can replace this condition on $D$ by an analogous condition on $D_s$
({\bf vi.}). However, this condition on $D$ and the self-adjointness of $D$
is related to the Euklidean signature of the metric of the underlying
manifold. As we shall see in sect.\ref{mink}, with the choice
$\gamma^0={\gamma^0}^\dagger$ one obtains an operator $D$ corresponding to an
underlying manifold with Minkowski metric.
\section{The Generalized Differential Algebra}
Having introduced the generalized differential algebra $\Omega \cA $ of $\cal A$
and a k-cycle $(\cal H ,D)$ over $\cal A$ we can now put these elements together
in order to define a generalized differential algebra as it was introduced
by A.~Connes \cite{cobuch}.
We begin with extending the $*$-representation $\pi$ to a $*$-representation
of the algebra $\Omega \cA $
\begin{equation}
\begin{array}{rcl}
\pi_D & : & \Omega_D \longrightarrow {\cal B}(\cH) \\
& & \\
\pi_D (A_0\delta A_1 \cdots \delta A_k) & = &
\pi(A_0)[D,\pi(A_1)]\cdots [D,\pi(A_k)]\;\; .
\end{array}\label{pd}
\end{equation}
However, there is another possibility to extend $\pi$ to a representation
of $\Omega \cA $ which is useful for our purpose:
\begin{equation}
\begin{array}{rcl}
\pi_{D_s} & : & \Omega_D \longrightarrow {\cal B}(\cH) \\
& & \\
\pi_{D_s}(A_0\delta A_1 \cdots \delta A_k) & = &
\pi(A_0)[D_s,\pi(A_1)]\cdots [D_s,\pi(A_k)]\;\; .
\end{array}\label{pds}
\end{equation}
Obviously the kernel of $\pi_{D_s}$ is much bigger than the kernel of $\pi_D$.
For instance $\delta C^\infty(I) \subset \Omega^1\cal A$ is contained in the
kernel of $\pi_{D_s}$ because only the ``space-like'' part $D_s$ of $D$ is used
in the definition eq.(\ref{pds}) of $\pi_{D_s}$.
On the images of $\pi_D$ resp.~$\pi_{D_s}$ the differential $\delta$ on $\Omega \cA $
does not induce well defined differentials. Therefore one has to devide
out two sided graded differential ideals. For $\pi_D$ such an ideal
is given by
\begin{equation}
\begin{array}{rcl}
\cal J ^k & = & \ker\pi_D \cap \Omega^k\cal A + \delta(\ker\pi_D \cap
\Omega^{k-1}\cal A)\\
& & \\
\cal J & = & \bigoplus^\infty_{k=1} \cal J ^k\;\; . \label{jdef}
\end{array}
\end{equation}
On the quotient algebra $\Omega_D \cal A$, which is defined as
\begin{equation}
\begin{array}{rcl}
\Omega_D ^k\cal A & = & {\Omega^k\cal A\over \cal J ^k}\\
& & \\
\Omega_D \cal A & = & \bigoplus_{k=1}^\infty \Omega_D ^k\cal A
\end{array}
\end{equation}
there is a differential $d$ with $d^2=0$ which is uniquely defined by the
differential $\delta$ on $\Omega \cA $ as
\begin{equation}
d(\sigma_D\pi_D(\omega))=
\sigma_D\pi_D(\delta\omega)\;\;,\;\omega\in\Omega_D
\end{equation}
where $\sigma_D$ denotes the map
\begin{equation}
\sigma_D:\pi_D(\Omega^k\cal A)\longrightarrow\Omega_D ^k\cal A\;\; .
\end{equation}
Thus $\Omega_D \cal A$ is a generalized graded differential algebra \cite{cobuch}.
Of course, it is now possible to define the differential ideal associated to
$\pi_{D_s}$ in a completely analogous way as for $\pi_D$. This would also lead to a
differential algebra with a differential which is uniquely defined by the
differential on $\Omega \cA $. However, such a differential algebra would not have an
interpretation as the ``space-like'' part of $\Omega_D \cal A$ in general. Therefore one
has to divide out a bigger differential ideal. One is led to the correct ideal
by the two lemmas following the next little preparing lemma.
\begin{lemm}\label{le0}
For $I= S^1$ there is an
\begin{equation}
\eta= \sum_i f^{(i)}\delta g^{(i)}\in\Omega^1\cal A\; , f^{(i)},g^{(i)}\in C(I)
\label{etas1}
\end{equation}
such that
\begin{equation}
\pi_D(\eta)=\gamma^0\;\; .\label{etas1a}
\end{equation}
If $I={I\kern -0.22em R\kern 0.30em}$ there is a sequence
\begin{equation}
\eta_n=\sum_if^{(i)}_n\delta g^{(i)}_n\in\Omega^1\cal A\; ,
f^{(i)}_n,g^{(i)}_n\in C(i)
\end{equation}
such that
\begin{equation}
\lim_{n\rightarrow\infty} \pi_D(\eta_n) = \gamma^0
\end{equation}
in the strong operator topology of ${\cal B}(\cH) $.
\end{lemm}
{\bf Proof:}
If $I=S^1$ let $U_i$ be a finite open cover of $S^1$, $f_i$ the corresponding
partition of unity and $g_i$ some smooth functions on $S^1$ with
$\partial_tg_i=1\; ,\forall t\in U_i$ then $\eta$ defined as in
eq.(\ref{etas1})
fulfills eq.(\ref{etas1a}).
If $I={I\kern -0.22em R\kern 0.30em}$ there are no bounded functions in $C^\infty_0({I\kern -0.22em R\kern 0.30em})$ such that
$\eta$ can be defined as in eq.(\ref{etas1}).
However let ${\{a_n\}}_{n\in{I\kern -0.22em N\kern 0.30em}}\; ,a_{n+1}>a_n>0$ be a sequence in ${I\kern -0.22em R\kern 0.30em}$
with $a_n\rightarrow\infty$ as $n\rightarrow\infty$. Define $U_n=\; ]-a_n, a_n[$ and choose
$f_n,g_n\in C^\infty_0({I\kern -0.22em R\kern 0.30em})$ such that $f_n(t)=1 ,\; t\in U_n$,
$\; |f_n(t)|\leq 1 ,\; t\kern 0.1 em \in\kern -0.80em / U_n $ and $\partial_tg_n(t)=1 , \; t\in U_n $,
$\; |\partial_tg_n(t)|\leq 1 ,\; t\kern 0.1 em \in\kern -0.80em / U_n$. For $\eta_n=f_n\delta g_n$ it is
\begin{equation}
\begin{array}{rcl}
(\Psi, (\pi_D(\eta_n)-\gamma^0)\Psi) & = & \int_{{I\kern -0.22em R\kern 0.30em}} dt (f_n\partial_tg_n-1)
{(\Psi, \gamma^0\Psi)}_s\\
& & \\
& \leq &
\int_{{I\kern -0.22em R\kern 0.30em}} dt{(\Psi,\Psi)}_s- \int_{-a_n}^{a_n} dt{(\Psi,\Psi)}_s
\end{array}
\;\;\;\; \Psi\in\cal H \;\;.
\end{equation}
Thus $\pi_D(\eta_n)$ converges in the strong operator topology to $\gamma^0$.
$\; \Box$
If $I={I\kern -0.22em R\kern 0.30em}$ we add a formal limit point $\eta$ of the sequence $\eta_n$ to
$\Omega^1\cal A$
with
\begin{equation}
\pi_D(\eta)=\gamma^0\;\; .
\end{equation}
This element is formal since we have not specified a topology
on $\Omega_D $ which would allow to consider convergence of $\eta_n$ in $\Omega_D $.
However, except for the definition of the map $T$,
the element $\eta$ will appear only as an argument of $\pi_D$
or $\pi_{D_s}$ and therefore the limit is well defined in ${\cal B}(\cH) $. Furthermore we
note that
\begin{equation}
\pi_{D_s}(\eta)= 0\;\; .
\end{equation}
\begin{lemm}\label{le1}
For $\omega \in \Omega^k\cal A$ it is
\begin{equation}
\pi_D(\omega)=\pi_{D_s}(\omega) + \pi_D(\alpha)\;\; ,\;\;
\alpha\in\ker \pi_{D_s} \cap\Omega^k\cal A\;\; .
\label{lemm1}
\end{equation}
\end{lemm}
{\bf Proof:}
We prove this lemma by defining an algebra homomorphism $T$ on $\Omega \cA $ which is
an projection, i.e.~$T^2=T$. $\Omega \cA $ is generated by the zeroth and first
degree and therefore it is sufficient to define $T$ on those spaces.
Since $\cal A=C^\infty(I,\cal A_s )$ it is $\partial_t A\in\cal A\; \forall A\in\cal A$.
We use this and the element $\eta$ to define
\begin{equation}
\begin{array}{rclc}
T(A) & =& A&\;\; A\in\cal A\\
& & & \\
T(\delta A)& =&\delta A-\partial_tA\eta\; ;\;\; T(\eta) = 0 &\;\; A\in\cal A\; .
\end{array}
\end{equation}
For an arbitrary degree $k>1$ one obtains
\begin{equation}
\begin{array}{rcl}
T & : & \Omega^k\cal A \longrightarrow \Omega^k\cal A\\
& & \\
T(A_0\delta A_1\cdots\delta A_k) & = & A_0 T(\delta A_1)\cdots T(\delta A_k)
\end{array}
\end{equation}
Since
\begin{equation}
\pi_D(\delta A) = [D_s,\pi(A)] + [D_t,\pi(A)]
= \pi_{D_s}(\delta A) + \pi_D(\partial_t A \eta) \label{tk1}
\end{equation}
this map has the useful property that for any
$\omega \in \Omega \cA $
\begin{equation}
\begin{array}{rcl}
\pi_{D_s}(\omega) & = & \pi_D(T(\omega))\\
& & \\
\pi_{D_s}((1-T)\omega) & = & 0
\end{array}
\end{equation}
It is $\pi_D(\Omega^0\cal A)=\pi_{D_s}(\Omega^0\cal A)=\pi(\cal A)$ and for any $k$ it is
\begin{equation}
\pi_D(\omega)=\pi_D(T(\omega)) + \pi_D((1-T)\omega)=\pi_{D_s}(\omega)+\pi_D(\alpha)
\; ,\;\;\forall \omega \in \Omega^k\cal A
\end{equation}
with $\alpha=(1-T)(\omega)\in\ker\pi_{D_s}$ and the lemma is proved.
$\; \Box$
\begin{lemm}\label{le2}
It is
\begin{equation}
\pi_D(\cal A) \subset \pi_D(\cal J ^2_D) \subset \pi_D(\Omega^2\cal A)
\end{equation}
and thus there is a filtration on $\pi_D(\Omega \cA )$:
\begin{equation}
\begin{array}{ccccccc}
\pi_D(\Omega^0\cal A) & \subset & \pi_D(\Omega^2\cal A) & \subset &
\pi_D(\Omega^4\cal A) & \subset & \cdots\\
& & & & & & \\
\pi_D(\Omega^1\cal A) & \subset & \pi_D(\Omega^3\cal A) & \subset &
\pi_D(\Omega^5\cal A) & \subset & \cdots \;\;\; .
\end{array}
\end{equation}
\end{lemm}
{\bf Proof:} Let us consider $\alpha =(f\otimes 1)\delta(g\otimes 1)
+(g\otimes 1)\delta(f\otimes1)-\delta(fg\otimes 1) \in \Omega^1\cal A$. It is
\begin{equation}
\pi_D(\alpha) = 0
\end{equation}
but
\begin{equation}
\pi_D(\delta\alpha)=2[D_t,\pi_t(f)][D_t,\pi_t(g)]=
2{\gamma^0}^2\partial_t f\partial_tg
=\pi(N)\pi(\partial_tf\partial_tg) \in \pi(\cal A)\;\; ,
\end{equation}
i.e. $\delta\alpha \in \cal J ^2_D$ and with such elements all of $\pi(\cal A)$
can be generated. Thus the lemma is proved. $\; \Box$
Suppose $\omega\in \ker\pi_D\cap\Omega^k\cal A$. Lemma \ref{le1} allows
us to write
\begin{equation}
0=\pi_D(\omega) = \pi_{D_s}(\omega) + \pi_D(\alpha)\;\; .
\end{equation}
Because of lemma \ref{le2} we cannot infer that $\pi_D(\omega)=0$ implies
$\pi_{D_s}(\omega) =0$. Thus if we would divide $\Omega \cA $ by the differential
ideal associated to $\pi_{D_s}$ in an analogous way as for $\pi_D$ in
eq.(\ref{jdef}) it may happen that the resulting differential algabra
contains elements which are not elements of $\Omega_D \cal A$ and therefore it
is not a subalgebra of $\Omega_D \cal A$. The correct differential ideal, which
leads to a graded subalgebra of $\Omega_D \cal A$ is constructed with the help
of the following ideal in $\Omega \cA $
\begin{equation}
\begin{array}{rcl}
\cal K ^{2k} & = & \{\omega\in\Omega^{2k}\cal A\; | \;\exists \alpha \in
\bigoplus^{k-1}_{j=0} \Omega^{2j}\cal A\; ,\; \pi_{D_s}(\omega+\alpha)=0\} \\
& & \\
\cal K ^{2k+1} & = & \{\omega\in\Omega^{2k+1}\cal A\; | \;\exists \alpha \in
\bigoplus^{k-1}_{j=0} \Omega^{2j+1}\cal A\; ,\; \pi_{D_s}(\omega+\alpha)=0\}\\
& & \\
\cal K & = & \bigoplus_{k=1}^\infty \cal K ^k\;\; .
\end{array}
\end{equation}
Let us also define the ideal $\cal K _0$
\begin{equation}
\begin{array}{rcl}
\cal K ^k_0 & = & \ker\pi_{D_s}\cap\Omega^k\cal A \\
& & \\
\cal K _0 & = & \bigoplus_{k=1}^\infty \cal K _0^k\;\; .
\end{array}
\end{equation}
A two sided differential ideal $\cal N $ is obtained as in eq.(\ref{jdef})
by including the image of $\delta$ on $\cal K $:
\begin{equation}
\begin{array}{rcl}
\cal N ^k &=& \cal K ^k + \delta \cal K ^{k-1}\\
& & \\
\cal N & = & \bigoplus_{k=1}^\infty \cal N ^k \;\; .
\end{array}
\end{equation}
The corresponding graded differential algebra $\Omega_{\cN}\cal A$ is then defined as
\begin{equation}
\begin{array}{rcl}
{\Omega_{\cN}}^k\cal A &=& {\Omega^k\cal A \over \cal N ^k}\\
& & \\
\Omega_{\cN} & = & \bigoplus_{k=0}^\infty {\Omega_{\cN}}^k\;\; .
\end{array}
\end{equation}
Let us denote by $\sigma_{\cal N }$ the map on the quotient space
\begin{equation}
\sigma_{\cal N }:\pi_{D_s}(\Omega^k\cal A)\longrightarrow\Omega_D ^k\cal A\;\; .
\end{equation}
The relation of $\Omega_D \cal A$ and $\Omega_{\cN}\cal A$ is determined by the relation of
$\cal N $ and $\cal J $ and therefore it is useful to prove the following lemma
\begin{lemm}\label{le3}
It is
\begin{equation}
\cal K ^k =(\ker\pi_D\cap\Omega^k\cal A) \cup \cal K _0^k\label{le3a}
\end{equation}
and
\begin{equation}
\cal N ^k=\cal J ^k \cup \cal K _0^k\;\; .\label{le3b}
\end{equation}
\end{lemm}
{\bf Proof:}
It is clear that $(\ker\pi_D\cap\Omega^k\cal A)\cup \cal K _0^k \subset \cal K ^k$. Thus
we have to consider elements $\omega\in\cal K ^{2k}$ with
\begin{equation}
0\neq \pi_{D_s}(\omega) =\sum_{j=0}^{k-1}\pi_{D_s}(\omega_{2j})\;\; ,\;\;
\omega_{2j}\in\Omega^{2j}\cal A\;\; .
\end{equation}
Because of lemma \ref{le1} there are $\alpha_{2j}\in\cal K _0^{2j}$ and
$\alpha\in\cal K _0^{2k}$ with
\begin{equation}
\begin{array}{rcl}
\pi_D(\omega_{2j}-\alpha_{2j}) & = & \pi_{D_s}(\omega_{2j})\;\; ,\\
& & \\
\pi_D(\omega-\alpha) & = & \pi_{D_s}(\omega)\;\; .
\end{array}
\end{equation}
We define $\omega^\prime\in\Omega^{2k}\cal A$ as
\begin{equation}
\omega^\prime= \omega-\alpha -
\sum_{j=0}^{k-1} {(N^{-1}\eta)}^{2(k-j)}(\omega_{2j}-\alpha_{2j})\;\; .
\end{equation}
Since
\begin{equation}
\pi_{D_s}(\omega-\omega^\prime)=0
\end{equation}
and
\begin{equation}
\pi_D(\omega^\prime)=0
\end{equation}
we infer that $\omega\in (\ker\pi_D\cap\Omega^{2k}\cal A)\cup\cal K _0^{2k}$. The
same is true for $\omega\in\cal K ^{2k+1}$ and therefore eq.(\ref{le3a}) is proved.
For the second part of the proof we compute $[\delta ,T]$:
\begin{equation}
\begin{array}{rcl}
\delta T (A_0\delta A_1\cdots \delta A_k) & = &
\delta(A_0 T(\delta A_1)\cdots T(\delta A_k))\\
& & \\
& = &\delta A_0T(\delta A_1)\cdots T(\delta A_k) \\
& & \\
& & + \sum_{j=1}^kA_0T(\delta A_1)\cdots
(\delta A_j +{(-1)}^j \delta(\partial_t A_j \eta))\cdots
T(\delta A_k)\;\; ,\\
& & \\
T\delta (A_0\delta A_1\cdots\delta A_k) &= &T(\delta A_0)
T(\delta A_1)\cdots T(\delta A_k)\;\;.
\end{array}
\end{equation}
Thus
\begin{equation}
\begin{array}{rcl}
[\delta ,T](A_0\delta A_1\cdots \delta A_k) & = &
\partial_tA_0\eta T(\delta A_1)\cdots T(\delta A_k)\\
& & \\
& & + \sum_{j=1}^kA_0T(\delta A_1)\cdots
(\delta A_j +{(-1)}^j \delta(\partial_t A_j \eta))\cdots T(\delta A_k) .
\end{array}
\end{equation}
Therefore, with $\pi_{D_s}(\delta\eta)=0$, we conclude that for any
$\omega\in\cal K _0$
\begin{equation}
\pi_{D_s}([\delta ,T](\omega))=0\;\; .\label{dtcommu}
\end{equation}
Furthermore it is for $\omega\in \cal K _0$
\begin{equation}
0=\pi_{D_s}(\omega)=\pi_D(T\omega)
\end{equation}
and therefore
\begin{equation}
\pi_D(\delta T\omega) \in \pi_D(\cal J )\;\; .
\end{equation}
On the other hand it is
\begin{equation}
\pi_{D_s}(\delta\omega)=\pi_D(T\delta\Omega)\;\; .
\end{equation}
Together with eq.(\ref{dtcommu}) this proves eq.(\ref{le3b}). $\; \Box$
We now state the main result of this section which shows that $\Omega_{\cN}\cal A$ is
the ``space-like'' part of $\Omega_D \cal A$ in the sense that there is a ``time''
differential $d_t$ and a ``time-like'' differential one-form $dt$ in
$\Omega_D \cal A$. We then denote by ``space-like'' forms such elements in $\Omega_D \cal A$
which do not contain $dt$.
\begin{theo}
There is an element $dt\in\Omega_D ^1\cal A$ such that for any $k$
\begin{equation}
dt\omega-{(-1)}^k\omega dt=0\;\; \forall\omega\in\Omega_D ^k \label{dtc}
\end{equation}
and
\begin{equation}
\Omega_D ^k\cal A = \Omega_{\cN}^k\cal A\; \oplus \; \Omega_{\cN}^{k-1}\cal A dt\;\; .
\end{equation}
The differential $d$ on $\Omega_D \cal A$ is given as a sum of the two differentials
$d_s$ and $d_t$:
\begin{equation}
d = d_s + d_t\;\; ,
\end{equation}
\begin{equation}
d_t(\sigma_D\pi_D(\omega))=
{(-1)}^k\sigma_D\pi_D(T(\partial_t\omega))dt
\;\; \omega\in\Omega^k\cal A \label{dtdeff}
\end{equation}
with
\begin{equation}
\begin{array}{rcl}
\partial_t (A_0\delta A_1\cdots \delta A_k) & = &\partial_t A_0\delta A_1
\cdots \delta A_k\\
& & \\
& & + \sum_{j=1}^k {(-1)}^{k-j}A_0\delta A_1\cdots \delta(\partial_tA_j)
\cdots\delta A_k\; .
\end{array}
\end{equation}
\end{theo}
{\bf Proof:}\newline
{}From lemma \ref{le1} we know that $\pi_{D_s}(\Omega \cA )$ is a subalgebra of
$\pi_D(\Omega \cA )=\pi_{D_s}(\Omega \cA )\cup\pi_D(\cal K _0)$
and hence
\begin{equation}
\bigoplus_{k=0}^\infty {\pi_{D_s}(\Omega^k\cal A)\over \pi_D(\cal J ^k)}\subset\Omega_D \cal A
\end{equation}
is a subalgebra of $\Omega_D \cal A$. Because of eq.(\ref{le3b}) we can conclude
that
\begin{equation}
\bigoplus_{k=0}^\infty {\pi_{D_s}(\Omega^k\cal A)\over \pi_D(\cal J ^k)}=
\bigoplus_{k=0}^\infty {\pi_{D_s}(\Omega^k\cal A)\over \pi_D(\cal N ^k)}= \Omega_{\cN}\cal A\;\; .
\end{equation}
{}From eq.(\ref{le3a}) we infer that
\begin{equation}
{\pi_{D_s}(\Omega^k\cal A)\over \pi_D(\cal J ^k)}\cap {\pi_D(\cal K )\over\pi_D(\cal J ^k)}
=\{ 0 \}\;\; .
\end{equation}
Thus we can decompose $\Omega_D \cal A$ as follows
\begin{equation}
\Omega_D ^k\cal A=\Omega_{\cN}^k\cal A \oplus {\pi_D(\cal K _0^k)\over \pi_D(\cal J ^k)}\;\; .
\end{equation}
We proceed by identifying $dt$ as
\begin{equation}
dt=\sigma_D\pi_D(\eta)\;\; .\label{dtdef2}
\end{equation}
Because of lemma \ref{le2} we know that $\eta^2\in\cal J ^2$ and hence $dt^2=0$.
For any \newline
$A_0\delta A_1\cdots A_k\in\cal K _0^k$ it is
\begin{equation}
\begin{array}{rcl}
\pi_D(A_0\delta A_1\cdots\delta A_k) & = &
\pi_D(A_0)([D_s,\pi_D(A_1)] +\pi_D(\partial_tA_1)\pi_D(\eta))\cdots\\
& & \\
& &\cdots ([D_s,\pi_D(A_k)] +\pi_D(\partial_tA_k)\pi_D(\eta)) \\
& &\\
&=&\sum_{j=1}^k {(-1)}^{k-j}\pi_{D_s}(A_0 \delta A_1\cdots\partial_tA_j
\cdots \delta A_k)\pi_D(\eta)+\pi_D(\alpha)
\end{array}\label{dt+}
\end{equation}
where $\pi_D(\alpha)\in\cal J ^k$ denotes the sum of terms with a factor
$\pi_D(\eta)^k ,\; k>1$. We also used property {\bf v.} of $D$ to anticommute
$\pi_D(\eta)$ to the left. From eq.(\ref{dt+}) we infer that
\begin{equation}
{\pi_D(\cal K ^k)\over \pi_D(\cal J ^k)} = \Omega_{\cN}^{k-1}\cal A dt
\end{equation}
Eq.(\ref{dtc}) is also a consequence of property {\bf v.} of $D$.
Since $\Omega_D \cal A$ is generated by $\Omega_D ^1\cal A$ and $dt^2=0$ it is sufficient
to show eq.(\ref{dtdeff}) for all $\mu\in \Omega_D ^1\cal A$. For any $w\in \Omega_{\cN}^1\cal A$
let $A_0\delta A_1\in \Omega^1\cal A$ be a representative, i.e.
\begin{equation}
\sigma_{\cal N }\pi_{D_s}(A_0\delta A_1)= w
\end{equation}
Let us first compute the action of $d_s$ on $\sigma_D\pi_D(T(A_0\delta A_1))$,
which is the image of $w$ in $\Omega_D \cal A$
\begin{equation}
d_s\sigma_D\pi_D(T(A_0\delta A_1))=\sigma_D\pi_D(T\delta A_0\delta A_1))=
\sigma_D([D_s,\pi(A_0)][D_s,\pi(A_1)])\;\; .
\end{equation}
We use this to compute the action of $d$ on $\sigma_D(A_0[D_s,\pi(A_1)])$
\begin{equation}
\begin{array}{rcl}
d\sigma_D\pi_D(A_0\delta A_1)&=&
\sigma_D\pi_D(\delta (A_0T(\delta A_1)))\\
& & \\
& = & \sigma_D(([D_s,\pi(A_0)]+[D_t,\pi(A_0)])\pi_D(T(\delta A_1)))
+\sigma_D\pi_D(A_0\delta T(\partial_t \eta))\\
& & \\
&=&d_s\sigma_D\pi_D(A_0\delta A_1)\\
& & + dt\sigma\pi_D(\partial_tA_0T(\delta A_1))+
\sigma_D\pi_D(T(A_0\delta(\partial_tA_1)))dt\;\; .
\end{array}
\end{equation}
This shows that
\begin{equation}
d_t(\sigma_D\pi_D(A_0\delta A_1))=
(d-d_s)\sigma_D\pi_D(T(A_0\delta A_1))
=\sigma_D\pi_D(\partial_t(A_0\delta A_1))
\end{equation}
and the theorem is proved.
$\; \Box$
\section{The Inner Product on $\Omega_D \cal A$ and the Lorentz Metric}\label{mink}
So far we have constructed a generalized differential algebra where we were
able
to identify the ``space-like'' and the ``time-like'' part because of the
structure $\cal A = C^\infty(I,\cal A_s )$ of the algebra and the special form of the
k-cycle $(\cal H , D)$. Following the lines presented by A.~Connes and J.~Lott
in \cite{colo,cobuch} it is now straightforward to construct a covariant
connection and curvature. However, there is still one important ingredient
missing which is neccessary to define an action or a Lagrange function
resp.~a Hamilton function, the objects we are interested in. In conventional
geometry one obtains an action or Lagrange function by integration over
appropriate differential forms. In \cite{co} A.~Connes showed that the correct
substitute for integration in non-commutative geometry is the Dixmier trace.
It is this trace which is used in the definition of actions in
\cite{colo,chams1,chams2}. However, we want to derive a
Hamilton function and therefore we do not have to integrate over the
non-commutative ``space-time'' but we have to integrate over a ``space-like''
surface. As before we will use the additional structure of $\cal A$ and $(\cal H , D)$
to define the correct operator theoretic substitute for integration
on ``space-like'' surfaces which will be the Dixmier trace on $\cal H_s $.
Let us first briefly recall the defintion of the Dixmier trace and some general
results about the inner product on $\Omega_D \cal A$ defined via Dixmier trace.
For a detailed account we refer to \cite{cobuch,Kbuch,GBV}.
The Dixmier trace \cite{dix} is the unique extension of the ususal trace to
the class $\cal L ^{(1,\infty)}(\cal H )$ which is an ideal in the algebra of bounded
operators. The elements of this ideal are characterized by the condition
that for any $T\in\cal L ^{(1,\infty)}(\cal H )$ the ordered eigenvalues $\lambda_i$
of $|T|$ satisfy
\begin{equation}
\sup_{N}{1\over\log N}\sum_{i=0}^N \lambda_i < \infty\;\; .
\end{equation}
On this ideal the Dixmier trace $Tr_\omega(\cdot )$ is defined as functional
with the property
\begin{equation}
Tr_\omega(T) = \lim_{N\rightarrow\infty}{1\over\log N}\sum_{i=0}^{N-1} \lambda_i\;\; .
\end{equation}
If $\cal A$ is an arbitrary subalgebra of a $C^*$-algebra with a finitely
summable k-cycle $(\cal H , D)$ then $|D|^{-d}$ is in $\cal L ^{(1,\infty)}(\cal H )$ for
some $d\in{I\kern -0.22em N\kern 0.30em}$, where $d$ corresponds to the dimension of the underlying
(non-commutative) space. Since
\begin{equation}
Tr_\omega(|D|^{-d}) > 0
\end{equation}
a inner product on $\pi_D(\Omega \cA )$ is obtained by defining for each $k$
\begin{equation}
\begin{array}{rcl}
(\cdot,\cdot)^k&:&\pi_D(\Omega^k\cal A)\times\pi_D(\Omega^k\cal A)\longrightarrow\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}\\
& & \\
(\pi_D(\omega_1),\pi_D(\omega_2))^k&=&
Tr_\omega(\pi_D(\omega_1^*)\pi_D(\omega_2)|D|^{-d})\;\;
\omega_1,\omega_2\in \Omega^k\cal A\;\; ,
\end{array}\label{topro}
\end{equation}
which is positive if $\pi_D(\omega^*)=\pi_D(\omega)^*,\; \forall \omega\in\Omega \cA $.
Let us denote by $\cal H_\pi ^k$ the Hilbert space completion of $\pi_D(\Omega^k\cal A)$
and let $P^{(k)}$ be the orthogonal projection of $\cal H_\pi ^k$ onto the orthogonal
complement of $\overline{\pi_D(\cal J ^k)}\subset \cal H_\pi ^k$ then an inner
product on $\Omega_D \cal A$ can be defined for each $k$ by
\begin{equation}
\begin{array}{rcl}
<\cdot,\cdot>^k&:&\Omega_D ^k\cal A\times\Omega_D ^k\cal A\longrightarrow\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}\\
& & \\
<\sigma_D(W_1),\sigma_D(W_2)>^k&=& (P^{(k)}W_1,P^{(k)}W_2)\;\; ,
W_1,W_2\in\pi_D(\Omega^k\cal A)\;\;,
\end{array}\label{toprof}
\end{equation}
which is positive if $(\cdot ,\cdot )$ is positive.
This allows to identify $\Omega_D ^k\cal A$ with a dense subspace of $\cal H_\pi ^k$ and
hence there is a map
\begin{equation}
\mbox{{\bf c}} : {\overline {\Omega_D ^k\cal A}} \longrightarrow \cal H_\pi ^k
\end{equation}
with $Im(\mbox{{\bf c}}) = {\overline {\pi(\cal J ^k)}}^\perp$.
In the case, where $\cal A=C^\infty(\cal M )$ is the algebra of smooth functions on
a compact spin-manifold $\cal M $ and $D=\kern 0.1 em \partial\kern -0.55em /$ is the Dirac operator, $\Omega_D \cal A$
is the usual de Rham algebra \cite{cobuch} and the inner product is
\begin{equation}
<w_1,w_2>=\int_{\cal M } *w_1\wedge w_2 \;\; ,\;\; w_1,w_2\in\Omega_D ^k\cal A
\end{equation}
where $*w_1$ is the Hodge dual of $w_1$.
Let us now turn to our case where the algebra is of the form
$\cal A=C^\infty(I,\cal A_s )$ where we would like to introduce a substitution for
integration on a space-like surface. However the ``space-like'' part of $\cal A$
and $\Omega_D \cal A$ is characterized by $\cal A_s$ and the smooth 1 parameter family
of k-cycles $(\cal H_s , D_s)$ over $\cal A_s $, which are finitely summable by
assumption. Therefore there is some $d$ (the dimension of the ``space-like''
part of the non-commutative space) such that for any $t\in I$ $\; |D_s|^{-d}$
is an operator on $\cal H _s$ with $|D_s|^{-d}\in \cal L ^{(1,\infty)}(\cal H_s )$ and
\begin{equation}
Tr_\omega(|D_s|^{-d})_s >0\;\; .
\end{equation}
Here $Tr_\omega(\cdot)_s$ denotes the Dixmier trace on
$\cal L ^{(1,\infty)}(\cal H_s )$. Since for any $t\in I$ any \linebreak
$W\in\pi_D(\Omega_D )$ is a bounded operator on $\cal H_s $ varying smoothly with $t$
\begin{equation}
\begin{array}{rcl}
(\cdot,\cdot)^k_s&:&\pi_D(\Omega^k\cal A)\times\pi_D(\Omega^k\cal A)\longrightarrow\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^\infty(I)\\
& & \\
(\pi_D(\omega_1),\pi_D(\omega_2))^k_s&=&
Tr_\omega(\pi_D(\omega_1^*)\pi_D(\omega_2)|D_s|^{-d})_s\;\; , \;\;
\omega_1,\omega_2\in\Omega^k\cal A
\end{array}\label{spro}
\end{equation}
defines a positive inner product on $\pi_D(\Omega \cA )$ for any $k$ and any (fixed)
$t\in I$ if \linebreak
$\pi_D(\omega^*)=\pi_D(\omega)^*,\;\forall \omega\in \Omega \cA $, a condition which
is met in our case (see {\bf vi.}), i.e.
\begin{equation}
(W,W)_s =f(t)\geq 0\;\; ,\;\; \forall W\in\pi_D(\Omega \cA )\; ,\forall t\in I
\end{equation}
With this inner product on $\pi_D(\Omega \cA )$ one can define an inner product on
$\Omega_D \cal A$ as in the general construction.
Let us denote by ${\cal H_s}_\pi ^k$ the completion\footnote{We call a sequence
convergent with respect to $(\cdot ,\cdot)_s$ if it converges pointwise
for all $t\in I$.} of $\pi_D(\Omega^k\cal A)$
with respect to $(\cdot,\cdot)_s$
and let $P^{(k)}_s$ be the orthogonal projection of ${\cal H_s}_\pi ^k$ onto the
orthogonal complement of $\pi_D(\cal J ^k)$ then for each $k$
\begin{equation}
\begin{array}{rcl}
<\cdot,\cdot>^k_s&:&\Omega_D ^k\cal A\times\Omega_D ^k\cal A\longrightarrow\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^\infty(I)\\
& & \\
<\sigma_D(W_1),\sigma_D(W_2)>^k_s&=& (P_s^{(k)}W_1,P_s^{(k)}W_2)_s\;\; ,
W_1,W_2\in\pi_D(\Omega^k\cal A)
\end{array}\label{sprof}
\end{equation}
defines a positive inner product on $\Omega_D \cal A$ for any $t\in I$. With this
product we will define Lagrange functions and the Hamilton formalism.
As in the general case there is a map
\begin{equation}
\mbox{{\bf c}}_s : {\overline {\Omega_D ^k\cal A}} \longrightarrow {\cal H_s}_\pi ^k
\end{equation}
and hence we can identify $\Omega_D ^k\cal A$ with a dense subspace of ${\cal H_s}_\pi ^k$.
The definition of the inner product in eq.(\ref{sprof}) allows for an
important freedom in the choice of the k-cycle over $\cal A$, which deserves
some discussion. For any \linebreak
$w_1\in\Omega_{\cN}^k\cal A\subset\Omega_D ^k\cal A$ and any
$w_2\in \Omega_{\cN}^{k-1}\cal A dt\subset\Omega_D ^k\cal A$ it is
\begin{equation}
<w_1,w_2>_s=\left(\mbox{{\bf c}}_s(w_1),\mbox{{\bf c}}_s(w_2)\right)_s=
Tr_\omega(\mbox{{\bf c}}_s(w_1)^*\mbox{{\bf c}}(w_2))_s= 0
\end{equation}
since $\mbox{{\bf c}}_s(w_1)^*\mbox{{\bf c}}(w_2)$ contains an odd number of commutators with $D_s$
which are off diagonal (with respect to the block diagonal structure of
eq.(\ref{blockdia})). This proves the following lemma:
\begin{lemm}\label{ordeco}
The decomposition
\begin{equation}
\Omega_D ^k\cal A=\Omega_{\cN}^k\cal A \oplus \Omega_{\cN}^{k-1}\cal A dt
\end{equation}
is orthogonal with respect to the inner product $<\cdot,\cdot>_s$.
\end{lemm}
We have seen that condition {\bf iii.} of $D$, i.e.~$D$ has a compact inverse
and \linebreak
$|D|^{-d}\in \cal L ^{(1,\infty)}(\cal H )$, is crucial for the definition
of the inner products (\ref{topro}) and (\ref{toprof}). However, we will
restrict ourselves to ``integration on space-like'' surfaces and hence use
the inner products defined by eq.(\ref{spro}) and (\ref{sprof}). Here we
only need that $D_s$ has a compact inverse and that
$|D_d|^{-d}\in\cal L ^{(1,\infty)}(\cal H_s )$ for some $d$ which is guaranteed by
condition {\bf vi.}. We can use this freedom and change the definition of
$D_t$ by choosing $\gamma^0$ self-adjoint. As a consequence we find for
any element $\omega\in\Omega_D \cal A$
\begin{equation}
\begin{array}{rcl}
{(d_s\omega)}^* &=& -d_s (\omega^*)\\
& & \\
{(d_t\omega)}^* &=& d_t (\omega^*)\; .
\end{array}\label{minkmet}
\end{equation}
Following A.~Chamseddine et.al.~\cite{CFF} we introduce a
generalized metric on $\Omega_D ^1\cal A$\footnote{Strictly speaking the metric is
introduced on $\overline{\Omega_D ^1\cal A}$ which is the Hilbert-space completion of
$\Omega_D ^1\cal A$. However, we assume that the construction holds on $\Omega_D ^1\cal A$.}.
In this context the $\cal A$-module $\Omega_D ^1\cal A$
is interpreted as the generalized cotangent bundle over a \linebreak
non-commutative space. We define the metric
\begin{equation}
g(\cdot,\cdot):\Omega_D ^1\cal A\times\Omega_D ^1\cal A\longrightarrow\cal A
\end{equation}
by the following equation
\begin{equation}
<A,g(v,w)>_s=-{Tr_\omega(A^*\mbox{{\bf c}}_s(v^*)\mbox{{\bf c}}_s(w)\|D_s|^{-d})}_s
\;\; ,\;\; \forall A\in\cal A ;\; v,w\in\Omega_D ^1\cal A\;\; .\label{metdef}
\end{equation}
This metric enjoys the property
\begin{equation}
g(Av,Bw)=A^*g(v,w)B\;\; ,\;\; \forall A,B\in\cal A ;\; v,w\in\Omega_D ^1\cal A\;\; .
\end{equation}
An important property of this metric is stated in the following theorem
\begin{theo}
If $\gamma^0$, as defined in eq.(\ref{etacon}), is self-adjoint, i.e.
\begin{equation}
\gamma_0 = \gamma_0^\dagger
\end{equation}
then $g(\cdot,\cdot)$, as defined in eq.(\ref{metdef}), is generalized
Minkowskian metric, which is positive definite on $\Omega_{\cN}^1\cal A$ and negative
definite on the $\cal A$-module generated by $dt$.
\end{theo}
{\bf Proof:}
Applying the arguments presented in \cite{CFF} to our case, we conclude that
$g(\cdot,\cdot)$ defines a positive definite Riemannian metric on
$\Omega_{\cN}^1\cal A\subset\Omega_D ^1\cal A$. From lemma~\ref{ordeco} we infer that
\begin{equation}
g(v,dt)=0\;\; , \;\; \forall v\in\Omega_{\cN}^1\cal A\;\; .
\end{equation}
{}From the definitions eq.(\ref{spro}), eq.(\ref{sprof}) and the definition
of $\gamma^0$ in eq.(\ref{etacon}) it follows that
\begin{equation}
g(dt,dt)=-\gamma^0\gamma^0=-N\in\cal A\label{timsign}
\end{equation}
and the theorem is proved. $\; \Box$
This theorem completely justifies the
terminology of ``space-like'' and ``time-like'' since with the choice
${\gamma^0}^\dagger=\gamma^0$ it is possible to identify time like elements
of $\Omega_D ^1\cal A$ as elements with negative norm, i.e.~elements $v\in\Omega_D ^1\cal A$
with
\begin{equation}
g(v,v)=|v|^2<0\;\; .
\end{equation}
For the rest of this article we will keep the choice
${\gamma^0}^\dagger=\gamma^0$, which means we are working on a non-commutative
Minkowski space.
We end this section with some furhter definitions and some assumption on
the algebra $\cal A$ which will be useful in the Hamiltonian framework.
The first definition is a slight generalization of eq.(\ref{metdef}).
We associate to any $v^{(l)} \in \Omega_D ^l\cal A,\; l\geq 0$ a map $i_l(v^{(l)})$,
which is defined for all $k\geq 0$ by
\begin{equation}
\begin{array}{rl}
<w_1,i_l(v^{(l)})w_2>_s=
Tr_\omega(\mbox{{\bf c}}_s(w_1^*)\mbox{{\bf c}}_s((v^{(l)})^*)\mbox{{\bf c}}_s(w_2)|D|^{-d})_s
&=<vw_1,w_2>_s\\
& \forall w_1\in\Omega_D ^k\cal A ,\;\forall w_2\in\Omega_D ^{k+l}\cal A\; .
\end{array}
\end{equation}
This map is well defined as can be seen by applying the arguments presented
in \cite{CFF} for the definition of the metric. Thus we have defined a map
which decreases the degree of forms
\begin{equation}
i_l(v^{(l)}) : \Omega_D ^{k+l}\cal A \longrightarrow {\overline{\Omega_D ^{k}\cal A}}\;\; .
\end{equation}
For the second definition we have to make a furhter assumption on the algebra
$\cal A$ and the k-cycle $(\cal H , D)$ over $\cal A$. Namely that for any
$v\in\Omega_D ^k\cal A$, $k>0$, there is a $C_v\in{I\kern -0.22em R\kern 0.30em}$ such that for all
$w\in\Omega_D ^{k-1}\cal A$
\begin{equation}
|<v, dw>_s|^2\leq C_v <w,w>_s\label{regcon}
\end{equation}
This condition is fulfilled, for example, if $\cal A=C^\infty(M)$ and $D$ is
the Dirac-operator on $M$ or if $\cal A$ is a finite dimensional algebra or
if $\cal A$ is a product of the first two cases. Thus eq.(\ref{regcon}) is
fulfilled for the class of algebras which has been used for model building
in physics so far. This condition ensures that there is a well defined
adjoint operator $d_s^*$ of $d_s$ on $\Omega_D ^k\cal A$
\begin{equation}
d_s^* : \Omega_D ^{k}\cal A \longrightarrow {\overline{\Omega_D ^{k-1}\cal A}}
\end{equation}
which is uniquely defined by
\begin{equation}
<d_s^*v,w>_s:= <v,d_sw>_s\;\; \forall v\in \Omega_D ^k\cal A,\; \forall w\in\Omega_D ^{k-1}\cal A
\;\; .
\end{equation}
Furhtermore we assume that the smooth 1-parameter family of k-cycles
$(\cal H_s , D_s)$ is tame \cite{GBV}, i.e.
\begin{equation}
Tr_\omega( [W_1,W_2] |D_s|^{-d})= 0\;\; , \;\; W_1,W_2\in\pi_D(\Omega \cA )
\end{equation}
and
\begin{equation}
\begin{array}{rcl}
i_l(v^{(l)})(\Omega_D \cal A)&\subset&\Omega_D \cal A\\
& & \\
d_s^*(\Omega_D \cal A) &\subset&\Omega_D \cal A
\end{array}\;\; .\label{gbm}
\end{equation}
These conditions are fulfilled in the above mentioned examples.
\section{Lagrange and Hamilton Function for Yang-Mills Theory}
Now we have all basic objects at hand which are necessary to define
a Lagrange function and the corresponding Hamilton function for
Yang-Mills theory in non-commutative geometry. However, we start with
a brief exposition of Yang-Mills theory in non-commutative geometry
as it was introduced A.~Connes and J.~Lott \cite{colo,cobuch}, which
allows us to fix our notation. A comprehensive presentation of this
subject can be found in \cite{cobuch,Kbuch,GBV}.
Yang-Mills theory is formulated on vector bundles. In the algebraic language
a vector bundle is a finitely generated projective module over $\cal A$ which
we denote by $\cal E $. Any finitely generated module $\cal E $ can be obtained from
a free module $\cal E _0=\cal A^N$ with the help of some idempotent
$e\in\cal A^{N\times N}$,
which means that we we can write $\cal E =e\cal A^N$. In our case, the structure
of $\cal A=C(I,\cal A_s )$ implies that $\cal E = C(I, \cal E_s )$, where $\cal E_s $ is a finitely
generated projective module over $\cal A_s $.
Furthermore we need a Hermitian
structure on $\cal E $,i.e., a sesquilinear form
\begin{equation}
(\cdot,\cdot)_{\cal E }:\cal E \times\cal E \longrightarrow \cal A
\end{equation}
with the following properties
\begin{itemize}
\item $(A\zeta,B\eta)_{\cal E }=A^*(\zeta,\eta)_{\cal E }B\;\; ,\;\;
\forall \zeta,\eta\in\cal E ,\; A,B\in\cal A$
\item $(\zeta,\zeta)_{\cal E }\geq 0\;\; ,\;\; \forall\zeta\in\cal E $
\item $\cal E $ is self dual for $(\cdot,\cdot)_{\cal E }$.
\end{itemize}
If we write $\cal E =e\cal A^n$ the hermitean structure requires that $e$ is
self-adjoint.
We extend $\cal E $ to a right module $\tilde{\cal E} $ over $\Omega_D \cal A$
\begin{equation}
\tilde{\cal E} ^k=\cal E \otimes_{\cal A}\Omega_D ^k\cal A\;\;\; ,\;\;\;
\tilde{\cal E} =\cal E \otimes_{\cal A}\Omega_D \cal A\;\;
\end{equation}
and also
\begin{equation}
(\cdot,\cdot)_{\tilde{\cal E} }:\tilde{\cal E} \times\tilde{\cal E} \longrightarrow \Omega_D \cal A\;\; .
\end{equation}
A connection is defined as a linear map
\begin{equation}
\nabla : \tilde{\cal E} ^k\longrightarrow\tilde{\cal E} ^{k+1}
\end{equation}
such that
\begin{equation}
\nabla(\zeta w)=\nabla(\zeta)w+(-1)^kdw\;\; ,\;\;
\zeta\in\tilde{\cal E} ^k ,\; w\in\Omega_D \cal A\;\; .
\end{equation}
One also requires that the connection is compatible with the metric
$(\cdot,\cdot)_{\tilde{\cal E} }$, which for Euklidean k-cycles, i.e., for
$D^\dagger=D$ is equivalent to the condition
\begin{equation}
(\zeta,\nabla\eta)_{\tilde{\cal E} }-(\nabla\zeta,\eta)_{\tilde{\cal E} } = d(\zeta,\eta)_{\tilde{\cal E} }
\;\; ,\;\; \zeta,\eta\in\cal E \;\; . \label{compcon}
\end{equation}
The set of compatible connections form an affine space and for any two
compatible connections $\nabla ,\; \nabla^\prime$ it is
\begin{equation}
\nabla-\nabla^\prime = \mbox{{\bf A}} \in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^1\cA)\;\; .
\end{equation}
Note, that the definition of a compatible connection depends on the
definition of the $*$-operation on $\Omega \cA $ and the choice of $D$ for the
k-cycle over $\cal A$. In our case we have $D_s^\dagger=D_s$ and
$D_t^\dagger = -D_t$. Thus condition (\ref{compcon}) is valid only on
the space-like part of the connection. For the time-like part of the connection
the compatibility condition reads
\begin{equation}
(\zeta,\nabla_t\eta)_{\tilde{\cal E} }+(\nabla_t\zeta,\eta)_{\tilde{\cal E} } =
d_t(\zeta,\eta)_{\tilde{\cal E} }
\;\; ,\;\; \zeta,\eta\in\cal E \;\; .
\end{equation}
One can check (see e.g.\cite{GBV}) that for $\cal E =e\cal A^N$
\begin{equation}
\nabla_0 \zeta=ed\zeta\;\; ,\;\; \zeta\in\cal E
\end{equation}
defines a compatible connection. Thus any compatible connection $\nabla$ can be
written as
\begin{equation}
\nabla=\nabla_0+\mbox{{\bf A}} \;\;, \;\; \mbox{{\bf A}} \in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^1\cA)\;\; .
\end{equation}
Here we used that the restriction on $\nabla$ to $\cal E $ already defines the
connection uniquely on $\tilde{\cal E} $. The curvature $\mbox{{\bf F}} $ is obtained by taking the
square of the connection
\begin{equation}
\mbox{{\bf F}} =\nabla^2=e(de)^2 +ede \alpha e +ed\alpha e -e\alpha de+ e\alpha e\alpha
\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^2\cA)
\end{equation}
with $e\alpha e= \mbox{{\bf A}} $ and $\alpha \in \cal A^{N\times N}\otimes_{\cal A}\Omega_D ^1\cal A$.
Connection and curvature transform covariantly under unitary transformations
\linebreak $U(\cal E )=\{ u\in\mbox{End}_{\cal A}(\cal E )| uu^*=u^*u = 1\}$, i.e.
\begin{equation}
\mbox{{\bf F}} ^\prime = u\mbox{{\bf F}} u^*\;\;\; , \;\;\; \nabla^\prime = u\nabla u^*
\end{equation}
from which we infer that the vector-potential $\mbox{{\bf A}} $ transforms as follows
\begin{equation}
\mbox{{\bf A}} ^\prime = u\mbox{{\bf A}} u^* + udu^*\;\; .
\end{equation}
The inner product on $\Omega_D \cal A$ and the hermitean structure on $\cal E $ induce
a natural inner product on $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)$ for any $k$. We want to construct this
product explicitely and therefore we note that any $\diva{T}\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)$ can be
written as
\begin{equation}
\diva{T} = \sum_{r,s=1}^N e_{ik}w_{rs}e_{lj}\;\; ,\;\; w_{kl}\in\Omega_D ^k\cal A\;\;.
\end{equation}
In this notation the inner product $(\cdot ,\cdot)$ on $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)$ can be defined
as
\begin{equation}
(\cdot ,\cdot) : \mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)\times\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA) \longrightarrow C^\infty(I)
\end{equation}
\begin{equation}
\begin{array}{rcl}
(\diva{T}^{(1)},\diva{T}^{(2)}) &=&
tr Tr_\omega(\mbox{{\bf c}}_s({\diva{T}^{(1)}}^\dagger)\mbox{{\bf c}}_s(\diva{T}^{(2)})
|D_s|^{-d})_s\\
& &\\
&=&\sum_{j,k=1}^N \sum_{r,s=1}^N \sum_{p,q=1}^N
<e_{rj}w_{sr}^{(1)}e_{ks},e_{jp}w_{pq}^{(2)}e_{qk}>_s
\end{array}
\;\; ,\;\; w_{rs}^{(1)},w_{pq}^{(2)}\in\Omega_D ^k\cal A,\label{hompro}
\end{equation}
We use this inner product to define the
Lagrange function $L$ for Yang-Mills theory in non-commutative geometry:
\begin{equation}
L(\mbox{{\bf A}} )=-\frac{1}{4}(\mbox{{\bf F}} ,\mbox{{\bf F}} )\in C^\infty(I)
\end{equation}
The action $S$ for Yang-Mills theory is obtained by integrating the Lagrange
function $L$ over time
\begin{equation}
S(\mbox{{\bf A}} )=\int_{t_1}^{t_2} dt L(\mbox{{\bf A}} ) = -\frac{1}{4}\int_{t_1}^{t_2} dt (\mbox{{\bf F}} ,\mbox{{\bf F}} )
\;\; .
\end{equation}
So far we have discussed the general case where $\cal E $ is a finitely generated
$\cal A$-module. However, we now will restrict ourselves to the case where
$\cal E =\cal A^N$ is a free module. However, note that the formalism which will
be presented in the following can be generalized to finitely generated
$\cal A$-modules. The reason for the restriction is just to avoid unecessary
complicated formulas.
Because of lemma \ref{ordeco} there is also an orthogonal decomposition of
$\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)$ with respect to the inner product $(\cdot ,\cdot)$:
\begin{equation}
\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)=\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^k\cA) \oplus \mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^{k-1}\cA dt)
\end{equation}
and therefore we can $\mbox{{\bf F}} $ decompose as follows
\begin{equation}
\begin{array}{rcl}
\mbox{{\bf F}} &=& \mbox{{\bf F}} _{st} + \mbox{{\bf B}} \\
& & \\
\mbox{{\bf F}} _{st} &=& d_t\mbox{{\bf A}} _s + \nabla_s\mbox{{\bf A}} _t\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)\\
& & \\
\mbox{{\bf B}} &=& d_s\mbox{{\bf A}} _s + \mbox{{\bf A}} _s^2\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^2\cA)
\end{array}
\end{equation}
where $\mbox{{\bf A}} _s\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA)$ is the space-like part of $\mbox{{\bf A}} $ and \newline
$\mbox{{\bf A}} _t\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\cA dt)$ is
the time-like part of $\mbox{{\bf A}} $. $\; \nabla_s= d_s +\mbox{{\bf A}} _s$ denotes the space-like
part of the connection. With this decomposition $L$ becomes
\begin{equation}
L=-\frac{1}{4}\left((\mbox{{\bf F}} _{st},\mbox{{\bf F}} _{st})+(\mbox{{\bf B}} ,\mbox{{\bf B}} )\right)\;\; ,
\end{equation}
where the first term on the right hand side is positive and the second term
is negative.
Now we define the canonical momenta in the usual way, namely the variation of
$L$ with respect to the time derivative of the variables at some fixed time
$t$.
In our case we have to vary $L$ with respect to $d_t\mbox{{\bf A}} $. We find that
\begin{eqnarray}
\mbox{{\bf E}} _s &=& {\delta L\over \delta dt\mbox{{\bf A}} _s}=-{1\over 2}<\mbox{{\bf F}} _{st},\cdot >
\in{\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)}_t^\star
\label{esdef}\\
\mbox{{\bf E}} _t &=& {\delta L\over \delta dt\mbox{{\bf A}} _t}= 0\;\; .\label{etdef}
\end{eqnarray}
Here ${\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)}_t^\star$ denotes the image of the map (at some fixed time $t$)
\begin{equation}
\begin{array}{rcl}
\star & : & {\cal T}_{st} \longrightarrow {\cal T}^\star_{st}\\
\star(T)& = & (T,\dot)\; ,\;\; T\in{\cal T}
\end{array}
\end{equation}
restricted to $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)$, where ${\cal T}_{st}$ is the Hilbert space completion
of $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)$ and ${\cal T}^\star_{st}$ is the dual Hilbert space of
${\cal T}_{st}$. However, we use the map $\star^{-1}$ to identify
${\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)}_t^\star$ with $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)_t$ and thus we consider the canonical
momentum $\cal E $ as an element of $\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)_t$. As in usual Yang-Mills theory
we see that there are no canonical momenta for $\mbox{{\bf A}} _t$. Thus eq.(\ref{etdef})
are primary constraints.
We define the Hamiltonian $H$ as
\begin{equation}
\begin{array}{rcl}
H &=& \mbox{{\bf E}} (d_t\mbox{{\bf A}} )-L=
\frac{1}{4}\left(-(\mbox{{\bf E}} _s ,\mbox{{\bf E}} _s) + (\mbox{{\bf B}} ,\mbox{{\bf B}} )\right) -
(\nabla_s^*\mbox{{\bf E}} _s,\mbox{{\bf A}} _t)\\
& & \\
& = &H_0 - \diva{G}(\mbox{{\bf A}} _t)
\end{array}
\label{ncgham}
\end{equation}
\begin{equation}
H_0= \frac{1}{4}\left(-(\mbox{{\bf E}} _s ,\mbox{{\bf E}} _s) + (\mbox{{\bf B}} ,\mbox{{\bf B}} )\right)\;\; ,\;\;
\diva{G}=(\nabla_s^\star\mbox{{\bf E}} _s,\mbox{{\bf A}} _t)\;\; ,
\end{equation}
where
\begin{equation}
\nabla_s^*: {\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)}\longrightarrow{\mbox{Hom}_{\cal A}(\cal E ,\cal E \otimes_{\cal A}\Omega_D ^{k-1}\cal A)}
\end{equation}
is defined by
\begin{equation}
(\diva{T}_1,\nabla_s\diva{T}_2)=(\nabla_s^*\diva{T}_1,\diva{T}_2)
,\;\;\diva{T}_1\in{\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)},\;\diva{T}_2\in
\mbox{Hom}_{\cal A}(\cal E ,\cal E \otimes_{\cal A}\Omega_D ^{k-1}\cal A) .
\end{equation}
Such a map exists because of the assumption (\ref{gbm}).
Note that $H_0$ is positive since it is $(\mbox{{\bf E}} _s,\mbox{{\bf E}} _s)\leq 0$.
As one may have expected, the Hamiltonian for Yang-Mills theory in
non-commutative geometry is formally exactly the same as for conventional
Yang-Mills theory. However, the Hamiltonian in eq.(\ref{ncgham})
is defined purely algebraic and therefore still makes sense in cases where
there is no space-time manifold.
\section{The Poisson Bracket and Time Evolution}
{}From the discussion of the previous section we infer that the canonical
phase-space $\Gamma_0$ of Yang-Mills theory in non-commutative geometry
is
\begin{equation}
\Gamma_0\subset {\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)}_t \oplus \mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA)_t\;\; ,
\end{equation}
where the subscript $t$ indicates that we have fixed the time $t$ when the
momenta were defined. Thus the elements of the phase-space $\Gamma_0$ do not
have any time dependence. More generally, we define for any $k$
\begin{equation}
{\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA)}_t= {\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD^k\cA) \over {\cal I}^k_t}
\end{equation}
where ${\cal I}_t$ is the graded ideal
\begin{equation}
{\cal I}_t=\{ \diva{z}\in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\OD\cA) | \diva{z}(t)= 0\}\;\; .
\end{equation}
Since from now on all objects are considered at some
fixed time $t$ we drop the subscript $t$ in order to simplify notation.
However, there are some restrictions on the elements of $\Gamma_0$. The
first one is a reality constraint on the variables which originates from
the condition that $\mbox{{\bf A}} $ is a compatible connection, i.e.
\begin{equation}
\mbox{{\bf A}} ^\dagger = \mbox{{\bf A}} \;\; .
\end{equation}
Since
\begin{equation}
\mbox{{\bf E}} =-d_t\mbox{{\bf A}} -\nabla_s\mbox{{\bf A}} _t
\end{equation}
the compatibility condition on $\mbox{{\bf A}} $ implies that
\begin{equation}
\mbox{{\bf E}} ^\dagger = -\mbox{{\bf E}}
\end{equation}
Thus the canonical phase-space of Yang-Mills theory in non-commutative
geometry is
\begin{equation}
\Gamma_0 =\{(\mbox{{\bf A}} ,\mbox{{\bf E}} )\in{\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA dt)}_t \oplus \mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA)_t |
(\mbox{{\bf A}} ^\dagger,\mbox{{\bf E}} ^\dagger)=(\mbox{{\bf A}} ,-\mbox{{\bf E}} )\} \; .\label{phase0}
\end{equation}
In eq.(\ref{phase0}) we also have used the fact that there is no
canonical momentum for $\diva{A}_t$ and hence this variable plays the role of
a Lagrange multiplier. Thus we can read off from eq.(\ref{ncgham}) the
secondary constraint on the elements $\mbox{{\bf A}} ,\; \mbox{{\bf E}} $ of $\Gamma_0$
(we suppressed the index $s$), namely
\begin{equation}
\diva{G}(\mbox{{\bf A}} _t)=(\nabla^*\mbox{{\bf E}} ,\mbox{{\bf A}} _t)=0\; ,\;\;
\forall \mbox{{\bf A}} _t \in\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\cA dt)\;\; . \label{Gauss}
\end{equation}
This is the Gau\ss-Law in non-commutative geometry.
However, we have not defined a Poisson bracket for this space so far. A
Poisson bracket is a antisymmetric linear map $\{\cdot ,\cdot\}$ on a
suitable space of functions $\cal C$ on $\Gamma_0$. Therefore we first
have to define $\cal C$.
We take for $\cal C$ the algebra of functions on $\Gamma_0$ which contain
arbitrary finite powers of the elements $\mbox{{\bf A}} , \mbox{{\bf E}} \in\Gamma_0$ and their
derivatives (of finite order). For any $w\in\Omega_D \cal A$ we define
\begin{equation}
\begin{array}{rclcrcl}
w^{(2k,0)}w &:=& (d_s^*d_s)^k w &\;\; ,\;\; &
w^{(0,2k)}w &:=& (d_sd_s^*)^k w \;\; , \\
& & & & & & \\
w^{(2k+1,0)}w &:=& (d_sd_s^*)^k d_s w &\;\; ,\;\; &
w^{(0,2k+1)}w &:=& (d_s^*d_s)^k d_s^* w \;\; .
\end{array}
\end{equation}
General combinations of derivatives are denoted by
$w^{(k,l)}=w^{(k,0)}+w^{(0,l)},\; w\in\Omega_D \cal A$.
Those elements are well defined because of assumption (\ref{gbm}).
Furhtermore we need the analogue of partial integration in non-commutative
geometry. For this purpose we define for any $k\geq 0$ the map $pr_k$
\begin{equation}
pr_k : \pi_D(\Omega \cA ) \longrightarrow \Omega_D ^k\cal A
\end{equation}
by the equation
\begin{equation}
<v,{pr}_k(W)>_s = Tr_\omega(\mbox{{\bf c}}_s(v)^* W |D_s|^{-d})_s\;\; ,\;\;
\forall v\in\Omega_D ^k\cal A\; , W\in\pi_D(\Omega \cA )\;\; .
\end{equation}
Again assumption (\ref{gbm}) ensures that this map exists. With the help
of this map we can define the analogue of partial integration for all
$v\in\Omega_D ^k\cal A, W\in\mbox{{\bf c}}_s(\Omega_D \cal A)$ by
\begin{equation}
\begin{array}{rcl}
Tr_\omega(\mbox{{\bf c}}_s(d_sv) W |D_s|^{-d})_s &=& -Tr_\omega(\mbox{{\bf c}}_s(v)
\mbox{{\bf c}}_s(d^*_s{pr}_{k+1}(W))|D_s|^{-d})_s\\
& & \\
Tr_\omega(\mbox{{\bf c}}_s(d_s^*v) W |D_s|^{-d})_s &=& -Tr_\omega(\mbox{{\bf c}}_s(v)
\mbox{{\bf c}}_s(d_s{pr}_{k-1}(W))|D_s|^{-d})_s
\end{array}\; .
\end{equation}
It is convenient to consider the subalgebra $\cal P (\Gamma_0)$ of the
algebra of continuous maps from $\Gamma_0$ to
$\mbox{Hom}_{\cal A}(\cal E ,\cal E \otimes_{\cal A}\pi_D(\Omega \cA ))$, which is generated by
elements $P_m^{(j,k)}$ of the form
\begin{equation}
P_m^{(j,k)}=\mbox{{\bf c}}_s({pr}_m(\mbox{{\bf c}}_s(\mbox{{\bf z}}_1)\cdots\mbox{{\bf c}}_s(\mbox{{\bf z}}_n))^{(j,k)})\;\;, \;\;
j,k\geq 0, n>0
\end{equation}
with
\begin{equation}
\mbox{{\bf z}}_l\in\{ \mbox{{\bf A}} ,\mbox{{\bf E}} ,\; N_0, N_s, N_t\}\;\; ,\;\; (\mbox{{\bf A}} ,\mbox{{\bf E}} )\in\Gamma_0\; .
\end{equation}
The elements $N, N_s, N_t$ with
\begin{equation}
\begin{array}{rclc}
N &\in&{\mbox{Hom}_{\cal A}(\cal E ,\cal E )} ,\;& N^\dagger=-N\\
& & & \\
N_s &\in&\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\ON^1\cA) ,\;& N_s^\dagger = N_s\\
& & & \\
N_t &\in&\mbox{Hom}_{\cA}(\cE,\cE\otimes_{\cA}\cA dt) , \;& N_t^\dagger=-N_t
\end{array}
\end{equation}
play the role of test functions.
We obtain $\cal C\subset C(\Gamma_0, \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex})$ by taking the trace of the elements
in $\cal P $
\begin{equation}
\cal C:=\{ F\in C(\Gamma_0,\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex})| F= tr Tr_\omega(P {|D_s|}^{-d})_s,\;P\in \cal P \}\; .
\end{equation}
Having specified the space of functions on $\Gamma_0$ we define the Poisson
bracket $\{\cdot,\cdot\}$ by the following set of rules
\begin{equation}
\begin{array}{rcl}
\{ tr Tr_\omega(P_1|D_s|^{-d})_s , tr Tr_\omega(P_2 |D_s|^{-d})_s\}& =&
tr Tr_\omega( G(P_1,P_2) |D_s|^{-d})_s\\
& & \\
&=& - tr Tr\omega( G(P_2,P_1) |D_s|^{-d})_s\;\; ,
\end{array}
\end{equation}
The functional $G(\cdot ,\cdot)$ is the non-commutative generalization of the
$\delta$-distribution.
For any $P_p,P^\prime_q\in\cal P ,\; 1\leq p\leq k,\; 1\leq q\leq l$ it is
\begin{equation}
\begin{array}{ll}
trTr_\omega(G(P_1\cdots P_k,P_1^\prime\cdots P_l^\prime)|D_s|^{-d})_s=& \\
& \\
=\sum_{cp_k}\sum_{cp_l}
trTr_\omega(P_{cp_k(1)}\cdots P_{cp_k(k-1)}G(P_{cp_k(k)},P^\prime_{cp_l(1)})
P^\prime_{cp_l(2)}\cdots P^\prime_{cp_l(l)})|D_s|^{-d})_s\; , &
\end{array}
\end{equation}
where $\sum_{cp_k}$ denotes the sum over the cyclic permutations of the first
$k$ indices and $\sum_{cp_l}$ denotes the sum over the cyclic permutations of
the last $l$ indices.
For any $\mbox{{\bf c}}_s(d_sv)\in \cal P ,\; v\in\Omega_D ^k\cal A$ and for any
$\mbox{{\bf c}}_s(d^*_sv)\in \cal P ,\; v\in\Omega_D ^k\cal A$ it is $\forall P_1,P_2\in\cal P $
\begin{equation}
\begin{array}{rcl}
trTr_\omega(G(P_1,\mbox{{\bf c}}_s(d_sv)) P_2 |D_s|^{-d})_s &=&
-trTr_\omega(G(P_1,\mbox{{\bf c}}_s(v))\mbox{{\bf c}}_s(d^*_s{pr}_{k+1}(P_2))|D_s|^{-d})_s\\
& & \\
trTr_\omega(G(P_1,\mbox{{\bf c}}_s(d_s^*v)) P_2 |D_s|^{-d})_s &=&
-trTr_\omega(G(P_1,\mbox{{\bf c}}_s(v)) \mbox{{\bf c}}_s(d_s{pr}_{k-1}(P_2))|D_s|^{-d})_s
\end{array}\; .
\end{equation}
And finally we define for the basic fields
$\mbox{{\bf z}}_1 ,\mbox{{\bf z}}_2 \in\{ \mbox{{\bf A}} ,\mbox{{\bf E}} ,\; N_0, N_s, N_t\}$
\begin{equation}
\!tr Tr_\omega( P_1 G(\mbox{{\bf c}}_s(\mbox{{\bf z}}_1),\mbox{{\bf c}}_s(\mbox{{\bf z}}_2)) P_2 |D_s|^{-d})_s =
\left\{
\begin{array}{lcr}
\!\! tr Tr_\omega(P_1{\gamma^0}^{-1}{id}_{\cal E }P_2|D_s|^{-d})_s &\mbox{if}&
\mbox{{\bf z}}_1=\mbox{{\bf A}} ,\mbox{{\bf z}}_2=\mbox{{\bf E}} \\
& & \\
\!\! -tr Tr_\omega(P_1{\gamma^0}^{-1}{id}_{\cal E }P_2|D_s|^{-d})_s &\mbox{if}&
\mbox{{\bf z}}_1=\mbox{{\bf E}} ,\mbox{{\bf z}}_2=\mbox{{\bf A}} \\
& & \\
0 & &\mbox{otherwise}
\end{array}\right.
\end{equation}
This completes the definition of the phase-space and the Poisson algebra.
The time evolution of the system is determined by the Hamiltonian $H$. For
any element $F\in\cal C$ it is
\begin{equation}
\dot{F} = \{ F, H\}
\end{equation}
where the dot denotes the time derivative of $F$. However, the Hamiltonian
is not uniquely defined for this system since for some arbitrary
$\Lambda \in\mbox{Hom}_{\cal A}(\cal E ,\cal E )$ we can add $\diva{G}(\Lambda dt)$
to the Hamiltonian without changing physics. This is possible because
$\diva{G}(\Lambda)dt$ has to vanish on the physical subspace of $\Gamma_0$.
Furhtermore, consistency requires that the condition eq.(\ref{Gauss}) is
time-independent which leads to the following equations
\begin{eqnarray}
\{\diva{G}(\Lambda dt), H_0\} &\approx& 0\; ,\label{GH}\\
\{\diva{G}(\Lambda_1 dt),\diva{G}(\Lambda_2 dt)\} &\approx& 0\;\; .\label{GG}
\end{eqnarray}
Here $\approx$ means that the equtions hold modulo constraints
This implies that the constraints have to form a closed algebra.
Let us check that eqs.(\ref{GH},\ref{GG}) are satisfied. We start with
eq.(\ref{GG}):
\begin{equation}
\begin{array}{rcl}
\{\diva{G}(\Lambda_1 dt),\diva{G}(\Lambda_2 dt)\} &=&
trTr_\omega(G(
(\mbox{{\bf c}}_s(\nabla_s\Lambda_1)\gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} ),
(\mbox{{\bf c}}_s(\nabla_s\Lambda_2)\gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} ))|D_s|^{-d})_s\\
& & \\
&=&-trTr_\omega(
\mbox{{\bf c}}_s(\nabla_s\Lambda_1)
(\Lambda_2\gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} ) - \gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} )\Lambda_2)
|D_s|^{-d})_s\\
& & \\
& &+trTr_\omega(
(\Lambda_1\gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} ) - \gamma^0\mbox{{\bf c}}_s(\mbox{{\bf E}} )\Lambda_1)
\mbox{{\bf c}}_s(\nabla_s\Lambda_2)
|D_s|^{-d})_s\\
& &\\
&=&-\diva{G}((\Lambda_1\Lambda_2-\Lambda_2\Lambda_1)dt)
\end{array}
\end{equation}
Before we turn to eq.(\ref{GH}) it is useful to compute the following
bracket
\begin{equation}
\begin{array}{rcl}
trTr_\omega
(\Lambda \gamma^0 G(\mbox{{\bf c}}_s(\mbox{{\bf E}} ), {1\over 2} \mbox{{\bf c}}_s(\diva{B})^2)
|D_s|^{-d})_s
&=& -trTr_\omega(\Lambda \mbox{{\bf c}}_s(d_s^*\diva{B}))|D_s|^{-d})_s\\
& & \\
& & +trTr_\omega(\mbox{{\bf c}}_s(\mbox{{\bf A}} )\mbox{{\bf c}}_s(\diva{B})
-\mbox{{\bf c}}_s(\diva{B})\mbox{{\bf c}}_s(\mbox{{\bf A}} ))|D_s|^{-d})_s\\
& & \\
&=& -trTr_\omega(\Lambda \mbox{{\bf c}}_s(\nabla_s^*\diva{B})|D_s|^{-d})_s
\end{array}\label{rotB}
\end{equation}
If we now insert $\Lambda=\mbox{{\bf c}}_s(\nabla_s\Lambda_0)$ in eq.(\ref{rotB})
we obtain
\begin{equation}
\{\diva{G}(\Lambda_0 dt), {1\over
2}trTr_\omega(\mbox{{\bf c}}_s(\diva{B})^2|D_s|^{-d})_s\}
=-trTr_\omega((\mbox{{\bf c}}_s(\nabla_s\Lambda)\mbox{{\bf c}}_s(\nabla_s^*\diva{B})|D_s|^{-d})_s=0 .
\end{equation}
The remaining part is
\begin{equation}
\{\diva{G}(\Lambda_0 dt), {1\over 2}trTr_\omega(\mbox{{\bf c}}_s(\mbox{{\bf E}} )^2|D_s|^{-d})_s\} =
trTr_\omega(\Lambda_0\mbox{{\bf c}}_s(\mbox{{\bf E}} )^2-\mbox{{\bf c}}_s(\mbox{{\bf E}} )^2\Lambda_0|D_s|^{-d})_s = 0\; .
\end{equation}
Hence the conditions eq.(\ref{GH}) and eq.(\ref{GG}) are fulfilled and
the constraints $\diva{G}(\Lambda dt)$ form a complete set of first-class
constraints generating the symmetry of the theory. Thus the observables of
the theory are elements $F\in \cal C$ with
\begin{equation}
\{\diva{G}(\Lambda dt), F\} = 0\;\; .
\end{equation}
The time evolution of the basic fields $\mbox{{\bf A}} , \mbox{{\bf E}} $ can be computed by
considering
\begin{equation}
\{ trTr_\omega(\mbox{{\bf c}}_s(\Lambda)\mbox{{\bf c}}_s(\mbox{{\bf A}} )|D_s|^{-d})_s, H_0\}=
trTr_\omega(\mbox{{\bf c}}_s(\Lambda){(\gamma^0)}^{-1}\mbox{{\bf c}}_s(\mbox{{\bf E}} )|D_s|^{-d})_s
\end{equation}
{}From this and eq.(\ref{rotB}) we infer that the time evolution of the
basic fields is (modulo gauge transformations)
\begin{equation}
\begin{array}{rcl}
\dot{\mbox{{\bf A}} } &=& -pr_1({(\gamma^0)}^{-1}\mbox{{\bf c}}_s(\mbox{{\bf E}} ))\\
& & \\
\dot{\mbox{{\bf E}} } &=& -pr_2{(\gamma^0})^{-1}\nabla_s^*\diva{B}\;\; .
\end{array}
\end{equation}
Equivalently, with $\mbox{{\bf E}} =\mbox{{\bf E}} _0dt$, we can write
\begin{equation}
\begin{array}{rcl}
\dot{\mbox{{\bf A}} } &=& \mbox{{\bf E}} _0\\
& & \\
\dot{\mbox{{\bf E}} _0} &=& -N^{-\frac{1}{2}}\nabla_s^*\diva{B}\;\; .
\end{array}
\end{equation}
\section{Examples}
In this section we apply the general contruction, presented in the previous
sections, to two examples, which are, more or less, standard (toy) examples in
non-commutative geometry applied to elementary particle physics. In first
one the algebra $\cal A_s $ is a sum of two identical finite dimensional algebras
of complex matrices. This is basicly the setting of the
``Two-Point Space'' as it was presented in \cite{cobuch}. The ``Yang-Mills''
on this discrete space generates a Higgs potential and spontaneous symmetry
breaking.
In the second example the algebra of the first example is enlarged by the
algebra of smooth functions on a compact Riemannian manifold. This leads to
a gauge theory with conventional gauge bosons and Higgs bosons. The gauge
symmetry of the model is $U(n)\times U(n)$ which is broken to $U(n)$.
One might intepret this example as a model with a left-right chiral symmetry
which is broken spontaneously to a vector symmetry. However, since we do not
yet have fermions included in our construction, such an interpretation might
be a little bit artificial.
\subsection{The Two-Point Space}
We start with the discrete space and take for $\cal A_s $
\begin{equation}
\cal A_s = \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n} \oplus \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n}
\end{equation}
which represents the space-like part of the algebra $\cal A$ in this example.
A general discussion of Connes' generalized differential algebra constructed
out of matrix-algebras can be found in \cite{KPPW}.
The complete algebra $\cal A$ over space-time is then
\begin{equation}
\cal A = C^\infty({I\kern -0.22em R\kern 0.30em},\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n} \oplus \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n})\; .
\end{equation}
The Hilbert-space $\cal H_s $ is
\begin{equation}
\cal H_s = (\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^n\oplus\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^n) \otimes \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^G\otimes \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^2
\end{equation}
where $\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^G$ denotes the ``generation-space'' with $G>1$ and the $\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^2$ factor
is needed for the construction on $\gamma^0$. The representation $\pi_s$ is
given for all $A=(A_1,A_2)\in\cal A_s $ as
\begin{equation}
\pi_s(A)=
\left( \begin{array}{cc}
A_1 & 0 \\
& \\
0 & A_2
\end{array}\right)
\otimes 1_{\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^G}\otimes 1_{\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^2}\;\; . \label{repmm}
\end{equation}
We take for the space-like operator $D_s$
\begin{equation}
D_s=\left(\begin{array}{cc}
0 & \tilde{D}_s \\
& \\
\tilde{D}_s & 0
\end{array}\right) \;\; , \;\;
\tilde{D}_s=\left(\begin{array}{cc}
0 & \mu \\
& \\
\mu^\dagger & 0
\end{array}\right)\otimes M
\end{equation}
where $M\in \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{G\times G}, M^2\neq \alpha 1_{\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{\!G}}\, , M^2\neq 0$ is a
matrix in generation space which guarantees that the representation of
two-forms on $\cal H_s $ is linear independent from the representation of $\cal A_s $.
We choose $\mu\in \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n}$ such that
$\mu\mu^\dagger=\mu^\dagger\mu=\lambda^2 1_{\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{\!n}}$. Thus the space-like
k-cycle $(\cal H_s , D_s)$ over $\cal A_s $ is defined and the extension to a k-cycle
$(\cal H , D)$ over $\cal A$ along the lines described in sect.3 is straightforward:
\begin{equation}
\begin{array}{rcl}
\cal H &=&L_2({I\kern -0.22em R\kern 0.30em} ,(\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^n\oplus\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^n) \otimes \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^G\otimes \mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^2)\;\; ,\\
& & \\
D &=& D_t + D_s \;\; , \;\; D_t=
\left(\begin{array}{cc}
1\partial_t & 0 \\
& \\
0 & -1\partial_t
\end{array}\right)\;\; ,
\end{array}
\end{equation}
where the $1$ in the definition of $D_t$ refers to the unit in
$\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{2n}\otimes\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^G$.
The representation $\pi$ maps elements of $\cal A$ onto time-dependent
blockdiagonal elements of the same form as in eq.(\ref{repmm}). The remaining
element in the general set-up which we have to specify is the $\cal A$-module
$\cal E $. We take the simplest choice, i.e., $\cal E =\cal A$.
Now we can write down the connection one form $\mbox{{\bf A}} =\mbox{{\bf A}} _t + \mbox{{\bf A}} _s$:
\begin{equation}
\mbox{{\bf A}} _t = \left(\begin{array}{cc}
\mbox{{\bf A}} _1 & 0 \\
& \\
0 & \mbox{{\bf A}} _2
\end{array}\right)\;\; ,\;\;
\mbox{{\bf A}} _s= \left(\begin{array}{cc}
0 & \phi \\
& \\
\phi^\dagger & 0
\end{array}\right)\;\; .
\end{equation}
$\mbox{{\bf A}} _1,\mbox{{\bf A}} _2$ are anti-hermitean $n\times n$ matrices multiplied by $dt$ and
$\phi$ is a complex $n\times n$ matrix of (matrix-) form degree 1, i.e., it
is a $n\times n$ matrix multiplied by $M$.
The curvature $\mbox{{\bf F}} =\mbox{{\bf F}} _{st} + \mbox{{\bf B}} $ of $\mbox{{\bf A}} $ is given by
\begin{equation}
\begin{array}{rcl}
\mbox{{\bf F}} _{st} & = &\left(\begin{array}{cc}
0 & \dot{\phi}dt + \mbox{{\bf A}} _1(\mu +\phi) + (\mu + \phi)\mbox{{\bf A}} _2\\
& \\
\dot{\phi}^\dagger dt +(\mu^\dagger +\phi^\dagger)\mbox{{\bf A}} _1 +
\mbox{{\bf A}} _2(\mu^\dagger +\phi^\dagger)& 0
\end{array}\right)\; ,\\
& & \\
\mbox{{\bf B}} & = & \left(\begin{array}{cc}
\phi\mu^\dagger + \mu\phi^\dagger +\phi\phi^\dagger & 0 \\
& \\
0 & \phi^\dagger\mu +\mu^\dagger\phi +\phi^\dagger\phi
\end{array}\right)\; .
\end{array}
\end{equation}
Since the space-like part $\cal A_s $ of the algebra $\cal A$ is finite dimensional
the Dixmier-trace in the definition of the Lagrange function reduces to the
normal trace and hence the Lagrange function $L$ is
\begin{equation}
\begin{array}{rcl}
L &=&-{1\over 4} tr (F^\dagger F)\\
& & \\
&=&\!\!\! {1\over 2}tr[(\dot{\phi}\gamma^0 + \mbox{{\bf A}} _1(\mu+\phi)+(\mu+\phi)\mbox{{\bf A}} _2)
(\dot{\phi}^\dagger \gamma^0 +(\mu^\dagger +\phi^\dagger)\mbox{{\bf A}} _1 +
\mbox{{\bf A}} _2(\mu^\dagger +\phi^\dagger))] - V(\phi)
\end{array}
\end{equation}
with
\begin{equation}
V(\phi)={1\over 4}tr[
\phi\mu^\dagger + \mu\phi^\dagger +\phi\phi^\dagger)
(\phi^\dagger\mu +\mu^\dagger\phi +\phi^\dagger\phi)]\; .\label{vpot}
\end{equation}
Now we turn to the Hamilton formalism and find for the momentum $\mbox{{\bf E}} $
\begin{equation}
\mbox{{\bf E}} =\left(\begin{array}{cc}
0 & -\pi^\dagger \\
& \\
\pi & 0
\end{array}\right)
\end{equation}
with
\begin{equation}
\pi= \dot{\phi}^\dagger dt + (\mu^\dagger+\phi^\dagger)\mbox{{\bf A}} _1 +
\mbox{{\bf A}} _2(\mu^\dagger +\phi^\dagger)\; . \label{phimpuls}
\end{equation}
Thus the Hamiltonian $H=H_0-G(\mbox{{\bf A}} _t)$ is given by
\begin{eqnarray}
H_0 &=& tr(\pi^\dagger\pi) + V(\phi)\\
G(\mbox{{\bf A}} _t) &=& tr [\mbox{{\bf E}} (D_s + \mbox{{\bf A}} _s)\mbox{{\bf A}} _t+ \mbox{{\bf E}} \mbox{{\bf A}} _t(D_s +\mbox{{\bf A}} _s)]
\end{eqnarray}
The Gau\ss-law constraints
\begin{equation}
G((\Lambda_1,\Lambda_2)dt)=0\; ,\; (\Lambda_1,\Lambda_2)=\Lambda\in\cal A_s ,\;
\Lambda^\dagger =-\Lambda
\end{equation}
generate the Lie-algebra of the $U(n)\times U(n)$ symmetry group.
The phase-space variables transform as follows
\begin{equation}
\begin{array}{rcl}
\delta\pi &=& \Lambda_2\pi - \pi\Lambda_1\\
& & \\
\delta\phi&=&\Lambda_1(\phi+\mu)-(\phi+\mu)\Lambda_2\; .
\end{array}\label{phitrafo}
\end{equation}
The inhomogeneous transformation property of $\phi$ is due to the fact that
$\phi$ is part of the connection in this formalism. However, a substitution
\begin{equation}
\varphi=\phi+\mu
\end{equation}
lead to a homogeneous transformation property
\begin{equation}
\delta\varphi = \Lambda_1 \varphi-\varphi\Lambda_2\;\; .\label{trvaph}
\end{equation}
The potential $V$ reads in this new variable
\begin{equation}
V(\varphi)={1\over 4}tr(\varphi\varphi^\dagger -\lambda^2)
(\varphi^\dagger\varphi-\lambda^2)\;\; .
\end{equation}
For the time-evolution of the system we find
\begin{equation}
\begin{array}{rcl}
\dot{\varphi}&=&\pi^\dagger\\
& & \\
\dot{\pi}&=& -{1\over 4}[\varphi^\dagger(\varphi^\dagger\varphi -\lambda^2)
+ (\varphi\varphi^\dagger -\lambda^2)\varphi^\dagger]
\end{array}
\end{equation}
We see that there are two configurations in phase-space, which are stable
under time evolution. The first one is $\pi=0, \varphi=0$, which is metastable
and $\pi=0, \varphi^\dagger\varphi=\lambda^2$ which is stable. The second
configuration is the vacuum expectation value of the Higgs-field. By
choosing for the vacuum expectation value $\varphi_0$
\begin{equation}
\varphi_0=1\lambda
\end{equation}
we infer from the transformation rule (\ref{trvaph}) of $\varphi$ that the
little group of $\varphi_0$ is the diagonal $U(n)$ subgroup of
$U(n)\times U(n)$. This shows that Yang Mills theory on discrete space
generates spontaneous symmetry breaking and thus we have translated this
appealing result of A.~Connes and J.~Lott \cite{colo} into the Hamilton
formalism.
\subsection{Yang-Mills Thoery on Space-Time $\times$ Two-Point Space}
In this second example we utilize the result of the previous example to
construct a Yang-Mills theory with spontaneously broken symmetry on a four
dimesional Minkowskian space-time. We assume that the space-time manifold
$M$ has the topology $M_3\times {I\kern -0.22em R\kern 0.30em}$ where $M_3$ is a compact manifold.
For this example let us take for $M_3$ the one point compactification of
${I\kern -0.22em R\kern 0.30em}^3$, i.e. $M_3=S^3$. The algebra $\cal A$ is of the form
\begin{equation}
\cal A= C^\infty(M)\otimes (\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n} \oplus\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n})\; .
\end{equation}
The space-like part of the algebra is
\begin{equation}
\begin{array}{rcl}
\cal A_s&=& C^\infty(S^3)\otimes (\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n} \oplus\mbox{\kern 0.20em \raisebox{0.09ex}{\rule{0.08ex}{1.22ex}^{n\times n})\\
& & \\
&=& C^\infty(S^3)\otimes \cal A_{mat}
\end{array}
\end{equation}
For $S^3$ there is a k-cycle $(\cal H _3, D_3)$ over $C^\infty(S^3)$, where $\cal H _3$
denotes the square integrable spin-sections over $S^3$ and $D_3$ denotes the
Dirac-operator on $S^3$, which leads to the usual de Rham algebra. The k-cycle
$(\cal H _{mat}, D_{mat})$ has been specified in the previous example (the
subscript $_{mat}$ is introduced in order to distinguish objects refering to
the discrete part of the algebra from the other objects). Usually one obtains
a k-cycle over an algebra which is a tensor product of two algebras by taking
the product k-cycle of the k-cycles over the factor algebras. However, there
is one difficulty in our case. For the definition of the operator $D$ of the
product k-cycle one needs a grading on one of the factor k-cycles. Since $S^3$
is odd-dimensional there is no such grading on the Clifford-bundle over $S^3$.
On the other hand for the Clifford-bundle over ${I\kern -0.22em R\kern 0.30em}\times S^3$ there is grading
given by $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$. Thus we can take the
product k-cycle $(\cal H , D)$ over $\cal A$ with
\begin{equation}
\begin{array}{rcl}
D&=&D_4\otimes 1_{mat} + \gamma^5\otimes D_{mat}\\
& & \\
\cal H &=&\cal H _4 \oplus\cal H _{mat}\;\; ,
\end{array}
\end{equation}
where $D_4=\gamma^\mu\partial_\mu$ denotes the Dirac operator on
$M={I\kern -0.22em R\kern 0.30em}\times S^3$ and $\cal H _4$ is the space of square integrable spin-sections
over $M$. Since a Dirac operator on a manifold with topology ${I\kern -0.22em R\kern 0.30em}\times M_3$
can always be decomposed in a time-like part $D_t$ and a space-like part $D_3$
the space-like k-cycle $(\cal H_s , D_s)$ over $\cal A_s $ is
\begin{equation}
\begin{array}{rcl}
D_s&=&D_3\otimes 1_{mat} + \gamma^5\otimes D_{mat}\\
& & \\
\cal H_s &=&\cal H _3 \otimes\cal H _{mat}\;\; .
\end{array}
\end{equation}
Again we choose for the $\cal A$-module $\cal E =\cal A$.
The connection $\cal A=\mbox{{\bf A}} _t+\mbox{{\bf A}} _s$ for this model is
\begin{equation}
\mbox{{\bf A}} _t = \left(\begin{array}{cc}
{\mbox{{\bf A}} _t}_1 & 0 \\
& \\
0 & {\mbox{{\bf A}} _t}_2
\end{array}\right)\;\; ,\;\;
\mbox{{\bf A}} _s= \left(\begin{array}{cc}
{\mbox{{\bf A}} _s}_1 & \phi \\
& \\
\phi^\dagger & {\mbox{{\bf A}} _s}_2
\end{array}\right)\;\; .
\end{equation}
$A_t$ is the same as in the previous example but on the block-diagonal of
$\mbox{{\bf A}} _s$ there are now the space-like parts of the conventional gauge
connections $\mbox{{\bf A}} _1$ and $\mbox{{\bf A}} _2$, i.e., ${\mbox{{\bf A}} _s}_1$ and ${\mbox{{\bf A}} _s}_2$ are
anti-hermitean matrices multiplied with space-like one forms.
The corresponding curvature is
\begin{equation}
\begin{array}{rcl}
\mbox{{\bf F}} _{st} & = &\left(\begin{array}{cc}
{\mbox{{\bf F}} _{st}}_1
& \dot{\phi}dt + {\mbox{{\bf A}} _t}_1(\mu +\phi) + (\mu + \phi){\mbox{{\bf A}} _t}_2\\
& \\
\dot{\phi}^\dagger dt +(\mu^\dagger +\phi^\dagger){\mbox{{\bf A}} _t}_1 +
{\mbox{{\bf A}} _t}_2(\mu^\dagger +\phi^\dagger)& {\mbox{{\bf F}} _{st}}_2
\end{array}\right)\\
& & \\
\mbox{{\bf B}} & = & \left(\begin{array}{cc}
\mbox{{\bf B}} _1+\phi\mu^\dagger + \mu\phi^\dagger +\phi\phi^\dagger &
\partial_i\phi dx^i + {\mbox{{\bf A}} _s}_1(\mu +\phi) + (\mu + \phi){\mbox{{\bf A}} _s}_2 \\
& \\
\partial_i\phi^\dagger dx^i +(\mu^\dagger +\phi^\dagger){\mbox{{\bf A}} _s}_1 +
{\mbox{{\bf A}} _s}_2(\mu^\dagger +\phi^\dagger)
& \phi^\dagger\mu +\mu^\dagger\phi +\phi^\dagger\phi
\end{array}\right)
\end{array}
\end{equation}
where $\mbox{{\bf B}} _i, i=1,2$ denotes the space-like curvature of $\mbox{{\bf A}} _i$,
${\nabla_s}_i$ is the corresponding covariant space-like derivative and
\begin{equation}
{\mbox{{\bf F}} _{st}}_i=-\partial_t{\mbox{{\bf A}} _s}_idt +{\nabla_s}_i{\mbox{{\bf A}} _t}_i\;\; .
\end{equation}
Due to A.~Connes' trace theorem the Dixmier trace is in this case equivalent
to an integration over $S^3$ and hence the Lagrange function is
\begin{equation}
\begin{array}{rcl}
L &=&-{1\over 4} tr\int d^3x (F^\dagger F)\\
& & \\
&=&{1\over 2}\int d^3x \left( -V(\phi) + tr[{\mbox{{\bf F}} _{st}}_1^2 + {\mbox{{\bf F}} _{st}}_2^2-
\mbox{{\bf B}} _1^2 -\mbox{{\bf B}} _2^2\right. \\
& & \\
& & +(\dot{\phi}\gamma^0 +{\mbox{{\bf A}} _t}_1(\mu+\phi)+(\mu+\phi){\mbox{{\bf A}} _t}_2)
(\dot{\phi}^\dagger \gamma^0 +(\mu^\dagger +\phi^\dagger){\mbox{{\bf A}} _t}_1 +
{\mbox{{\bf A}} _t}_2(\mu^\dagger +\phi^\dagger))\\
& & \\
& &\left. -(\partial_i\phi\gamma^i +{\mbox{{\bf A}} _s}_1(\mu+\phi)+(\mu+\phi){\mbox{{\bf A}} _s}_2)
(\partial_i\phi^\dagger \gamma^i +(\mu^\dagger +\phi^\dagger){\mbox{{\bf A}} _s}_1 +
{\mbox{{\bf A}} _s}_2(\mu^\dagger +\phi^\dagger))]\right)
\end{array}
\end{equation}
with $V(\phi)$ given by eq.(\ref{vpot}).
The canonical momenta for this system are
\begin{equation}
\mbox{{\bf E}} =\left(\begin{array}{cc}
\mbox{{\bf E}} _1 & -\pi^\dagger \\
& \\
\pi & \mbox{{\bf E}} _2
\end{array}\right)
\end{equation}
with $\pi$ defined in eq.(\ref{phimpuls}) and
\begin{equation}
\mbox{{\bf E}} _i={\mbox{{\bf F}} _{st}}_i\;\; ,\;\; i=1,2\;\; .
\end{equation}
Thus we can determine the Hamiltonian $H_0 - G(\mbox{{\bf A}} _t)$ to be
\begin{equation}
\begin{array}{rcl}
H_0 &=& \int d^3x \left(V(\phi) +tr[ \mbox{{\bf E}} _1^2 + \mbox{{\bf E}} _2^2+ \pi^\dagger\pi
+\mbox{{\bf B}} _1^2 +\mbox{{\bf B}} _2^2 \right.\\
& & \\
& & \left. +(\partial_i\phi\gamma^i +{\mbox{{\bf A}} _s}_1(\mu+\phi)+(\mu+\phi){\mbox{{\bf A}} _s}_2)
(\partial_i\phi^\dagger \gamma^0 +(\mu^\dagger +\phi^\dagger){\mbox{{\bf A}} _s}_1 +
{\mbox{{\bf A}} _s}_2(\mu^\dagger +\phi^\dagger))]\right).
\end{array}
\end{equation}
Again the Gau\ss-law can be summarized as
\begin{equation}
G(\mbox{{\bf A}} _t) = \int d^3x tr [\mbox{{\bf E}} (D_s + \mbox{{\bf A}} _s)\mbox{{\bf A}} _t+ \mbox{{\bf E}} \mbox{{\bf A}} _t(D_s +\mbox{{\bf A}} _s)]\; .
\end{equation}
The phase-space variables transform as follows
\begin{equation}
\delta\mbox{{\bf E}} _i = \Lambda_i\mbox{{\bf E}} _i-\mbox{{\bf E}} _i\Lambda_i \; ,\; i=1,2 .
\end{equation}
The transformation rule for the fields $\pi$ and $\phi$ are determined by
eq.(\ref{phitrafo}). By shifting $\phi$ to $\varphi=\phi+\mu$ we obtain a
field which transforms homogeneously under gauge transformations. For
$\varphi^\dagger\varphi= 1\lambda^2 $ the potential is minimized and thus
the symmetry is spontaneously broken. In the gauge
\begin{equation}
\varphi = 1\lambda
\end{equation}
we see that $\mbox{{\bf A}} _+ =\mbox{{\bf A}} _1 +\mbox{{\bf A}} _2$ correspond to the massless modes of the
gauge fields and $\mbox{{\bf A}} _-=\mbox{{\bf A}} _1-\mbox{{\bf A}} _2$ correspond to the massive modes.
\section{Conclusions}
We have derived the Hamilton formalism for Yang-Mills theory in non-commutative
geometry. For this purpose we exploited the special structure of
$\cal A=C(I,\cal A_s )$ which seems to be very natural since the topology of space-time
in the conventional Hamilton formalism is $M={I\kern -0.22em R\kern 0.30em}\times\Sigma$. The first step
was to show that the structure of the algebra together with an appropriately
choosen k-cycle allows to identify the time-like part of the generalized
differential algebra. Thus the notion of time obtains a well defined meaning in
this context.
The next step was to introduce the non-commutative generalization of
integration over space-like surfaces via the Dixmier trace. This opened the
possibility to apply the formalism to Minkowskian space-time by abandoning
the ellipticity of the operator $D$ of the k-cycle $(\cal H ,D)$ over $\cal A$
but maintaining the ellipticity of the space-like part $D_s$ of $D$.
However, in this case one is restricted to the non-commutative counterpart of
integration over space-like surfaces. For the definition of Lagrange functions
and Hamilton functions integration over space-like surfaces is sufficient.
For the definition of actions one may use a hybrid formalism, i.e., one
performs integration over the (possibly non-commutative) space-like surface
via Dixmier trace and for the time variable one uses conventional integration.
The structure $C(I,\cal A_s )$ of the algebra ensures that this is possible.
For the definition of the Poisson bracket we had to make some additional
assumptions which we introduced at the end of sect.~5. Especially the
assumption which allowed us to define the adjoint of the operator $d$ seems
to be a brute force assumption. Although all assumptions we made are
fulfilled for the examples we presented, a finer criterion for the existence
of an adjoint of $d$ seems to be desirable.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,118 |
{"url":"https:\/\/www.beatthegmat.com\/ask-a-duke-fuqua-student-t73276-60.html","text":"\u2022 Award-winning private GMAT tutoring\nRegister now and save up to $200 Available with Beat the GMAT members only code \u2022 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code \u2022 Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code \u2022 FREE GMAT Exam Know how you'd score today for$0\n\nAvailable with Beat the GMAT members only code\n\n\u2022 5-Day Free Trial\n5-day free, full-access trial TTP Quant\n\nAvailable with Beat the GMAT members only code\n\n\u2022 5 Day FREE Trial\nStudy Smarter, Not Harder\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Get 300+ Practice Questions\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Free Trial & Practice Exam\nBEAT THE GMAT EXCLUSIVE\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Magoosh\nStudy with Magoosh GMAT prep\n\nAvailable with Beat the GMAT members only code\n\n\u2022 Free Veritas GMAT Class\nExperience Lesson 1 Live Free\n\nAvailable with Beat the GMAT members only code\n\n## Ask a Duke Fuqua Student\n\nThis topic has 21 expert replies and 75 member replies\nGoto page\nai86 Newbie | Next Rank: 10 Posts\nJoined\n03 Jan 2012\nPosted:\n7 messages\nFollowed by:\n1 members\n2\nFri Jun 08, 2012 6:26 pm\nHi everyone,\n\nI am so sorry for my lack of response. I know this is not a good reason but emails from this thread somehow didn't get into my inbox earlier. Thanks for your understanding. I'll make sure I reply quickly from now on.\n\n-Irene-\n\n_________________\nIrene\nMBA, Class of 2013\nThe Fuqua School of Business, Duke University\n\nssinghrajora Newbie | Next Rank: 10 Posts\nJoined\n26 Jun 2012\nPosted:\n3 messages\nTue Jun 26, 2012 7:56 pm\nHello Irene\n\nI have worked in Business Development and Operations for coal mining projects and power projects in India and Indonesia. My work included acquisition of new targets, project development and management. I want to change carrer and get into consulting (if possible into energy vertical\/ acquisitions).\n\nI am looking for a program which has corporate strategy as its strength and I find that Duke's strategy area is one of the leading in the world. I would like to know whether a combination of my experience and a Duke MBA can get me where I want to go?\n\nShailendra\nIndia\/ Indonesia\n750\/4.5, 5 yrs (2.5 yrs international)\n\nai86 Newbie | Next Rank: 10 Posts\nJoined\n03 Jan 2012\nPosted:\n7 messages\nFollowed by:\n1 members\n2\nTue Jun 26, 2012 8:17 pm\nHi Shailendra,\n\nMost of my friends who land a consulting internship\/full-time came from various different backgrounds. So, I think that you would also be able to enter consulting as long as you can make a compelling story into why you would be a good fit in the consulting company. Fuqua is one of the core schools for Deloitte, BCG and I think McKInsey also (but please check again). And other top consulting firms also recruit at Duke.\n\nAs to your specific background and entry into the consulting field, I don't think that I have the capability to give you a good assessment. =(\n\nI hope this helps.\n\nCheers,\nIrene\n\nPS: it's interesting that we have somewhat similar background - I used to live in Indonesia and my pre-MBA experience is related to the mining industry. =)\n\n_________________\nIrene\nMBA, Class of 2013\nThe Fuqua School of Business, Duke University\n\nssinghrajora Newbie | Next Rank: 10 Posts\nJoined\n26 Jun 2012\nPosted:\n3 messages\nWed Jun 27, 2012 10:10 pm\nHi Irene\n\nThank you for your reply. What was your work profile in mining? Since our careers share two important traits, I believe that you might be the perfect person to answer my queries.\n\nDoes Fuqua\u00e2\u20ac\u2122s vastly diverse class consist of many students with backgrounds similar as yours (ours)? I mean students from the verticals of energy and extractive minerals. Have you recently seen any GATE programs directed towards these verticals?\n\nThank you.\n\nShailendra\nIndia\/ Indonesia\n750\/4.5, 5 yrs (2.5 yrs international)\n\nai86 Newbie | Next Rank: 10 Posts\nJoined\n03 Jan 2012\nPosted:\n7 messages\nFollowed by:\n1 members\n2\nThu Jun 28, 2012 12:58 am\nHi Shailendra,\n\nI worked as a Project Engineer at a manufacturing company. We produce pulleys which is typically used in mines and oil sands. I know of a few people who are interested in entering the alternative energy industry post-MBA. Fuqua had an Energy focused Career Symposium last year - so I am sure it's a big post-MBA career at Fuqua. As far as pre-MBA experience, only one friend came to mind. He came from a PE firm which focuses on energy & alternative energy. I am sure there's more than one - I just happened not to know many of them..\n\nAnd as for GATE, I don't know of any GATE programs which is specifically directed towards these verticals. However, if most of the students enrolled in the GATE program is interested, they most likely could arrange a visit to those companies. I did GATE Lat-AM; at the start of semester, our GATE leader made a poll to see which industries people are interested in and arranged visit to companies in those specific industries.\n\nHope this helps. =)\n\n_________________\nIrene\nMBA, Class of 2013\nThe Fuqua School of Business, Duke University\n\nssinghrajora Newbie | Next Rank: 10 Posts\nJoined\n26 Jun 2012\nPosted:\n3 messages\nFri Jun 29, 2012 11:44 pm\nHi Irene\n\nAgain, thank you for the very useful information. Right now I am researching about MBA schools for applying in this fall. There are few schools with strong strategy departments, and among them Fuqua clearly stands out in both 'strategy' and 'energy'. That's why I am very inquisitive about it.\n\nSincere thanks.\n\nShailendra\nIndia\/ Indonesia\n750\/4.5, 5 yrs (2.5 yrs international)\n\nkwilson219 Newbie | Next Rank: 10 Posts\nJoined\n05 Aug 2011\nPosted:\n2 messages\nWed Aug 01, 2012 9:59 am\n\nI have a few questions that I was hoping you could answer.\n\n1. One reason I want to pursue an MBA is to develop quantitative skills related to business. What specific quantitative classes have you taken during your MBA and what valuable skills and lessons have you taken away from them that you will use in your post MBA career?\n2. Over the summer, what internship(s) did you have a chance to participate in and how did the career development center at Fuqua support you in finding an internship?\n3. I read that the Fuqua student body is a very tight knit community. Could you elaborate on this reputation and what makes the Fuqua community unique?\n4. What events or classes have allowed you to expand your networking base? What type of activities have you participated in, through the Fuqua MBA network, that have helped you develop professional relationships and connections?\n5. If you could give one piece of advice to someone applying to Fuqua, what would that be?\n\nKurt\n\nai86 Newbie | Next Rank: 10 Posts\nJoined\n03 Jan 2012\nPosted:\n7 messages\nFollowed by:\n1 members\n2\nWed Aug 01, 2012 6:32 pm\nHi Kurt,\n\nThose are very good questions and I will try by best to answer them.\n\n1. I took Managerial Finance, Accounting, Statistics, and Operations (some Quant - not much). Given my Engineering background, I don't have much knowledge in Finance or Accounting. Thus, I found the classes useful and will certainly help me in my future career, especially if I decide to pursue a career in Finance.\n\n2. I am currently finishing up my internship at Accenture and throughout the recruiting process, the Career Management Center (CMC) at Fuqua has always been there to support me. They helped me out with my resume, mock interviews or simply there to listen in to my worries. However, one thing to note is that you have to be pro-active to make the most of the CMC. They are always there to help you out but they will not check on you everyday (or so to speak) to see how you're doing. There're 440 students afterall =)\n\n3. Fuqua is a very tight knit community. Everyone knows everyone (or almost... ). Most people are also very helpful. During my first week at Fuqua, friends whom I just met then, helped me assemble my furnitures, drove me to Walmart to get appliances, etc. It's really amazing. Any question that you have about anything related to Fuqua or general stuff, you can post to the FB group and someone will answer within the next hour (or minutes even). Questions from: how to get driving license to where the party's at tonight.\n\n4. There are a lot of activities that you can participate in at Fuqua. There's company presentations and networkin nights which allows you to meet some of the professionals from the companies you're interested in. There's also student lead initiatives such as the Consulting Symposium, Marketing Symposium, etc. I am not sure if this is what you were asking about - if not, please do let me know and I'll try to address your question better.\n\n5. Talk to current students and visit Fuqua if you can =). The teamwork and community at Fuqua is something that has to be experienced as it's really hard to explain.\n\n_________________\nIrene\nMBA, Class of 2013\nThe Fuqua School of Business, Duke University\n\nbecnil MBA Student\nJoined\n15 Dec 2009\nPosted:\n170 messages\nFollowed by:\n8 members\n20\nTest Date:\nMarch 12th, 2010\nGMAT Score:\n740\nFri Aug 03, 2012 5:07 pm\nI fully agree with ai86 on this topic; she has already summed up the key points, but I will try to add in a bit.\n\n1. Our core courses in Fuqua are designed to provide a balance of strong quantitative and qualitative understanding to a diverse community of students. So, irrespective of your background, you can be assured that you will get a solid quantitative base in your first two terms. And then, through your electives, you can choose as much quant focus as you want - in terms of modeling, you can take Decision Models, Strategic Modeling; if you love Operations, you can go with Supply Chain Management; if you like quantitative Finance\/Accounting, there is a whole array of classes that you can take and may also concentrate in one of these areas.\n\n2. I am interning this summer with Xylem Inc., a water technology and services company, in their Strategy & Business Development group. I am working on building a business case for Xylem to enter an evolving market and launch a new product line. As ai86 mentioned, the CMC was there for us all along, helping with applications, resumes, cover letters etc. However, I will also say that my classmates and alum were great assets throughout my recruiting efforts. Leveraging each other's backgrounds and experiences, we were able to collectively achieve success in securing the desired summer positions.\n\nGood luck.\n\n_________________\nMBA 2013, Duke University, Fuqua School of Business\n\nmappleby285 Newbie | Next Rank: 10 Posts\nJoined\n16 Jul 2011\nPosted:\n9 messages\nSat Aug 11, 2012 5:58 pm\nCan a current student comment on what the schedule is like with the 6 week terms? My wife is in grad school right now on the quarters system, my undergrad was done on semesters. It seems like even with the 10 week long quarters the classes feel very rushed for my wife. How does it work having only 6 weeks long for each term?\n\nbecnil MBA Student\nJoined\n15 Dec 2009\nPosted:\n170 messages\nFollowed by:\n8 members\n20\nTest Date:\nMarch 12th, 2010\nGMAT Score:\n740\nSat Aug 11, 2012 8:29 pm\nIn short, the Terms are going to fly by very fast, so be prepared for it. There is literally no comparison with the Semester system (which I had both in my UG and my MS). But at the same time, you will be learning a ton of topics, which will allow you to determine where you want to specialize in your second year. This is an area where Fuqua differentiates heavily from other top business schools.\n\nYou will be busy with a lot of teamwork, individual assignments, mid-terms, and of course socializing ! The best way to cope up with this is to get ready and not be shocked. Everything will be just fine !\n\n_________________\nMBA 2013, Duke University, Fuqua School of Business\n\ncoreaspirant Newbie | Next Rank: 10 Posts\nJoined\n02 Sep 2012\nPosted:\n3 messages\nMon Sep 03, 2012 3:53 am\nHi\n\nI want to get into Corporate Finance, with no prior experience in it.\n\n1) While reading and understanding the roles, I have got a good understanding of the roles in Corporate Finance. In the short term goal, do we have to be VERY SPECIFIC abt the profile (such as a particular profile name - for example investment management analyst ), and roles and responsibilities?\n\n2) According to you, which courses and clubs can be most helpful in the transition?\n\n3) What are the various things which you feel to be the best part of Duke's culture and which you feel had an impact on you as a person?\n\n4) This question might sound stupid - but every MBA has leadership courses. While going through the courses on Fuqua Business School's site, I couldn't get their details?\n\nRegards\n\nbecnil MBA Student\nJoined\n15 Dec 2009\nPosted:\n170 messages\nFollowed by:\n8 members\n20\nTest Date:\nMarch 12th, 2010\nGMAT Score:\n740\nMon Sep 03, 2012 4:43 pm\nHello,\n\nHere are my responses (in italics):\n\nI want to get into Corporate Finance, with no prior experience in it.\n\n1) While reading and understanding the roles, I have got a good understanding of the roles in Corporate Finance. In the short term goal, do we have to be VERY SPECIFIC abt the profile (such as a particular profile name - for example investment management analyst ), and roles and responsibilities?\n\n[i]I don't think so. My answer is going to be subjective, but I believe if you can demonstrate your understanding of the industry and a basic focus, that should be fine. A specific position is probably not needed. Since you do not have a background in Corp Finance, make sure you answer your rationale and a logical progression to this new industry - e.g. tie your intended roles to your background, interest etc., and answer why you want to go there\n[\/i]\n2) According to you, which courses and clubs can be most helpful in the transition?\n\nOf course Corporate Finance, Adv Corporate Finance and other specific finance electives that you will delve into once you start the program. There is a Finance Club at Fuqua which will get you started, and then based on your specific interests within Finance, you can focus on other specialized clubs within Finance (e.g. EVCC)\n\n3) What are the various things which you feel to be the best part of Duke's culture and which you feel had an impact on you as a person?\n\nThe spirit of collaboration through \"Team Fuqua\". We live by this principle day in and day out. The way I have seen my classmates and the senior year look after one another is amazing. Not only within Fuqua, but even outside, I have seen Fuquans upholding their collaborative, inclusive qualities, and I have felt very proud. Accessibility is also something I have found to be amazing - from the Dean to the Professors to alum, I have found a lot of warmth; people want to hear from you, and are open to feedback and suggestions for improvement. Finally, the focus on leadership development at Fuqua has been a unique experience for me. Everything here is student-run: you will have much more leadership opportunities than you can possibly undertake, and the environment truly inspires you to lead. COLE Fellowship is a very special program at Fuqua that focuses on leadership development by associating SYs with the FY teams.\n\n4) This question might sound stupid - but every MBA has leadership courses. While going through the courses on Fuqua Business School's site, I couldn't get their details?\n\nYou don't necessarily need to take Leadership courses to learn to be a leader. In fact, I think you will learn it better by actually doing it. Referring to my answer above, Fuqua has tremendous opportunities for you to lead in many different avenues. That being said, we do have leadership courses. Refer to http:\/\/www.fuqua.duke.edu\/student_resources\/academics\/concentrations\/leadership_ethics\/\n\nRegards\n\n_________________\nMBA 2013, Duke University, Fuqua School of Business\n\ncoreaspirant Newbie | Next Rank: 10 Posts\nJoined\n02 Sep 2012\nPosted:\n3 messages\nTue Sep 04, 2012 4:39 am\nThanks a lot Becnil\n\nRegards\n\nai86 Newbie | Next Rank: 10 Posts\nJoined\n03 Jan 2012\nPosted:\n7 messages\nFollowed by:\n1 members\n2\nTue Sep 04, 2012 5:11 am\nHi,\n\nI would like to second Becnil on his \"Team Fuqua\" experience. I think Fuqua does a great job in creating an atmosphere that fosters a teamwork and collaborative environment, inside and outside Fuqua. Looking back in my past year, I am completely thankful for my first year and second year friends (now 2012 alumni) who have helped me through everything, from getting settled into Durham to school and career questions. In regards with the faculty members, they are all open to suggestions and replies to your emails in a matter of hours or so (at least in my experience). The Associate Dean and Career Center take part in our skits (Fuqua Vision) and other student activities.\n\nIn regards with leadership courses, I completely agree with Becnil also that there are a lot of avenues at Fuqua in which you could practice and improve your leadership skills. Some of these are the various Fellows, GATE leaders, student clubs and many others.\n\nCheers,\nai\n\n_________________\nIrene\nMBA, Class of 2013\nThe Fuqua School of Business, Duke University\n\n### Top First Responders*\n\n1 GMATGuruNY 67 first replies\n2 Rich.C@EMPOWERgma... 44 first replies\n3 Brent@GMATPrepNow 40 first replies\n4 Jay@ManhattanReview 25 first replies\n5 Terry@ThePrinceto... 10 first replies\n* Only counts replies to topics started in last 30 days\nSee More Top Beat The GMAT Members\n\n### Most Active Experts\n\n1 GMATGuruNY\n\nThe Princeton Review Teacher\n\n132 posts\n2 Rich.C@EMPOWERgma...\n\nEMPOWERgmat\n\n112 posts\n3 Jeff@TargetTestPrep\n\nTarget Test Prep\n\n95 posts\n4 Scott@TargetTestPrep\n\nTarget Test Prep\n\n92 posts\n5 Max@Math Revolution\n\nMath Revolution\n\n91 posts\nSee More Top Beat The GMAT Experts","date":"2018-04-25 02:33:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17717097699642181, \"perplexity\": 3346.2473525138416}, \"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-2018-17\/segments\/1524125947690.43\/warc\/CC-MAIN-20180425022414-20180425042414-00215.warc.gz\"}"} | null | null |
Biografia
Figlia di Valentino Bompiani, per la cui casa editrice ideò la collana di letteratura fantastica "Pesanervi", ha trascorso diversi anni a Parigi e a Londra per poi trasferirsi a Roma e nei dintorni di Siena. Ha insegnato per una ventina d'anni letteratura inglese all'Università di Siena.
Come traduttrice ha lavorato su opere di Antonin Artaud, Louis-Ferdinand Céline, Gilles Deleuze, Leonora Carrington e Marguerite Yourcenar.
Ha fondato nel 2002, con Roberta Einaudi (nipote di Giulio Einaudi), la casa editrice nottetempo, con sede a Roma.
Opere
Narrativa
Bàrtelemi all'ombra, Milano: Mondadori, 1967 (trad. tedesca di Iris von Kaschnitz, Bartelemi im Schatten, Reinbek bei Hamburg: Rowohlt, 1969).
Piazza pulita, illustrazioni di Maria Enrica Agostinelli, Milano: Bompiani, 1968
Le specie del sonno, Milano: Franco Maria Ricci, 1975; con introduzione di Italo Calvino, Macerata: Quodlibet, 1998 ISBN 88-86570-18-X (trad. francese di Eliane Formentelli, Les Régnes du sommeil, Lagrasse: Verdier, 1990).
Mondanità, Milano: La Tartaruga, 1980 (trad. tedesca Die Schwarzen Lackschuhe, Klett-Cotta, 2001).
L'attesa, Milano: Feltrinelli, 1988 ISBN 88-07-05055-2, Milano, Et al. edizioni, 2011.
L'incantato, Milano: Garzanti, 1987 (finalista Premio letterario nazionale per la donna scrittrice 1988; trad. francese L'Étourdi, Paris: Gallimard, 1989).
Vecchio cielo, nuova terra, Milano: Garzanti, 1988 ISBN 88-11-67121-3 (trad. francese Ciel ancien, terre nouvelle, Paris: Gallimard, 1990; trad. spagnola di Isabel Cervello, Viejo cielo, nueva tierra, Barcelona, Editorial Lumen, 1994).
L'amorosa avventura di una pelliccia e di un'armatura, Palermo: Sellerio, 2000 ISBN 88-389-1573-3.
Ritratto di Sarah Malcolm, Vicenza: Neri Pozza, 2005 ISBN 88-545-0068-2.
La stazione termale, Palermo, Sellerio, 2012 (trad. francese (Jean-Luc Defromont) La station thermale, Paris, Liana Levi, 2012).
Mela Zeta, Milano, Nottetempo, 2016.
L'altra metà di Dio, Milano, Feltrinelli, 2019.
La penultima illusione, Milano, Feltrinelli, 2022.
Saggistica e traduzioni
trad. (con Giorgio Agamben) di Antonin Artaud, Il monaco da Matthew Gregory Lewis, Milano: Bompiani, 1967.
trad. di Louis-Ferdinand Céline, Rigodon, Milano: Bompiani, 1970; Garzanti, 1974.
introduzione e trad. di Emily Brontë, Poesie, Torino: Einaudi, 1971.
trad. di Leonora Carrington, Giù in fondo, Milano: Adelphi, 1979.
introduzione a Virginia Woolf, La signora dell'angolo di fronte, trad. di Masolino D'Amico, Milano: Il Saggiatore, 1979
Lo spazio narrante: Jane Austen, Emily Brontë, Sylvia Plath, Milano: La Tartaruga, 1978 ISBN 88-7738-043-8, Milano, Et al. edizioni, 2012.
cura di Emily Dickinson, Poesie, La Spezia: Melita, 1981.
Perché scrivete?, 1992, in Gianni Celati (a cura di), Narratori delle riserve, Milano: Feltrinelli, 1992, pp. 46–50.
Per Elsa, in Per Elsa Morante. Saggi e testimonianze, Milano: "Linea d'ombra", 1993, pp. 199–203.
Scrivere è una cosa ignobile, l'importante è scribacchiare, in Viola Papetti (a cura di), Le foglie messaggere. Scritti in onore di Giorgio Manganelli, Roma: Editori Riuniti, 2000.
trad. Marguerite Yourcenar, I trentatré nomi di Dio: tentativo di un diario senza data e senza pronome personale, Roma: Nottetempo, 2003.
trad. di Colette, Cheri, Roma: Nottetempo, 2005.
Metamorfosi: prose scelte, tra "fabula" ed "essai", con un saggio di Antonio Prete, Verona: Anterem, 2005 (Premio di poesia Lorenzo Montano).
Note
Bibliografia
Dizionario della letteratura italiana del Novecento, Torino, Einaudi, 1992, ad vocem
Altri progetti
Collegamenti esterni
intervista su Wuz
risposte a un questionario di Nazione Indiana
video intervista a "La Repubblica"
Traduttori dal francese
Traduttori dall'inglese
Professori dell'Università degli Studi di Siena
Scrittori per ragazzi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,947 |
\section{Introduction}
\subsection{Directed polymers}
The {\it directed polymer model} is a stochastic model of a random path that interacts with a random environment. In its simplest formulation on an integer lattice $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$, positive random weights $\{Y_x\}_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ are assigned to the lattice vertices and the quenched probability of a finite lattice path $\pi$ is declared to be proportional to the product $\prod_{x\tspa\in\tspa\pi} Y_x$.
In the usual Boltzmann-Gibbs formulation we take $Y_x=e^{-\beta \omega_x}$ so that the energy of a path is proportional to the potential $\sum_{x\tspa\in\tspa\pi} \omega_x$ and the strength of the coupling between the path $\pi$ and the environment $\omega$ is modulated by the inverse temperature parameter $\beta$.
The directedness of the model means that some spatial direction $\mathbf{u}}\def\vvec{\mathbf{v}\in\R^d$ represents time and the admissible paths $\pi$ are required to be $\mathbf{u}}\def\vvec{\mathbf{v}$-directed. One typical example would be to require that the steps of $\pi$ are of the form $(\pme_i,1)\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ for $i\in\{1,\dotsc,d-1\}$. In this example the time direction is $\mathbf{u}}\def\vvec{\mathbf{v}=e_d$, space is the $(d-1)$-dimensional lattice $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^{d-1}$, and $\pi$ is a simple random walk path in space-time. Another common choice is to restrict the steps of $\pi$ to directed basis vectors $\{e_i\}_{1\le i\le d}$ so that time proceeds in the diagonal direction $\mathbf{u}}\def\vvec{\mathbf{v}=e_1+\dotsm+e_d$.
This model was introduced in the statistical
physics literature by Huse and Henley in 1985 \cite{huse-henl} as a model of the domain wall in an Ising model with impurities. Since the polymer can be viewed as a perturbation of a simple random walk, a natural question to investigate is whether the walk is diffusive across large scales. The early rigorous mathematical work by Imbrie and Spencer
\cite{imbr-spen} and Bolthausen \cite{bolt-cmp-89} in the late 1980s established that in dimensions $d\ge 4$ (one time dimension plus at least three spatial dimensions) the path behaves diffusively for small enough $\beta$. This behavior is now known as {\it weak disorder}. Later work \cite{come-varg-06, laco-10} established that in lower dimensions $d\in\{2,3\}$ or if $\beta$ is large enough, the polymer model exhibits {\it strong disorder}, characterized by localization. Excellent reviews of this development can be found in \cite{come-16, denholl-polymer}.
Since the early interest in the phase transition between weak and strong disorder, the study of directed polymers has branched out in several directions. The discovery of exactly solvable 1+1 dimensional models, the first of which were the O'Connell-Yor Brownian directed polymer \cite{oconn-yor-01} and the inverse-gamma, or log-gamma, polymer \cite{sepp-12-aop-corr}, led to rigorous proofs that directed polymers are members of the Kardar-Parisi-Zhang (KPZ) universality class \cite{boro-corw-ferr-14, boro-corw-reme,sepp-valk-10}. This had been expected since directed polymers are positive temperature analogues of directed last-passage percolation, for which predictions of KPZ universality were first rigorously verified \cite{baik-deif-joha-99, joha}. On KPZ we refer the reader to the recent reviews \cite{corw-16-rev, corw-18-ams, quas-14, quas-spoh-15}.
Through Feynman-Kac-type representations, directed polymers provide solutions to stochastic partial differential equations. Early work in this direction by Kifer \cite{kife-97} connected a polymer in the weak disorder regime with a stochastic Burgers equation. The significant current example of this, which also takes us back to the study of KPZ universality, is the connection between the continuum directed random polymer and the stochastic heat equation with multiplicative noise, whose logarithm is the Hopf-Cole solution of the KPZ equation. We refer to Corwin's review \cite{corw-12-rev}.
\subsection{Infinite polymers}
Another natural direction of polymer research is the limit as the path length is taken to infinity. This limit can be readily taken in weak disorder. This can be found in the work of Comets and Yoshida \cite{come-yosh-aop-06}.
In strong disorder the existence of limiting infinite quenched polymer measures was first proved in 1+1 dimensions for the inverse-gamma polymer in \cite{geor-rass-sepp-yilm-15}.
The limiting quenched probability distributions on infinite-length polymer paths can be naturally described as the Gibbs measures whose finite-dimensional conditional distributions are given by the quenched point-to-point polymer distributions $Q_{x,y}(\pi)=Z_{x,y}^{-1} \prod_{x\tspa\in\tspa\pi} Y_x$. Here $\pi$ is a path between points $x$ and $y$ and the partition function $Z_{x,y}=\sum_\pi \prod_{x\tspa\in\tspa\pi} Y_x$ normalizes $Q_{x,y}$ to be a probability distribution on the paths between $x$ and $y$. (This notion is developed precisely in Section \ref{sec:cif}.)
This Gibbsian point of view arose prominently in the work of Bakhtin and Li \cite{bakh-li-19} who studied a 1+1 dimensional model with a Gaussian random walk. They used polymer Gibbs measures to construct global solutions to a stochastic Burgers equation on the line, subject to random kick forcing at discrete time intervals. Their sequel \cite{bakh-li-18} showed that as the temperature is taken to zero, the Gibbs measures concentrate around the geodesic of the corresponding directed percolation model.
Janjigian and Rassoul-Agha \cite{janj-rass-20-aop} developed aspects of a general theory of polymer Gibbs measures for i.i.d.\ vertex weights and directed nearest-neighbor paths on the discrete planar square lattice $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$. We work in their setting, with a specialized choice of weight distribution.
\subsection{Bi-infinite polymers}
The work cited above
addressed the existence and uniqueness of {\it semi-infinite} Gibbs measures.
These are measures on semi-infinite, or one-sided infinite, paths, with fixed initial point. The existence of {\it bi-infinite} Gibbs measures was left open. These would be measures on bi-infinite paths that satisfy the Gibbs property.
Bi-infinite polymer Gibbs measures would be special cases of the general theory of Gibbs measures as developed in Georgii's monograph \cite{geor}. The polymer specification is a Markovian one because the distribution $Q_{x,y}$ on paths from $x$ to $y$ depends only on the boundary points $x$ and $y$. However, this specification is not shift-invariant and hence the general theory of Chapters 10-11 of \cite{geor} is not helpful here.
In this paper we assume that the i.i.d.\ vertex weights $\{Y_x\}_{x\tsp\in\tsp\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ on the planar lattice $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ have inverse-gamma distribution. Then we prove that, for almost every choice of weights, nontrivial bi-infinite Gibbs measures do not exist. Trivial bi-infinite Gibbs measures do exist, by which we mean ones that are supported on bi-infinite straight lines.
The key tools of the nonexistence proof are the following.
\begin{enumerate} [{(i)}] \itemsep=3pt
\item Planar comparison inequalities, reviewed and proved in Appendix \ref{app:genpol}.
\item
KPZ wandering exponent $2/3$ of the polymer path, quoted in Appendix \ref{sec:kpz5} from \cite{sepp-12-aop-corr}.
\item A jointly stationary bivariate inverse-gamma polymer from the forthcoming work \cite{fan-sepp-20+} of the second author and W.~L.~Fan, developed in full detail in Appendix \ref{sec:stat-pol}.
\end{enumerate}
From these ingredients and coupling arguments we derive a bound on the speed of decay of the probability that a polymer path from far away in the southwest to far away in the northeast goes through the origin. This bound is given in Theorem \ref{thm:ub} at the end of Section \ref{sec:estim}. The KPZ fluctuation bounds on polymer paths enable us to deduce this result from local point-to-point estimates and a coarse-graining step.
Item (iii) above is the joint distribution of two {\it Busemann functions} of the polymer process. We do not use the Busemann functions themselves in this paper and hence do not develop them. We refer the reader to \cite{bakh-li-19,geor-rass-sepp-yilm-15, janj-rass-20-aop}.
A methodological point to emphasize is that our proof does not rely on any integrable probability features of the inverse-gamma polymer, such as those developed in \cite{boro-corw-reme, corw-ocon-sepp-zygo}. The KPZ fluctuation estimates of Appendix \ref{sec:kpz5} were proved in \cite{sepp-12-aop-corr} with techniques that are the same in spirit as the arguments in the present paper.
It is reasonable to expect that non-existence of bi-infinite Gibbs measures extends to general weight distributions, since the present proof boils down to path fluctuations which are expected to be universal in 1+1 dimensions under mild hypotheses. However, currently available techniques do not appear to yield sufficiently sharp estimates to prove this result in general polymer models. Specifically, items (ii) and (iii) from the list above force us to work with an exactly solvable model.
The zero-temperature counterpart of our result is the non-existence of bi-infinite geodesics in first-passage or last-passage percolation models. This has been proved for the planar exponential directed last-passage percolation model \cite{basu-hoff-sly-arxiv-18, bala-busa-sepp-arxiv}. The organization of our estimates mimics our zero-temperature proof in \cite{bala-busa-sepp-arxiv}.
\subsection{Organization of the paper} Section \ref{sec:cif} develops enough of the general polymer theory from \cite{janj-rass-20-aop} so that in Section \ref{sec:inv-ga-thm} we can state the main result Theorem \ref{thm:noex} on the nonexistence of bi-infinite inverse-gamma polymer Gibbs measures. Along the way we apply results from \cite{janj-rass-20-aop} to prove for general weights that infinite polymers have to be directed into the open quadrant, unless they are rigid straight lines (Theorem \ref{thm:e_i-mu}). This result will also contribute to the proof of the main Theorem \ref{thm:noex}.
Section \ref{sec:invga} gives a quick description of the ratio-stationary inverse-gamma polymer and derives one estimate.
The heart of the proof is in Section \ref{sec:estim}. A coarse-graining argument decomposes the southwest boundary of a large $2N\times 2N$ square into blocks of size $N^{2/3}$. Two separate estimates are developed.
\begin{enumerate}[(a)]
\item The first kind is for the probability that a polymer path from an $N^{2/3}$-block denoted by $\mathcal{I}$ goes through the origin and reaches the diagonally opposite block $\rim\mathcal{I}$ of size $N^{19/24}$.
This probability is shown to decay by controlling it with random walks that come from the ratio-stationary polymer processes (Lemma \ref{lm:close}).
\item The second estimate (Lemma \ref{lm:far}) controls the paths from $\mathcal{I}$ through the origin that miss $\rim\mathcal{I}$. Such paths are rare due to KPZ bounds according to which the typical path remains within a range of order $N^{2/3}$ around the straight line between its endpoints.
\end{enumerate}
Section \ref{sec:estim} culminates in Theorem \ref{thm:ub} that combines the estimates.
Section \ref{sec:pf-main} combines Theorem \ref{thm:ub} with the earlier Theorem \ref{thm:e_i-mu} to complete the proof of Theorem \ref{thm:noex}. The estimates for paths that go through the origin are generalized to other crossing points on the $y$-axis by suitably shifting the environment.
Since the background polymer material will be at least partly familiar to some readers, we have collected these facts in the appendix. Appendix \ref{app:genpol} covers polymers on $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ with general vertex weights and Appendix \ref{app:inv-gam} specializes to inverse-gamma weights. Appendix \ref{sec:rw} states a positive lower bound on the running maximum of a random walk with a small negative drift that we use in a proof. This result is quoted from the technical note \cite{busa-sepp-rw} that we have published separately.
\subsection{Notation and conventions} Subsets of reals and integers are denoted by subscripts, as in
$\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0}=\{1,2,3,\dotsc\}$ and $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0}^2=(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})^2$.
$\lzb a,b\rzb$ denotes the integer interval $[a,b]\cap\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN$ if $a,b\in\R$, and the integer rectangle $([a_1,b_1]\times[a_2,b_2])\cap\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ if $a,b\in\R^2$.
For points $x=(x_1,x_2)$ and $y=(y_1,y_2)$ in $\R^2$, the $\ell^1$ norm is $\abs{x}_1=\abs{x_1}+\abs{x_2}$, the inner product is $x\cdot y=x_1y_1+x_2y_2$, the origin is $\zevec=(0,0)$, and the standard basis vectors are $e_1=(1,0)$ and $e_2=(0,1)$.
We utilize two partial orders: \\[-15pt]
\begin{enumerate} [(i)] \itemsep=2pt
\item the {\it coordinatewise order}: $(x_1,x_2)\le(y_1,y_2)$ if $x_r\le y_r$ for $r\in\{1,2\}$, and
\item the {\it down-right order}: $(x_1,x_2)\preccurlyeq(y_1,y_2)$ if $x_1\le y_1$ and $x_2\ge y_2$.
\end{enumerate}
Their strict versions mean that the defining inequalities are strict: $(x_1,x_2)<(y_1,y_2)$ if $x_r<y_r$ for $r\in\{1,2\}$, and $(x_1,x_2)\prec(y_1,y_2)$ if $x_1< y_1$ and $x_2> y_2$.
Sequences are denoted by $x_{m:n}=(x_i)_{i=m}^n$ and $x_{m:\infty}=(x_i)_{i=m}^\infty$ for integers $m\le n<\infty$ and also generically by $x_\bbullet$. An admissible path $x_\bbullet$ in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ satisfies $x_k-x_{k-1}\in\{e_1,e_2\}$.
Limit velocities of these paths lie in the simplex
$[e_2,e_1]=\{(u,1-u): u\in[0,1]\}$, whose relative interior is the open line segment $\,]e_2,e_1[\,$.
$\bE$ and $\P$ refer to the random weights (the environment) $\omega$, and otherwise
$E^\mu$ denotes expectation under probability measure $\mu$.
The usual gamma function for $\rho>0$ is
$\Gamma(\rho)=\text{int}_0^\infty x^{\rho-1}e^{-x}\,dx$, and the digamma and trigamma
functions are $\psi_0=\Gamma'/\Gamma$ and $\psi_1=\psi_0'$.
$X\sim{\rm Ga}(\rho)$ if the random variable $X$ has the density function $f(x)=\Gamma(\rho)^{-1} x^{\rho-1}e^{-x}\,dx$ on $\R_{>0}$, and $X\sim{\rm Ga}^{-1}(\rho)$ if $X^{-1}\sim{\rm Ga}(\rho)$.
\section{Polymer Gibbs measures}
\label{sec:cif}
\subsection{Directed polymers}
Let $(\wgtd_x)_{x\tspa\in\tspa\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ be an assignment of strictly positive real weights on the vertices of $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$.
For vertices $o\le p$ in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ let $\pathsp_{o,p}$ denote the set of admissible
lattice paths $x_\bbullet=(x_i)_{0\le i\le n}$ with $n=\abs{p-o}_1$ that satisfy $x_0=o$,
$x_i-x_{i-1}\in\{e_1,e_2\}$, $x_n=p$.
Define point-to-point
polymer partition functions
between vertices $o\le p$ in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ by
\begin{align}\label{h:Z}
Z_{o,p}=\sum_{x_\brbullet\tspa\in\tspa\pathsp_{o,p}} \prod_{i=0}^{\abs{p-o}_1} \wgtd_{x_i} .
\end{align}
We use the convention $Z_{o,p}=0$ of $o\le p$ fails.
The quenched polymer probability distribution on the set $\pathsp_{o,p}$ is defined by
\begin{equation}\label{h:Q}
Q_{o,p}\{x_\bbullet\} =\frac1{Z_{o,p}} \prod_{i=0}^{\abs{p-o}_1} \wgtd_{x_i} ,
\quad x_\bbullet\in\pathsp_{o,p}.
\end{equation}
When the weights $\omega=(Y_x)$ are random variables on some probability space $\OAbP$, the averaged or annealed polymer distribution $P_{o,p}$ on $\pathsp_{o,p}$ is defined by
\begin{equation}\label{h:P}
P_{o,p}(A)=\text{int}_\Omega \sum_{x_\brbullet\tspa\in\tspa A} Q^\omega_{o,p}(x_\bbullet)\,\P(d\omega)
\qquad\text{for } A\subset\pathsp_{o,p}.
\end{equation}
The notation $Q^\omega_{o,p}$ highlights the dependence of the quenched measure on the weights.
It is also convenient to use the unnormalized quenched polymer measure, which is simply the sum of path weights:
\begin{equation}\label{h:Z(A)} Z_{o,p}(A)= \sum_{x_\brbullet\tspa\in\tspa A} \prod_{i=0}^{\abs{p-o}_1} \wgtd_{x_i}
= Z_{o,p} \hspace{0.9pt} Q_{o,p}(A) \qquad\text{for } A\subset\pathsp_{o,p}. \end{equation}
A basic law of large numbers object of this model is the limiting {\it free energy density}. Assume now the following:
\begin{equation}\label{mom-ass}
\text{the weights $(Y_x)_{x\tsp\in\tsp\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ are i.i.d.\ random variables and } \
\bE[\tspa\abs{\log Y_0}^p\tspa]<\infty \ \text{ for some } \ p>2.
\end{equation}
Then there exists a concave, positively homogeneous, nonrandom continuous function $\Lambda:\R_{\ge0}^2\to\R$ that satisfies this {\it shape theorem}:
\begin{equation}\label{lln}
\lim_{n\to\infty} \sup_{x\tsp\in\tsp\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{\ge0}^2: \hspace{0.9pt} \abs{x}_1\ge n}
\frac{\log Z_{\zevec,x} - \Lambda(x)}{\abs{x}_1} =0
\qquad \P\text{-almost surely.}
\end{equation}
(See Section 2.3 in \cite{janj-rass-20-aop}.) In general, further regularity of $\Lambda$ is unknown. In certain exactly solvable cases, including the inverse-gamma polymer we study in this paper, the following properties are known:
\begin{equation}\label{reg-ass}
\text{the function $\Lambda$
is differentiable and strictly concave on the open interval $\,]e_2,e_1[\,$.}
\end{equation}
Fix the base point $o=\zevec$ (the origin) and consider sending the endpoint $p$ to infinity in the quenched measure $Q_{\zevec,p}$. Fix a finite path $x_{0:n}\in\pathsp_{\zevec,y}$ where $\zevec\le y\le p$ and $n=y\cdot(e_1+e_2)$. To understand what happens as $\abs p_1\to\infty$ it is convenient to write $Q_{\zevec,p}$ as a Markov chain:
\begin{equation}\label{h:Q5}
Q_{\zevec,p}\{X_{0:n}=x_{0:n}\} =\frac1{Z_{\zevec,p}} \biggl(\; \prod_{i=0}^{n-1} \wgtd_{x_i}\biggr) Z_{x_n,p} = \prod_{i=0}^{n-1} \frac{Z_{x_{i+1},p}\wgtd_{x_i}}{Z_{x_i,p}}
\end{equation}
with initial state $X_0=\zevec$, transition probability $\pi^{\zevec,p}(x,x+e_i)=Z_{x,p}^{-1}\tsp {Z_{x+e_i,p}\wgtd_{x}}$ for $p\ne x\in\lzb\zevec,p\rzb$, and absorbing state $p$. The formulation above reveals that when the limit of the ratio $Z_{x+e_i,p}/Z_{x,p}$ exists for each fixed $x$ as $p$ tends to infinity, then $Q_{\zevec,p}$ converges weakly to a Markov chain. When $p$ recedes in some particular direction, this can be proved under local hypotheses on the regularity of $\Lambda$. See Theorem 3.8 of \cite{janj-rass-20-aop} for a general result and Theorem 7.1 in \cite{geor-rass-sepp-yilm-15} for the inverse-gamma polymer.
The limiting Markov chains are examples of rooted semi-infinite polymer Gibbs measures, which we discuss in the next section.
\subsection{Infinite Gibbs measures}
In this section we adopt mostly the terminology and notation of \cite{janj-rass-20-aop}.
To describe semi-infinite and bi-infinite polymer Gibbs measures, introduce the spaces of semi-infinite and bi-infinite polymer paths in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$:
\begin{align*}
\pathsp_u&=\{x_{m:\infty}:x_m=u,\, x_i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2,\, x_i-x_{i-1}\in\{e_1, e_2\}\}\\
\text{and}\qquad \pathsp&=\{x_{-\infty:\infty}: x_i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2,\, x_i-x_{i-1}\in\{e_1, e_2\}\}.
\end{align*}
$\pathsp_u$ is the space of paths rooted or based at the vertex $u\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$. The indexing of the paths is immaterial. However, it adds clarity to index unbounded paths so that $x_k\cdot(e_1+e_2)=k$, as done in \cite{janj-rass-20-aop}. We follow this convention in the present section. So in the definition of $\pathsp_u$ above take $m=u\cdot(e_1+e_2)$. The projection random variables on all the path spaces are denoted by $X_i(x_{m:n}) =x_i$ for all choices $-\infty\le m\le n\le \infty$ and $i$ in the correct range.
Fix $\omega\in\Omega$ and $m\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN$. Define a family of stochastic kernels $\{\kappa^\omega_{k,l}: l\ge k\ge m\}$ on semi-infinite paths $x_{m:\infty}$ through the integral of a bounded Borel function $f$:
\begin{equation}\label{kernel}\begin{aligned}
\kappa^\omega_{k,l}f(x_{m:\infty})&=\text{int} f(y_{m,\infty})\,\kappa^\omega_{k,l}(x_{m:\infty},dy_{m,\infty})\\
&=\sum_{y_{k:l}\in\pathsp_{x_k, x_l}} f(x_{m:k}\,y_{k:l}\, x_{l:\infty})\,Q^\omega_{x_k,x_l}(y_{k:l}).
\end{aligned}\end{equation}
In other words, the action of $\kappa^\omega_{k,l}$ amounts to replacing the segment $x_{k:l}$ of the path with a new path $y_{k:l}$ sampled from the quenched polymer distribution $Q^\omega_{x_k,x_l}$. The argument $x_{m:k}\,y_{k:l}\, x_{l:\infty}$ inside $f$ is the concatenation of the three path segments. There is no inconsistency because $y_k=x_k$ and $y_l=x_l$ $Q^\omega_{x_k,x_l}$-almost surely. The key point is that the measure $\kappa^\omega_{k,l}(x_{m:\infty})$ is a function of the subpaths $(x_{m:k},x_{l:\infty})$.
Note that the same kernel $\kappa^\omega_{k,l}$ works on paths $x_{m:\infty}$ for any $m\le k$ and also on the space $\pathsp$ of bi-infinite paths by replacing $m$ with $-\infty$ in the expressions above. With these kernels one defines semi-infinite and bi-infinite polymer Gibbs measures. Let $\mathcal{F}_I=\sigma\{X_i:i\in I\}$ denote the $\sigma$-algebra generated by the projection variables indexed by the subset $I$ of indices.
\begin{definition}
Fix $\omega\in\Omega$ and $u\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ and let $m=u\cdot(e_1+e_2)$. Then a Borel probability measure $\nu$ on $\pathsp_u$ is a {\it semi-infinite polymer Gibbs measure rooted at $u$ in environment $\omega$} if for all integers $l\geq k\geq m$ and any bounded Borel function $f$ on $\pathsp_u$ we have $E^\nu[ \tsp f \tspa\vert\tspa \mathcal{F}_{\lzb m,k\rzb\cup\lzb l,\infty\lzb}\tspa]=\kappa^\omega_{k,l}f$. This set of probability measures is denoted by $\text{DLR}^\omega_u$.
\end{definition}
\begin{definition}
Fix $\omega\in\Omega$. Then a Borel probability measure $\mu$ on $\pathsp$ is a {\it bi-infinite Gibbs measure in environment $\omega$} if for all integers $k\leq l$ and any bounded Borel function $f$ on $\pathsp$ we have $E^\mu[ \tspa f \tspa\vert\tspa \mathcal{F}_{\rzb -\infty,k\rzb\cup\lzb l,\infty\lzb}\tspa]=\kappa^\omega_{k,l}f$. This set of probability measures is denoted by $\overleftrightarrow{\text{\rm DLR}}^\omega$.
\end{definition}
An equivalent way to state $\mu\in\overleftrightarrow{\text{\rm DLR}}^\omega$ is to require
\[ \text{int}_{\pathsp} f(X_{-\infty:k}) \hspace{0.9pt} g(X_{k:l}) \hspace{0.9pt} h(X_{l:\infty}) \,d\mu =
\text{int}_{\pathsp} f(X_{-\infty:k}) \hspace{0.9pt} (\kappa^\omega_{k,l}g)(X_{-\infty:k}, X_{l:\infty}) \hspace{0.9pt} h(X_{l:\infty}) \,d\mu \]
for all bounded Borel functions on the appropriate path spaces. For $\mu\in\text{DLR}^\omega_u$ the requirement is the same with $\pathsp$ replaced by $\pathsp_u$ and with $-\infty$ replaced by $m$.
The issue addressed in our paper is the nonexistence of nontrivial bi-infinite Gibbs measures. For the sake of context, we state an existence theorem for semi-infinite Gibbs measures.
\begin{theorem}{\rm\cite[Theorem 3.2]{janj-rass-20-aop}}
Assume \eqref{mom-ass} and \eqref{reg-ass}. Then there exists an event $\Omega_0$ such that $\P(\Omega_0)=1$ and for every $\omega\in \Omega_0$ the following holds. For each $u\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ and interior direction $\xi\in\,]e_2,e_1[\,$ there exists a Gibbs measure $\Pi^{\omega, \xi}_u\in\text{\rm DLR}^\omega_u$ such that $X_n/n\to\xi$ almost surely under $\Pi^{\omega, \xi}_u$. Futhermore, these measures can be chosen to satisfy this consistency property: if $u\cdot(e_1+e_2)\le y\cdot(e_1+e_2)=n\le z\cdot(e_1+e_2)=r$, then for any path $x_{n:r}\in\pathsp_{y,z}$,
\[ \Pi^{\omega, \xi}_u(X_{n:r}=x_{n:r}\,\vert\,X_n=y) = \Pi^{\omega, \xi}_y(X_{n:r}=x_{n:r}). \]
\end{theorem}
Uniqueness of Gibbs measures is a more subtle topic, and we refer the reader to \cite{janj-rass-20-aop}.
Since the Gibbs measure $\Pi^{\omega, \xi}_u$ satisfies the strong law of large numbers $X_n/n\to\xi$, we can call it (strongly) {\it $\xi$-directed}. In general, a path $x_{m:\infty}$ is $\xi$-directed if $x_n/n\to\xi$ as $n\to\infty$.
We turn to
bi-infinite Gibbs measures. First we observe that there are trivial bi-infinite Gibbs measures supported on straight line paths.
\begin{definition}
A path $x_\brbullet$
is a {\it straight line} if for a fixed $i\in\{1,2\}$, $x_{n+1}-x_n=e_i$ for all path indices $n$.
\end{definition}
If $x_{\brbullet}$ is a bi-infinite straight line then $\mu=\delta_{x_{\brbullet}}$ is a bi-infinite Gibbs measure because the polymer distribution $Q_{u, u+me_i}$ is supported on the straight line from $u$ to $u+me_i$. More generally, any probability measure supported on bi-infinite straight lines is a bi-infinite Gibbs measure.
The next natural question is whether there can be bi-infinite polymer paths that are not merely straight lines but still directed into $e_i$. That this option can be ruled out is essentially contained in the results of \cite{janj-rass-20-aop}. We make this explicit in the next theorem. It says that under both semi-infinite and bi-infinite Gibbs measures, up to a zero probability event, $e_i$-directedness even along a subsequence is possible only for straight line paths. Note that \eqref{dlr:70} covers both $e_i$- and $(-e_i)$-directedness.
\begin{theorem} \label{thm:e_i-mu} Assume \eqref{mom-ass}. There exists an event $\Omega_0\subseteq \Omega$ such that $\P(\Omega_0)=1$ and for every $\omega\in \Omega$ the following statements hold for both $i\in\{1,2\}$: \smallskip
{\rm (a)} For all $u\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ and $\nu\in{\text{\rm DLR}}^\omega_u$, with $m=u\cdot(e_1+e_2)$,
\begin{equation}\label{dlr:65} \nu\bigl\{ \tsp\varliminf_{n\to\infty} n^{-1}\abs{X_n\cdote_{3-i}} = 0 \bigr\} = \nu\{ X_n=u+(n-m)e_i \text{ for } n\ge m \}.
\end{equation}
{\rm (b)}
For all $\mu\in\overleftrightarrow{\text{\rm DLR}}^\omega$,
\begin{equation}\label{dlr:70} \mu\bigl\{ \tsp\varliminf_{\abs n\to\infty} \abs{n^{-1} X_n\cdote_{3-i}} = 0 \bigr\} = \mu\{ \text{$X_{-\infty:\infty}$ is an $e_i$-directed bi-infinite straight line} \}.
\end{equation}
\end{theorem}
\begin{proof}
Let the event $\Omega_0$ of full $\P$-probability be the intersection of the events specified in Lemma 3.4 and Theorem 3.5 of \cite{janj-rass-20-aop}.
\medskip
Part (a). We can assume that
the left-hand side of \eqref{dlr:65} is positive because the event on the right is a subset of the one on the left. Since $A=\{ \varliminf_{n\to\infty} n^{-1}\abs{X_n\cdote_{3-i}} = 0\}$ is a tail event, it follows that $\widetilde} \def\wh{\widehat} \def\wb{\overline\nu=\nu(\cdot\hspace{0.9pt}\vert\,A)\in \text{DLR}^{\omega}_u$. Since $\pathsp_u$ is compact, $\widetilde} \def\wh{\widehat} \def\wb{\overline\nu$ is a mixture of extreme members of $\text{DLR}^{\omega}_u$. (This is an application of Choquet's theorem, discussed more thoroughly in Section 2.4 of \cite{janj-rass-20-aop}.) This mixture can be restricted to extreme Gibbs measures that give the event $A$ full probability.
By Lemma 3.4 and Theorem 3.5 of \cite{janj-rass-20-aop}, an extreme member of $\text{DLR}^{\omega}_u$ that is not directed into the open interval $]e_2,e_1[$ must be a degenerate point measure
$\Pi^{e_i}_u$, which is the probability measure supported on the single straight line path $(u+(n-m)e_i)_{n:n\ge m}$. We conclude that $\widetilde} \def\wh{\widehat} \def\wb{\overline\nu=\Pi^{e_i}_u$.
From this we deduce \eqref{dlr:65}. Let $B^{e_i}_u=\{ X_n=u+(n-m)e_i \text{ for } n\ge m\} $ be the event that from $u$ onwards the path is an $e_i$-directed line. Then by conditioning,
\begin{align*}
\nu(B^{e_i}_u) = \nu(B^{e_i}_u\cap A) = \widetilde} \def\wh{\widehat} \def\wb{\overline\nu(B^{e_i}_u) \tspa\nu(A) =\nu(A).
\end{align*}
\medskip
Part (b). Consider first the case $n\to\infty$. Let $m\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN$ and $x\cdot(e_1+e_2)=m$.
Suppose $\mu(X_m=x)>0$. Then, by Lemma 2.4 in \cite{janj-rass-20-aop},
$\mu_x=\mu(\cdot\hspace{0.9pt}\vert\,X_m=x) \in \text{DLR}^{\omega}_x$.
Part (a) applied to $\mu_x$ shows that
\begin{equation}\label{dlr:72} \mu\{ X_m=x,\, \,\varliminf_{n\to\infty} n^{-1}\abs{X_n\cdote_{3-i}} = 0 \} = \mu\{ X_n=x+(n-m)e_i \text{ for } n\ge m\}.
\end{equation}
By summing over the pairwise disjoint events $\{X_m=x\}$ gives, for each fixed $m\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN$,
\[ \mu\{ \tsp\varliminf_{n\to\infty} n^{-1}\abs{X_n\cdote_{3-i}} = 0 \tsp\} = \mu\{ X_n=X_m+(n-m)e_i \text{ for } n\ge m\}. \]
The events on the right decrease as $m\to-\infty$, and in the limit we get
\[ \mu\{ \tsp\varliminf_{n\to\infty} n^{-1}\abs{X_n\cdote_{3-i}} = 0\tsp \} = \mu\{ X_n=X_m+(n-m)e_i \text{ for all } n, m\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN\} \]
which is exactly the claim \eqref{dlr:70} the case $n\to\infty$.
The case $n\to-\infty$ of \eqref{dlr:70} follows by reflection across the origin. Let $\omega=(Y_x)_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ and define reflected weights $\widetilde} \def\wh{\widehat} \def\wb{\overline\omega=(\widetilde} \def\wh{\widehat} \def\wb{\overline Y_x)_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ by $\widetilde} \def\wh{\widehat} \def\wb{\overline Y_x=Y_{-x}$. Given $\mu\in\overleftrightarrow{\text{\rm DLR}}^\omega$, define the reflected measure $\widetilde} \def\wh{\widehat} \def\wb{\overline\mu$ by setting, for $m\le n$ and $x_{m:n}\in\pathsp_{x_m,x_n}$,
$\widetilde} \def\wh{\widehat} \def\wb{\overline\mu(X_{m:n}=x_{m:n}) =\mu(X_i=-x_{-i}\ \text{ for } i=-n,\dotsc,-m)$.
Then $\widetilde} \def\wh{\widehat} \def\wb{\overline\mu\in\overleftrightarrow{\text{\rm DLR}}^{\widetilde} \def\wh{\widehat} \def\wb{\overline\omega}$.
Directedness towards $-e_i$ under $\mu$ is now directedness towards $e_i$ under $\widetilde} \def\wh{\widehat} \def\wb{\overline\mu$, and we get the conclusion by applying the already proved part to $\widetilde} \def\wh{\widehat} \def\wb{\overline\mu$.
\end{proof}
Moving away from the $e_i$-directed cases, the non-existence problem was resolved by Janjigian and Rassoul-Agha in the case of Gibbs measures directed towards a fixed interior direction:
\begin{theorem} {\rm\cite[Thm.~3.13]{janj-rass-20-aop}} \label{thm:jra6}
Assume \eqref{mom-ass} and \eqref{reg-ass}. Fix $\xi\in\,]e_2,e_1[\,$. Then there exists an event $\Omega_{\text{\rm bi},\xi}\subseteq\Omega$ such that $\P(\Omega_{\text{\rm bi},\xi})=1$ and for every $\omega\in \Omega_{\text{\rm bi},\xi}$ there exists no measure $\mu\in\overleftrightarrow{\text{\rm DLR}}^\omega$ such that as $n\to\infty$, $X_n/n\to\xi$ in probability under $\mu$.
\end{theorem}
We assumed \eqref{reg-ass} above to avoid introducing technicalities not needed in the rest of the paper.
The global regularity assumption \eqref{reg-ass} can be weakened to local hypotheses, as done in
Theorem 3.13 in \cite{janj-rass-20-aop}.
The results above illustrate how far one can presently go without stronger assumptions on the model. The hard question left open is whether bi-infinite Gibbs measures can exist in random directions in the open interval $\,]e_2, e_1[\,$. To rule these out we restrict our treatment to the exactly solvable case of inverse-gamma distributed weights.
That only directed Gibbs measures would need to be considered in the sequel is a consequence of Corollary 3.6 of \cite{janj-rass-20-aop}. However, we do not need to assume this directedness a priori and we do not use Theorem \ref{thm:jra6}. At the end we will appeal to Theorem \ref{thm:e_i-mu} to rule out the extreme slopes. As stated above, Theorem \ref{thm:e_i-mu} does not seem to involve the regularity of $\Lambda$. But in fact through appeal to Theorem 3.5 of \cite{janj-rass-20-aop}, it does rely on the nontrivial (but provable) feature that $\Lambda$ is not affine on any interval of the type $\,]\zeta,e_1]$ (and symmetrically on $[e_2, \eta[\,$). This is the positive temperature counterpart of Martin's shape asymptotic on the boundary \cite{mart-04} and can be deduced from that (Lemma B.1 in \cite{janj-rass-20-aop}).
\subsection{Bi-infinite Gibbs measures in the inverse-gamma polymer}
\label{sec:inv-ga-thm}
A random variable $X$ has the {\it inverse gamma distribution} with parameter $\theta>0$, abbreviated $X\sim\text{\rm Ga}^{-1}(\theta)$, if its reciprocal $X^{-1}$ has the standard gamma distribution with parameter $\theta$, abbreviated $X^{-1}\sim\text{\rm Ga}(\theta)$. Their density functions for $x>0$ are
\begin{equation}\label{invga} \begin{aligned}
f_{X^{-1}}(x)&= \frac1{\Gamma(\theta)} \tsp x^{\theta-1} e^{-x}
\qquad\text{for the gamma distribution ${\rm Ga}(\theta)$} \\
\text{ and }\quad
f_X(x)&=\frac1{\Gamma(\theta)} \tsp x^{-1-\theta} e^{-x^{-1}}
\qquad\text{for the inverse gamma distribution ${\rm Ga}^{-1}(\theta)$.}
\end{aligned} \end{equation}
Here $\Gamma(\theta)=\text{int}_0^\infty s^{\theta-1} e^{-s}\,ds$ is the gamma function.
Our basic assumption is:
\begin{equation}\label{ass-invga} \begin{aligned}
&\text{The weights $(Y_x)_{x\tspa\in\tspa\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ are i.i.d.\ inverse-gamma distributed random variables} \\
&\text{on some probability space $\OAbP$.}
\end{aligned} \end{equation}
The main result is stated as follows.
\begin{theorem}\label{thm:noex}
Assume \eqref{ass-invga}. Then for $\P$-almost every $\omega$, every bi-infinite Gibbs measure
is supported on straight lines: that is, $\mu\in\overleftrightarrow{\text{\rm DLR}}^\omega$ implies that $\mu(\text{$X_{-\infty:\infty}$ is a bi-infinite straight line})=1$.
\end{theorem}
Due to Theorem \ref{thm:e_i-mu}(b), to prove Theorem \ref{thm:noex} we only need to rule out the possibility of bi-infinite polymer measures that are directed towards the open segments $\,]-e_2, -e_1[\,$ and $\,]e_2, e_1[\,$. The detailed proof is given in Section \ref{sec:pf-main}, after the development of preliminary estimates. For the proof we take $Y_x$ to be a Ga$^{-1}(1)$ variable.
For the interested reader, we mention that
the semi-infinite Gibbs measures of the inverse-gamma polymer are described in the forthcoming work \cite{fan-sepp-20+}. Earlier results appeared in \cite{geor-rass-sepp-yilm-15} where such measures were obtained as almost sure weak limits of quenched point-to-point and point-to-line polymer distributions.
\section{Stationary inverse-gamma polymer}
\label{sec:invga}
The proof of Theorem \ref{thm:noex} relies on the fact that the inverse-gamma polymer possesses a stationary version with accessible distributional properties, first constructed in \cite{sepp-12-aop-corr}.
This section gives a brief description of the stationary polymer and proves an estimate. Further properties of the stationary polymer are developed in the appendixes.
Let $(Y_x)_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ be i.i.d.\ Ga$^{-1}(1)$ weights.
A stationary version of the inverse-gamma polymer is defined in a quadrant by choosing suitable boundary weights
on the south and west boundaries of the quadrant. For a parameter $0<\alpha<1$ and a base vertex $o$, introduce independent boundary weights on the $x$- and $y$-axes emanating from $o$:
\begin{equation}\label{IJ1} I^\alpha_{o+ie_1}\sim{\rm Ga}^{-1}(1-\alpha)
\qquad\text{and}\qquad
J^\alpha_{o+je_2}\sim{\rm Ga}^{-1}(\alpha)
\qquad\text{for } \ i,j\ge 1.
\end{equation}
The above convention, that the horizontal edge weight $I^\alpha$ has parameter $1-\alpha$ while the vertical $J^\alpha$ has $\alpha$, is followed consistently and it determines various formulas in the sequel.
For vertices $p\ge o$ define the partition functions
\begin{align}\label{g:Z1}
Z^\alpha_{o,p}=\sum_{x_\brbullet\tspa\in\tspa\pathsp_{o,p}} \prod_{i=0}^{\abs{p-o}_1} \widetilde} \def\wh{\widehat} \def\wb{\overline\wgtd_{x_i}
\quad\text{with weights}\quad
\widetilde} \def\wh{\widehat} \def\wb{\overline\wgtd_x
=\begin{cases}
1, &x=o\\[2pt]
Y_x, &x\in o+\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2_{>0}\\[2pt]
I^\alpha_{x}, &x\in o+(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_1\\[2pt]
J^\alpha_{x}, & x\in o+(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_2.
\end{cases}
\end{align}
Note that now a weight at $o$ does not count. The superscript $\alpha$ distinguishes $Z^\alpha_{o,p}$ from the generic partition function $Z_{o,p}$ of \eqref{h:Z}.
The stationarity property is that the joint distribution of the ratios $Z^\alpha_{o,x} /Z^\alpha_{o,x-e_i}$ is invariant under translations of $x$ in the quadrant $o+\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{\ge0}^2$. See Appendix \ref{sec:stat-pol} for more details.
The quenched polymer distribution corresponding to \eqref{g:Z1} is given by
$Q^\alpha_{o,p}(x_\bbullet)= (Z^\alpha_{o,p})^{-1} \prod_{i=0}^{\abs{p-o}_1} \widetilde} \def\wh{\widehat} \def\wb{\overline\wgtd_{x_i}$ for $x_\bbullet\in\pathsp_{o,p}$, and the annealed measure is $P^\alpha_{o,p}(x_\bbullet)=\bE[Q^\alpha_{o,p}(x_\bbullet)]$.
It will be convenient to consider also backward polymer processes whose paths proceed in the southwest direction and the stationary version starts with boundary weights on the north and east. For vertices $o\ge p$ let $\rim{\pathsp}_{o,p}$ be the set of down-left paths starting at $o$ and terminating at $p$. As sets of vertices and edges, paths in $\rim{\pathsp}_{o,p}$ are exactly the same as those in $\pathsp_{p,o}$. The difference is that in $\rim{\pathsp}_{o,p}$ paths are indexed in the down-left direction.
For $o\ge p$, backward partition functions are then defined with i.i.d.\ bulk weights as
\begin{align}
\rim{Z}_{o,p}=\sum_{x_\brbullet\tspa\in\tspa\rim{\pathsp}_{o,p}} \prod_{i=0}^{\abs{o-p}_1} \wgtd_{x_i}
\end{align}
and in the stationary case as
\begin{align}\label{g:Z1rev}
\rim Z^\alpha_{o,p}=\sum_{x_\brbullet\tspa\in\tspa\rim\pathsp_{o,p}} \prod_{i=0}^{\abs{o-p}_1} \widetilde} \def\wh{\widehat} \def\wb{\overline\wgtd_{x_i}
\quad\text{with weights}\quad
\widetilde} \def\wh{\widehat} \def\wb{\overline\wgtd_x
=\begin{cases}
1, &x=o\\[2pt]
Y_x, &x\in o-\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2_{>0}\\[2pt]
I^\alpha_{x}, &x\in o-(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_1\\[2pt]
J^\alpha_{x}, & x\in o-(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_2.
\end{cases}
\end{align}
The independent boundary weights $I^\alpha_{o-ie_1}$ and $J^\alpha_{o-je_2}$ ($i,j\ge 1$) have the distributions \eqref{IJ1}.
We define functions that capture the wandering of a path $x_\bbullet\in\pathsp_{o,p}$.
The (signed) exit point or exit time ${\tau}_{o,p}={\tau}_{o,p}(x_\bbullet)$ marks the position where the path $x_\bbullet$ leaves the southwest boundary and moves into the bulk, with the convention that a negative value indicates a jump off the $y$-axis. More generally, for 3 vertices $o\le v<p$, ${\tau}_{o,v,p}={\tau}_{o,v,p}(x_\bbullet)$ marks the position where $x_\bbullet\in\pathsp_{o,p}$ enters the rectangle $\lzb v+e_1+e_2, p\rzb$, again with a negative sign if this entry happens on the east edge $\{v+e_1+je_2: 1\le j\le (p-v)\cdote_2\}$. Here is the precise definition:
\begin{equation}\label{exit77}
{\tau}_{o,v,p}(x_\bbullet)
=\begin{cases} -\max\{j\ge 1: v+je_2\in x_\bbullet\}, &\text{if } x_\bbullet\cap(v+(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_2)\ne\emptyset\\[2pt]
\max\{i\ge 1: v+ie_1\in x_\bbullet\}, &\text{if } x_\bbullet\cap(v+(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_1)\ne\emptyset.
\end{cases}
\end{equation}
Exactly one of the two cases above happens for each path $x_\bbullet\in\pathsp_{o,p}$. The exit point from the boundary is then defined by ${\tau}_{o,p}={\tau}_{o,o,p}$.
An analogous definition is made for the backward polymer. For $o\ge v> p$ and $x_\bbullet\in\rim{\pathsp}_{o,p}$,
\[
\rim{\tau}_{o,v,p}(x_\bbullet)
=\begin{cases} -\max\{j\ge 1: v-je_2\in x_\bbullet\}, &\text{if } x_\bbullet\cap(v-(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_2)\ne\emptyset\\[2pt]
\max\{i\ge 1: v-ie_1\in x_\bbullet\}, &\text{if } x_\bbullet\cap(v-(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_1)\ne\emptyset.
\end{cases}
\]
The signed exit point from the northeast boundary is $\rim{\tau}_{o,p}=\rim{\tau}_{o,o,p}$.
\medskip
The remainder of this section is devoted to an estimate needed in the body of the proof.
First recall that the {\it digamma function} $\psi_0=\Gamma'/\Gamma$ is strictly concave and strictly increasing on $(0,\infty)$,
with $\psi_0(0+)=-\infty$ and $\psi_0(\infty)=\infty$. Its derivative, the {\it trigamma function} $\psi_1=\psi_0'$, is positive, strictly convex, and strictly decreasing, with
$\psi_1(0+)=\infty$ and $\psi_1(\infty)=0$. These functions appear as means and variances:
\begin{equation}\label{invga7}
\text{for }\ \eta\sim \text{\rm Ga}^{-1}(\alpha), \ \ \bE[\log \eta]=-\psi_0(\alpha) \quad\text{and}\quad \Vvar(\log \eta)=\psi_1(\alpha).
\end{equation}
In the stationary polymer $Z^\alpha_{o,p}$ in \eqref{g:Z1}, the boundary weights are stochastically larger than the bulk weights. Consequently the polymer path prefers to run along one of the boundaries, its choice determined by the direction $(p-o)/\abs{p-o}_1\in[e_2, e_1]$. For each parameter $\alpha\in(0,1)$ there a particular {\it characteristic direction} $\xi(\alpha)\in\,]e_2, e_1[$ at which the attraction of the two boundaries balances out. For $\rho\in[0,1]$ this function is given by
\begin{align}\label{XtR}
\xi(\rho)=\Bigl(\frac{\psi_1(\rho)}{\psi_1(\rho)+\psi_1(1-\rho)} \, ,\,
\frac{\psi_1(1-\rho)}{\psi_1(\rho)+\psi_1(1-\rho)}\Bigr)\in[e_2,e_1].
\end{align}
The extreme cases are interpreted as $\xi(0)=e_1$ and $\xi(1)=e_2$.
The inverse function $\rho=\rho(\xi)$ of a direction
$\xi=(\xi_1, \xi_2)\in[e_2,e_1]$ is defined by
$\rho(e_2)=1$, $\rho(e_1)=0$, and
\begin{align*}
-\xi_1\hspace{0.9pt}\trigamf(1-\rho(\xi))+\xi_2\hspace{0.9pt}\trigamf(\rho(\xi))=0
\quad\text{for $\xi \in \,]e_2,e_1[\,$ }.
\end{align*}
The function $\rho(\xi)$ is a strictly decreasing bijective
mapping of $\xi_1\in[0,1]$ onto $\rho\in[0,1]$, or, equivalently, a strictly decreasing mapping of $\xi$ in the down-right order.
The significance of the characteristic direction for fluctuations is that ${\tau}_{o,p}$ is of order $\abs{p-o}_1^{2/3}$ if and only if $p-o$ is directed towards $\xi(\alpha)$, and of order $\abs{p-o}_1$ in all other directions. These fluctuation questions were first investigated in \cite{sepp-12-aop-corr}.
We insert here a lemma on the regularity of the characteristic direction.
\begin{lemma}\label{lem:psi}
There exist functions $\phi>0$ and $B>0$ on $(0,1)$ such that, whenever $\rho_0\in(0,1)$
and
$|\delta-\rho_0|<\rho_1=\frac{1}{2}(\rho_0\wedge(1-\rho_0))$,
\begin{align}\label{psi1}
\frac{\xi_2(\rho_0+\delta)}{\xi_1(\rho_0+\delta)}-\frac{\xi_2(\rho_0)}{\xi_1(\rho_0)}&= \phi(\rho_0)\delta+f(\rho_0, \delta)
\end{align}
where the function $f$ satisfies
\begin{align}
|f(\rho_0, \delta)|&\leq B(\rho_0)\delta^2\label{psi2}.
\end{align}
The functions $\phi$, $\phi^{-1}$ and $B$ are bounded on any compact subset of $(0,1)$.
\end{lemma}
\begin{proof}
As the function $\psi_1$ is smooth on $(0,\infty)$
\begin{align*}
\frac{\xi_2(\rho_0+\delta)}{\xi_1(\rho_0+\delta)}-\frac{\xi_2(\rho_0)}{\xi_1(\rho_0)}
&=\frac{\psi_1(1-(\rho_0+\delta))}{\psi_1(\rho_0+\delta)}-\frac{\psi_1(1-\rho_0)}{\psi_1(\rho_0)}\\
&=-\, \delta\tspa\frac{\psi'_1(1-\rho_0)\psi_1(\rho_0)+\psi_1'(\rho_0)\psi_1(1-\rho_0)}{\psi_1(\rho_0)^2}+f(\rho_0,\delta)
\end{align*}
where $\psi_1'<0$ and $f$ satisfies \eqref{psi2}.
\end{proof}
Recall that to prove Theorem \ref{thm:noex}, our intention is to rule out bi-infinite polymer measures whose forward direction is into the open first quadrant, and whose backward direction is into the open third quadrant. The main step towards this is that, as $N$ becomes large, a polymer path from southwest to northeast across the square $\lzb -N,N\rzb^2$, with slope bounded away from $0$ and $\infty$, cannot cross the $y$-axis anywhere close to the origin.
To achieve this we control partition functions from the southwest boundary of the square $\lzb-N,N\rzb^2$ to the interval $\cJ=\lzb -N^{{2}/{3}}e_2,N^{{2}/{3}}e_2 \rzb$ on the $y$-axis, and backward partition functions from the northeast boundary of the square $\lzb-N,N\rzb^2$ to the interval $\rim\cJ=e_1+\cJ$ shifted one unit off the $y$-axis.
Let $\varepsilon>0$. We establish notation for the southwest portion of the boundary of the square $\lzb -N,N\rzb^2$ that is bounded by the lines of slopes $\varepsilon$ and $\varepsilon^{-1}$. With $\mathcal W$ for west and $\mathcal{S}} \def\cT{\mathcal{T}$ for south, let $\partial^N_\mathcal{W} =\{-N\}\times \lzb -N,-\varepsilon N \rzb$, $\partial^N_\mathcal{S}=\lzb -N,-\varepsilon N\rzb \times \{-N\}$, and then $\partial^N=\partial^{N, \varepsilon}=\partial^N_\mathcal{W}\cup \partial^N_{S}$.
The parameter $\varepsilon>0$ stays fixed for most of the proof, and hence will be suppressed from much of the notation.
We also let $o_i=(-N,-\varepsilon N)$ and $o_f=(-\varepsilon N,-N)$. A lattice point $o=(o_1, o_2)\in \partial^N$ is associated with its (reversed) direction vector $ \xi(o)=(\xi_1(o), 1-\xi_1(o))\in \,]e_2,e_1[\,$ and parameter $\rho(o)\in (0,1)$ through the relations
\begin{align}
\xi(o)&= \left(\frac{o_1}{o_1+o_2},\,\frac{o_2}{o_1+o_2}\right)\label{xi-o}
\\
\intertext{and indirectly via \eqref{XtR}:}
\rho(o)&=\rho(\xi(o))
\ \iff\ \xi(\rho(o)) =\xi(o)
\label{rho-o}
\end{align}
For all $o\in \partial^N$ we have the bounds
\[ \xi(o)\in \Bigl[ \Bigl(\frac{1}{1+\varepsilon},\frac{\varepsilon}{1+\varepsilon}\Bigr), \Bigl(\frac{\varepsilon}{1+\varepsilon},\frac{1}{1+\varepsilon}\Bigr)\Bigr]=[\xi_i,\xi_f].
\]
If we define the extremal parameters (for a given $\varepsilon>0$) by
\begin{align*}
\rho_i=\rho(o_i)=\rho\biggl( \frac{1}{1+\varepsilon},\frac{\varepsilon}{1+\varepsilon}\biggr)
\qquad\text{and}\qquad
\rho_f=\rho(o_f)=\rho\biggl(\frac{\varepsilon}{1+\varepsilon},\frac{1}{1+\varepsilon}\biggr)
\end{align*}
then we have the uniform bounds
\begin{equation}\label{2089}
0< \rho_i \le \rho(o)\le \rho_f < 1
\qquad\text{ for all } o\in\partial^N=\partial^{N\!,\tspa\varepsilon}.
\end{equation}
For $o\in \partial^N$ define perturbed parameters (with dependence on $r, N$ suppressed from the notation):
\begin{equation}\label{rho*}
\rhodown{\rho}(o)=\rho(o)-rN^{-\frac{1}{3}}
\qquad\text{and}\qquad
\rhoup{\rho}(o)=\rho(o)+rN^{-\frac{1}{3}}.
\varepsilon
The variable $r$ can be a function of $N$ and become large but always $r(N)N^{-1/3}\to0$ as $N\to\infty$. Then for $N\ge N_0(\varepsilon)$ the perturbed parameters are bounded uniformly away from $0$ and $1$:
\begin{equation}\label{2095}
0< \rho_0(\varepsilon) \le \rhodown{\rho}(o)
< \rhoup{\rho}(o) \le \rho_1(\varepsilon) <1
\quad\text{ for all } o\in\partial^N=\partial^{N\!,\tspa\varepsilon}\text{ and } N\ge N_0(\varepsilon).
\end{equation}
We consider the stationary processes $Z^{\rhoup{\rho}(o)}_{o,\bbullet}$ and $Z^{\rhodown{\rho}(o)}_{o,\bbullet}$. Our next lemma shows that the perturbation $r$ can be taken such that, for all $o\in \partial^N$ and $x\in \cJ=\lzb -N^{{2}/{3}}e_2,N^{{2}/{3}}e_2 \rzb$, on the scale $N^{2/3}$ the exit point under $Q^{\rhoup{\rho}(o)}_{o,x}$ is far enough in the $e_1$ direction, and under $Q^{\rhodown{\rho}(o)}_{o,x}$ far enough in the $e_2$ direction, with high probability.
\begin{lemma}\label{lem-lb1} For each $\varepsilon>0$ there exist finite positive constants
$c(\varepsilon), C_0(\varepsilon), C_1(\varepsilon)$ and $N_0(\varepsilon)$ such that, whenever $1\le d\le c(\varepsilon)N^{1/3}$, $C_0(\varepsilon)d\le r\le c(\varepsilon)N^{1/3}$, $N\ge N_0(\varepsilon)$, $o\in\partial^N$, and $y>0$, we have the bounds
\begin{equation}\label{lb-1} \P\Big\{\sup_{x\in\cJ}Q^{\rhodown{\rho}(o)}_{o,x}\big({\tau}_{o,x}\ge -dN^{\frac{2}{3}}\big)>y \Big\} \le C_1(\varepsilon)y^{-1}r^{-3}
\end{equation}
and
\begin{equation}\label{lb-2} \P\Big\{\sup_{x\in\cJ}Q^{\rhoup{\rho}(o)}_{o,x}\big({\tau}_{o,x}\le dN^{\frac{2}{3}}\big)>y \Big\} \le C_1(\varepsilon)y^{-1}r^{-3}.
\end{equation}
\end{lemma}
\begin{proof}
We prove \eqref{lb-2} as \eqref{lb-1} is similar.
We turn the quenched probability into a form to which we can apply fluctuation bounds. The justifications of the steps below go as follows.
\begin{enumerate} [(i)]
\item The first inequality below is from \eqref{pmon}.
\item Observe that the path leaves the boundary to the left of the point $o+dN^{2/3}e_1$ if and only if it intersects the vertical line $o+dN^{2/3}e_1 +je_2$ at some $j\ge 1$.
\item Move the base point from $o$ to $o+dN^{2/3}e_1$ and apply \eqref{m:830}. By the stationarity, the new boundary weights on the axes emanating from $o+dN^{2/3}e_1$ have the same distribution as the original ones. This gives the equality in distribution.
\item Choose an integer $\ell$ so that the vector from $o+dN^{2/3}e_1-\elle_2$ to $N^{{2}/{3}}e_2$ points in the characteristic direction $\xi(\rhoup{\rho}(o))$. Apply \eqref{m:830} and stationarity.
\end{enumerate}
\begin{align*}
&\sup_{x\in\cJ}Q^{\rhoup{\rho}(o)}_{o,x}\big({\tau}_{o,x}< dN^{\frac{2}{3}}\big) \leq Q^{\rhoup{\rho}(o)}_{o,\tsp N^{2/3}e_2}\big({\tau}_{o,\tsp N^{2/3}e_2}< dN^{\frac{2}{3}}\big) \\
&\qquad =
Q^{\rhoup{\rho}(o)}_{o,\tsp N^{2/3}e_2}\big({\tau}_{o, \tspa o+dN^{2/3}e_1, \tspa N^{2/3}e_2}< 0 \big)
\deq Q^{\rhoup{\rho}(o)}_{o+dN^{2/3}e_1,\tspa N^{2/3}e_2}\big({\tau}_{o+dN^{2/3}e_1, \tspa N^{2/3}e_2}< 0 \big) \\[3pt]
&\qquad
= Q^{\rhoup{\rho}(o)}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}\big({\tau}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}< -\ell\tsp\big).
\end{align*}
We show that
$\ell\ge c_0(\varepsilon)rN^{2/3}$ for a constant $c_0(\varepsilon)$. Let $o=-(Na, Nb)$, with $\varepsilon\le a,b\le 1$.
Lemma \ref{lem:psi} gives the next identity. The $O$-term hides an $\varepsilon$-dependent constant that is uniform for all $\rho(o)$ because, as observed in \eqref{2089}, the assumption $o\in\partial^N$ bounds $\rho(o)$ away from $0$ and $1$.
\begin{align*}
\frac{N^{2/3}+Nb+\ell}{Na-dN^{2/3}} = \frac{\xi_2(\rhoup{\rho}(o))}{\xi_1(\rhoup{\rho}(o))} = \frac{b}a + \phi(\rho(o)) rN^{-1/3} +O(r^2N^{-2/3}) .
\end{align*}
From this we deduce
\begin{align*}
\ell = \phi(\rho(o)) arN^{2/3} - \frac{b}a d N^{2/3} -N^{2/3} - \phi(\rho(o)) rdN^{1/3} +O(r^2N^{1/3}) +O(r^2d) .
\end{align*}
Recall from Lemma \ref{lem:psi} that $\phi(\rho(o))>0$ is uniformly bounded away from zero for $o\in\partial^N$.
For a small enough constant $c(\varepsilon)$ and large enough constants $C_0(\varepsilon)$ and $N_0(\varepsilon)$, if we have $1\le d\le c(\varepsilon)N^{1/3}$, $C_0(\varepsilon)d\le r\le c(\varepsilon)N^{1/3}$ and $N\ge N_0(\varepsilon)$, the above simplifies to $\ell\ge c_0(\varepsilon)rN^{2/3}$.
We can derive the final bound.
\begin{align*
&\P\Big\{\sup_{x\in\cJ}Q^{\rhoup{\rho}(o)}_{o,x}\big({\tau}_{o,x}< dN^{\frac{2}{3}}\big)>y \Big\}\\
&\qquad \leq \P\Big\{ Q^{\rhoup{\rho}(o)}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}\big({\tau}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}< -\ell\tsp\big) >y \Big\} \\
&\qquad \le y^{-1} \tspa \bE\Big[Q^{\rhoup{\rho}(o)}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}\big({\tau}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}< -c(\varepsilon)rN^{2/3}\tsp\big) \Big]\\
&\qquad
=y^{-1} \tsp P^{\rhoup{\rho}(o)}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}\big({\tau}_{o+dN^{2/3}e_1-\elle_2, \tspa N^{2/3}e_2}< -c_0(\varepsilon)rN^{2/3}\tsp\big)\leq C_1(\varepsilon)y^{-1} \tspa r^{-3}.
\end{align*}
The final inequality comes from Theorem \ref{thm:kpz3}.
\end{proof}
\section{Estimates for paths across a large square} \label{sec:estim}
After the preliminary work above we turn to develop the estimates that prove the main theorem.
Throughout, $\mathbf{d}=(d_1,d_2)\in \bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{\ge 1}^2$ denotes a pair of parameters that control the coarse graining on the southwest and northeast boundaries of the square $\lzb-N,N\rzb^2$. For $o\in \partial^N$ let
\begin{align*}
\mathcal{I}_{o,\mathbf{d}}=\{u \in \partial^N: \abs{u-o}_1
\leq \tfrac12{d_1}N^\frac{2}{3}\}.
\end{align*}
Let $o_c\in\mathcal{I}_{o,\mathbf{d}}$ denote the minimal point of $\mathcal{I}_{o,\mathbf{d}}$ in the coordinatewise partial order, that is, defined by the requirement that
\[ o_c\in\mathcal{I}_{o,\mathbf{d}} \quad\text{and}\quad o_c\le u \ \ \forall u\in\mathcal{I}_{o,\mathbf{d}}. \]
This setting is illustrated in Figure \ref{fig:points}.
On the rectangle $\lzb o_c, Ne_2\rzb$ we define coupled polymer processes. For each $u\in\mathcal{I}_{o,\mathbf{d}}$ we have the bulk process $Z_{u,\brbullet}$ that uses $\text{\rm Ga}^{-1}(1)$ weights $Y$. Two stationary comparison processes based at $o_c$ have parameters $\rhodown{\rho}(o_c)$ and $\rhoup{\rho}(o_c)$ defined as in \eqref{rho*}. Their basepoint is taken as $o_c$ so that we get simultaneous control over all the processes based at vertices $u\in \mathcal{I}_{o,\mathbf{d}}$.
Couple the boundary weights on the south and west boundaries of the rectangle $\lzb o_c, Ne_2\rzb$ as described in Theorem \ref{thm:st-lpp} in Appendix \ref{sec:stat-pol}. In particular, for $k,\ell\ge 1$ we have the inequalities
\begin{equation}\label{2160}
Y_{o_c+ke_1} \le I^{\rhodown{\rho}(o_c)}_{o_c+ke_1}\le I^{\rhoup{\rho}(o_c)}_{o_c+ke_1}
\quad\text{ and }\quad
Y_{o_c+\elle_2} \le J^{\rhoup{\rho}(o_c)}_{o_c+\elle_2}\le J^{\rhodown{\rho}(o_c)}_{o_c+\elle_2}.
\end{equation}
For all these coupled processes we define ratios of the partition functions from the base point to the $y$-axis, for all $u\in \mathcal{I}_{o,\mathbf{d}}$ and $i\in\lzb-N^{2/3}, N^{2/3}\rzb $:
\begin{equation}\label{2165}
J^u_i=\frac{Z_{u,ie_2}}{Z_{u,(i-1)e_2}} \,,
\quad J^{\rhodown{\rho}(o_c)}_i=\frac{Z^{\rhodown{\rho}(o_c)}_{o_c,ie_2}}{Z^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}}
\qquad\text{and}\qquad
J^{\rhoup{\rho}(o_c)}_i=\frac{Z^{\rhoup{\rho}(o_c)}_{o_c,ie_2}}{Z^{\rhoup{\rho}(o_c)}_{o_c,(i-1)e_2}}.
\end{equation}
Recall that $\cJ=\lzb -N^\frac{2}{3}e_2,N^\frac{2}{3}e_2 \rzb$.
\begin{lemma}\label{lem ge1}
For $0<y<1$, define the event
\begin{equation}\label{Aod}
A_{o_c,\mathbf{d},y}
= \left\{\, \inf_{x\in\cJ}Q^{\rhodown{\rho}(o_c)}_{o_c,\tsp x}\big({\tau}_{o_c,\tsp x}<-d_1N^{\frac{2}{3}}\big)\ge 1-y\, , \; \inf_{x\in\cJ}Q^{\rhoup{\rho}(o_c)}_{o_c,\tsp x}\big({\tau}^{\rhoup{\rho}(o_c)}_{o_c,\tsp x}> d_1N^{\frac{2}{3}}\big)\ge 1-y \right\}.
\end{equation}
Under the assumptions of Lemma \ref{lem-lb1} for $d=d_1$ we have the bound
\begin{equation}\label{Aod2} \P\bigl(A_{o_c,\mathbf{d},y}\bigr) \ge 1-C_1(\varepsilon)y^{-1} r^{-3}. \end{equation}
On the event $A_{o_c,\mathbf{d},y}$, for any $m,n\in\lzb-N^{2/3}, N^{2/3}\rzb$ such that $m< n$ we have the inequalities
\begin{equation}\label{Aod1}
(1-y)\prod_{i=m+1}^{n}J^{\rhoup{\rho}(o_c)}_i \; \leq \; \prod_{i=m+1}^{n}J^u_i \; \leq \; \frac{1}{1-y}\prod_{i=m+1}^{n}J^{\rhodown{\rho}(o_c)}_i\quad \forall u\in \mathcal{I}_{o,d}.
\end{equation}
\end{lemma}
\begin{proof} Bound \eqref{Aod2} comes by switching to complements in Lemma \ref{lem-lb1}.
We show the second inequality of \eqref{Aod1}. The first inequality follows similarly.
Let $u\in \mathcal{I}_{o,\mathbf{d}}$. The first inequality in the calculation \eqref{Aod13} below is justified as follows in two cases. Recall the notation \eqref{h:Z(A)} for restricted partition functions $Z_{o,p}(A)$.
\medskip
(i) Suppose $u=o_c+je_2$ for some $0\le j\le d_1N^{2/3}$.
Apply \eqref{m:770} in the following setting. Take $Z^{(2)}_{u, \cbullet}$ to be $Z_{u, \cbullet}$. Let $Z^{(1)}_{u, \cbullet}$ use the same bulk weights $Y$. On the boundary $Z^{(1)}_{u, \cbullet}$ takes
$Y^{(1)}_{u+\elle_2}=J^{\rhodown{\rho}(o_c)}_{u+\elle_2}$ on the $y$-axis, and on the $x$-axis takes any $Y^{(1)}_{u+me_1}< Y_{u+me_1}$ for $1\le m\le -u\cdote_1$. Then the second inequality of \eqref{m:770} followed by the second inequality of \eqref{m:772} gives
\[ \frac{Z_{u,ie_2}}{Z_{u,(i-1)e_2}} \leq \frac{Z^{(1)}_{u,ie_2}}{Z^{(1)}_{u,(i-1)e_2}} \leq \frac{Z^{(1)}_{u,ie_2}\bigl({\tau}_{u,ie_2}< j-d_1N^{\frac{2}{3}}\bigr)}{Z^{(1)}_{u,(i-1)e_2}\bigl({\tau}_{u, (i-1)e_2}< j-d_1N^{\frac{2}{3}}\bigr)}.
\]
Next observe that the condition
${\tau}_{u,\brbullet}< j-d_1N^{\frac{2}{3}}<0$ renders the boundary weights on the $x$-axis $u+(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0})e_1$ irrelevant. Therefore we can replace $Y^{(1)}_{u+me_1}$ with the stationary boundary weights $I^{\rhodown{\rho}(o_c)}_{u+me_1}$ without changing the restricted partition functions on the right-hand side. This gives the first equality below:
\begin{align*}
\frac{Z^{(1)}_{u,ie_2}\bigl({\tau}_{u,ie_2}< j-d_1N^{\frac{2}{3}}\bigr)}{Z^{(1)}_{u,(i-1)e_2}\bigl({\tau}_{u, (i-1)e_2}< j-d_1N^{\frac{2}{3}}\bigr)}
&= \frac{Z^{\rhodown{\rho}(o_c)}_{u,ie_2}\bigl({\tau}_{u,ie_2}< j-d_1N^{\frac{2}{3}}\bigr)}{Z^{\rhodown{\rho}(o_c)}_{u,(i-1)e_2}\bigl({\tau}_{u,(i-1)e_2}< j-d_1N^{\frac{2}{3}}\bigr)}\\
&=
\frac{Z^{\rhodown{\rho}(o_c)}_{o_c,ie_2}\bigl({\tau}_{o_c,ie_2}<-d_1N^{\frac{2}{3}}\bigr)}{Z^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}\bigl({\tau}_{o_c,(i-1)e_2}<-d_1N^{\frac{2}{3}}\bigr)}.
\end{align*}
The second equality comes by multiplying upstairs and downstairs with the boundary weights $J^{\rhodown{\rho}(o_c)}_{o_c+\elle_2}$ for $1\le\ell\le j= (u-o_c)\cdote_2$.
\medskip
(ii) On the other hand, if $u=o_c+ke_1$ for some $0\le k\le d_1N^{2/3}$, then first by \eqref{in2} and then by applying the argument of the previous paragraph to $u=o_c$:
\begin{align*}
\frac{Z_{u,ie_2}}{Z_{u,(i-1)e_2}}\leq
\frac{Z_{o_c,ie_2}}{Z_{o_c,(i-1)e_2}}
\le
\frac{Z^{\rhodown{\rho}(o_c)}_{o_c,ie_2}\bigl({\tau}_{o_c,ie_2}<-d_1N^{\frac{2}{3}}\bigr)}{Z^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}\bigl({\tau}_{o_c,(i-1)e_2}<-d_1N^{\frac{2}{3}}\bigr)}.
\end{align*}
\medskip
Now for the derivation.
\begin{equation}\label{Aod13}\begin{aligned}
\prod_{i=m+1}^{n}J^u_i&=\prod_{i=m+1}^{n}\frac{Z_{u,ie_2}}{Z_{u,(i-1)e_2}}\leq \prod_{i=m+1}^{n}\frac{Z^{\rhodown{\rho}(o_c)}_{o_c,ie_2}\bigl({\tau}_{o_c,ie_2}<-d_1N^{\frac{2}{3}}\bigr)}{Z^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}\bigl({\tau}_{o_c,(i-1)e_2}<-d_1N^{\frac{2}{3}}\bigr)} \\
&=
\prod_{i=m+1}^{n}
\frac{Q^{\rhodown{\rho}(o_c)}_{o_c,ie_2}\big({\tau}_{o_c,ie_2}< -d_1N^{\frac{2}{3}}\big)}{Q^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}\big({\tau}_{o_c,(i-1)e_2}<-d_1N^{\frac{2}{3}}\big)}
\cdot \prod_{i=m+1}^{n}\frac{Z^{\rhodown{\rho}(o_c)}_{o_c,ie_2}}{Z^{\rhodown{\rho}(o_c)}_{o_c,(i-1)e_2}}\\
&= \frac{Q^{\rhodown{\rho}(o_c)}_{o_c,ne_2}\big({\tau}_{o_c,ne_2}< -d_1N^{\frac{2}{3}}\big)}{Q^{\rhodown{\rho}(o_c)}_{o_c,me_2}\big({\tau}_{o_c,me_2}<-d_1N^{\frac{2}{3}}\big)} \prod_{i=m+1}^{n}J^{\rhodown{\rho}(o_c)}_i
\le \frac{1}{1-y}\prod_{i=m+1}^{n}J^{\rhodown{\rho}(o_c)}_i .
\end{aligned}\end{equation}
\end{proof}
Next we define the analogous construction reflected across the origin. Define east (${\mathcal E}$) and north ($\mathcal{N}$) portions of the boundary by $\partial^N_\mathcal{E} =\{N\}\times \lzb \varepsilon N,N \rzb$ and $\partial^N_\mathcal{N}=\lzb \varepsilon N,N\rzb \times \{N\}$, and combine them into $\rim{\partial}^N=\rim{\partial}^{N\!,\tspa\varepsilon}=\partial^N_\mathcal{E}\cup \partial^N_\mathcal{N}$.
Each point $\rim{o}=(\rim{o}_1,\rim{o}_2)\in \rim{\partial}^N$ is associated with a parameter $\rho(\rim{o})\in (0,1)$ and a direction $ \xi(\rim{o})\in \,]e_2,e_1[\,$ through the relations in \eqref{rho-o} and \eqref{xi-o}.
For each point $\rim{o}\in \rim{\partial}^N$ define the set
\begin{align*}
\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}=\bigl\{v \in \rim{\partial}^N:\distance(v,\rim{o})\leq \tfrac12{d_2}N^\frac{2}{3}\bigr\}
\end{align*}
and the maximal point $\rim{o}_c\in \rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$
in the coordinatewise partial order, defined by the requirement that
\[ \rim o_c\in\rim\mathcal{I}_{\rim o,\mathbf{d}} \quad\text{and}\quad v\le\rim o_c \ \ \forall v\in\rim \mathcal{I}_{\rim o,\mathbf{d}}. \]
As previously for sets $\mathcal{I}_{o,\mathbf{d}}$ on the southwest boundary, given now a northeast boundary point $\rim o\in\rim\partial^N$ we construct a family of coupled backward partition functions from $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ to points on the shifted $y$-axis $e_1+\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bNe_2$. From each
$v\in \rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ we have the backward bulk partition functions $\rim{Z}_{v,\bbullet}$ that use the i.i.d.\ Ga$^{-1}(1)$ weights $Y$. From the base point $\rim o_c$ we define two stationary backward polymer processes $\rim{Z}^{\rhodown{\rho}(\rim{o}_c)}_{\rim{o}_c,\bbullet}$ and $\rim{Z}^{\rhoup{\rho}(\rim{o}_c)}_{\rim{o}_c, \bbullet}$ with parameters $\rhodown{\rho}(\rim{o}_c)=\rho(\rim{o}_c)-rN^{-\frac{1}{3}}$ and $\rhoup{\rho}(\rim{o}_c)=\rho(\rim{o}_c)+rN^{-\frac{1}{3}}$. Weights are coupled on the northeast boundary according to
Theorem \ref{thm:st-lpp}: for $k,\ell\ge 1$,
\begin{equation}\label{2168}
Y_{\rim o_c-ke_1} \le I^{\rhodown{\rho}(\rim o_c)}_{\rim o_c-ke_1}\le I^{\rhoup{\rho}(\rim o_c)}_{\rim o_c-ke_1}
\quad\text{ and }\quad
Y_{\rim o_c-\elle_2} \le J^{\rhoup{\rho}(\rim o_c)}_{\rim o_c-\elle_2}\le J^{\rhodown{\rho}(\rim o_c)}_{\rim o_c-\elle_2}.
\end{equation}
The boundary weights in \eqref{2160} and in \eqref{2168} above are taken independent of each other.
Ratio weights on the shifted $y$-axis are defined by
\begin{equation}\label{2172}
\rim{J}^{\hspace{0.9pt} v}_i=\frac{\rim{Z}_{v,e_1+(i-1)e_2}}{\rim{Z}_{v,e_1+ie_2}}\,,
\quad
\rim{J}^{\hspace{0.9pt}\rhodown{\rho}(\rim{o}_c)}_i=\frac{\rim{Z}^{\rhodown{\rho}(\rim{o}_c)}_{\rim{o}_c,\,e_1+(i-1)e_2}}{\rim{Z}^{\rhodown{\rho}(\rim{o}_c)}_{\rim{o}_c,\,e_1+ie_2}}
\quad\text{and}\quad
\rim{J}^{\hspace{0.9pt}\rhoup{\rho}(\rim{o}_c)}_i=\frac{\rim{Z}^{\rhoup{\rho}(\rim{o}_c)}_{\rim{o}_c,\,e_1+(i-1)e_2}}{\rim{Z}^{\rhoup{\rho}(\rim{o}_c)}_{\rim{o}_c,\,e_1+ie_2}}.
\end{equation}
The collection of ration weights in \eqref{2165} is independent of the collection in \eqref{2172} above because they are constructed from independent inputs.
We have this analogue of Lemma \ref{lem ge1}. $\rim{\cJ}=e_1+\cJ= \lzb e_1-N^\frac{2}{3}e_2\,, \,e_1+N^\frac{2}{3}e_2 \rzb$ is the shift of the interval $\cJ$ in \eqref{Aod}.
\begin{lemma}\label{lem ge2}
For $0<y<1$, define the event
\begin{equation}\label{Bod}
B_{\rim{o}_c,\mathbf{d},y}
= \left\{\,\inf_{x\in \wh{\cJ}}\rim{Q}^{\rhodown{\rho}(\rim{o}_c)}_{\rim{o}_c,x}\big(\rim{{\tau}}_{\rim{o}_c,x}<-d_2N^{\frac{2}{3}}\big)\ge 1- y , \; \inf_{x\in\wh{\cJ}}\rim{Q}^{\rhoup{\rho}(\rim{o}_c)}_{\rim{o}_c,x}\big(\hat{{\tau}}_{\rim{o}_c,x}> d_2N^{\frac{2}{3}}\big)\ge 1- y \right\}.
\end{equation}
Under the assumptions of Lemma \ref{lem-lb1} for $d=d_2$ we have the bound
\begin{equation}\label{Bod2} \P\bigl(B_{\rim{o}_c,\mathbf{d},y}\bigr) \ge 1-C_1(\varepsilon)y^{-1} r^{-3}. \end{equation}
On the event $B_{\rim{o}_c,\mathbf{d},y}$, for any $m<n$ in $\lzb-N^{2/3}, N^{2/3}\rzb$ we have the inequalities
\begin{equation}\label{Bod1}
(1-y)\prod_{i=m+1}^{n}\rim{J}^{\hspace{0.9pt}\rhoup{\rho}(\rim{o}_c)}_i\leq \prod_{i=m+1}^{n}\rim{J}^{\hspace{0.9pt} v}_i \leq \frac{1}{1-y}\prod_{i=m+1}^{n}\rim{J}^{\hspace{0.9pt}\rhodown{\rho}(\rim{o}_c)}_i\quad \forall v\in \rim{\mathcal{I}}_{\rim{o}_c,\mathbf{d}}.
\end{equation}
\end{lemma}
\smallskip
Now we use partition functions from the southwest and northeast together. Let $o\in \partial^N,\rim{o}\in\rim{\partial}^N$ and consider the polymers from points $u\in \mathcal{I}_{o,\mathbf{d}}$ to the interval $\cJ$ on the $y$-axis and reverse polymers from points $v\in \rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ to the shifted interval $\wh{\cJ}=e_1+\cJ$. Abbreviate the parameters for the base points as
\begin{equation}\label{2186} \rhoup{\rho}=\rhoup{\rho}(o_c), \quad \rhodown{\rho}=\rhodown{\rho}(o_c), \quad
\rhoup{\lambda}=\rhoup{\rho}(\rim{o}_c), \quad\text{and}\quad \rhodown{\lambda}=\rhodown{\rho}(\rim{o}_c). \end{equation}
For $i\in\lzb-N^{2/3}, N^{2/3}\rzb $, take the $Z$-ratios from \eqref{2165} and \eqref{2172} and define
\begin{equation}\label{2188}
X^{u,v}_i=\frac{J^u_i}{\rim{J}^{\hspace{0.9pt} v}_i}, \quad
Y'_i=\frac{J^{\rhodown{\rho}}_i}{\rim{J}^{\hspace{0.9pt}\rhoup{\lambda}}_i}
\quad\text{and}\quad
Y_i=\frac{J^{\rhoup{\rho}}_i}{\rim{J}^{\hspace{0.9pt}\rhodown{\lambda}}_i}.
\varepsilon
A two-sided multiplicative walk $\mwalk(X)$ with steps $\{X_j\}$ is defined by
\begin{align}\label{mw}
\mwalk_n(X)=
\begin{cases}
\prod_{j=1}^{n}X_j & n \geq 1 \\
1 & n=0\\
\prod_{j=n+1}^{0}X^{-1}_j & n \le-1.
\end{cases}
\end{align}
The ratios from \eqref{2188} above define the walks
\begin{equation}\label{2192}
\mwalk^{u,v}=\mwalk(X^{u,v})\,,\quad
\mwalk'=\mwalk(Y{'}) \quad\text{and}\quad
\mwalk=\mwalk(Y).
\end{equation}
Specialize the parameter $y$ in the events in \eqref{Aod} and \eqref{Bod} to set
\begin{align*}
A_{o,\mathbf{d}}=A_{o,\mathbf{d},\frac{\sqrt{2}-1}{\sqrt{2}}}
\qquad\text{and}\qquad
B_{\rim{o},\mathbf{d}}=B_{\rim{o},\mathbf{d},\frac{\sqrt{2}-1}{\sqrt{2}}}.
\end{align*}
\begin{lemma}\label{lm:sw}
The processes
\begin{equation} \label{cor b-rw1}
\{\mwalk'_m : m\in \lzb-N^{2/3},0\rzb\,\} \quad\text{and}\quad
\{\mwalk_n: n\in \lzb0,N^{2/3} \rzb\,\}
\quad\text{are independent.}
\end{equation}
On the event $A_{o,\mathbf{d}}\cap B_{\rim{o},\mathbf{d}}$, for all $u\in \mathcal{I}_{o,\mathbf{d}}$ and $v\in \rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$,
\begin{equation}\label{sw5} \begin{aligned}
\tfrac12 \mwalk'_n &\leq \mwalk_n^{u,v}\leq 2\mwalk_n \quad\text{for } \ n\in \lzb-N^\frac{2}{3},-1\rzb\\
\text{and}\quad \tfrac12 \mwalk_n &\leq \mwalk_n^{u,v}\leq 2\mwalk'_n \quad\text{for } \ n\in \lzb1,N^\frac{2}{3}\rzb.
\end{aligned}\end{equation}
\end{lemma}
\begin{proof} To prove the independence claim \eqref{cor b-rw1}, observe first from the construction itself that
the collection $\{J^{\rhodown{\rho}}_i, J^{\rhoup{\rho}}_i\}_{i\tspa\in\tspa\lzb -N^{2/3}, N^{2/3}\rzb}$ is independent of the collection $\{\rim J^{\hspace{0.9pt}\rhodown{\lambda}}_i, \rim J^{\hspace{0.9pt}\rhoup{\lambda}}_i\}_{i\tspa\in\tspa\lzb -N^{2/3}, N^{2/3}\rzb}$, as pointed out below \eqref{2172}. Then within these collections, Theorem \ref{thm:st-lpp}(i) implies the independence of
$\{J^{\rhodown{\rho}}_i\}_{i\le0}$ and $\{J^{\rhoup{\rho}}_i\}_{i\ge1}$, and the independence of
$\{\rim J^{\hspace{0.9pt}\rhodown{\lambda}}_i\}_{i\ge 1}$ and $\{\rim J^{\hspace{0.9pt}\rhoup{\lambda}}_i\}_{i\le0}$. With boundary weights on the southwest, the independence of $\{J^{\rhodown{\rho}}_i\}_{i\le0}$ and $\{J^{\rhoup{\rho}}_i\}_{i\ge1}$ is a direct application of Theorem \ref{thm:st-lpp}(i) with the choice $(\lambda, \rho, \sigma)=(\rhodown{\rho}, \rhoup{\rho}, 1)$. After reflection of the entire setting of Theorem \ref{thm:st-lpp} across its base point $u$, the boundary weights reside on the northwest, as required for $\{\rim J^{\hspace{0.9pt}\rhodown{\lambda}}_i\}_{i\ge 1}$ and $\{\rim J^{\hspace{0.9pt}\rhoup{\lambda}}_i\}_{i\le0}$, and the direction $e_2$ has been reversed to $-e_2$. Hence the inequalities $i\le0$ and $i\ge1$ in the independence statement must be switched around.
To summarize, the collections $\{J^{\rhodown{\rho}}_i, \rim J^{\hspace{0.9pt}\rhoup{\lambda}}_i\}_{i\le0}$ and $\{J^{\rhoup{\rho}}_i, \rim J^{\hspace{0.9pt}\rhodown{\lambda}}_i\}_{i\ge1}$ are independent of each other, which implies the independence of $\{Y'_i\}_{i\le0}$ from $\{Y_i\}_{i\ge1}$.
\medskip
We show the case $n\in \lzb 1, N^{2/3}\rzb$ of \eqref{sw5}.
\begin{align*}
\mwalk_n^{u,v}=\prod_{i=1}^{n}X^{u,v}_i=\prod_{i=1}^{n}J^u_i\cdot \prod_{i=1}^{n}(\rim{J}^{\hspace{0.9pt} v}_i)^{-1}
\begin{cases}
\leq \sqrt{2}\prod_{i=1}^{n}J^{\rhodown{\rho}}_i \cdot \sqrt{2}\prod_{i=1}^{n}(\rim{J}^{\hspace{0.9pt}\rhoup{\lambda}}_i)^{-1}=2\prod_{i=1}^{n} Y'_i = 2\mwalk'_n\,; \\[5pt]
\geq \frac 1{\sqrt{2}}\prod_{i=1}^{n}J^{\rhoup{\rho}}_i \cdot \frac 1{\sqrt{2}}\prod_{i=1}^{n}(\rim{J}^{\hspace{0.9pt}\rhodown{\lambda}}_i)^{-1}=\frac12\prod_{i=1}^{n} Y_i = \frac12 \mwalk_n.
\end{cases}
\end{align*}
An analogous argument gives the case $n\in \lzb -N^{2/3}, -1\rzb$.
\end{proof}
Each path that crosses the $y$-axis leaves the axis along a unique edge $e_i=(ie_2,ie_2+e_1)$.
Decompose the set of paths between $u\in\partial^N$ and $v\in\rim\partial^N$ according to the edge taken:
\[ \pathsp_{u,v}=\bigcup_{i\in \bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN} \pathsp^i_{u,v}
\]
where the sets
\begin{align}\label{tz}
\pathsp^i_{u,v}=\{\pi\in\pathsp_{u,v}:e_i\in\pi\}
\end{align}
satisfy $\pathsp^i_{u,v}\cap \pathsp^j_{u,v}=\emptyset $ for $i\neq j$.
Let
\begin{align}\label{pi}
p^{u,v}_i=Q_{u,v}(\pathsp^i_{u,v})=\frac{Z_{u,ie_2}Z_{ie_2+e_1,v}}{Z_{u,v}}
\end{align}
be the quenched probability of paths going through the edge $e_i$.
For all $n\in\lzb-N^{2/3}, N^{2/3}\rzb$ we claim that
\begin{equation}\label{pi2}
p_0^{u,v}\leq (\mwalk^{u,v}_n)^{-1}.
\end{equation}
Here is the verification for $n\ge1$:
\begin{equation}\begin{aligned}\nonumber
p^{u,v}_0&\leq \frac{p^{u,v}_0}{p^{u,v}_n}
=\frac{Z_{u,\zevec}Z_{e_1,v}}{Z_{u,ne_2}Z_{ne_2+e_1,v}}
= \prod_{i=1}^n\frac{Z_{u,(i-1)e_2}Z_{(i-1)e_2+e_1,v}}{Z_{u,ie_2}Z_{ie_2+e_1,v}}
= \prod_{i=1}^n\frac{\rim{J}^{\hspace{0.9pt} v}_i}{J^u_i}= \prod_{i=1}^n (X^{u,v}_i)^{-1}=(\mwalk^{u,v}_n)^{-1}.
\end{aligned}\end{equation}
The case $n\le-1$ goes similarly.
We come to the key estimates. The first one controls the quenched probability of paths between $\mathcal{I}_{o,\mathbf{d}}$ and $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ that go through the edge $e_0$ from $\zevec$ to $e_1$.
\begin{lemma}\label{lm:close}
Let $r=N^{\frac2{15}}$ and $\mathbf{d}=(d_1,d_2)=(1,N^\frac18)$. There exist finite positive constants $C(\varepsilon)$ and $N_0(\varepsilon)$ such that, for all $N\ge N_0(\varepsilon)$ and $o\in \partial^N$ with $\rim o=-o$,
\begin{align*}
\P\Big(\sup_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\,\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}}p^{u,v}_0>N^{-1}\Big)\leq C(\varepsilon) (\log N)^6N^{-2/5}.
\end{align*}
\end{lemma}
\begin{proof}
For any $u\in\mathcal{I}_{o,\mathbf{d}}$ and $v\in\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$, by \eqref{pi2} and \eqref{sw5},
\begin{equation}\begin{aligned}\label{ine}
&\{p^{u,v}_0>N^{-1}\}\cap \{A_{o,\mathbf{d}}\cap B_{\rim{o},\mathbf{d}}\}
\subseteq \{\max_{n\tspa\in\tspa\lzb-N^{2/3}, N^{2/3}\rzb}\mwalk^{u,v}_n<N\}\cap (A_{o,\mathbf{d}}\cap B_{\rim{o},\mathbf{d}}) \\
&\qquad\qquad\qquad
\subseteq\bigl\{ \max_{-N^{2/3}\leq n\leq-1 }\mwalk'_n< 2N,\, \max_{1\leq n\leq N^{2/3}}\mwalk_n<2N\bigr\}\cap (A_{o,\mathbf{d}}\cap B_{\rim{o},\mathbf{d}}).
\end{aligned}\end{equation}
By the independence in \eqref{cor b-rw1},
\begin{equation}\label{2205} \begin{aligned}
\P\bigl(\max_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\,\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}}p^{u,v}_0>N^{-1}\bigr)
&\leq \P\bigl(\,\max_{-N^{2/3}\leq n\leq-1 }\mwalk'_n< 2N\bigr)\, \P\bigl(\,\max_{1\leq n\leq N^{2/3}}\mwalk_n<2N\bigr)\\
&\qquad\qquad
+2\tspa\P(A_{o,\mathbf{d}}^c\cup B_{\rim{o},\mathbf{d}}^c).
\end{aligned}\end{equation}
To apply the random walk bound from Appendix \ref{sec:rw}, we convert the multiplicative walks into additive walks. For given steps $\xi=\{\xi_i\}$ define the two-sided walk $S(\xi)$ by
\begin{align*}
\swalk_n(\xi)=
\begin{cases}
\sum_{i=1}^{n}\xi_i & n \geq 1 \\
0 & n=0\\
-\sum_{i=n+1}^{0}\xi_i & n < 0.
\end{cases}
\end{align*}
Recall the parameters defined in \eqref{2186}.
With reference to \eqref{2188} and \eqref{2192}, define the additive walks \begin{align*}
\swalk_n&=\log\mwalk_n
\qquad\text{with steps }
\ \xi_i=\log{J^{\rhoup{\rho}}_i}-\log {\rim{J}^{\hspace{0.9pt}\rhodown{\lambda}}_i} \,, \\
\swalk'_n&=\log\mwalk'_n \qquad\text{with steps }
\ \xi'_i=\log{J^{\rhodown{\rho}}_i}-\log {\rim{J}^{\hspace{0.9pt}\rhoup{\lambda}}_i}.
\end{align*}
With the bounds \eqref{Aod2} and \eqref{Bod2}, \eqref{2205} becomes
\begin{equation}\label{2208} \begin{aligned}
\P\bigl(\max_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\,\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}}p^{u,v}_0>N^{-1}\bigr)
&\leq \P\bigl(\,\max_{-N^{2/3}\leq n\leq-1 }\swalk'_n< \log(2N)\bigr)\, \P\bigl(\,\max_{1\leq n\leq N^{2/3}}\swalk_n<\log(2N)\bigr)\\
&+ Cr^{-3}.
\end{aligned}\end{equation}
We use Theorem \ref{thm:lm2} to bound $\P(\,\max_{1\leq n\leq N^{2/3}}\swalk_n<\log(2N))$.
Since
\[
\rhoup{\rho}= \rho(o_c)+rN^{-1/3}= \rho(o_c)+N^{-1/5}
\quad\text{and}\quad
\rhodown{\lambda}= \rho(\rim{o}_c) -rN^{-1/3}=\rho(\rim{o}_c) -N^{-1/5},
\]
we can establish constants $0<\rho_{\rm min}<\rho_{\rm max}<1$ and $N_0(\varepsilon)\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{>0}$ such that
$\rhoup{\rho}, \rhodown{\lambda} \in [\rho_{\rm min},\rho_{\rm max}]$ for all $o\in\partial^N$ and $N\ge N_0(\varepsilon)$.
As $|o-o_c|\leq \tfrac12d_1N^{2/3}$ and $|\rim{o}-\rim{o}_c|\leq \tfrac12d_2N^{2/3}$, the restriction of the slope to $[\varepsilon, \varepsilon^{-1}]$ implies that there is a constant $C=C(\varepsilon)$ such that
\begin{align*}
\abs{\tspa \rho(o_c)-\rho(o)\tspa} \leq Cd_1N^{-1/3}
\qquad\text{and}\qquad
\abs{\tspa \rho(\rim{o}_c)-\rho(\rim{o})} \leq Cd_2N^{-1/3}.
\end{align*}
Then, since $\rho(o)=\rho(-o)=\rho(\rim{o})$,
\begin{align*}
\abs{\hspace{0.9pt} \rho(\rim{o}_c)-\rho(o_c)\tsp}
\le \abs{\tspa \rho(\rim{o}_c)-\rho(\rim{o})}
+ \abs{\tspa \rho(o_c)-\rho(o)\tspa} \leq Cd_2N^{-1/3} + Cd_1N^{-1/3}
\le CN^{-5/24}.
\end{align*}
Hence
\begin{align*}
\rhodown{\lambda} - \rhoup{\rho}= \rho(\rim{o}_c)-\rho(o_c)-2rN^{-1/3}
\begin{cases}
\le -2N^{-1/5} ( 1 - CN^{-1/120}\hspace{0.9pt}) \\[3pt]
\ge -2N^{-1/5} ( 1 + CN^{-1/120}\hspace{0.9pt}) .
\end{cases}
\end{align*}
We conclude that for $N\ge N_0(\varepsilon)$, the mean step of $S_n$ satisfies
\begin{align*}
\bE(S_1)= \bE\bigl[ \log{J^{\rhoup{\rho}}_i}-\log {\rim{J}^{\hspace{0.9pt}\rhodown{\lambda}}_i}\bigr]
=\psi_0(\rhodown{\lambda}) -\psi_0(\rhoup{\rho})
\in [-CN^{-/1/5}, 0]
\end{align*}
where the (new) constant $C=C(\varepsilon)$ works for all $o\in\partial^N$.
In Theorem \ref{thm:lm2} set $x=(\log N)^2$
to conclude that for $N\ge N_0(\varepsilon)$
\begin{align}\label{la}
\P\bigl\{\sup_{1\leq n\leq N^{2/3}}2\swalk_n<(\log N)^2\bigr\}\leq C(\log N)^3 N^{-1/5} .
\end{align}
This bound with the same constant $C=C(\varepsilon) $ works for all points $o\in\partial^N$ and all $N\ge N_0(\varepsilon)$.
Similarly one can show that
\begin{align}\label{la2}
\P\bigl\{\sup_{-N^{2/3}\leq n\leq-1 }2\swalk'_n<(\log N)^2\bigr\}\leq
C(\log N)^3 N^{-1/5}.
\end{align}
The lemma follows by inserting these bounds and $r=N^{2/15}$ into \eqref{2208}.
\end{proof}
The next lemma controls the quenched probability of paths from points $u\in\mathcal{I}_{o,\mathbf{d}}$ that go through the edge $e_0$ from $\zevec$ to $e_1$ but miss the interval $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ on the northeast side of the square $\lzb-N,N\rzb^2$.
The complement of $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ on $\rim{\partial}^N$ is denoted by
\[
\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}=\{v\in \rim{\partial}^N: \abs{v-\rim{o}}_1>\tfrac12{d_2}N^{\frac{2}{3}}\}.
\]
\begin{figure
\begin{subfigure}{.5\textwidth}
\begin{tikzpicture}[scale=0.5, every node/.style={transform shape}]
\draw (0,0) rectangle (10,10);
\draw [dashed][line width=0.01cm] (0,5) -- (10,5);
\draw [dashed][line width=0.01cm] (5,0) -- (5,10);
\foreach 100 in {0,...,6}
{ \draw [fill] (0,100*1/5) circle [radius=0.07];
};
\foreach 100 in {0,...,3}
{ \draw [fill] (100*1/5,0) circle [radius=0.07];
};
\node [scale=2][above] at (1.2,0.3) {$\mathcal{I}_{o,\mathbf{d}}$};
\node [scale=2][above][red] at (-0.5,-0.7) {$o_c$};
\node [scale=2][above][cyan] at (-0.5,0.3) {$o$};
\draw [fill] (0,3/5) circle [radius=0.07][cyan];
\draw [fill] (0,0) circle [radius=0.07][red];
\draw [dashed][line width=0.01cm] (0,1/5) -- (0,3/5);
\foreach 100 in {0,...,8}
{ \draw [fill] (10,10-100*1/5) circle [radius=0.07];
};
\foreach 100 in {0,...,8}
{ \draw [fill] (10-100*1/5,10) circle [radius=0.07];
};
\draw [fill] (10,10) circle [radius=0.07][red];
\draw [fill] (10,9.4) circle [radius=0.07][cyan];
\node [scale=2][above][red] at (10,10) {$\rim{o}_c$};
\node [scale=2][above][cyan] at (10.5,9) {$\rim{o}$};
\node [scale=2][above] at (8.9,8) {$\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$};
\node [scale=2][above][blue] at (6.8,8.2) {$\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}$};
\node [scale=2][above][blue] at (8.8,5.6){$\rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}$};
\foreach 100 in {9,...,23}
{ \draw [fill] (10-100*1/5,10) circle [radius=0.07][blue];
};
\foreach 100 in {9,...,23}
{ \draw [fill] (10,10-100*1/5) circle [radius=0.07][blue];
};
\end{tikzpicture}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\begin{tikzpicture}[scale=0.5, every node/.style={transform shape}]
\draw (0,0) rectangle (10,10);
\draw [dashed][line width=0.01cm] (0,5) -- (10,5);
\draw [dashed][line width=0.01cm] (5,0) -- (5,10);
\foreach 100 in {1,...,9}
{ \draw [fill] (0,2+100*1/5) circle [radius=0.07];
};
\node [scale=2][above] at (1.2,2) {$\mathcal{I}_{o,\mathbf{d}}$};
\node [scale=2][above][red] at (-0.6,1.7) {$o_c$};
\node [scale=2][above][cyan] at (-0.6,2.6) {$o$};
\draw [fill] (0,3) circle [radius=0.07][cyan];
\draw [fill] (0,2.2) circle [radius=0.07][red];
\draw [dashed][line width=0.01cm] (0,1/5) -- (0,3/5);
\foreach 100 in {1,...,9}
{ \draw [fill] (10,6+100*1/5) circle [radius=0.07];
};
\draw [fill] (10,7.8) circle [radius=0.07][red];
\draw [fill] (10,7) circle [radius=0.07][cyan];
\node [scale=2][above][red] at (10.6,7.3) {$\rim{o}_c$};
\node [scale=2][above][cyan] at (10.6,6.5) {$\rim{o}$};
\node [scale=2][above] at (9,6.5) {$\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$};
\node [scale=2][above][blue] at (7.2,8.2) {$\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}$};
\node [scale=2][above][blue] at (8.8,4.8) {$\rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}$};
\foreach 100 in {0,...,23}
{ \draw [fill] (10-100*1/5,10) circle [radius=0.07][blue];
};
\foreach 100 in {1,...,4}
{ \draw [fill] (10,5.2+100*1/5) circle [radius=0.07][blue];
};
\foreach 100 in {1,...,10}
{ \draw [fill] (10,10-100*1/5) circle [radius=0.07][blue];
};
\end{tikzpicture}
\end{subfigure}
\caption{\small The square $\lzb-N,N\rzb^2$ with two possible arrangements of the segments ${\mathcal{I}}_{o,\mathbf{d}}$, $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ and $\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}=\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}\cup\rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}$ on the boundary of the square. In both cases $\rim{o}=-o$.}
\label{fig:points}
\end{figure}
\begin{lemma} \label{lm:far}
Let $\mathbf{d}=(d_1,d_2)=(1,N^\frac{1}{8})$. There are finite constants $C(\varepsilon)$ and $N_0(\varepsilon)$ such that, for all $\delta>0$, $N\ge N_0(\varepsilon)$ and $o\in \partial^N$ with $\rim{o}=-o\in\rim\partial^N$,
\begin{equation}\label{far7}
\P\Big(\sup_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\,\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}} p^{u,v}_0>\delta\Big)\leq C(\varepsilon)\delta^{-1} N^{-\frac{3}{8}}.
\end{equation}
\end{lemma}
\begin{proof}
Define the sets of boundary points
\begin{align*}
&\partial \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}=\{v\in \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}:\exists u\in \rim{\mathcal{I}}_{\rim{o},\mathbf{d}} \text{ such that } \abs{v-u}_1=1\}\\
&\partial \mathcal{I}_{o,\mathbf{d}}=\{v\in \mathcal{I}_{o,\mathbf{d}} :\exists u\in \partial^N\setminus \mathcal{I}_{o,\mathbf{d}} \text{ such that } \abs{v-u}_1=1\},
\end{align*}
Their cardinalities satisfy $1\le |\partial \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}|\leq |\partial \mathcal{I}_{o,\mathbf{d}}| \leq 2$. (For example, $\partial \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}$ is a singleton if $\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}$ contains one of the endpoints $(N, \fl{\varepsilon N})$ or $(\fl{\varepsilon N}, N)$ of $\rim{\partial}^N$.)
We denote the points of $\partial \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}$ by $q^1,q^2$ and those of $\partial \mathcal{I}_{o,\mathbf{d}}$ by $h^1,h^2$,
labeled so that
\[ q^1\preccurlyeq \rim{o} \preccurlyeq q^2
\qquad\text{and}\qquad
h^2 \preccurlyeq o_1 \preccurlyeq h^1. \]
Geometrically, starting from the north pole $Ne_2$ and traversing the boundary of the square $\lzb-N,N\rzb^2$ clockwise, we meet the points (those that exist) in this order: $q^1\to \rim{o}\to q^2\to h^1\to o\to h^2$ (Figure \ref{fig:bigdev}). The set $\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}$ can be decomposed into two disjoint sets
\begin{align*}
\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}=\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}\cup \rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}
\end{align*}
where
\begin{align*}
\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}=\{v\in \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}:v\preccurlyeq q^1\} \qquad\text{and}\qquad
\rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}=\{v\in \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}: v\succcurlyeq q^2\}.
\end{align*}
We show that
\begin{equation}\label{2206}
\P\Big(\sup_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\,\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}} p^{u,v}_0>\delta\Big)\leq C(\varepsilon)\delta^{-1} N^{-\frac{3}{8}}.
\end{equation}
The same bound can be shown for $\rim{\mathcal{F}}^2_{\rim{o},\mathbf{d}}$ and the lemma follows from a union bound.
Recall the definition of $\pathsp^i_{u,v}$ in \eqref{tz} and define the set
\begin{align}\label{pm}
\pathsp^-_{u,v}=\bigcup_{i\leq 0}\pathsp^i_{u,v}.
\end{align}
For all $u\in\mathcal{I}_{o,\mathbf{d}}$ and $v\in\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}$, the pairs $(u,v)$ and $(h^1,q^1)$ satisfy the relation $(u,v)\preccurlyeq (h^1,q^1)$ defined in \eqref{def:ogg}.
By Lemma \ref{lem:opm} we can couple random paths $\pi^{u,v}\sim Q_{u,v}$ and $\pi^{h^1,q^1}\sim Q_{h^1,q^1}$ so that $\pi^{u,v}\preccurlyeq\pi^{h^1,q^1}$
in the path ordering defined in Appendix \ref{sec:p-ord},
simultaneously for all $u\in\mathcal{I}_{o,\mathbf{d}}$ and $v\in\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}$. Then $\pi^{u,v}\in \pathsp^0_{u,v}$ forces $\pi^{h^1,q^1}\in\pathsp^-_{h^1,q^1}$, and we conclude that
\begin{align*}
p^{u,v}_0=Q_{u,v}(\pathsp^0_{u,v})
\leq Q_{h^1,q^1}\big(\pathsp^-_{h^1,q^1}\big)
\qquad\text{for all } \
u\in\mathcal{I}_{o,\mathbf{d}},\,v\in\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}.
\end{align*}
Hence
\[ \text{the probability on the left of \eqref{2206}} \;\le\; \P\{ Q_{h^1,q^1}(\pathsp^-_{h^1,q^1})>\delta\}. \]
The last probability will be shown to be small by appeal to a KPZ wandering exponent bound from \cite{sepp-12-aop-corr} stated in Appendix \ref{sec:kpz5}. To this end we check that the line segment $[h^1,q^1]$ from $h^1$ to $q^1$ crosses the vertical axis far above the origin on the scale $N^{2/3}$.
For $o\in \partial^N$ and $\rim{o}=-o\in\rim\partial^N$, decompose $h^j=o+l^j$ and $q^j=\rim{o}+r^j$. These vectors $l^j=(l^j_1, l^j_2)$ and $r^j=(r^j_1, r^j_2)$ satisfy
\begin{equation}\label{e^uv} \abs{l^j}_1= \tfrac12d_1N^\frac{2}{3}, \quad \abs{r^j}_1= \tfrac12d_2N^\frac{2}{3},
\quad\text{and}\quad r^j_1r^j_2\leq 0. \end{equation}
Use first the definition of $h^j$ and then $q^j_i-h^j_i= \rim{o}_i+r^j_i-(o_i+l_i^j) = -2o_i +r^j_i- l^j_i$ to obtain
\begin{equation} \label{234} \begin{aligned}
h^j_2 - \frac{q^j_2-h^j_2}{q^j_1-h^j_1}h^j_1
&=o_2- \frac{q^j_2-h^j_2}{q^j_1-h^j_1}o_1 +l^j_2 - \frac{q^j_2-h^j_2}{q^j_1-h^j_1} l^j_1 \\[4pt]
&= \frac{o_2r^j_1-o_1r^j_2}{q^j_1-h^j_1} - \frac{o_2l^j_1-o_1l^j_2}{q^j_1-h^j_1} +l^j_2 - \frac{q^j_2-h^j_2}{q^j_1-h^j_1} l^j_1 .
\end{aligned}\end{equation}
The first term on the last line is of order $\Theta(d_2N^{2/3})$ because there is no cancellation in the numerator. It is positive if $j=1$ and negative if $j=2$. This term dominates because $d_2=N^\frac{1}{8}>>1=d_1$.
Let $y^1e_2\in[h^1,q^1]$, that is, $y^1$ is the distance from the origin to the point where the line segment $[h^1,q^1]$ crosses the $y$-axis. We bound this quantity from below. In addition to \eqref{e^uv}, utilize $\;-N\le o_i\le -\varepsilon N$, $2N\varepsilon\le q^j_i-h^j_i\le 2N$ and the slope bound $\varepsilon\le \frac{q^j_2-h^j_2}{q^j_1-h^j_1}\le \varepsilon^{-1}$. The last line of \eqref{234} gives
\begin{equation}\label{237-1} \begin{aligned}
y^1=
h^1_2 +\frac{q^1_2-h^1_2}{q^1_1-h^1_1}(-h^1_1)
&\ge \frac{\varepsilon N \abs{r^1}_1}{ 2N} - \Bigl( \frac{N}{2N\varepsilon} +1+\varepsilon^{-1}\Bigr) \abs{l^1}_1 \\[4pt]
&\ge \tfrac14\varepsilon d_2N^\frac{2}{3} - 2\varepsilon^{-1} d_1N^\frac{2}{3}
\ge \tfrac1{8}\varepsilon d_2N^\frac{2}{3}.
\end{aligned} \end{equation}
The last inequality used $(d_1, d_2)=(1,N^{1/8})$ and took $N\ge (16\varepsilon^{-2})^8$.
The wandering exponent bound stated in Theorem \ref{thm:kpz5} gives
\begin{align*}
P_{h^1,q^1}(\pathsp^-_{h^1,q^1})\leq C(\varepsilon)d_2^{\,-3}
\end{align*}
for a constant $C(\varepsilon)$ that works for all $o\in\partial^N$ and $N\ge N_0(\varepsilon)$.
By Markov's inequality
\begin{align}\label{ub}
\P\{ Q_{h^1,q^1}(\pathsp^-_{h^1,q^1})>\delta\} \leq C(\varepsilon)\delta^{-1}d_2^{\,-3}=C(\varepsilon)\delta^{-1}N^{-3/8}.
\end{align}
The proof of \eqref{2206} is complete.
\end{proof}
We combine the estimates from above to cover all vertices on $\partial^N$ and $\rim\partial^N$.
\begin{figure}
\includegraphics[width=8.3cm]{Fig_bigdev.pdf}
\caption{\small Illustration of the proof of Lemma \ref{lm:far}. The path $\pi^{u,v}$ connects $\mathcal{I}_{o,\mathbf{d}}$ and $\rim{\mathcal{F}}^1_{\rim{o},\mathbf{d}}$ through the edge $e_0=((0,0),(1,0))$. The path $\pi^{\,h^1\!, \,q^1}$ lies below $\pi^{u,v}$ and hence well below the $[h^1, q^1]$ line segment (dashed line).}
\label{fig:bigdev}
\end{figure}
\begin{theorem}\label{thm:ub} There exist constants $C(\varepsilon), N_0(\varepsilon)$ such that for $\delta\in(0,1)$ and $N\ge \delta^{-1}\vee N_0(\varepsilon)$,
\[
\P\Big(\; \sup_{u\,\in\,\partial^N,\,v\,\in\,\rim{\partial}^N} p^{u,v}_0>\delta \Big)\leq C(\varepsilon)\delta^{-1}N^{-\frac{1}{24}}.
\]
\end{theorem}
\begin{proof}
As before, $\mathbf{d}=(1,N^\frac{1}{8})$.
We first claim that for any $o\in \partial^N$,
\begin{equation}\label{ub43}
\P\Big(\sup_{u\hspace{0.9pt}\in\hspace{0.9pt}\mathcal{I}_{o,\mathbf{d}}\tspa, \, v\,\in\,\rim{\partial}^N
} p^{u,v}_0>\delta\Big)\leq C(\varepsilon)\delta^{-1} N^{-\frac{3}{8}}.
\end{equation}
This comes from a combination of Lemmas \ref{lm:close} and \ref{lm:far}: since $\rim{\partial}^N=\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}\cup \rim{\mathcal{F}}_{\rim{o},\mathbf{d}}$,
\begin{align*}
\P\Big(\sup_{u\hspace{0.9pt}\in\hspace{0.9pt}\mathcal{I}_{o,\mathbf{d}},v\,\in\,\rim{\partial}^N
} p^{u,v}_0>\delta\Big)
&\leq \P\Big(\sup_{u\hspace{0.9pt}\in\hspace{0.9pt}\mathcal{I}_{o,\mathbf{d}},v\,\in\,\rim{\mathcal{I}}_{\rim{o},\mathbf{d}}
} p^{u,v}_0>\delta\Big)+\P\Big(\sup_{u\hspace{0.9pt}\in\hspace{0.9pt}\mathcal{I}_{o,\mathbf{d}},v\,\in\,\rim{\mathcal{F}}_{\rim{o},\mathbf{d}}
} p^{u,v}_0>\delta\Big) \\
&\le C(\varepsilon)(\log N)^6N^{-\frac25}
+ C(\varepsilon)\delta^{-1} N^{-\frac{3}{8}}
\leq C(\varepsilon)\delta^{-1} N^{-\frac{3}{8}}.
\end{align*}
Next we coarse grain the southwest boundary $\partial^N$. Let
\begin{align*}
\mathcal{O}^N= \partial^N \cap \Big(\big\{(-N +id_1\fl{N^{\frac{2}{3}}}\,,-N)\big\}_{i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{\ge0}} \; \bigcup \,\big\{(-N,-N+jd_1\fl{N^{\frac{2}{3}}})\big\}_{j\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_{\ge0}}\Big)
\end{align*}
so that
\begin{align*}
\Big\{\sup_{u\,\in\,\partial^N,\,v\,\in\,\rim{\partial}^N} p^{u,v}_0>\delta\Big\} \subseteq \Big\{\sup_{o\,\in\, \mathcal{O}^N} \sup_{u\,\in\,\mathcal{I}_{o,\mathbf{d}},\,v\,\in\, \rim{\partial}^N} p^{u,v}_0>\delta\Big \}.
\end{align*}
As $|\mathcal{O}^N|\leq C(\varepsilon) d_1^{-1}N^{1-\frac{2}{3}}=C(\varepsilon)N^\frac{1}{3}$, a union bound and \eqref{ub43} give the conclusion:
\begin{align*}
\P\Big(\sup_{u\,\in\,\partial^N,\,v\,\in\,\rim{\partial}^N} p^{u,v}_0>\delta\Big)
&\leq \sum_{o\,\in\,\mathcal{O}^N} \P\Big(\; \sup_{u\,\in\,\mathcal{I}_{o,\mathbf{d}}, \,v\,\in\,\rim{\partial}^N} p^{u,v}_0>\delta\Big)\\
&\leq C(\varepsilon)N^{\frac{1}{3}}\delta^{-1}N^{-\frac{3}{8}}=C(\varepsilon)\delta^{-1}N^{-\frac{1}{24}}.
\qedhere
\end{align*}
\end{proof}
%
\medskip
\section{Proof of the main theorem}
\label{sec:pf-main}
\begin{proof}[Proof of Theorem \ref{thm:noex}]
By Theorem \ref{thm:e_i-mu}(b), for almost every $\omega$ every bi-infinite Gibbs measure $\mu$ satisfies
\begin{equation}\label{3067} \begin{aligned}
&\bigl\{ \tsp\varliminf_{\abs n\to\infty} \abs{n^{-1} X_n\cdote_1} =0\bigr\}
\cup
\bigl\{ \tsp\varliminf_{\abs n\to\infty} \abs{n^{-1} X_n\cdote_2} =0\bigr\} \\
&\qquad\qquad =
\{\text{$X_\bbullet$ is a bi-infinite straight line}\}
\qquad\text{$\mu$-almost surely}
\end{aligned}\end{equation}
where $X_\bbullet=X_{-\infty:\infty}$ is the bi-infinite polymer path under the measure $\mu$.
This equality follows because Theorem \ref{thm:e_i-mu}(b) has these consequences for \eqref{3067}: the union on the left is disjoint, the event on the right is a subset of the union on the left, and their $\mu$-probabilities are equal. The complement of the union on the left is the following event: the limit points of $\abs{n}^{-1}X_n$ lie in $\,]-e_2, -e_1[$ when $n\to-\infty$ and in $\,]e_2, e_1[$ when $n\to\infty$.
Thus to complete the proof we show the existence of an event $\Omega'$ such that $\P(\Omega')=1$ and for each $\omega\in\Omega'$, no $\mu\in \overleftrightarrow{\text{\rm DLR}}^\omega$ assigns positive probability to this last property of the limit points of $\abs{n}^{-1}X_n$.
\medskip
We put $\varepsilon$ back into the notation. For $\varepsilon>0$ let
\begin{align*}
\cD^\varepsilon=\{\xi\in\,]e_2,e_1[\,:\varepsilon^{1/2} \leq \xi_2/\xi_1\leq {\varepsilon}^{-1/2}\}.
\end{align*}
Say that a bi-infinite path $x_\bbullet$ is $(-\cD^{\varepsilon})\times \cD^{\varepsilon}$-directed if the limit points of $\abs{n}^{-1}x_n$ lie in $-\cD^{\varepsilon}$ when $n\to-\infty$ and in $\cD^{\varepsilon}$ when $n\to\infty$.
Recall the definition of the edges $e_i=(ie_2,ie_2+e_1)$ and define these sets of bi-infinite paths:
\begin{align*}
\pathsp^{\varepsilon, i}=\big\{x_\bbullet\in\pathsp: \text{ $x_\bbullet$ is $(-\cD^{\varepsilon})\times \cD^{\varepsilon}$-directed and $x_\bbullet$ goes through $e_i$}
\big\}.
\end{align*}
We show the existence of an event $\Omega'$ of full $\P$-probability such that, for $\omega\in\Omega'$, $\mu\in \overleftrightarrow{\text{\rm DLR}}^\omega$, $\varepsilon>0$, and $i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN$,
\begin{align}\label{c00}
\mu(\pathsp^{\varepsilon, i})=0.
\end{align}
Assume this proved.
Let $\varepsilon_k=2^{-k}$. Then for $\omega\in\Omega'$ and $\mu\in \overleftrightarrow{\text{\rm DLR}}^\omega$,
\begin{align*}
\mu\{\text{$X_\bbullet$
is $\,]-e_2,-e_1[\,\times\,]e_2,e_1[\,$-directed}\}
&\le \sum_{k\ge1} \mu \{ \text{$X_\bbullet$ is $(-\cD^{\varepsilon_k})\times\cD^{\varepsilon_k}$-directed}\} \\
&\le \sum_{k\ge1}\sum_{i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN}\mu\big(\pathsp^{\varepsilon_k, i}\big)
=0,
\end{align*}
which is the required result.
\medskip
It remains to define the event $\Omega'$ and verify \eqref{c00}. Recall the definition \eqref{pi} of $p^{u,v}_i$. Define translations $T_x$ on weight configurations $\omega=(Y_x)$ by $(T_x\omega)_y=Y_{x+y}$. Define
\[ \xi^{\varepsilon}_N= \sup_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N\!,\tsp\varepsilon}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim{\partial}^{N\!,\tsp\varepsilon}}p^{u,v}_0\,,
\qquad
\Omega''_{\varepsilon}=\bigl\{ \varliminf_{N\to\infty} \xi^{\varepsilon}_{N+\ce{N^{2/3}}}=0\bigr\}
\qquad\text{and}\qquad
\Omega'=\bigcap_{k\ge 1} \bigcap_{i\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN} T_{ie_2}\Omega''_{\varepsilon_k} . \]
By Theorem \ref{thm:ub}, $\xi^{\varepsilon}_{N}\to0$ in probability as $N\to\infty$, and hence $\P(\Omega')=\P(\Omega''_{\varepsilon})=1$.
A $(-\cD^\varepsilon)\times\cD^{\varepsilon}$-directed bi-infinite path intersects both $ \partial^{N\!,\tspa\varepsilon}$ and $\rim\partial^{N\!,\tspa\varepsilon}$ for all large enough $N$. (This is because $\cD^\varepsilon$ bounds the slopes by $\varepsilon^{1/2}$ which is larger than $\varepsilon$.) Thus if we let
\begin{equation*}
\pathsp^{N,\varepsilon, i}=\{x_\bbullet\in\pathsp^{\varepsilon, i}:x_\bbullet\cap\partial^{N\!,\tspa\varepsilon}\neq\emptyset, \,x_\bbullet\cap\rim\partial^{N\!,\tspa\varepsilon}\neq\emptyset\}
\end{equation*}
then
\begin{equation}\label{c45} \pathsp^{\varepsilon, i} = \bigcup_{m\ge 1} \bigcap_{N\ge m} \pathsp^{N,\varepsilon, i}. \end{equation}
\begin{figure}
\includegraphics[width=10cm]{double_rec_crop.pdf}
\caption{\small The inner $N\times N$ square is centered at $\zevec$ while the outer $N_1\times N_1$ square is centered at $ie_2$. The (thick, dark) boundary segments of the outer square cover the (thick, light) boundary segments of the inner square. Thus the path through $ie_2$ that crosses $\partial^{N,\varepsilon}$ and $\rim\partial^{N,\varepsilon}$ is forced to also cross $ie_2+\partial^{N_1,\tspa\varepsilon'}$ and $ie_2+\rim\partial^{N_1,\tspa\varepsilon'}$.}
\label{fig:2sq}
\end{figure}
Let $\varepsilon=2^{-k}$ for some $k\ge 1$, $\varepsilon'=\varepsilon/2$, and abbreviate $N_1=N+\ce{N^{2/3}}$.
In the scale $N_1$ consider the translated square $ie_2+\lzb -N_1, N_1\rzb^2$ centered at $ie_2$, with its boundary portions $ie_2+\partial^{N_1,\tspa\varepsilon'}$ in the southwest and $ie_2+\rim\partial^{N_1,\tspa\varepsilon'}$ in the northeast. This translated $N_1$-square contains $\lzb-N,N\rzb^2$ for all $i\in\lzb-N^{2/3}, N^{2/3}\rzb$.
There exists a finite constant $N_0(\varepsilon)$ such that $\abs i+\varepsilon'N_1\le \varepsilon N$ for all $i\in\lzb-N^{2/3}, N^{2/3}\rzb$ and $N\ge N_0(\varepsilon)$.
Then every path $x_\bbullet\in\pathsp^{\varepsilon, \tsp N, \tsp i}$ necessarily goes through both $ie_2+\partial^{N_1,\tspa\varepsilon'}$ and $ie_2+\rim\partial^{N_1,\tspa\varepsilon'}$. In other words, $x_\bbullet$ is a member of the translate $ie_2+\pathsp^{\varepsilon', \tsp N_1, \tsp 0}$ of the class of paths that go through the edge $e_0$. This is illustrated in Figure \ref{fig:2sq}.
On the event $\pathsp^{\varepsilon, \tsp N, \tsp i}$ let, in the coordinatewise ordering, $X_\partial=\inf\{X_\bbullet\cap (ie_2+\partial^{N_1,\tspa\varepsilon'})\}$ be the first vertex of the path $X_\bbullet$ in $ie_2+\partial^{N_1,\tspa\varepsilon'}$ and
$X_{\rim{\partial}}=\sup\{X_\bbullet\cap (ie_2+\rim\partial^{N_1,\tspa\varepsilon'})\}$ the last vertex of the path in $ie_2+\rim\partial^{N_1,\tspa\varepsilon'}$. Note that for $u\in(ie_2+\partial^{N_1,\tspa\varepsilon'})$ and
$v\in(ie_2+\rim\partial^{N_1,\tspa\varepsilon'})$, the event $\{X_\partial=u, X_{\rim{\partial}}=v\}$ depends on the entire path $X_\bbullet$ only through its edges outside $ie_2+\lzb-N_1,N_1\rzb^2$.
Suppose $\mu(\pathsp^{N,\varepsilon, i})>0$ for some $\mu\in \overleftrightarrow{\text{\rm DLR}}^\omega$. Below we apply the Gibbs property, recall the definition \eqref{tz} of $\pathsp_{u,v}^0$ as the set of paths from $u$ to $v$ that take the edge $e_0=(\zevec, e_1)$, and write $Q^\omega$ so that we can include explicitly translation of the weights $\omega$.
\begin{align*}
\mu(\pathsp^{N,\varepsilon, i})&\le \mu(ie_2+\pathsp^{\varepsilon', \tsp N_1, \tsp 0})
\\
&\le \sum_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N_1\!,\tspa\varepsilon'}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim\partial^{N_1\!,\tspa\varepsilon'}}
\mu(ie_2+\pathsp_{u,v}^0\,\vert\,X_\partial=ie_2+u,X_{\rim{\partial}}=ie_2+v)\hspace{0.9pt}\mu(X_\partial=ie_2+u,X_{\rim{\partial}}=ie_2+v)\\
&=\sum_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N_1\!,\tspa\varepsilon'}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim\partial^{N_1\!,\tspa\varepsilon'}} Q^\omega_{ie_2+u,\tsp ie_2+v}(ie_2+\pathsp_{u,v}^0)
\hspace{0.9pt}\mu(X_\partial=ie_2+u,X_{\rim{\partial}}=ie_2+v)\\
&\le \max_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N_1\!,\tspa\varepsilon'}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim\partial^{N_1\!,\tspa\varepsilon'}} Q^\omega_{ie_2+u,\tsp ie_2+v}(ie_2+\pathsp_{u,v}^0)
=\max_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N_1\!,\tspa\varepsilon'}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim\partial^{N_1\!,\tspa\varepsilon'}} Q^{T_{ie_2}\omega}_{u,\tsp v}(\pathsp_{u,v}^0) \\
&=
\max_{u\hspace{0.9pt}\in\hspace{0.9pt}\partial^{N_1\!,\tspa\varepsilon'}\!,\,v\hspace{0.9pt}\in\hspace{0.9pt}\rim\partial^{N_1\!,\tspa\varepsilon'}}p^{u,v}_0(T_{ie_2}\omega) = \xi^{\varepsilon'}_{N_1}(T_{ie_2}\omega).
\end{align*}
Then \eqref{c45} gives, on the event $\Omega'$,
\[
\mu(\pathsp^{\varepsilon, i}) \le \varliminf_{N\to\infty} \mu(\pathsp^{N,\varepsilon, i}) \le \varliminf_{N\to\infty} \xi^{\varepsilon'}_{N_1}\circ T_{ie_2} =0.
\]
\eqref{c00} has been verified. This completes the proof of the main result Theorem \ref{thm:noex}.
\end{proof}
\smallskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 88 |
Q: C++ type based caching without using static storage I'm using something like:
struct VectorCache
{
template<typename T>
std::vector<T>& GetTs()
{
static std::vector<T> ts;
return ts;
}
};
to create/access some vectors based on the contained type. This works fine as long as I have only one object of type VectorCache, but when I use multiple objects I will get same vectors from all instances of VectorCache as the vectors are static variables.
I tried to move the vectors as member variables using something similar to boost::any and access them using std::type_index of T, but this is somehow slower than the direct access I used before.
Another options is to transform struct VectorCache to something like template<int index> struct VectorCache, but the problem is still there - I will have to be careful to have only one instance/index to have correct behaviour.
Is it possible to access the vectors directly based on T and also have the caching instance based instead of class based?
A: You could try an unchecked analogue of Boost.Any. See if that's fast enough for you (though I don't believe it would make a big difference):
#include <memory>
#include <type_traits>
#include <typeindex>
#include <unordered_map>
#include <vector>
class AnyCache
{
struct TEBase
{
virtual ~TEBase() {}
virtual void * get() = 0;
};
template <typename T> struct TEObject : TEBase
{
T obj;
virtual void * get() override { return static_cast<void *>(&obj); }
};
std::unordered_map<std::type_index, std::unique_ptr<TEBase>> cache;
public:
AnyCache(AnyCache const &) = delete;
AnyCache & Operator=(AnyCache const &) = delete;
template <typename T> decltype(auto) get()
{
using U = std::decay_t<T>;
using C = std::vector<U>;
std::unique_ptr<TEBase> & p = cache[typeid(U)];
if (!p) { p = std::make_unique<TEObject<C>>(); }
return *static_cast<C *>(p->get());
}
};
Usage:
AnyCache ac;
ac.get<int>().push_back(20);
ac.get<std::string>().push_back("Hello");
for (auto const & x : ac.get<Foo>()) { std::cout << x << '\n'; }
A: If - and it's a big if - your VectorCache-using code isn't threaded, you can do this:
struct VectorCache
{
VectorCache() : instance_counter_(++s_instance_counter_) { }
template<typename T>
std::vector<T>& GetTs()
{
static std::vector<std::vector<T>> tss;
if (tss.size() <= instance_counter_)
tss.resize(instance_counter_);
return tss[instance_counter_];
}
size_t instance_counter_;
static size_t s_instance_counter_;
};
// and define size_t VectorCache::s_instance_counter_;
implementation on ideone.com
With a little synchronisation you can make it thread safe, or even thread specific if that suits. Add deletion of copy construction / assignment etc. if that makes sense in your intended usage.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 563 |
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More Information: Documentation, Community Support.\n\n\u2022 I have tried to expand the Idea-section at orbit method a little.\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 promted by demand from my Basic-Course-On-Category-Theory-Students I expanded the entry 2-category:\n\n\u2022 mentioned more relations to other concepts in the Idea-section;\n\n\u2022 added an Examples-section with a bunch of (classes of) examples;\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 I have received an email asking for clarification at the (old) entry equivalence of 2-categories, as to the meaning of \u201cessentially full\u201d. I have briefly added a parenthetical \u201ci.e. essentially surjective on hom-categories\u201d. But the entry deserves to be expanded a bit more, maybe somebody feels inspired to do so?\n\n\u2022 Added actual definition for pseudofunctor and modified notions, moved discussion from idea section to new discussion section at bottom of page.\n\n\u2022 brief category:people-entry for hyperlinking references at tricategory\n\n\u2022 starting something, for the moment mainly to record the other result of Brown & Szczarba (dg-algebraic rational homotopy theory for general connected spaces)\n\n\u2022 Added to the entry fuzzy dark matter pointer to Lee 17 which appeared today on the preprint server. This is just a concise 2.5 page survey of all the available literature, but as such is very useful. For instance it points out this Nature-article:\n\n\u2022 Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nature Physics 10, 496\u2013499 (2014) (doi:10.1038\/nphys2996)\n\nwhich presents numerical simulation of the fuzzy dark matter model compared to experimental data.\n\n\u2022 Create a stub page.\n\n\u2022 Thanks to Karol Szumi\u0142o\u2019s answer to my MO question, I have added to Brown representability theorem a mention of the counterexamples for nonconnected pointed spaces and for unpointed spaces (plus a mention of Brown\u2019s abstract categorical version).\n\n\u2022 Page created, but author did not leave any comments.\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 a category:reference-entry\n\n\u2022 Page created. Idempotent monoids should be to monoids as idempotent monads are to monads.\n\nI\u2019ve added the examples of idempotent elements in (ordinary) monoids (1), idempotent morphisms in categories (2), solid rings (3), idempotent monads (4), idempotent $1$-morphisms in bicategories (5), and \u201csolid ring spectra\u201d (6) \u2015What are other examples?\n\nAlso, should idempotent monoids have a unit? The examples 1 and 2 I mentioned above don\u2019t, but 3, 4, and 6 do, while whether 5 does or doesn\u2019t seems to vary a bit among the literature (AFAIU).\n\n\u2022 Maximilian Dor\u00e9, Samson Abramsky, Towards Simplicial Complexes in Homotopy Type Theory (pdf)\n\n\u2022 starting something\n\n\u2022 tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.\n\n\u2022 I have added pointers to Mikhail Kapranov\u2019s talks on the sphere spectrum in relation to super-algebra, and added some words at the beginning that this was the original motivation for the proposed definition of spectral supergeometry in the entry.\n\nAlso I fixed the link to the video recording of Krapranov\u2019s 2013 talk. The previous link no longer worked but there is a YouTube copy of the video. Fixed this also at superalgebra, see there at Kaprananov 13\n\n\u2022 Baez and Dolan mainly did the periodic table of k-tuply monoidal n-categories; this article was written like all we did was \u201cslightly distort\u201d some existing table.\n\n\u2022 A couple of properties.\n\n\u2022 Explain the connection with enriched monads\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 brief category:people-entry for hyperlinking references\n\n\u2022 starting something. There is nothing to be seen yet, but I need to save.\n\n\u2022 added a brief section (here) on the original \u201cConner-Floyd orientation\u201d\n\n$\\array{ M SU &\\longrightarrow& K \\mathrm{O} \\\\ \\big\\downarrow && \\big\\downarrow \\\\ M \\mathrm{U} &\\longrightarrow& K \\mathrm{U} }$\n\u2022 I noticed that the entry classifying space is in bad shape. I have added a table of contents and tried to structure it slightly, but much more needs to be done here.\n\nI have added a paragraph on standard classifying spaces for topological principal bundles via the geometric realization of the simplicial space associated to the given topological group.\n\nIn the section \u201cFor crossed complexes\u201d there is material that had been provided by Ronnie Brown which needs to be harmonized with the existing Idea-section. 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\section{Introduction}
In the first two papers of this series (Anguige and Tod 1998a,b, henceforth ATI, ATII) consideration was given to the Cauchy problem for the conformal Einstein equations near an isotropic singularity for two different matter models: the perfect fluid (ATI) and the collisionless gas (ATII). This problem was solved in full generality for the perfect fluid but for the collisionless gas we had to assume spatial homogeneity to make progress. In this paper we again take the collisionless gas as matter model, the aim being to extend the existence and uniqueness results of (ATII) to the inhomogeneous case.
Recall that a spacetime~$(\tilde{M},\tilde{g}_{ab})$~has an isotropic singularity if there exists a manifold~$M\supset\tilde{M}$, a smooth Lorentz metric~$g_{ab}$~on~$M$, and a function~$Z$~defined on~$M$~such that
\begin{equation}\tilde{g}_{ab}=Z^{2}g_{ab}~~~\textrm{for}~~Z>0\end{equation}
\begin{equation}Z\rightarrow 0~~~\textrm{on}~\Sigma\end{equation}
for some spacelike hypersurface~$\Sigma$~in~$M$. Given some matter model one seeks to generate cosmologies with isotropic singularities by solving the Cauchy problem for the conformal Einstein equations, with data given for the unphysical metric~$g_{ab}$~and the matter at the singularity~$\Sigma$.
In (ATI) it was shown that with a polytropic perfect fluid as matter source the free data for the conformal field equations consist of just the 3-metric of the singularity: the second fundamental form of the singularity has to vanish and there is no data for the matter. Every choice of 3-metric then determines a unique~$\gamma-$law cosmology with an isotropic singularity.
With a collisionless gas as source the field equations of general relativity are known as the Einstein-Vlasov (EV) equations, for the metric~$\tilde{g}_{ab}$~and the particle distribution function~$\tilde{f}$. In (ATII) it was shown that the massless EV system transforms nicely under a conformal rescaling as in (1)-(2) and that one may identify the free data for the field equations at the singularity surface~$\Sigma$. The situation is rather different from the fluid case. One is free to prescribe the limiting particle distribution function~$f^{0}$~at the singularity, subject to a single integral constraint. This function then determines the first and second fundamental forms of the singularity surface. The second fundamental form need not be zero here.
Under the assumption of spatial homogeneity for the metric and the matter it was proved in (ATII) that each~$f^{0}$~determines a unique EV cosmology with an isotropic singularity. In the present paper we show that the same result holds in the absence of any symmetries.
With a convenient choice of conformal gauge and in harmonic coordinates, the inhomogeneous field equations may be written as a symmetric hyperbolic system of integro-differential equations with a~$\frac{1}{Z}$~forcing term, for certain carefully chosen matter and metric variables. These equations are essentially of the form studied in (Claudel and Newman 1998), but the matter integrals present in the Einstein equations prevent a direct application of the existence and uniqueness result obtained there. It is however possible to combine certain techniques used in this work with the standard method of energy estimates to obtain the desired result. We note that the semi-group theory used by Claudel and Newman is entirely avoided in our approach.
The paper is organised as follows:
In section 2 a brief review of the Einstein-Vlasov sytem and its behaviour under conformal rescaling is given.
In section 3 we collect a few results from (ATII) on the initial data for the field equations and on conformal gauge fixing.
Section 4 contains a proof of the main result, which is summarised in Theorem 4.1 at the end of the paper.
\section{The massless Einstein-Vlasov system and conformal rescaling}
A collisionless gas in GR is described by a positive function~$f(x^{a},p_{b})$~on the spacetime cotangent bundle, which represents the number of particles near the point~$x^{a}$~with 4-momentum near~$p_{b}$~(Ehlers 1971). The condition that the matter be collision-free is equivalent to requiring that~$f$~be constant along the geodesic flow, which is the vector field defined, in local coordinates, by\begin{equation}\mathcal{L}=g^{ab}p_{a}\frac{\partial}{\partial
x^{b}}-\frac{1}{2}p_{a}p_{b}\frac{\partial g^{ab}}{\partial
x^{c}}\frac{\partial}{\partial p_{c}}\end{equation}
The statement~$\mathcal{L}f=0$~is known as the Vlasov equation.
For massless particles the function~$f$~is supported on the seven-dimensional submanifold of the cotangent bundle given by the equation~$g^{ab}p_{a}p_{b}=0$. The stress-energy-momentum tensor due to massless particles is given by
\begin{equation}T_{ab}=\int_{\mathbb{R}^{3}}fp_{a}p_{b}\frac{(-g)^{-1/2}}{p^{0}}d^{3}p\end{equation}
with the positive quantity~$p^{0}$~being determined by the relation~$g^{ab}p_{a}p_{b}=0$. Note that the massless condition implies~$T^{a}_{~a}=0$.
If the Vlasov equation holds then this stress tensor is divergence-free independently of the Einstein equations being satisfied.
The coupled Einstein-Vlasov equations, for the
metric~$g_{ab}$~and the particle distribution~$f$~are
\begin{equation}G_{ab}=8\pi\int_{\mathbb{R}^{3}}fp_{a}p_{b}\frac{(-g)^{-1/2}}{p^{0}}d^{3}p\end{equation}
\begin{equation}\mathcal{L}_{g}(f)=0\end{equation}
Suppose now that we have a physical spacetime~$(\tilde{M},\tilde{g}_{ab})$~and a massless particle distribution function~$\tilde{f}$ so that
~$\tilde{g}_{ab}$~and~$\tilde{f}$~satisfy tilded versions of
(5)-(6). Suppose also that we have an unphysical spacetime~$(M,g_{ab})$~defined by
\begin{equation}\tilde{g}_{ab}=~Z^{2}g_{ab}\end{equation}
for some conformal factor~$Z$~on~$M\supset\tilde{M}$.
As discussed in (AT) it is a consequence of the conformal invariance of null geodesics that the massless Vlasov equation is also conformally invariant. This is to say that if we simply write~$f\equiv\tilde{f}$~then the Vlasov equation for~$\tilde{f}$~in~$\tilde{M}$~implies the Vlasov equation for~$f$~in~$M$:
\begin{equation}\mathcal{L}_{g}f=0\end{equation}
If we now use the unphysical metric~$g_{ab}$~to define an unphysical stress tensor via
\begin{equation}T_{ab}=\int~fp_{a}p_{b}\frac{(-g)^{-1/2}}{p^{0}}d^{3}p\end{equation}
then
\begin{equation}\tilde{T}_{ab}=\frac{1}{Z^{2}}T_{ab}\end{equation}
and one may now use the unphysical Vlasov equation (8) to show that\linebreak[4]$\nabla^{a}T_{ab}=0$~in~$M$, just as~$\tilde{\nabla}^{a}\tilde{T}_{ab}=0$~in~$\tilde{M}$ .
The conformal EV equations for~$g_{ab}$~and~$f$~are now
\begin{displaymath}R_{ab}=2\nabla_{a}\nabla_{b}\log Z-2\nabla_{a}\log Z\nabla_{b}\log Z\end{displaymath}
\begin{equation}+g_{ab}(\square
\log Z+2\nabla_{c}\log Z\nabla^{c}\log Z)+\frac{8\pi}{Z^{2}}T_{ab}\end{equation}
and
\begin{equation}\mathcal{L}_{g}(f)=0\end{equation}
From now on we assume that a rescaling of the spacetime metric as in (1)-(2) exists, and our aim will be to solve equations (11)-(12) for~$f$~and~$g_{ab}$~in the neighbourhood of the singularity surface~$\Sigma$.
\\
In (ATII) it was noted that the massless condition on the matter implies that the conformal factor~$Z$~must remain smooth at~$\Sigma$. In what follows we will require~$Z$~to be a good coordinate in~$M$, and we thus make the following extra assumption:
\\\\\textbf{Assumption~2.1}~\textit{The conformal
factor}~$Z$~\textit{is such that}~$\nabla_{a}Z\neq 0$~\textit{at}~$\Sigma$.\\\\
\section{Initial data and conformal gauge fixing}
The singular nature of the conformal field equations (11),(12) imposes constraints on the data at an isotropic singularity~$\Sigma$~,which are rather different from the usual constraints on a regular hypersurface. To say what these are we first make the following definitions:\\
Let~$h^{0}_{ij}$~be the 3-metric of the singularity surface~$\Sigma$~and
let~$K^{0}_{ij}$~be the second fundamental form of~$\Sigma$~in~$M$.~$f^{0}$~will stand for the orthogonal projection of~$f$~onto~$\Sigma$~:~$f^{0}(x^{i},p_{b}^{~\perp})=f(0,x^{i},p_{b})$~and~$V^{0}=(g^{ab}\nabla_{a}Z\nabla_{b}Z)^{1/2}$~, as evaluated at~$\Sigma$.\\\\
The constraints then read as follows:
\begin{equation}\int f^{0}(x^{i},p_{j})p_{k}~d^{3}p=0\end{equation}
\begin{equation}(V^{0})^{2}h^{0}_{ij}=\frac{8\pi}{\sqrt{\textrm{det}h^{0}}}\int\frac{f^{0}p_{i}p_{j}}{((h^{0})^{mn}p_{m}p_{n})^{1/2}}~d^{3}p\end{equation}
\begin{equation}\Big(2\delta^{i}_{~k}\delta^{j}_{~l}-\chi^{ij}_{~~kl}\Big)(K^{0})^{kl}=D^{k}(\chi^{ij}_{~~k})+\frac{1}{2}(K^{0})(h^{0})^{ij}\end{equation}
where~$K^{0}=(h^{0})^{ij}K^{0}_{ij}$~,
\begin{equation}\chi_{ijkl}\equiv\frac{4\pi}{(V^{0})^{2}\sqrt{h^{0}}}\int\frac{f^{0}p_{i}p_{j}p_{k}p_{l}}{((h^{0})^{mn}p_{m}p_{n})^{3/2}}~d^{3}p\end{equation}
\begin{equation}\chi_{ijk}\equiv
-\frac{4\pi}{(V^{0})^{2}\sqrt{h^{0}}}\int\frac{f^{0}p_{i}p_{j}p_{k}}{((h^{0})^{mn}p_{m}p_{n})}~d^{3}p\end{equation}
and~$D_{i}$~is the covariant derivative operator associated with~$h^{0}_{ij}$.\\
Now (13) is just an integral constraint on~$f^{0}$~and from (ATII) we have the following theorem:
\textbf{Theorem 3.1} Let~$f^{0}(x^{i},p_{j})$~be a smooth, positive function on~$U\times \mathbb{R}^{3}$, where~$U$~is an open subset of~$\mathbb{R}^{3}$. Suppose that for each~$x\in U$
\begin{enumerate}
\item~~$f^{0}$~is compactly supported in~$p$.
\item~~$f^{0}$~is supported outside some open ball containing~$p=0$.
\item~~$f^{0}$~is not identically zero in~$p$.
\end{enumerate}
Then, given a smooth strictly positive function~$V^{0}$~on~$U$, there exists a unique, smooth, positive definite 3-metric~$h^{0}_{ij}$~on~$U$~satisfying (14) and a unique smooth trace-free symmetric tensor~$K^{0}_{ij}$~satisfying (15)\\\\
Also from (ATII) we have that the freedom in the choice of unphysical metric~$g_{ab}$~may be used to obtain the following:
\textbf{Theorem 3.2} If~$(\tilde{M},\tilde{g}_{ab})$~is a solution of the
massless Einstein-Vlasov equations, with an isotropic singularity, and
Assumption 2.1 holds, then the conformal factor~$Z$~may be chosen so
that~$V^{0}=1,~K^{0}=0$, and~$Z$~is a harmonic function
in~$M$.\\\\
We will henceforth refer to the gauge choice of Theorem 3.2 as the \linebreak\textit{harmonic}~gauge.\\
In summary the initial data set for the conformal EV equations at~$Z=0$, in the harmonic gauge, is as follows:
\\
\begin{itemize}
\item~$f^{0}$~is free data, subject only
to the integral constraint (13).
\item The initial
3-metric~$h^{0}_{ij}$~is determined by ~$f^{0}$~via~(14) \linebreak with ~$V^{0}\equiv 1$.
\item The initial second fundamental form~$K^{0}_{ij}$~is determined by
~$f^{0}$~via (15) with~$V^{0}\equiv 1$,~$K^{0}\equiv 0$
\end{itemize}
\section{The Cauchy problem for the conformal EV equations near~$\Sigma$}
In this section we show that the harmonic gauge conformal EV equations may be solved by a combination of the standard method of energy estimates with certain of the techniques used in the proof of the Newman-Claudel theorem. This theorem cannot be applied directly because of the presence of matter integrals in the field equations.
\subsection{The reduced Einstein equations as a symmetric hyperbolic system}
The first step in solving the field equations is to impose the harmonic coordinate condition which leads to a system of reduced equations.
Given a coordinate system ~$x^{\mu}$~ we make the following definition
\begin{equation}R^{H}_{\mu\nu}\equiv
R_{\mu\nu}+g_{\alpha(\mu}\partial_{\nu)}H^{\alpha}\end{equation}
where ~$H^{\alpha}=\square x^{\alpha}$.
\\
The reduced conformal EV equations are then defined to be
\begin{equation}R^{H}_{\mu\nu}-\frac{2}{Z}\nabla_{\mu}\nabla_{\nu}Z+\frac{4}{Z^{2}}\nabla_{\mu}Z\nabla_{\nu}Z-\frac{V^{2}}{Z^{2}}g_{\mu\nu}=\frac{1}{Z^{2}}T_{\mu\nu}\end{equation}
\begin{equation}\mathcal{L}_{g}f=0\end{equation}
If we
put~$x^{0}=Z$~and~$h_{\alpha\beta\gamma}=g_{\alpha\beta,\gamma}$, then
equations (19)-(20) can be decomposed as
\begin{equation}\partial_{z}g_{\alpha\beta}=h_{\alpha\beta 0}\end{equation}
\begin{equation}-g^{ij}\partial_{z}h_{\alpha\beta
j}=-g^{ij}\partial_{j}h_{\alpha\beta 0}\end{equation}
\begin{equation}\partial_{z}g^{\alpha\beta}=-g^{\alpha\mu}g^{\beta\nu}h_{\mu\nu
0}\end{equation}
\begin{displaymath}g^{00}\partial_{z}h_{\alpha\beta
0}=-g^{ij}\partial_{i}h_{\alpha\beta
j}-2g^{0i}\partial_{i}h_{\alpha\beta 0}+F_{\alpha\beta}(g,g^{-1},h)\end{displaymath}
\begin{equation}+\frac{1}{Z}\{Z_{\alpha\beta}+2g^{0\gamma}(h_{\alpha\gamma\beta}+h_{\beta\gamma\alpha}-h_{\alpha\beta\gamma})\}\end{equation}
\begin{equation}p^{0}\frac{\partial f}{\partial Z}+p^{i}\frac{\partial
f}{\partial
x^{i}}-\frac{1}{2}(\partial_{i}g^{\alpha\beta})p_{\alpha}p_{\beta}\frac{\partial
f}{\partial p_{i}}=0\end{equation}
where
\begin{equation}Z_{\mu\nu}\equiv\frac{2}{Z}\{4\nabla_{\mu}Z\nabla_{\nu}Z-g^{00}g_{\mu\nu}-T_{\mu\nu}\}\end{equation}
and ~$F_{\alpha\beta}$~is polynomial in its arguments.
\\
It follows that the reduced Einstein equations (21)-(24) are, for fixed ~$f$, in symmetric hyperbolic form.
\subsection{Initial data for the reduced equations}
We would like to solve the conformal EV equations in
the harmonic gauge. So, as initial data for the reduced equations first choose a positive smooth function~$f^{0}$~on~$T^{\star}\Sigma$~satisfying the constraint (13). We also require~$f^{0}(x^{i},\cdot)$\linebreak[4]to have fixed compact support and to be supported away from a fixed neighbourhood of the origin for~$x^{i}$~in a compact set. Then put~$(g^{0})^{00}=(V^{0})^{2}=1$~and choose~$g^{0}_{0i}=0$, so that the~$x^{i}$~are `initially comoving'. We may now use Theorem 3.1 to determine~$g^{0}_{ij}$~and~$h^{0}_{ij0}=2K^{0}_{ij}$~from (14) and (15) with ~$K^{0}$~set to zero.
\\
From (19) one must have that~$\square Z=0$~initially. But by
definition
\begin{equation}\square Z=\partial_{\alpha}g^{\alpha
0}+\frac{1}{2}g^{\alpha
0}g^{\rho\sigma}\partial_{\alpha}g_{\rho\sigma}\end{equation}
so that ~$h_{000}=0$~initially.
\\
Equation (24) implies
\begin{equation}Z^{0}_{\alpha\beta}=2\{h^{0}_{\alpha\beta
0}-h^{0}_{\alpha 0\beta}-h^{0}_{\beta 0\alpha}\}\end{equation}
so that~$Z^{0}_{ij}=2h^{0}_{ij0},~Z^{0}_{00}=Z^{0}_{0i}=0$.
\\
In order to recover the full harmonic gauge equations from the reduced equations we
will have to show that~$H^{\alpha}\equiv\square
x^{\alpha}=0$~in~$M$. We therefore pick~$h^{0}_{i00}=0$~and choose
spatial coordinates~$x^{i}$~satisfying the following condition on~$\Sigma$:
\begin{equation}g^{mn}\Gamma^{i}_{mn}=0\end{equation}
where~$\Gamma^{i}_{jk}$~are the Christoffel symbols
of~$g^{0}_{ij}$. These choices ensure that~$\square x^{i}=0$~on~$\Sigma$.
\subsection{Recovering the harmonic gauge equations}
We now use the contracted Bianchi identities to show that if ~$(f,g_{\alpha\beta})$~is a solution of the
reduced equations (19)-(20) with data as prescribed in 4.2,
then~$\square Z=\square x^{i}=0$~and~$(f,g_{\alpha\beta})$~solves the
conformal EV equations (11)-(12).\\
First write
\begin{equation}S_{\mu\nu}\equiv
\frac{1}{Z^{2}}T_{\mu\nu}+\frac{2}{Z}\nabla_{\mu}\nabla_{\nu}Z+\frac{V^{2}}{Z^{2}}g_{\mu\nu}-\frac{4}{Z^{2}}\nabla_{\mu}Z\nabla_{\nu}Z\end{equation}
We know from (20) that~$\nabla^{\mu}T_{\mu\nu}=0$~and this leads to
\begin{equation}\nabla^{\mu}S_{\mu\nu}=\frac{2}{Z}(\nabla^{\alpha}Z)(R_{\alpha\nu}-S_{\alpha\nu})+\frac{2}{Z}\partial_{\nu}H^{0}-\frac{4}{Z^{2}}(\partial_{\nu}Z)H^{0}\end{equation}
and now (19) gives
\begin{equation}\nabla^{\mu}R^{H}_{\mu\nu}=\frac{2}{Z}(\nabla^{\alpha}Z)(-g_{\beta(\alpha}\partial_{\nu)}H^{\beta})+\frac{2}{Z}\partial_{\nu}H^{0}-\frac{4}{Z^{2}}(\partial_{\nu}Z)H^{0}\end{equation}
From (19) one has~$R^{H}=\frac{2}{Z}H^{0}$~, so that
\begin{equation}R=\frac{2}{Z}H^{0}-g^{\alpha\beta}g_{\gamma(\alpha}\partial_{\beta
)}H^{\gamma}\end{equation}
The contracted Bianchi identities,~$\nabla^{\mu}G_{\mu\nu}=0$,~now imply
\begin{displaymath}0=\frac{2}{Z}(\nabla^{\alpha}Z)(-g_{\beta(\alpha}\partial_{\nu)}H^{\beta})+\frac{2}{Z}\partial_{\nu}H^{0}-\frac{4}{Z^{2}}(\partial_{\nu}Z)H^{0}\end{displaymath}
\begin{equation}-\nabla^{\alpha}(g_{\beta(\alpha}\partial_{\nu)}H^{\beta})-\frac{1}{2}\nabla_{\nu}\Big(\frac{2}{Z}H^{0}-g^{\alpha\beta}g_{\gamma(\alpha}\partial_{\beta)}H^{\gamma}\Big)+F_{\nu}(g,g^{-1},h,H,\partial
H)\end{equation}
where~$F_{\nu}$~is polynomial in its arguments and homogeneous
in~$H,~\partial H$.
\\
Now multiply (34) by~$g^{\nu\lambda}$~to get
\begin{displaymath}0=-\frac{1}{2}g^{\rho\mu}\partial^{2}_{\rho\mu}H^{\lambda}-\frac{2}{Z}(\nabla^{\alpha}Z)g^{\nu\lambda}g_{\beta(\alpha}\partial_{\nu)}H^{\beta}\end{displaymath}
\begin{equation}+\frac{1}{Z}g^{\nu\lambda}\partial_{\nu}H^{0}-\frac{3}{Z}g^{\nu\lambda}(\partial_{\nu}Z)P^{0}+\hat{F}^{\lambda}\end{equation}
where~$P^{0}\equiv\frac{1}{Z}H^{0}$.
\\
From (35) one gets the following at~$Z=0$:
\begin{equation}-2(\nabla^{\alpha}Z)g^{\nu\lambda}g_{\beta(\alpha}\partial_{\nu)}H^{\beta}+g^{\nu\lambda}\partial_{\nu}H^{0}-3g^{\nu\lambda}(\partial_{\nu}Z)P^{0}=0\end{equation}
Now~$P^{0}(0)=\partial_{z}H^{0}(0)$, so (36) implies
that~$P^{0}=\partial_{z}H^{0}=0$~at$~Z=0$. Equation (36) also
implies that~$\partial_{z}H^{i}=0$~at~$Z=0$.\\
From the definition of ~$P^{0}$~it follows that
\begin{equation}\frac{\partial P^{0}}{\partial
Z}=\frac{1}{Z}\Big(-P^{0}+\frac{\partial H^{0}}{\partial
Z}\Big)\end{equation}
If we put~$h_{\alpha}^{\beta}\equiv \partial_{\alpha}H^{\beta}$~then
(35) and (37) can be written in first order form as
\begin{equation}-g^{ij}\frac{\partial h_{j}^{\alpha}}{\partial
Z}=-g^{ij}\frac{\partial h_{0}^{\alpha}}{\partial x^{j}}\end{equation}
\begin{equation}\frac{\partial H^{\alpha}}{\partial
Z}=h_{0}^{\alpha}\end{equation}
\begin{equation}\frac{\partial P^{0}}{\partial
Z}=-\frac{1}{Z}P^{0}+\frac{1}{Z}h_{0}^{0}\end{equation}
\begin{displaymath}(g^{00})\frac{\partial h_{0}^{\lambda}}{\partial
Z}=2g^{0i}\partial_{i}h_{0}^{\lambda}-g^{ij}\partial_{i}h_{j}^{\lambda}\end{displaymath}
\begin{equation}-\frac{2}{Z}\{g^{0\alpha}h_{\alpha}^{\lambda}+3g^{0\lambda}P^{0}\}+\tilde{F}^{\lambda}\end{equation}
The system (38)-(41) can be written in an obvious way as:
\begin{equation}a^{0}(u)\frac{\partial u}{\partial
Z}=a^{i}(u)\frac{\partial u}{\partial
x^{i}}+b(u)u+\frac{1}{Z}c(u)u\end{equation}
with~$a^{0}$~+ve definite and~$a^{\alpha}$~symmetric.
\\
The eigenvalues ~$\lambda$~of~$((a^{0})^{-1}c)(0)$~satisfy either
\begin{equation}\lambda=0\end{equation}
or
\begin{equation}\lambda P^{0}=-P^{0}+h_{0}^{0}\end{equation}
\begin{equation}\lambda h_{0}^{\alpha}=-2g^{0\beta}(0)h_{\beta}^{\alpha}-6g^{0\alpha}(0)P^{0}\end{equation}
Putting~$\alpha=i$~in (45) gives
\begin{displaymath}\lambda h_{0}^{i}=-2h_{0}^{i}\end{displaymath}
while putting~$\alpha=0$~in (45) gives
\begin{equation}\lambda h_{0}^{0}=-2h_{0}^{0}-6P^{0}\end{equation}
So if ~$\lambda\neq~0,-2$~then
\begin{equation}(2+\lambda)(1+\lambda)+6=0\end{equation}
Thus~$(a^{0})^{-1}c(0)$~has no positive integer eigenvalues. Now, as remarked in section 4.2, the choice of initial data for the reduced equations forces the quantities~$\square x^{\alpha}$~to vanish at~$Z=0$. Therefore (42) has the trivial solution~$u\equiv 0$~and by Theorem A.1 (see Appendix) this is the only smooth solution with the given data. It follows that any smooth solution of the reduced equations (19)-(20), with data as in 4.2, is a solution of the harmonic gauge EV equations .
\subsection{Solving the reduced field equations}
We aim to solve the reduced EV equations on a manifold of the form~$M\times [0,T]$, with~$M$~a paracompact 3-manifold. We will use the method of energy estimates to obtain local (in space and time) solutions of the field equations, which can then be patched together using the finite domain of dependence to obtain
a solution in a neighbourhood of the singularity~$\Sigma=M\times \{0\}$.
\subsubsection{Localisation}
First choose a locally finite cover of~$\Sigma$~by relatively compact harmonic coordinate charts~$O_{\alpha}$. Now construct a second cover~$\{O'_{\alpha}\}$~in such a way that~$\bar{O}'_{\alpha}\subset O_{\alpha}$.\\
Restrict attention to a single chart~$O_{\alpha}$~and consider the reduced equations (19)-(20) on~$O_{\alpha}\times [0,T]$. In order to obtain energy estimates it is desirable to have data in~$C_{0}^{\infty}(O_{\alpha})$. We thus cut off~$f^{0}$~to zero smoothly, along with~$g_{ij}+\delta_{ij}$. We now work with~$\bar{g}_{\alpha\beta}=g_{\alpha\beta}-\eta_{\alpha\beta}$~as field variable ($\eta_{\alpha\beta}=\textrm{diag}(1,-1,-1,-1)$). Obtain~$K^{0}_{ij}$~from the new~$f^{0},~\bar{g}_{\alpha\beta}$~via equation (15). This cutting-off procedure is done in such a way that the original data is maintained on~$O_{\alpha}''\supset\bar{O'}_{\alpha}$~and is zero outside~$O_{\alpha}'''\supset\bar{O}_{\alpha}''$.
Now note that the singular field equation (26) will no longer be consistent at~$Z=0$~with the localised~$C_{0}^{\infty}$~data. To remedy this we introduce a function~$F(x^{i})\in C_{0}^{\infty}(O_{\alpha})$~such that~$0\leq F(x^{i})\leq 1$~and
\begin{equation}F(x)=\left\{\begin{array}{ll}0&~~~x\in O_{\alpha}\setminus O_{\alpha}''\\1&~~~x\in O_{\alpha}'''':~O_{\alpha}''\supset O_{\alpha}''''\supset\bar{O}_{\alpha}'\end{array}\right.\end{equation}
We use the function~$F$~to obtain the following localised modification of (26)-(27):
\begin{equation}R^{H}_{\mu\nu}=F(x^{i})\left\{\frac{2}{Z}\nabla_{\mu}\nabla_{\nu}Z+\frac{1}{Z^{2}}(T_{\mu\nu}+g^{00}g_{\mu\nu}-4\nabla_{\mu}Z\nabla_{\nu}Z)\right\}\end{equation}
\begin{equation}L_{g}f=0\end{equation}
We will solve (49)-(50) on~$O_{\alpha}\times [0,T_{\alpha}]$~and then use the finite domain of dependence to argue that this gives rise to a solution of conformal Einstein-Vlasov on~$O_{\alpha}'\times [0,T_{\alpha}']$, determined by the right data.
\subsubsection{Approximate solutions of the field equations}
An important step in solving the reduced field equations is, as in the proof of the Newman-Claudel theorem, to write them in such a way that they are manifestly formally solvable. One then obtains approximate finite Taylor series solutions which solve the equations up to~$O(Z^{q})$,~$q$~arbitrary. We note that the hyperbolicity of the field equations is not used at this stage.
\\\\
First define
\begin{equation}Z_{\mu\nu}=\frac{2}{Z}\left\{4\nabla_{\mu}Z\nabla_{\nu}Z-g^{00}g_{\mu\nu}-T_{\mu\nu}-(4\nabla_{\mu}Z\nabla_{\nu}Z-g^{00}g_{\mu\nu}-T_{\mu\nu})(0)\right\}\end{equation}
Then (49) is just
\begin{equation}R^{H}_{\mu\nu}=F(x^{i})\left\{\frac{2}{Z}\nabla_{\mu}\nabla_{\nu}Z+\frac{1}{2Z}Z_{\mu\nu}\right\}\end{equation}
(F was chosen to be zero wherever~$(4\nabla_{\mu}Z\nabla_{\nu}Z-g^{00}g_{\mu\nu}-T_{\mu\nu})(0)\neq 0$)\\
Define also
\begin{equation}\bar{Z}_{\alpha\beta}=Z_{\alpha\beta}-Z_{\alpha\beta}(0)~~~~\bar{h}_{\alpha\beta\gamma}=h_{\alpha\beta\gamma}-h_{\alpha\beta\gamma}(0)\end{equation}
In terms of these variables, equations (49)-(50) become
\begin{equation}-g^{ij}\partial_{z}\bar{h}_{\alpha\beta
j}=-g^{ij}\partial_{j}(\bar{h}_{\alpha\beta 0}+h_{\alpha\beta
0}(0))\end{equation}
\begin{equation}\partial_{z}\bar{g}_{\alpha\beta}=\bar{h}_{\alpha\beta
0}+h_{\alpha\beta 0}(0)\end{equation}
\begin{equation}\partial_{z}\bar{g}^{\alpha\beta}=-g^{\beta\nu}g^{\alpha\mu}(\bar{h}_{\mu\nu
0}+h_{\mu\nu 0})\end{equation}
\begin{displaymath}g^{00}\partial_{z}\bar{h}_{\alpha\beta
0}=-g^{ij}\partial_{i}(\bar{h}_{\alpha\beta j}+h_{\alpha\beta
j}(0))-2g^{0i}\partial_{i}(\bar{h}_{\alpha\beta 0}+h_{\alpha\beta
0}(0))\end{displaymath}
\begin{displaymath}+2\widehat{F}_{\alpha\beta}+2F(x^{i})(F_{1})^{\gamma}(\bar{h}_{\alpha\gamma\beta}+\bar{h}_{\beta\gamma\alpha}-\bar{h}_{\alpha\beta\gamma})\end{displaymath}
\begin{equation}+\frac{F(x^{i})}{Z}\{\bar{Z}_{\alpha\beta}+(H_{1})^{\gamma}(0)(\bar{h}_{\alpha\gamma\beta}+\bar{h}_{\beta\gamma\alpha}-\bar{h}_{\alpha\beta\gamma})\}\end{equation}
and
\begin{equation}L_{g}f=0\end{equation}
where
\begin{equation}(F_{1})^{\gamma}(Z)\equiv
\int_{0}^{1}((H_{1})^{\gamma})'(SZ)~dS~,~~~(H_{1})^{\gamma}\equiv g^{0\gamma}\end{equation}
The prime denotes differentiation with respect to time, and~$F_{1}$~ can be computed from (53) and (56).\\
Meanwhile we calculate the following evolution for~$\bar{Z}_{\mu\nu}$
\begin{displaymath}\partial_{z}\bar{Z}_{\mu\nu}=-F_{2}(\bar{h}_{\mu\nu
0}+h_{\mu\nu
0}(0))+(F_{3})^{\alpha\beta}_{~~~\mu\nu}(\bar{h}_{\alpha\beta
0}+h_{\alpha\beta 0}(0))\end{displaymath}
\begin{displaymath}+(F_{4})^{\alpha\beta}_{~~~\mu\nu}(\bar{h}_{\alpha\beta
0}+h_{\alpha\beta
0}(0))+(F_{5})_{\mu\nu}+(F_{6})^{\alpha\beta}_{~~~\mu\nu}(\bar{h}_{\alpha\beta
i}+h_{\alpha\beta i})\end{displaymath}
\begin{displaymath}-(F_{7})^{\alpha\beta}_{~~~\mu\nu}(\bar{h}_{\alpha\beta
0}+h_{\alpha\beta 0})\end{displaymath}
\begin{displaymath}+\frac{1}{Z}\Big\{-\bar{Z}_{\mu\nu}-H_{2}(0)\bar{h}_{\mu\nu
0}+(H_{3})^{\alpha\beta}_{~~~\mu\nu}(0)\bar{h}_{\alpha\beta
0}\end{displaymath}
\begin{equation}+(H_{4})^{\alpha\beta}_{~~~\mu\nu}(0)\bar{h}_{\alpha\beta
0}+(H_{6})^{\alpha\beta}_{~~~\mu\nu}(0)\bar{h}_{\alpha\beta
i}-(H_{7})^{\alpha\beta}_{~~~\mu\nu}(0)\bar{h}_{\alpha\beta
0}\Big\}\end{equation}
where ~$F_{i}$~is to~$H_{i}$~as ~$F_{1}$~is to~$H_{1}$~and
\begin{equation}H_{2}\equiv 2g^{00}\end{equation}
\begin{equation}(H_{3})^{\alpha\beta}_{~~~\mu\nu}\equiv
2g_{\mu\nu}g^{0\alpha}g^{0\beta}\end{equation}
\begin{equation}(H_{4})^{\alpha\beta}_{~~~\mu\nu}\equiv
(\textrm{det}g)^{-1/2}g^{\alpha\beta}\int\frac{fp_{\mu}p_{\nu}}{p^{0}}~d^{3}p\end{equation}
\begin{equation}(H_{5})_{\mu\nu}\equiv
2(\textrm{det}g)^{-1/2}\int\frac{p_{\mu}p_{\nu}p^{i}}{(p^{0})^{2}}\frac{\partial
f}{\partial x^{i}}~d^{3}p\end{equation}
\begin{equation}(H_{6})^{\alpha\beta}_{~~~\mu\nu}\equiv
(\textrm{det}g)^{-1/2}\int\frac{p_{\mu}p_{\nu}p^{\alpha}p^{\beta}}{(p^{0})^{2}}\frac{\partial
f}{\partial p_{i}}~d^{3}p\end{equation}
\begin{equation}(H_{7})^{\alpha\beta}_{~~~\mu\nu}\equiv (\textrm{det}g)^{-1/2}\int\frac{f}{(p^{0})^{2}}\Big(p^{\alpha}p^{\beta}((p_{0})^{-1}p_{\mu}p_{\nu}+p_{\mu}\delta^{0}_{\nu}+p_{\nu}\delta^{0}_{\mu})\Big)~d^{3}p\end{equation}
The~$F_{i}$~are regular terms, and can be calculated from (54),
(55), (58).\\
The whole system of equations (54)-(58), (60) can be written in the form
\begin{equation}A^{0}(u)\partial_{z}u=A^{i}(u)\partial_{i}u+B(u)u+\frac{1}{Z}C(x^{i},u)u\end{equation}
(i=1,2\ldots 6) for some matrices~$A^{\alpha}, B, C$~with~$A^{0}$~positive definite. The vector~$u$~stands for~$(f,\bar{g}_{\alpha\beta},\bar{g}^{\alpha\beta},\bar{h}_{\alpha\beta\gamma},\bar{Z}_{\alpha\beta}).$\\\\
~~~\textbf{Lemma 4.1}~~~The matrix~$C(x^{i},u)$~in (74) satisfies~$C(x^{i},u)u^{0}=0~\forall u$~and\linebreak[4]$(A^{0})^{-1}C(x^{i},u^{0})$~has no positive integer eigenvalues.
\textit{Proof}~Since~$\bar{Z}_{\alpha\beta}(0)=\bar{h}_{\alpha\beta\gamma}(0)=0$~the first part follows. Now note that the eigenvalues of~$((A^{0})^{-1}C)(0)$~satisfy either~$\lambda=0$~or
\begin{equation}\lambda h_{\alpha\beta 0}=F(x^{i})\left\{Z_{\alpha\beta}+\{h_{\alpha
0\beta}+h_{\beta 0\alpha}-h_{\alpha\beta 0}\}\right\}\end{equation}
\begin{displaymath}\lambda
Z_{\alpha\beta}=-Z_{\alpha\beta}+2\Bigg\{-h_{\alpha\beta0}+g_{\alpha\beta}(0)g^{0\mu}(0)g^{0\nu}(0)h_{\mu\nu
0}\end{displaymath}
\begin{displaymath}+\frac{1}{2}(\textrm{det}g(0))^{-1/2}g^{\mu\nu}(0)h_{\mu\nu
0}\int\frac{f^{0}p_{\alpha}p_{\beta}}{p^{0}}~d^{3}p\end{displaymath}
\begin{equation}-\frac{1}{2}(\textrm{det}g(0))^{-1/2}h_{\mu\nu
0}\int\frac{f^{0}p^{\mu}p^{\nu}}{(p^{0})^{2}}\{(p_{\alpha}p_{\beta}/p_{0})+p_{\alpha}\delta^{0}_{\beta}+p_{\beta}\delta^{0}_{\beta}\}~d^{3}p\Bigg\}\end{equation}
where~$p_{0}=p^{0}=(-g^{ij}(0)p_{i}p_{j})^{1/2}$~and~$h_{\alpha\beta i}=0$.\\
Now if~$F(x^{i})=0$~then either~$\lambda=0$~or~$h_{\alpha\beta 0}=0$. If~$\lambda\neq 0$~then from (68)-(69) we must have~$\lambda=-1$. It is therefore enough to consider those~$x$~at which~$F(x)\neq 0$. At such~$x$~the relation between~$f^{0}$~and~$g_{ij}(0)$~is given by (14).\\
Equations (68)-(69) can be expanded as follows:
\begin{equation}\lambda h_{000}=F(Z_{00}+2h_{000})\end{equation}
\begin{equation}\lambda
Z_{00}=-Z_{00}-6h_{000}+6g^{ij}(0)h_{ij0}\end{equation}
\begin{equation}\lambda h_{i00}=F(Z_{i0})\end{equation}
\begin{equation}\lambda
Z_{i0}=-Z_{i0}-2h_{i00}-2(\textrm{det}g(0))^{-1/2}h_{jk0}\int\frac{f^{0}}{(p_{0})^{2}}p^{j}p^{k}p_{i}~d^{3}p\end{equation}
\begin{equation}\lambda h_{ij0}=F(Z_{ij}-2h_{ij0})\end{equation}
\begin{displaymath}\lambda
Z_{ij}=-Z_{ij}-2h_{ij0}+2g_{ij}(0)h_{000}-g_{ij}(0)(h_{mn0}g^{mn}(0))\end{displaymath}
\begin{equation}-h_{mn0}\Big(\chi^{mn}_{~~~ij}\Big)\end{equation}
where
\begin{equation}\chi^{mn}_{~~~ij}=(\textrm{det}g(0))^{-1/2}\int\frac{f^{0}}{(p_{0})^{3}}p^{m}p^{n}p_{i}p_{j}~d^{3}p\end{equation}
Taking the trace of (74)-(75) leads to
\begin{equation}(g^{mn}(0)h_{mn0})((\lambda+1)(\lambda+2)+6F)=6Fh_{000}\end{equation}
while (70)-(71) give
\begin{equation}((\lambda+1)(\lambda-2F)+6F)h_{000}=6Fg^{mn}(0)h_{mn0}\end{equation}
and hence
\begin{equation}(g^{mn}(0)h_{mn0})(((\lambda+1)(\lambda-2F)+6F)((\lambda+1)(\lambda+2)+6F)-36F^{2})=0\end{equation}
Now if~$\lambda$~were a positive integer then the polynomial in~$F$~and~$\lambda$~on the left hand side of (79) would be greater than or equal to~$12(1+2F(1-F))$. Since~$0\leq F\leq 1$~it follows that if~$g^{mn}(0)h_{mn0}\neq 0$~then~$\lambda$~is not a positive integer.\\
Suppose now that~$g^{mn}(0)h_{mn0}=0$. Equations (74)-(75) then
become
\begin{equation}\lambda h_{ij0}=F(Z_{ij}-2h_{ij0})\end{equation}
\begin{equation}\lambda
Z_{ij}=-Z_{ij}-2h_{ij0}+2g_{ij}(0)h_{000}-h_{mn0}\Big(\chi^{mn}_{~~~ij}\Big)\end{equation}
and (70)-(71) become
\begin{equation}\lambda h_{000}=F(Z_{00}+2h_{000})\end{equation}
\begin{equation}\lambda Z_{00}=-Z_{00}-6h_{000}\end{equation}
which imply
\begin{equation}((\lambda+1)(\lambda-2)+6F)h_{000}=0\end{equation}
Suppose~$(\lambda+1))(\lambda-2)+6F\neq 0$, so that~$h_{000}=0$. Then (74)-(75)
imply
\begin{equation}((\lambda+1)(\lambda+2)+2F)h_{ij0}=-Fh_{mn0}\chi^{mn}_{~~~ij}\end{equation}
But clearly the eigenvalues of~$\chi$~are positive and hence~$\lambda$~cannot be a positive integer.\\
Suppose finally that~$h_{mn0}=0$. Then (72)-(73) are
\begin{equation}\lambda h_{i00}=F(Z_{i0})\end{equation}
\begin{equation}\lambda Z_{i0}=-Z_{i0}-2h_{i00}\end{equation}
which imply
\begin{equation}(\lambda^{2}+\lambda+2F)h_{i00}=0\end{equation}
and we conclude from all this that~$((A^{0})^{-1}C)(0)$~has no positive
integer eigenvalues.
Lemma 4.1 now allows us to construct approximate solutions~$w$~of the field equations (67). Specifically we make the following ansatz
\begin{equation}w=\sum_{p=0}^{q}\frac{Z^{p}}{p!}w^{(p)}~~~~\textrm{with}~~w(0)=w^{(0)}=u^{0}\end{equation}
and now note that
\begin{displaymath}Z(A^{0}(w)\partial_{z}w-A^{i}(w)\partial_{i}w-B(w)w)-C(w)w=\end{displaymath}
\begin{equation}\sum_{p=0}^{q}\frac{Z^{p}}{p!}(pA^{0}(w)-C(w))w^{(p)}-Z(A^{i}(w)\partial_{i}w+B(w)w)\end{equation}
Let the~$w^{(p)}$~be determined by demanding that the~$Z$-derivatives up to order~$q$~of the right hand side of (90) vanish at Z=0. To see that this prescription is well-defined note for example that the first derivative is
\begin{displaymath}\sum_{p=1}^{q}\frac{Z^{p-1}}{(p-1)!}(pA^{0}(w)-C(w))w^{(p)}+\sum_{p=0}^{q}\frac{Z^{p}}{p!}(pA^{0}(w)-C(w))'w^{(p)}\end{displaymath}
\begin{displaymath}-Z(A^{i}(w)\partial_{i}w+B(w)w)'-(A^{i}(w)\partial_{i}w+B(w)w)\end{displaymath}
Setting~$Z=0$~and equating the resulting expression to zero gives
\begin{displaymath}(A^{0}(u^{0})-C(u^{0}))w^{(1)}-(C(w))'(0)u^{0}-(A^{i}(u^{0})\partial_{i}u^{0}+B(u^{0})u^{0})=0\end{displaymath}
Lemma 4.1 now implies that~$w^{(1)}$~is uniquely determined by the initial data~$u^{0}$. Equating the pth derivative of (90) to zero at~$Z=0$~gives an equation of the form
\begin{displaymath}(pA^{0}(u^{0})-C(u^{0}))w^{(p)}+\phi(w^{(p-1)},\ldots)=0\end{displaymath}
Thus the~$w^{(p)}$~are inductively well-defined and are~$C_{0}^{\infty}$~by the choice of initial data.\\
By Taylor's theorem we must now have that
\begin{equation}Z(A^{0}(w)\partial_{z}w-A^{i}(w)\partial_{i}w-B(w)w)-C(w)w=\zeta\end{equation}
for some~$\zeta\in C_{0}^{\infty}$~admitting an estimate~$\|\zeta\|_{s}\leq\kappa_{s}Z^{q+1}$, say. It follows that
\begin{equation}A^{0}(w)\partial_{z}w-A^{i}(w)\partial_{i}w-B(w)w-\frac{1}{Z}C(w)w=\xi\end{equation}
where~$\xi\in C_{0}^{\infty}$~for each~$Z$~and admits an estimate~$\|\xi\|_{s}\leq\kappa_{s}Z^q$.\\
We now wish to check that the approximate solution~$w=(f_{q},\bar{g}_{q},\bar{g}^{-1}_{q},\partial\bar{g}_{q},\bar{Z}_{q})$\linebreak[4] of equations (67) gives rise to an approximate solution of the original localised reduced equations (56)-(57). First a calculation shows that the process of writing
\begin{displaymath}Z^{-1}(H_{i}(Z)-H_{i}(0))=\int_{0}^{1}H_{i}'(SZ)dS\end{displaymath}
which led to (67) can be inverted to obtain an approximate solution of equations (50) and (52) with error~$O(Z^{q})$. Now one needs to check that the part~$Z_{\alpha\beta}$~of the approximate solution is close to the original definition (51). To do this define~$\hat{Z}_{\mu\nu}$~by the right hand side of (51) expressed in terms of the appropriate parts of our existing approximate solution. Then one calculates that
\begin{equation}\partial_{z}(\hat{Z}_{\mu\nu}-Z_{\mu\nu}(0))=-\frac{1}{Z}(\hat{Z}_{\mu\nu}-Z_{\mu\nu}(0))+\frac{2}{Z}\{\star+O(Z^{q})\}\end{equation}
where~$\star$~is most of the right hand side of (60):
\begin{equation}\partial_{z}(Z_{\mu\nu}-Z_{\mu\nu}(0))=-\frac{1}{Z}(Z_{\mu\nu}-Z_{\mu\nu}(0))+\frac{2}{Z}\{\star\}+O(Z^{q})\end{equation}
Subtracting (93) from (94) gives
\begin{equation}\partial_{z}(Z(\hat{Z}_{\mu\nu}-Z_{\mu\nu}))=O(Z^{q})\end{equation}
Hence~$\hat{Z}_{\mu\nu}-Z_{\mu\nu}=O(Z^{q})$.
\subsubsection{The reduced EV equations as a coupled symmetric hyperbolic system}
While the form (61)-(65), (67) of the reduced equations is useful for demonstrating formal solvability, it is not appropriate for the application of the method of energy estimates. The presence of the~$\frac{\partial f}{\partial x^{i}}$~terms in the equation for~$Z_{\mu\nu}$~spoils the symmetry of the system as a whole. On the other hand, while the field equations in the form (49)-(50) are symmetric, the~$\frac{1}{Z^{2}}$~terms are too strong to be handled by the contraction mapping technique of Claudel and Newman. We need to choose variables in such a way that the field equations are symmetric as a whole and have singular terms which are no worse than~$\frac{1}{Z}$. This can be achieved by defining the following new variable:
\begin{equation}P_{\alpha\beta}=\phi(4\nabla_{\alpha}Z\nabla_{\beta}Z-g^{00}g_{\alpha\beta})-\frac{(\textrm{det}g)^{-1/2}}{p^{0}}fp_{\alpha}p_{\beta}\end{equation}
where~$\phi=\phi(p_{i})\in C_{0}^{\infty}$~and~$\int_{\mathbb{R}^{3}}\phi~d^{3}p=1$. We also suppose that~$\textrm{supp}\phi\subset\textrm{supp}f^{0}(x,\cdot)$. Now define
\begin{equation}Q_{\alpha\beta}=\frac{2}{Z}(P_{\alpha\beta}-P_{\alpha\beta}(0))\end{equation}
with~$Q_{\alpha\beta}(0)=2\partial_{z}P_{\alpha\beta}(0)$~determined by the localised data. In particular~$Q_{\mu\nu}(0)\in C_{0}^{\infty}(O_{\alpha}\times\mathbb{R}^{3})$~and~supp$Q_{\mu\nu}(0,x^{i},\cdot)\subset\textrm{supp}f^{0}(x^{i},\cdot)$.
The reduced field equations now decompose as
\begin{equation}\partial_{z}\bar{g}_{\alpha\beta}=h_{\alpha\beta 0}\end{equation}
\begin{equation}-g^{ij}\partial_{z}h_{\alpha\beta j}=-g^{ij}\partial_{j}h_{\alpha\beta 0}\end{equation}
\begin{equation}\partial_{z}\bar{g}^{\alpha\beta}=-g^{\alpha\mu}g^{\beta\nu}h_{\mu\nu 0}\end{equation}
\begin{displaymath}g^{00}\partial_{z}h_{\alpha\beta 0}=-g^{ij}\partial_{j}h_{\alpha\beta i}-2g^{0i}\partial_{i}h_{\alpha\beta 0}+F_{\alpha\beta}(g,h)\end{displaymath}
\begin{equation}+\frac{F(x^{i})}{Z}\left\{2g^{0\gamma}(h_{\alpha\gamma\beta}+h_{\beta\gamma\alpha}-h_{\alpha\beta\gamma})+\int_{\mathbb{R}^{3}}Q_{\alpha\beta}~d^{3}p\right\}\end{equation}
\begin{equation}\frac{\partial f}{\partial Z}=-\frac{p^{i}}{p^{0}}\frac{\partial f}{\partial x^{i}}+\frac{1}{2}(\partial_{i}g^{\alpha\beta})\frac{p_{\alpha}p_{\beta}}{p^{0}}\frac{\partial f}{\partial p_{i}}\end{equation}
Using the Vlasov equation one calculates the following evolution for~$Q_{\alpha\beta}$
\begin{displaymath}\partial_{z}Q_{\alpha\beta}=-\frac{p^{i}}{p^{0}}\frac{\partial Q_{\alpha\beta}}{\partial x^{i}}+\frac{1}{2}(\partial_{i}g^{\mu\nu})\frac{p_{\mu}p_{\nu}}{p^{0}}\frac{\partial Q_{\alpha\beta}}{\partial p_{i}}\end{displaymath}
\begin{displaymath}+\frac{1}{Z}\Bigg\{-Q_{\alpha\beta}+2\phi(g_{\alpha\beta}g^{0\mu}g^{0\nu}h_{\mu\nu 0}-g^{00}h_{\alpha\beta 0}-(p^{i}/p^{0})\partial_{i}(g^{00}g_{\alpha\beta}))\end{displaymath}
\begin{displaymath}f\Bigg(-2\partial_{z}\left(\frac{p_{\alpha}p_{\beta}}{p^{0}}\textrm{det}g^{-1/2}\right)-2\frac{p^{i}}{p^{0}}\partial_{i}\left(\frac{p_{\alpha}p_{\beta}}{p^{0}}\textrm{det}g^{-1/2}\right)\end{displaymath}
\begin{displaymath}+(\partial_{i}g^{\mu\nu})p_{\mu}p_{\nu}\frac{\partial~}{\partial p_{i}}\left(\frac{p_{\alpha}p_{\beta}}{p^{0}}\right)\textrm{det}g^{-1/2}\Bigg)+\frac{p_{\mu}p_{\nu}}{p^{0}}(\partial_{i}g^{\mu\nu})\frac{\partial\phi}{\partial p_{i}}(\bar{g}^{00}g_{\alpha\beta}+\bar{g}_{\alpha\beta})\end{displaymath}
\begin{equation}-2\frac{p^{i}}{p^{0}}\partial_{i}P_{\alpha\beta}(0)-\frac{p_{\mu}p_{\nu}}{p^{0}}(\partial_{i}g^{\mu\nu})\frac{\partial~}{\partial p_{i}}\left(\frac{fp_{\alpha}p_{\beta}}{p^{0}}\textrm{det}g^{-1/2}\right)(0)\Bigg\}\end{equation}
Clearly the system (98)-(103) is symmetric hyperbolic.
We now note that the terms containing~$p^{0}$~in equations (102)-(103) blow up at the vertex of the light cone and thus cause technical problems there. For this reason we multiply the right hand sides of these equations by a function~$\psi(p_{i})\in C_{0}^{\infty}(\mathbb{R}^{3})$~which is equal to unity on a set containing the support of~$f^{0}$~and equal to zero in a neighbourhood of the origin. Call the resulting equations $(102)',~(103)'$.
We now wish to show that the approximate solution~$(f_{q},g_{q},h_{q})$~of the field equations obtained earlier gives rise to an approximate solution\linebreak[4]~$(f_{q},g_{q},h_{q},Q_{q})$~of (98)-(101), $(102)'-(103)'$. To this end define~$Q_{\alpha\beta}=2Z^{-1}(P_{\alpha\beta}-P_{\alpha\beta}(0))$~with~$P_{\alpha\beta}$~constructed from~$f_{q},g_{q}$~via equation (96). If~$(f_{q},g_{q},h_{q})$~satisfies (56)-(57) up to~$O(Z^{q})$~then one has (dropping the subscript~$q$~for the moment)
\begin{displaymath}g^{00}\partial_{z}h_{\alpha\beta 0}=-g^{ij}\partial_{i}h_{\alpha\beta j}-2g^{0i}\partial_{i}h_{\alpha\beta 0}+F_{\alpha\beta}\end{displaymath}
\begin{equation}\frac{F(x)}{Z}\left\{\int_{\mathbb{R}^{3}}Q_{\alpha\beta}~d^{3}p+2g^{0\gamma}(h_{\alpha\gamma\beta}+h_{\beta\gamma\alpha}-h_{\alpha\beta\gamma})\right\}+O(Z^{q})\end{equation}
A calculation gives that this~$Q_{\alpha\beta}$~satifies
\begin{equation}\partial_{z}Q_{\alpha\beta}= rhs+O(Z^{q-1})\end{equation}
where~`$rhs$'~is the right hand side of equation $(141)'$. It follows that we have an approximate solution~$(f_{q-1},g_{q-1},h_{q-1},Q_{q-1})$~of the field equations with error~$O(Z^{q-1})$.
\subsubsection{The field equations in matrix form}
Let the vector~$u=u(Z,x^{i})$~stand for~$(\bar{g}_{\alpha\beta},\bar{g}^{\alpha\beta},h_{\alpha\beta\gamma})$. Then we may write (98)-(101) in the following form
\begin{equation}A^{0}(u)\frac{\partial u}{\partial Z}=\sum_{i=1}^{3}A^{i}(u)\frac{\partial u}{\partial x^{i}}+G(u)+\frac{F(x^{i})}{Z}\left\{C(u)+\int_{\mathbb{R}^{3}}\left(\begin{array}{l}0\\Q\end{array}\right)~d^{3}p\right\}\end{equation}
where~$A^{\alpha}, G, C$~are polynomial in the components of~$u$,~$A^{\alpha}$~are symmetric matrices and~$A^{0}$~is positive definite.
Let~$v=v(Z,x^{i},p_{j})$~stand for~$(f,Q_{\alpha\beta})$. Then we may write $(102)'-(103)'$ as
\begin{equation}\frac{\partial v}{\partial Z}=\sum_{i=1}^{6}B^{i}(p_{j},u)\partial_{i}v+\frac{1}{Z}D(u,v,p_{j})\end{equation}
where the~$B^{i}$~are symmetric matrices and~$B^{i}, D$~depend smoothly on~$p_{j}, v$~and on~$u$~away from det$g=0$.
Now let~$u_{q}$~and~$v_{q}$~stand for the appropriate parts of the approximate solution we obtained earlier. Then we have
\begin{displaymath}A^{0}(u_{q})\frac{\partial u_{q}}{\partial Z}=\sum_{i=1}^{3}A^{i}(u_{q})\frac{\partial u_{q}}{\partial x^{i}}+G(u_{q})+\frac{F(x^{i}}{Z}\left\{C(u_{q})+\int_{\mathbb{R}^{3}}\left(\begin{array}{l}0\\Q_{q}\end{array}\right)~d^{3}p\right\}\end{displaymath}
\begin{equation}+\xi_{1}\end{equation}
\begin{equation}\frac{\partial v_{q}}{\partial Z}=\sum_{i=1}^{6}B^{i}(p_{j},u_{q})\partial_{i}v_{q}+\frac{1}{Z}D(u_{q},v_{q},p_{j})+\xi_{2}\end{equation}
where~$\xi_{1}\in C_{0}^{\infty}(\mathbb{R}^{3})$~and~$\xi_{2}\in C_{0}^{\infty}(\mathbb{R}^{6})$~are~$O(Z^{q})$.
By considering the differences between (106), (108) and (107), (109) one is lead to the following linear system of equations
\begin{displaymath}A^{0}(U)\partial_{z}(u-u_{q})=A^{i}(U)\partial_{i}(u-u_{q})-(A^{0}(U)-A^{0}(u_{q}))\partial_{z}u_{q}\end{displaymath}
\begin{displaymath}+(A^{i}(U)-A^{i}(u_{q}))\partial_{i}u_{q}+(G(U)-G(u_{q}))\end{displaymath}
\begin{equation}\frac{F(x^{j})}{Z}\left\{C(U)-C(u_{q})+\int_{\mathbb{R}^{3}}\left(\begin{array}{cl}0\\Q(v)-Q_{q}\end{array}\right)~d^{3}p\right\}-\xi_{1}\end{equation}
\begin{displaymath}\partial_{z}(v-v_{q})=\sum_{i=1}^{6}B^{i}(p_{j},U)\partial_{i}(v-v_{q})+(B^{i}(p_{j},U)-B^{i}(p_{j},u_{q}))\partial_{i}v_{q}\end{displaymath}
\begin{equation}\frac{1}{Z}(D(U,V,p_{j})-D(u_{q},v_{q},p_{j}))-\xi_{2}\end{equation}
for the unknowns~$v-v_{q}, u-u_{q}$~with initial data~$v-v_{q}=u-u_{q}=0$. The quantities~$U$~and~$V$~are taken to be known smooth functions.~$Q_{q}$~stands for the appropriate part of~$v_{q}$.
\subsubsection{Solving the field equations by a contraction mapping technique}
By the theory of regular, linear symmetric hyperbolic PDE (Racke 1992) we can, given smooth~$(U,V)$~solve (111) for smooth~$v-v_{q}$~and then (110) for smooth~$u-u_{q}$~as long as~$U-u_{q}, V-v_{q}$~are~$O(Z^{q})$. These solutions exist as long as~det$g$~remains positive, where the elements~$g_{\alpha\beta}$~are extracted from the vector~$U$. Equations (110)-(111) thus generate a map~$\Phi:(U,V)\rightarrow (u,v)$. We will define a metric space~$(d,S)$~of functions in such a way that~$\Phi$~is a contraction mapping on~$S$~with respect to~$d$. Some standard techniques from the theory of regular quasilinear symmetric hyperbolic systems (Racke 1992) are used to achieve this.
Let~$U$~defined on~$\mathbb{R}^{3}$~and~$V$~defined on~$\mathbb{R}^{6}$~belong to the space~$S$~iff the following hold:
\begin{enumerate}
\item~$V(Z,x^{i},\cdot)$~is supported in~$\Omega$~for~$Z\leq T$, where~$\Omega$,~relatively compact, is slightly larger than supp$f^{0}(x^{i},\cdot)$~and supported outside~$B_{\epsilon}(0)$~for$Z\leq T$~with~$B_{\epsilon}(0)$~slightly smaller than supp$f^{0}$.
\item~The elements~$g_{\alpha\beta}$~of~$U$~are such that~$|\textrm{det}g(Z,x^{i})|\geq\delta~~\forall Z\leq T$.
\item~$U\in C_{0}^{\infty}(O_{\alpha}),~V\in C_{0}^{\infty}(O_{\alpha}\times\Omega)$.
\item max$\{\|U-u_{q}\|_{s},~\|V-v_{q}\|_{s}\}\leq \rho Z^{q}$~for~$Z\leq T,~\rho$~const,~$s$~large.
\item~$\|\partial_{z}U\|_{s-1}\leq L$
\end{enumerate}
Here~$\|~\|_{s}$~is the~$L^{2}$~type Sobolev norm of order~$s$~(Adams 1975). We choose~$s$~so large that the Sobolev imbedding theorem may be applied wherever needed in the sequel.
\textbf{Lemma 4.2} If~$T$~is chosen sufficiently small and~$L$~sufficiently large then~$\Phi:S\rightarrow S$.
~~~~\textit{Proof}~First we deal with condition 4. Apply~$x^{i}$~derivatives of order~$\alpha$~to (110) and~$x^{i}, p_{j}$~derivatives of order~$\alpha'$~to (111) to get, after some rearrangement
\begin{equation}A^{0}(U)\partial_{z}\nabla^{\alpha}(u-u_{q})=A^{i}(U)\partial_{i}\nabla^{\alpha}(u-u_{q})+F_{1}^{\alpha}-\nabla^{\alpha}\xi_{1}\end{equation}
\begin{equation}\partial_{z}\nabla^{\alpha'}(v-v_{q})=B^{i}(x)\partial_{i}\nabla^{\alpha'}(v-v_{q})+F_{2}^{\alpha'}-\nabla^{\alpha'}\xi_{2}\end{equation}
where the~$F_{i}$~contain all the singular~$\frac{1}{Z}$~terms as well as terms like
\begin{displaymath}-A^{0}(U)\nabla^{\alpha}(A^{0}(U)^{-1}A^{j}(U)\partial_{j}(u-u_{q}))\end{displaymath}
etc.
Now take the inner product of (112) with~$\nabla^{\alpha}(u-u_{q})$~and of (113) with~$\nabla^{\alpha'}(v-v_{q})$. Using the symmetry of~$A^{\alpha}, B^{i}$~and the uniform equivalence of the~$L^{2}$~norm of~$\nabla^{\alpha}(u-u_{q})$~to
\begin{displaymath}\int A^{0}(U)\nabla^{\alpha}(u-u_{q})\cdot\nabla^{\alpha}(u-u_{q})~d^{3}x^{i}\end{displaymath}
in a standard way (Racke 1992) we sum derivatives of order~$\leq s$~to obtain
\begin{equation}\|v-v_{q}\|_{s}(Z)\leq c\int_{0}^{Z}\Big(\|v-v_{q}\|_{s}+\sum_{|\alpha|\leq s}\|F_{2}^{\alpha}\|_{2}\Big)(t)+t^{q}~dt\end{equation}
\begin{equation}\|u-u_{q}\|_{s}(Z)\leq c\int_{0}^{Z}\Big(\|u-u_{q}\|_{s}+\sum_{|\alpha'|\leq s}\|F_{1}^{\alpha'}\|_{2}\Big)(t)+t^{q}~dt\end{equation}
where we also used~$\|\xi_{i}\|_{s}\leq cZ^{q}$.
For the terms in~$F_{2}^{\alpha}$~coming from~$B^{i}$~one may readily use a Moser type estimate (Racke 1992) to obtain an~$L^{2}$~bound in terms of the~$H^{s}$~norm of~$v-v_{q}, U-u_{q}$. For the terms coming from~$D$~it is possible to obtain an~$L^{2}$~bound in terms of the~$H^{s}$~norm of~$U-u_{q}, V-v_{q}$~multiplied by~$\frac{1}{Z}$. In this way (114) leads to
\begin{equation}\|v-v_{q}\|_{s}\leq c\int_{0}^{Z}\|v-v_{q}\|_{s}+\|U-u_{q}\|_{s}+\frac{1}{t}(\|U-u_{q}\|_{s}+\|V-v_{q}\|_{s})+t^{q}~dt\end{equation}
Similarly the terms in~$F_{1}^{\alpha}$~coming from~$A^{\alpha}, G$~may be bounded in a standard way by~$\|u-u_{q}\|_{s}, \|U-u_{q}\|_{s}$. Terms coming from~$C$~can, by Moser and Sobolev inequalities be bounded by a constant times~$Z^{-1}\|U-u_{q}\|_{s}$. The bound (iv) on~$U$~gives a bound for the speed of propagation in equation (111) and thus a bound on the support of~$Q(v)$~in the~$p_{i}$~variables. This entails that the momentum integral of~$Q$~may be estimated by~$\|v-v_{q}\|_{s}$~, via the Cauchy-Schwarz inequality. In this way (115) leads to
\begin{equation}\|u-u_{q}\|_{s}\leq c\int_{0}^{Z}\|u-u_{q}\|_{s}+\|U-u_{q}\|_{s}+\frac{1}{t}(\|v-v_{q}\|_{s}+\|U-u_{q}\|_{s})+t^{q}~dt\end{equation}
Now write~$X=\|v-v_{q}\|_{s}+\|u-u_{q}\|_{s}$~and~$Y=\|U-u_{q}\|_{s}+\|V-v_{q}\|_{s}$. Adding (116) and (117) gives
\begin{equation}X(Z)\leq C\int_{0}^{Z}X+Y+t^{q}+\frac{1}{t}(X+Y)~dt\end{equation}
From (111) we see that
\begin{displaymath}\frac{\partial^{n}(v-v_{q})}{\partial Z^{n}}\Bigg\vert_{Z=0}=0\end{displaymath}
for~$n<q$. Thus~$v-v_{q}$~is~$O(Z^{q})$. Similarly~$u-u_{q}$~is~$O(Z^{q})$.\\Now put
\begin{equation}X'(Z)\equiv Z^{-q}X(Z)\qquad\|X\|'_{s}\equiv\sup_{0\leq t\leq T}t^{-q}\|X\|_{s}(t)\end{equation}
with similar definitions for~$Y$. Equation (118) now implies
\begin{equation}X'(Z)\leq C\left(Z^{-q}\int_{0}^{Z}X'(t^{q}+t^{q-1}+Y'(t^{q}+t^{q-1})~dt~+~Z\right)\end{equation}
and thus
\begin{equation}\|X\|'_{s}\leq C\left\{\left(\frac{T}{q+1}+\frac{1}{q}\right)\|X\|'_{s}+\left(\frac{T}{q+1}+\frac{1}{q}\right)\rho+T\right\}\end{equation}
By choosing~$q$~large and~$T$~small we can therefore arrange that~$\|X\|'_{s}\leq\rho$. It follows that~$\Phi$~preserves condition 4. Conditions 1, 2 and 3 follow from standard theory for linear hyperbolic PDE and from the finite speed of propagation inherent in the equations.
Now given condition 4 for~$(U,V)$~it is elementary, using a Moser estimate to show that
\begin{equation}\|\partial_{z}(u-u_{q})\|_{s-1}\leq C_{1}\left(\|u-u_{q}\|_{s}+(1+Z^{-1})\|v-v_{q}\|_{s}\right)+C_{2}\end{equation}
But by the preceeding result the right hand side can be bounded by a constant. Thus if~$L$~is chosen large enough then~$\Phi$~preserves condition 5. Hence~$\Phi:S\rightarrow S$.
Now define a distance function~$d$~on~$S$~according to
\begin{equation}d(p,p')=\sup_{0\leq t\leq T}t^{-q}\|p-p'\|_{2}\end{equation}
\textbf{Lemma 4.3} For small~$T$~and large~$q$~the mapping~$\Phi$~is a contraction on~$S$~with respect to~$d$.
~~~~\textit{Proof}~Let~$(U,V),~(U',Y')\in S$~and put~$(u,v)=\Phi(U,V),~(u',v')=\Phi(U',V')$. Also write~$\Delta U=U-U'$~etc. Then by a standard energy argument one gets
\begin{equation}\|\Delta v\|_{2}\leq C\int_{0}^{Z}\|\Delta v\|_{2}+\|\Delta U\|_{2}+\frac{1}{t}(\|\Delta U\|_{2}+\|\Delta V\|_{2})~dt\end{equation}
and
\begin{equation}\|\Delta u\|_{2}\leq c\int_{0}^{Z}\|\Delta u\|_{2}+\|\Delta U\|_{2}+\frac{1}{t}\|\Delta v\|_{2}~dt\end{equation}
Now put~$X=\|\Delta v\|_{2}+\|\Delta u\|_{2},~Y=\|\Delta U\|_{2}+\|\Delta V\|_{2}$. Then
\begin{equation}X\leq C\int_{0}^{Z}X+Y+\frac{1}{t}(X+Y)~dt\end{equation}
If we now define
\begin{equation}\|X\|'_{2}=\sup_{0\leq t\leq T}t^{-q}\|u\|_{2}\end{equation}
with a similar definition for~$Y$~then (126) leads to
\begin{equation}\|X\|'_{2}\leq C\left(\frac{Z}{q+1}+\frac{1}{q}\right)\|X\|'_{2}+C\left(\frac{Z}{q+1}+\frac{1}{q}\right)\|Y\|'_{2}\end{equation}
For small~$T$~and large~$q$~one may thus arrange that~$\|X\|'_{2}\leq (1-\delta)\|Y\|'_{2}$~for some~$\delta > 0$. Hence the Lemma.
Now it is elementary to show that the set~$S^{\dagger}\supset S$~of functions~$p$~satisfying the condition
\begin{displaymath}\sup_{0\leq t\leq T}t^{-q}\|p\|_{2}\leq\rho\end{displaymath}
is complete with respect to the metric~$d$. Hence there exists~$p\in\bar{S}$~such that~$p=\Phi(p)$. Now let~$p^{k}=\Phi^{k}(p^{0})$~for some~$p^{0}\in S$. Then~$d(p^{k},p)\rightarrow 0$~as~$k\rightarrow\infty$~and hence~$p^{k}$~is Cauchy with respect to~$d$.
By the Gagliardo-Nirenberg inequality (Racke 1992) one now obtains
\begin{equation}\|Z^{-q}(p^{k}-p^{m})\|_{s'}\leq \|Z^{-q}(p^{k}-p^{m})\|_{s}^{s'/s}\|Z^{-q}(p^{k}-p^{m})\|_{2}^{1-s'/s}\end{equation}
for~$s'<s$.\\
But~$\Phi:S\rightarrow S$~and so the~$\|Z^{-q}p^{k}\|_{s}$~are uniformly bounded. It follows that if we define the distance function~$d_{s'}$~by
\begin{displaymath}d_{s'}(p,p')=\sup_{0\leq t\leq T}t^{-q}\|p-p'\|_{s'}\end{displaymath}
then~$p^{k}$~is Cauchy with respect to~$d_{s'}$. Hence~$\|p^{k}-p\|'_{s'}\rightarrow 0$~as~$k\rightarrow\infty$~and~$\|p\|'_{s'}\leq\rho$.\\
From the field equations one sees that the~$\partial_{z}p^{k}$~converge in~$C([0,T], H^{s'-1})$\linebreak[4]and therefore~$p\in C([0,T],H^{s'})\cap C^{1}([0,T],H^{s'-1})$. By the Sobolev imbedding theorem~$p$~is classically~$C^{1}$~and satifies the field equations (~$p$~stands for the whole solution here). Since~$p$~is~$O(Z^{q})$~we can repeatedly differentiate the field equation to get~$p\in\bigcap_{p=0}^{s'}C^{p}([0,T],H^{s'-p})$. It follows that the parts of~$p$~corresponding to~$f, g_{\alpha\beta}$~and~$h_{\alpha\beta\gamma}$~belong to~$\bigcap_{p=0}^{s'}C^{p}([0,T],H^{s'-p})$. We could of course have chosen~$s'$~arbitrarily large and we thus get that~$f, g_{\alpha\beta}$~and\linebreak[4]$h_{\alpha\beta\gamma}$~belong to~$C([0,T_{s}],H^{s})$, for~$T_{s}\leq T, s>0$. The data at~$T_{s}$~can then be used to construct data for the regular Einstein-Vlasov equations. Now it is well-known that the evolution of~$H^{s}$~data for the regular EV equations stays in~$C([0,Z],H^{s})$~as long as the~$C^{1}$~norm remains bounded (Rendall 1997). In this way we see that our solution of the conformal field equations lies in~$C([0,T],H^{s})$~for every $s$~and is therefore~$C^{\infty}$.
It remains to show that the part~$Q_{\alpha\beta}$~of our solution agrees with the original definition (96)-(97). If we use the parts~$f$~and~$g_{\alpha\beta}$~of our solution to construct a quantity~$\hat{Q}_{\alpha\beta}$~via (96)-(97) then a calculation gives that
\begin{equation}\partial_{z}(\hat{Q}_{\alpha\beta}-Q_{\alpha\beta})=\sum_{i=1}^{6}M^{i}(u, p_{j})\partial_{i}(\hat{Q}_{\alpha\beta}-Q_{\alpha\beta})-\frac{1}{Z}(\hat{Q}_{\alpha\beta}-Q_{\alpha\beta})\end{equation}
where the~$M^{i}$~are symmetric.
Since the singular part of this equation is `negative definite' an~$L^{2}$~energy estimate gives that~$Q_{\alpha\beta}=\hat{Q}_{\alpha\beta}$~as required.
Also note that for small~$T'$~we have~$F(x)=\Phi(p)=1$~on the domain of dependence of~$O'_{\alpha}\times [0,T']$. Thus~$u$~solves the conformal Einstein-Vlasov equations on~$O'_{\alpha}\times [0,T']$~and is determined by the right data.
\subsubsection{Uniqueness of solutions}
We now show that the solution~$p$~constructed above is the only smooth solution of the conformal field equations with the given data.
Suppose then that we have two solutions~$(f, g_{\alpha\beta}),~(\hat{f}, \hat{g}_{\alpha\beta})$~of the localised equations (49)-(50) with the same~$C_{0}^{\infty}$~data. These two solutions will then satisfy equations (96)-(97), (106)-(107). Let~$u$~stand for~$(g, \partial g)$~and let~$v$~stand for~$(f, Q)$~with similar definitions for~$\hat{u},~\hat{v}$. Since the system (106)-(107) is symmetric hyperbolic and the supports of~$f$~and~$\hat{f}$~remain bounded on any closed time interval of existence we may obtain the following~$L^{2}$~energy estimates for the differences~$\Delta u$~and~$\Delta v$:
\begin{equation}\|\Delta u\|_{2}(Z)\leq C\int_{0}^{Z}\|\Delta u\|_{2}+\frac{1}{s}(\|\Delta u\|_{2}+\|\Delta v\|_{2})~ds\end{equation}
\begin{equation}\|\Delta v\|_{2}(Z)\leq C\int_{0}^{Z}\|\Delta u\|_{2}+\|\Delta v\|_{2}+\frac{1}{s}(\|\Delta u\|_{2}+\|\Delta v\|_{2})~ds\end{equation}
Now letting~$X=\|\Delta u\|_{2}+\|\Delta v\|_{2}$~we get that
\begin{equation}X(Z)\leq C\int_{0}^{Z}(1+s^{-1})X~ds\end{equation}
We know from section 4.4.2 that the Taylor series of a solution to the field equations is determined by the initial data. Thus our two solutions~$(u, v)$~and\linebreak[4]$(\hat{u}, \hat{v})$~must agree up to arbitrary order in~$Z$. This is to say that~$X\leq\lambda_{q}Z^{q}$~for arbitrary~$q$. Plugging this into (133) gives
\begin{equation}X(Z)\leq C\lambda_{q}Z^{q}\left(\frac{1}{q}+\frac{Z}{q+1}\right)\end{equation}
Now choose~$q>C$~and choose~$T$~small enough so that the right hand side of (134) is less than~$(1-\epsilon)\lambda_{q}Z^{q}$~on~$[0,T]$~for some~$\epsilon >0$. We see iteratively that~$X\leq (1-\epsilon)^{n}\lambda_{q}Z^{q}$~on~$[0,T]$. Thus~$\Delta u=\Delta v=0$~as required.
\subsubsection{Dependence of solutions on initial data}
Suppose now that we have a 1-parameter family of small perturbations\linebreak[4]$f^{0}_{\epsilon}(x,p)$~of some initial datum~$f^{0}_{0}$~depending smoothly on~$\epsilon, x$~and~$p$~for~$\epsilon\in(-1,1)$. We wish to show that the corresponding solutions~$(f_{\epsilon}, g^{\epsilon}_{ij})$~of the conformal field equations depend smoothly on the parameter~$\epsilon$.
First it follows from an analogue of Theorem 3.1 that the family of initial metrics~$h^{\epsilon}_{ij}$~depends smoothly on~$\epsilon$~and~$x^{i}$. Now we may reconsider the field equations (136)-(141) for~$f, g, h, Q$~with~$\epsilon$~simply acting as an extra coordinate. By replacing the spaces~$H^{s}(\mathbb{R}^{3})$~and~$H^{s}(\mathbb{R}^{6})$~respectively with\linebreak[4]$H^{s}(\mathbb{R}^{3}\times (-1,1))$~and~$H^{s}(\mathbb{R}^{6}\times (-1,1))$~we can now repeat the analysis of section 4.4.5 to get that the solution~$(f, g_{\alpha\beta})$~depends smoothly on~$\epsilon$.
\\\\\\
The results of section 4.4 may now be summarised as follows:\\\\
\textbf{Theorem 4.1} Given a smooth positive function~$f^{0}$~on the cotangent bundle of a paracompact 3-manifold~$M$~satisfying the integral constraint (13) and such that for~$x$~in a compact set
\begin{enumerate}
\item supp$f^{0}(x,\cdot)\subset\Omega_{1}$,~for some relatively compact~$\Omega_{1}\subset\mathbb{R}^{3}$
\item~$\mathbb{R}^{3}\setminus$supp$f^{0}(x,\cdot)\supset\Omega_{2}$, for some neighbourhood of the origin~$\Omega_{2}\subset\mathbb{R}^{3}$.
\end{enumerate}
there exists a unique solution~$(M\times(0,T), f,g_{ab})$~of the Einstein-Vlasov equations with an isotropic singularity having~$f^{0}$~as the orthogonal projection of the limiting particle distribution function at the singularity surface. The solution obtained depends smoothly on the choice of initial data.
\section*{Appendix}
\textbf{Theorem A.1.} Consider a PDE of the form
\begin{gather}A^{0}(u)\partial_{t}u=A^{i}(u)\partial_{i}u+B(u)u+\frac{1}{t}C(u)u\tag{1}\\
u(0)=u^{0}\in C_{0}^{\infty}(\mathbb{R}^{n})\tag{2}\end{gather}
with~$A^{\alpha}$~symmetric and~$A^{0}$~positive definite.\\
If~$C(u)u^{0}=0$~and~$(A^{0})^{-1}C(u^{0})$~has no positive integer eigenvalues
then there exists at most one smooth solution~$u$~of (1) with~$u(0)=u^{0}$.
\textit{Proof.}~Suppose we have two smooth solutions~$u,~v$~of (1) with~$u(0)=v(0)=u^{0}$. One may now subtract the equation satisfied by~$u$~from that satisfied by~$v$~and obtain the following energy estimate:
\begin{gather}\|u-v\|_{2}(t)\leq K\int_{0}^{t} \|u-v\|_{2}+\frac{1}{s}\|u-v\|_{2}~ds\tag{3}\end{gather}
on the time interval~$[0,T]$~for some constant~$K$.\\
Next note that the Taylor series of any smooth solution of (1), (2) is determined by~$u^{0}$, by the following argument:\\
Write~$u=u^{0}+tu^{1}+t^{2}u^{2}+\ldots +t^{q}R(t,x^{i})$. Then (1) implies
\begin{gather}A^{0}(u)\{u^{1}+2tu^{2}+\ldots +qt^{q-1}R+t^{q}R'\}\notag\end{gather}
\begin{gather}-A^{i}(u)\partial_{i}u-B(u)u-\{u^{1}+tu^{2}+t^{q-1}R\}=0\tag{4}\end{gather}
Evaluating (4) at~$t=0$~gives
\begin{gather}(A^{0}(u^{0})-C(u^{0}))u^{1}= A^{i}(u^{0})\partial_{i}u^{0}+B(u^{0})u^{0}\notag\end{gather}
and thus~$u^{1}$~is determined by~$u^{0}$.\\
Now we differentiate (4)~$p$~times with respect to time and evaluate at~$t=0$~to get an equation of the form
\begin{gather}(pA^{0}(u^{0})-C(u^{0}))u^{p}= F(u^{p-1},u^{p-2}\ldots)\notag\end{gather}
and thus~$u^{p}$~is determined inductively by~$u^{0}$.\\
Since our two solutions~$u, v$~of (1) have the same Taylor series it follows that we must have the inequality~$\|u-v\|_{2}\leq \lambda_{q}t^{q}$~for~$t\in [0,T]$~and~$q$~arbitrary. Substituting this into the right hand side of (3) leads to
\begin{gather}\|u-v\|_{2}\leq K\lambda_{q}t^{q}\left(\frac{1}{q}+\frac{t}{q+1}\right)\tag{5}\end{gather}
Now choose~$q> K$~and choose~$T$~so small that the right hand side of (5) is less than~$(1-\epsilon)\lambda_{q}t^{q}$~for some~$\epsilon>0$. Iteratively one gets~$\|u-v\|_{2}\leq (1-\epsilon)^{n}\lambda_{q}t^{q}$~on~$[0,T]$. Thus~$u=v$~as required.
\pagebreak
\section*{References}
\begin{description}
\item~K. Anguige, K. P. Tod, 1998, ~\emph{Isotropic Cosmological Singularities I}, \\gr-qc/9903008
\item~K. Anguige, K. P. Tod, 1998, ~\emph{Isotropic Cosmological Singularities II}, \\gr-qc/9903009
\item~C. M. Claudel, K. P. Newman, 1998~\emph{The Cauchy problem for quasilinear hyperbolic evolution problems with a singularity in the time, Proc. R. Soc. Lond}~454, 1073-1107
\item~R. A. Adams, 1975,~\emph{Sobolev spaces}~(New York: Academic Press)
\item~A. D. Rendall, 1997,~\emph{An introduction to the Einstein-Vlasov system}, in~\emph{Mathematics of Gravitation}, ed. P. Chrusciel (Banach Center Publications, Warszawa) vol. 41, part 1
\item~J. Ehlers, 1971,~in~\emph{General Relativity and cosmology}, ed. R. K. Sachs,~\emph{Varenna Summer School XLVII}~(Acad. Press N. Y)
\item~R. Racke, 1992,~\emph{Lectures on nonlinear evolution equations, Aspects of Mathematics}~vol. E19 (Vieweg)
\end{description}
\end{document}
| {
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Comebacks, OT & Sailor Bear magic: Baylor football's most unbelievably believable finishes
November 7, 2019 // Posted In Athletics, History, Videos
Five years ago last month, Baylor and TCU squared off in what was recently recognized by ESPN as one of the 150 greatest games in college football history. Today, it's known simply by the final score: 61-58.
But that's not the only crazy finish in recent Baylor football history; in honor of Baylor football's recent fourth-quarter dominance under head coach Matt Rhule, here are our top 5 best Baylor finishes of the last 15 years:
Baylor 35, Texas A&M 34 (OT) — October 30, 2004
Ten years before 61-58, Baylor squared off against another in-state rival — the Texas A&M Aggies, who hadn't lost to the Bears in almost 20 years. Hosting the 16th-ranked Aggies, Baylor trailed 13-3 at halftime. But a spirited second half led by quarterback Shawn Bell sent the game to overtime. After A&M scored on its first possession of overtime, Bell hit Dominique Zeigler to pull within one. Eschewing a game-tying extra point, head coach Guy Morriss called for a gutsy two-point conversion. Bell found Zeigler again to win the game, the goalposts came down, and the Bears picked up perhaps their most memorable win of the millennium's first decade. (The video highlights are below, but the Texas A&M radio call is also a classic.)
Baylor 45, Oklahoma 38 — Nov. 19, 2011
In a season filled with superlatives, this was Robert Griffin III's Heisman moment. The Bears had never defeated Oklahoma, who came into the game ranked fifth in the nation. But Baylor established itself with a strong first quarter, and tied the game in the third on a stupendous caromed pass that turned into an 87-yard Kendall Wright touchdown. After OU tied the game with less than a minute remaining, the Bears mounted a drive that led to an electrifying game-winning RG3 touchdown pass to Terrance Williams with seven seconds remaining. Less than a month later, RG3 earned the program's first Heisman Trophy.
Baylor 61, TCU 58 — Oct. 11, 2014
61-58 is a part of Baylor lore, and it's a substantially better number than 58-37, the score by which the No. 5-ranked Bears trailed No. 9 TCU with less than 12 minutes remaining. Three straight touchdown drives tied the game up in less than eight minutes of play, and a huge defensive stop by cornerback Ryan Reid gave the Bears the ball back for a final march downfield. Freshman kicker Chris Callahan hit a 28-yard field goal with no time left to give McLane Stadium its first signature moment.
Baylor 35, Oklahoma State 31 — Nov. 3, 2018
Celebrating Homecoming 2018 with Sailor Bear's first-ever appearance on their helmet, the Bears made sure they came through for him. But it wasn't easy; Baylor trailed by 10 with just under nine minutes to go. Needing a touchdown and a defensive stop to have a chance, the Bears got both, setting up quarterback Charlie Brewer and the offense for one final drive with 90 seconds left. Quickly driving into range for a potential game-tying field goal, Brewer had time for one more shot to the end zone, hitting Denzel Mims from six yards out for a pivotal Homecoming victory as "Sailor Bear magic" was born.
Baylor 33, Texas Tech 30 (2OT) — Oct. 12, 2019
Maybe there's something about Homecoming of recent vintage. After nearly erasing a 25-point deficit in Coach Rhule's first year against West Virginia, the Bears followed up 2018's thriller with a double-overtime Homecoming victory over Texas Tech. This time around, Texas Tech claimed the lead with just over 90 seconds remaining, but the Bears reeled off 11 plays to march to the 2-yard line, tying the game on regulation's final play with a John Mayers field goal. After trading touchdowns in the first overtime ever played at McLane Stadium, the Bears held Tech to a field goal in the second OT period. Moments later, they scored on a JaMycal Hasty touchdown run to win the game and remain undefeated.
In a competitive Big 12 and with a team that's built a reputation as a tough out in the fourth quarter, we wouldn't be surprised to have even more amazing Baylor wins to talk about before too long.
Sic 'em, Bears! | {
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Customize your results by using the narrow options to see the collections that best fit your sofa or click on a collection to see all fit types available. | {
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Masthead Bermudan sloop with deck stepped twin spreader rig by Z-Spars. Stainless steel standing rigging. Terylene running rigging (replaced as needed). In-mast furling mainsail system. Profurl headsail furling system. Winches: 2 x Harken 44 2-speed self-tailing sheet winches. 2 x Harken 40 2-speed self-tailing halyard winches. Sails: Furling mainsail - Kemp Performance Cruise Premium cloth- good condition - new 2011 Furling genoa - Kemp Performance Cruise Premium cloth- good condition - new 2011 Asymmetrical spinnaker with snuffer Mechanical: Yanmar 4JH3E 56hp 4 cylinder inboard diesel engine. Shaft drive with three bladed fixed propeller. Plastic fuel tank. The engine has completed 1357 hours as at 6th November 2018. The engine is about to be serviced. 4HP through tunnel bowthruster with own dedicated battery. Electrical: 4 x 12-volt batteries (2 x domestic and 1 x engine and 1 x bowthruster) age not known, charged from alternator on the engine. 220-volt battery charger, shore power system and immersion heater on the calorifier. | {
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{"url":"https:\/\/learn.careers360.com\/engineering\/question-help-me-please-peptization-is-a\/","text":"# Peptization is a :Option 1)process of bringing colloidal molecule into solutionOption 2)process of converting soluble particles to form colloidal solutionOption 3)process of converting precipitate into colloidal solutionOption 4)process of converting a colloidal solution into precipitate\n\nAnswers (1)\n\nPreparation of Colloids -\n\nPeptization\n\n- wherein\n\nDuring peptization, the precipitate absorbs one of the ions of the electrolyte on its surface. This causes the development of positive or negative charge on precipitates, which ultimately break up into smaller particles of the size of a colloid.\n\nPeptization is a process of converting precipitate into colloidal solution .\n\n$\\therefore$\u00a0option (3) is correct\n\nOption 1)\n\nprocess of bringing colloidal molecule into solution\n\nOption 2)\n\nprocess of converting soluble particles to form colloidal solution\n\nOption 3)\n\nprocess of converting precipitate into colloidal solution\n\nOption 4)\n\nprocess of converting a colloidal solution into precipitate\n\nExams\nArticles\nQuestions","date":"2020-06-06 23:58:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.5359435081481934, \"perplexity\": 13990.624708419478}, \"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-2020-24\/segments\/1590348521325.84\/warc\/CC-MAIN-20200606222233-20200607012233-00446.warc.gz\"}"} | null | null |
{"url":"http:\/\/tex.stackexchange.com\/questions\/89057\/pgfplots-gives-spurious-lines-in-external-mode-with-geometry","text":"# Pgfplots gives spurious lines in external mode with geometry\n\nThis is not a major problem but all the same I would like to confirm that I am not doing anything wrong. Geometry package with showframe=true option is used to see if all the figures are within the margins. Also, tikz external library is used and the plot shows some spurious horizontal and vertical lines.\n\nThis happens only if showframe=true is set in geometry and the tikz external library is used. If external is switched off, the plot is drawn correctly. As I said earlier, since I am not going to display frame in the final draft, this is not a major issue.\n\nMWE\n\n\\documentclass{report}\n\\usepackage[showframe=true,hmargin=3cm,vmargin=2cm]{geometry}\n\\usepackage{tikz}\n\\usepackage{pgfplots}\n\\pgfplotsset{compat=1.7}\n\n%Comment out to not use external library and get correct figure.\n\\usetikzlibrary{external}\n\\tikzexternalize\n\n\\begin{document}\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}\n\\end{axis}\n\\end{tikzpicture}\n\\end{figure}\n\\end{document}\n\n-\nI think the comment of @JLDiaz should become an answer. Apparently, both packages change the TeX \\shipout routine. As long as the resulting bounding box is correct... \u2013\u00a0Christian Feuers\u00e4nger Jan 3 '13 at 19:34\n@JLDiaz Can you convert your comment to an answer? \u2013\u00a0percusse Mar 11 '13 at 21:05\n@percusse Done. \u2013\u00a0JLDiaz Mar 12 '13 at 0:33\n\nI had the same problem and fixed it by writing the following after the \\tikzexternalize command:\n\\tikzifexternalizing{\\setkeys{Gm}{showframe=false}}{}","date":"2016-05-30 08:50:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9256951808929443, \"perplexity\": 1122.7805125126629}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464050955095.64\/warc\/CC-MAIN-20160524004915-00105-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
Brock & Kiraz English honey.
How to cite this entry? Toma Audo, Treasure of the Syriac Language: A Dictionary of Classical Syriac (Mosul: Imprimerie des pères dominicains, 1897-. Reprints: Chicago, 1978; Stockholm, 1979; Glane/Losser, 1985; Piscataway, NJ, 2008) Vol 1, p. 174 [from sedra.bethmardutho.org, tagged by George A. Kiraz, accessed on Apr. 23, 2019].
How to cite this entry? Sebastian P. Brock & George A. Kiraz, Gorgias Concise Syriac-English, English-Syriac Dictionary (Piscataway, NJ: Gorgias Press, 2015) ܕܒܫܐ [from sedra.bethmardutho.org, accessed on Apr. 23, 2019].
How to cite this entry? Robert Payne Smith, Thesaurus Syriacus (Oxford: The Calerndon Press, 1879) Vol 1, p. 408 [from sedra.bethmardutho.org, tagged by Cosimo Paravano, accessed on Apr. 23, 2019].
How to cite this entry? George A. Kiraz, SEDRA 3 ܕܒܫܐ [from sedra.bethmardutho.org, accessed on Apr. 23, 2019].
How to cite this entry? J. Payne Smith (Mrs. Margoliouth), A Compendious Syriac Dictionary (Oxford: The Clarendon Press, 1903) p. 83 [from sedra.bethmardutho.org, tagged by Michael Wingert, accessed on Apr. 23, 2019].
How to cite this entry? J. Payne Smith (Mrs. Margoliouth), Supplement to the Thesaurus Syriacus of R. Payne Smith (Oxford: The Clarendon Press, 1927) p. 83 [from sedra.bethmardutho.org, tagged by Sebastian Kenoro Kiraz, accessed on Apr. 23, 2019]. | {
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Is immunotherapy the wrong choice for some lung cancer patients?
Amidst all of the glowing reports about immunotherapy for lung and many other cancers, it would be understandable for patients and physicians to be tempted to rush toward prioritizing immunotherapy as the first treatment strategy to pursue. In fact, a highly publicized trial called KEYNOTE-024 was just presented at the ESMO meeting in Copenhagen and demonstrated a significant improvement in progression-free and overall survival over standard chemotherapy doublet treatment as the first line approach for patients with high level expression of the PD-L1 protein on their tumor (about 30% of patients). But there is also converging evidence that some patients are consistently less likely to benefit from immunotherapy -- specifically, those patients with an EGFR mutation and perhaps others with another "driver mutation" such as an ALK or ROS1 rearrangement. This is an important issue to know, because I and some other lung cancer specialist colleagues see patients with one of these highly targetable lesions sometimes being mistakenly recommended immunotherapy over the optimal targeted therapy for their cancer, or patients deflect a recommendation for an EGFR or ALK inhibitor in favor of immunotherapy based largely or completely on the hype around the latest new idea in cancer treatment.
It is only in the last decade that we learned that patients with EGFR mutations, ALK rearrangements, ROS1 rearrangements, and perhaps other specific mutations can be the key "driver" of a lung cancer. Such markers are most typically seen in younger patients who often have little or no smoking history, and the idea is that these random developments can be sufficient to generate the chaotic, rapid cell growth and division of a cancer, even without just about any other mutation. This is in contrast to most lung cancers and many other kinds of cancer, which often develop as a product of a wide range of dozens to hundreds of mutations that collect over a long time, which is why so many cancers are associated with advancing age and exposure to various environmental toxins, most notably tobacco smoke for lung cancer.
Though there are certainly differences among targeted therapies, these are now routinely recommended as first line treatment when we identify an EGFR mutation or ALK or ROS1 rearrangement after a patient is diagnosed with advanced NSCLC. These oral medications are typically associated with a response rate of 60-75% and a better side effect profile than standard chemotherapy, which is still also of potential value but usually a deferred until after patients have exhausted the benefit of the more effective and better tolerated targeted therapy. Though several immunotherapy agents have been tested compared with second line Taxotere (docetaxel) and have been shown to be more active and better tolerated for previously treated advanced NSCLC, these same trials have consistently shown that never-smokers show less benefit than smokers, and patients with EGFR mutations are a subset of patients who benefit a bit less from immunotherapy than from standard chemo (the differences aren't statistically significant, but they are conspicuous for being the only clinical variable that trends in the wrong direction from a benefit for immunotherapy). Though smaller subgroups such as patients with an ALK or ROS1 rearrangement have never been looked at in these trials as a distinct subgroup, it has been hard for lung cancer specialists to identify a case of any such patient who has responded very dramatically and for a long time to an immune checkpoint inhibitor. We suspect that most or all patients with driver mutations are unlikely to be major beneficiaries of immunotherapy.
At the same time, there have also been reports of potentially increased toxicity, specifically pneumonitis (inflammation in the lungs) from a combination of the third generation EGFR inhibitor Tagrisso (osimertinib) with the immune checkpoint inhibitor durvalamab that has raised our concern about presuming that immunotherapy is always safe.
Though we don't yet know as much as we'd like to about why immunotherapy seems to be relatively less active for patients with EGFR mutations and perhaps other driver mutations as well, I think it is related to the concept of "mutational load". In addition to the probability of a good response to immunotherapy being associated with high levels of expression of the protein PD-L1, protein helps to decrease the reactivity of the immune system, on tumor cells, tumors that have a large number of individual mutations also often tend to respond better to immunotherapies. I think of it simply as a "landscape" that has far more features for the immune system to recognize. The converse of this is that the tumors that are driven by a single very important mutation, often arising with very few other mutations, don't provide much for the immune system to recognize and go after.
Importantly, this isn't to say that someone with a driver mutation shouldn't receive an immune checkpoint inhibitor at some point. After all, these agents have tended to perform about comparably to standard second line chemotherapy. But that is a lower bar than standard doublet chemo and far less than the usually very impressive efficacy of the right targeted therapy for the right target, if we are fortunate enough to find one. So please don't believe that immunotherapy is the right answer for everyone. The concept of precision medicine means that individual patients have different best treatments for them. For those patients with an EGFR mutation at least, and very likely for other patients with other driver mutations, the best therapy by far is still likely to be their oral targeted therapy, and it would be a grave mistake to deflect that choice because they are swept by the tide of excitement for immunotherapy. | {
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It provided great insight and oppurtunities, and a realistic preiview of whats excpected from employers in a business environment.
Nice Place to work, very friendly people to work with. Didn't find all of the work captivating but still a variety of work. | {
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} | 1,199 |
\section{Numerical Integration}
For the complex Ginzburg-Landau equation, we use a pseudospectral integration method. We take a periodic domain of size of size $L=32\pi$ in each direction and discretize using $N_x=N_y=128$ grid points in each spatial direction. Derivatives are calculated using fast Fourier transforms, and the discretized system is integrated with a 4(5) Runge-Kutta-Fehlberg method (which is also used for the other equations, with relative and absolute error tolerances of $10^{-6}$). To produce states in the dynamical phases of interest, we take random initial conditions $A_0 = \sum_{nm} \alpha_{nm} e^{i\mathbf{k}_{nm}\cdot \mathbf{x}} + \epsilon e^{i\mathbf{k}_{2\,2}\cdot \mathbf{x}}$, where $\alpha_{nm}$ are complex random Gaussian amplitudes with mean zero and variance $\sigma^2/(1+n^2+m^2)$, $\mathbf{k}_{nm} = 2\pi(n\hat{\mathbf{x}}+m\hat{\mathbf{y}})/L$, the sum ranges over $-2\leq n,m \leq 2$, and $\epsilon$ is the scale of an initial plane wave perturbation with wavevector $k_{2\,2}$. The mode amplitudes are determined by $\sigma=0.1,0.1,0.1,1.0$ and $\epsilon=0.01,0.01,1.0,0.01$ for the four trajectories used in the main text. The system is allowed to approach an attractor for the first $90$ time units, then the trajectory is formed by the next $10$ time units, in steps of $0.01$. We also provide an animation showing the phase and amplitude for longer runs of $100$ time units (Fig.~\ref{figs1}). A similar pseudospectral approach was used for the Oregonator and Swift-Hohenberg examples, but, in the Swift-Hohenberg case, with $N_x=256$ discretization points, a domain of size $L=64 \pi$, an integration time of $5$ time units, and random initial condition given by the real part of $u_0=\sum_{n=-20}^{20} \alpha_n e^{ik_nx}$ with $k_n=2\pi n/L$ and $\alpha_n$ complex random Gaussian amplitudes with mean zero and variance $1.0/(1+\sqrt{|n|})^2$.
\begin{figure}[hb]
\includegraphics[width=0.5\columnwidth]{figs1.png}
\caption{Snapshot of the animation showing the phase $\phi$ and amplitude $r$ of the trajectories, where $A=re^{i\phi}$.\label{figs1}}
\end{figure}
\clearpage
\section{Demonstrations}
Demonstrations of SINDyCP in discrete maps, ODEs and PDEs are shown in Fig.~\ref{figs2}. The left panels illustrate the logistic map,
\begin{equation}
x_{n+1}=rx_n(1-x_n),
\end{equation}
which is a discrete-time system with a single dependent variable $x_n$ and a single parameter $r$. This equation is the model for a universal period-doubling route to chaos as the parameter $r$ increases past $3.56995$. We perform the SINDyCP fit using four sample trajectories of $1000$ iterations, corresponding to parameter values $r=3.6,3.7,3.8,3.9$ (red dotted lines in Fig.~\ref{figs2}). We employ a library consisting of polynomials up to third order in the dependent variable $x_n$ and linear functions of the control parameter $r$, and the SINDyCP approach correctly identifies the parameterized equation. The middle panels illustrate the Lorenz system,
\begin{align}
\dot{x} = \sigma (y - x),
\dot{y} = x(\rho - z) - y,
\dot{z} = x y - \beta z,
\end{align}
which consists of three ordinary differential equations in three dependent variables $x$, $y$, and $z$ and three parameters $\sigma$, $\rho$ and $\beta$. This equation exhibits the iconic butterfly-shaped Lorenz attractor for certain parameter values. We perform the SINDyCP fit using five sample trajectories that have converged to their attractors, corresponding to the randomly selected parameter values $\sigma=10.0, 9.8, 9.9, 10.3, 9.5$, $\rho=27.6, 28.2, 28.3, 27.6, 28.1$, and $\beta =3.1, 2.4, 2.4, 2.3, 2.4$, respectively. We use feature and parameter libraries consisting of polynomials up to fourth order in the dependent variables $(x,y,z)$ and linear functions in the parameters $(\sigma, \rho, \beta)$, and the SINDyCP approach again correctly identifies the parameterized equation. Finally, the right panels illustrate the CGLE described in the main text.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{figs2}
\caption{Demonstrations of the SINDyCP approach for three models (top row) of nonlinear dynamics. Several trajectories produced from different parameter values (middle row) are supplied as input, and the SINDyCP fit (bottom row) correctly identifies the governing equations in each case. \label{figs2}}
\end{figure}
\clearpage
\section{Oregonator model and normal form transformation}
We mainly follow the analyses of the Oregonator model in Refs.~[30,32], with realistic parameter values shown in Table \ref{tabs1}. The fixed point $(C_X,C_Y,C_Z)=(C_X^0, C_Y^0, C_Z^0)$ undergoes a Hopf bifurcation as $\mu$ increases from zero, leading to oscillatory chemical dynamics. For small $\mu$, the weakly nonlinear theory follows from a perturbative expansion of the model. Take $\mathbf{x} \equiv (C_X,C_Y,C_Z)-(C_X^0, C_Y^0, C_Z^0)$ and express Eqs.~(4)-(6) as $\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})$. Define the multilinear operators of partial derivatives $\mathbf{F}_{\mathbf{x}^n}(\mathbf{e}_{i_1},\cdots, \mathbf{e}_{i_n}) = {\partial^n \mathbf{F}}/{\partial x_{i_1}\cdots\partial x_{i_n}}$ with $\mathbf{e}_{i}$ the $i$th component unit vector. Then the Taylor expansion for the system is
\begin{equation}
\dot{\mathbf{x}} = \left(\partial \mathbf{F}/\partial \mu\right) \mu + \mathbf{F}_{\mathbf{x}^1}(\mathbf{x}) + \left(\partial \mathbf{F}/\partial \mu\right)_{\mathbf{x}_1}(\mathbf{x})\mu + \frac{1}{2} \mathbf{F}_{\mathbf{x}^2}(\mathbf{x},\mathbf{x})+\frac{1}{6} \mathbf{F}_{\mathbf{x}^3}(\mathbf{x},\mathbf{x},\mathbf{x})+D\cdot \nabla^2\mathbf{x} + \cdots,
\end{equation}
where $D$ is a diagonal matrix with elements $D_X$, $D_Y$ and $D_Z$. We develop a transformation $\mathbf{x}=\mathbf{y} + \mathbf{h}(\mathbf{y},\mu)$ perturbatively, where $\mathbf{y} \equiv A e^{i\omega_0 t} \mathbf{u} + \bar{A} e^{-i\omega_0 t} \bar{\mathbf{u}}$. Here $\mathbf{u}$ is one of the critical eigenvectors of the Jacobian matrix $\mathbf{F}_{\mathbf{x}^1}$ with eigenvalue $\lambda = i\omega_0$ (with zero real part for $\mu=0$) and overbars represent complex conjugates, and we also define the corresponding left eigenvector at $\mathbf{u}^\perp$. The near-identity transformation function $\mathbf{h}(\mathbf{y},\mu)$ is selected so as to eliminate the non-resonant terms in the evolution equation of $A$, which can be accomplished under general conditions. This results in an amplitude equation $\dot{A} = \mu \sigma A + g|A|^2A + d\nabla^2 A$,
where
\begin{align}
\sigma &= \mathbf{u}^\perp \cdot \left(\partial \mathbf{F}/\partial \mu \right)_{\mathbf{x}^1}(\mathbf{u})- \mathbf{u}^\perp \cdot \mathbf{F}_{\mathbf{x}^2}\left[\mathbf{u},\left(\mathbf{F}_{\mathbf{x}^1}\right)^{-1}\left(\partial \mathbf{F}/\partial \mu \right)\right], \\
g&=\frac{1}{2}\mathbf{u}^\perp \cdot \mathbf{F}_{\mathbf{x}^3}\left(\mathbf{u},\mathbf{u},\bar{\mathbf{u}}\right)-\mathbf{u}^\perp \cdot \mathbf{F}_{\mathbf{x}^2}\left\{\mathbf{u},\left[\mathbf{F}_{\mathbf{x}^1}\right]^{-1}\left[\mathbf{F}_{\mathbf{x}^2}\left(\mathbf{u},\bar{\mathbf{u}}\right)\right]\right\} \nonumber \\
&\quad -\frac{1}{2}\mathbf{u}^\perp \cdot \mathbf{F}_{\mathbf{x}^2}\left\{\bar{\mathbf{u}},\left[\mathbf{F}_{\mathbf{x}^1}-\left(\lambda-\bar{\lambda}\right)I\right]^{-1}\left[\mathbf{F}_{\mathbf{x}^2}\left(\mathbf{u},\mathbf{u}\right)\right]\right\},\\
d&=\mathbf{u}^\perp \cdot D \cdot \mathbf{u}.
\end{align}
By rescaling the amplitude by a factor of $\mu^{1/2}$, time by a factor of $1/\mu$, and space by a factor of $1/\mu^{1/2}$ and employing additional rescalings to unitize the real components and eliminate the mean rotation, we can arrive at the CGLE in Eq.(2), where $b\equiv\mathrm{Im}(d)/\mathrm{Re}(d)=b_0=0.173$ and $c\equiv-\mathrm{Im}(g)/\mathrm{Re}(g)=c_0=2.379$. As expected, these parameter values correspond to the amplitude turbulence regime of the CGLE.
For our numerical simulations, we use a spatial domain of length $L=0.4/\mu^{1/2}\text{ cm}$ and an integration time of $T=200/\mu\text{ s}$, where we scaled by $\mu$ to ensure the trajectories have corresponding scales. We strobe the time in steps of $5.94804\text{ s}$, which corresponds to the critical frequency of the instability. We then interpolate the time series in steps of $T/1000$ to generate the trajectories. The first 200 time steps are discarded as the trajectories relax to their attractors. The next 400 time steps are used to train the SINDyCP model, while the remaining 400 steps are used as test trajectories to evaluate the $R^2$ scores. We finally employ the normal form transformation described above for the SINDyCP model to evaluate the parameterized $b(\mu)$ and $c(\mu)$ shown in Fig.~2(b) of the main text. Consistently, the normal form parameters very closely approximate the analytic results $b(0)\approx b_0$ and $c(0)\approx c_0$, but significant variations emerge for larger $\mu$.
\begin{table}[h]
\begin{tabular}{c | c | c | c | c | c | c | c | c | c | c | c}
$k_1$ & $k_2$ & $k_3$ & $k_4$ & $k_5$ & $D_X$ & $D_Y$ & $D_Z$ & $C_H$ & $C_A$ & $C_B/(1-\mu)$ & $\nu$ \\
\hline
2 & $10^6$ & $10$ & $2 \times 10^3$ & $1$ &$10^{-5}$ & $1.6 \times 10^{-5}$ & $0.6 \times 10^{-5}$ & $0.5$ & $1$ & $0.787$ & $1$
\end{tabular}
\caption{Parameter values for the Oregonator model, in cgs units (suppressed for brevity).\label{tabs1}}
\end{table}
\clearpage
\section{Weak formulation implementation}
We refer the reader to Refs. [25-28] for the theory of the weak formulation of SINDy. Here, we only briefly describe our efficient numerical integration method for the weak formulation used in pysindy. We suppose that the spatial grid is one-dimensional, for the moment, and the values of the coordinates on the grid points are $x_i$. The weak form requires us to calculate the integral of interpolated data $f(x)$ weighted by the $d$th derivatives of test function $\phi(x)$,
\begin{equation}
I^{(d)} \equiv \int_{x_0}^{x_N}f(x)\phi^{(d)}(x)dx.
\end{equation}
We choose to use test functions $\phi(x)=(x^2-1)^p$ in our implementation, and thus their $d$th derivatives are
\begin{align}
\phi^{(d)}(x) &= \frac{\partial}{\partial{x^d}}(x^2-1)^p = \sum_{k=0}^{p}\begin{pmatrix} p \\ k \end{pmatrix}(-1)^k \frac{(2(p-k))!}{(2(p-k) -d)!} x^{2(p-k)-d}.
\end{align}
We are provided with some feature values $u_i$ at the grid points, and we consider the value of a library function $f$ applied to that feature, $f_i \equiv f(u_i)$. We linearly interpolate the function as $f(x) = f_i + \frac{x-x_i}{x_{i+1}-x_i}(f_{i+1}-f_i)$ where $x_i \leq x \leq x_{i+1}$.
Expanding the interpolation, and integrating the $x\phi^{(d)}(x)$ terms by parts,
\begin{align}
I^{(d)} &= \sum_{i=0}^{N-1}\int_{x_i}^{x_{i+1}}\left[f_i + \frac{x-x_i}{x_{i+1}-x_i}(f_{i+1}-f_i) \right] \phi^{(d)}(x)dx \nonumber \\
&= \sum_{i=0}^{N-1} \frac{f_i x_{i+1}-f_{i+1}x_i}{x_{i+1}-x_i}\left[\Phi^{(d)}(x_{i+1})-\Phi^{(d)}(x_i) \right]\nonumber \\
&+ \frac{f_{i+1}-f_i}{x_{i+1}-x_{i}}\left[\Phi^{(d-1)}(x_{i+1})-\Phi^{(d-1)}(x_i) \right], \label{sum}
\end{align}
where $\Phi^{(d)}(x)$ are the antiderivatives of $\phi^{(d)}$ [i.e. $\Phi^{(d)}(x) = \phi^{(d-1)}(x)$ for $d>0$].
By relabelling the dummy summation variables, we can recast Eq.~\eqref{sum} as a dot product between the input data $f_j$ and a weight $w_j$
\begin{align}
I^{(d)} &= \sum_{j=0}^{N-1} w_j \cdot f_j, \label{dot}
\end{align}
with
\begin{align}
w_j &\equiv \frac{x_{j+1}\left[\Phi^{(d)}(x_{j+1})-\Phi^{(d)}(x_j) \right]}{x_{j+1}-x_j} - \frac{x_{j-1}\left[\Phi^{(d)}(x_{j})-\Phi^{(d)}(x_{j-1}) \right]}{x_{j}-x_{j-1}} \nonumber \\
&+ \frac{\Phi^{(d-1)}(x_j)-\Phi^{(d-1)}(x_{j-1})}{x_j - x_{j-1}} -
\frac{\Phi^{(d-1)}(x_{j+1})-\Phi^{(d-1)}(x_{j})}{x_{j+1} - x_{j}}, \label{iweights}
\end{align}
where $0<j<N-1$. At the left and right sides of the domain (for $j=0$ and $j=N-1$), we must adjust the weights to correct for boundary effects,
\begin{align}
w_0 &\equiv \frac{x_{1}\left[\Phi^{(d)}(x_{1})-\Phi^{(d)}(x_0) \right]}{x_{1}-x_0} -
\frac{\Phi^{(d-1)}(x_{1})-\Phi^{(d-1)}(x_{0})}{x_{1} - x_{0}}, \label{lweights} \\
w_{N-1} &\equiv - \frac{x_{N-2}\left[\Phi^{(d)}(x_{N-1})-\Phi^{(d)}(x_{N-2}) \right]}{x_{N-1} - x_{N-2}} + \frac{\Phi^{(d-1)}(x_{N-1})-\Phi^{(d-1)}(x_{N-2})}{x_{N-1} - x_{N-2}}. \label{rweights}
\end{align}
Expressing the integrals along each dimension as dot products [Eq.~\eqref{dot}] enables efficient vectorization with BLAS operations, and the integration weights [Eq.~\eqref{iweights}-\eqref{rweights}] only need to be evaluated a single time when the library is first initialized (in a vectorized fashion). We further vectorize the code by forming tensor products over all integration dimensions to calculate multidimensional integrals using a single tensor dot product.
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,748 |
Chemical and Material Engineering Department Coming to Concordia / March 14, 2017 by Carl Bindman
Published in The Link 37.24 on March 14, 2017
Less than 13 per cent of practicing, licensed engineers across Canada are women, according to the regulatory body Engineers Canada.
But in a 2013 report from that same group, they found that female enrolment in specific fields of engineering were much higher. Chemical Engineering was one such field, at 33 per cent female enrolment. As Concordia will soon have its own Chemical and Materials Engineering department, there is hope that the school's numbers will rise.
"Right now the average at Concordia in engineering is 17 per cent [of women enrolled]," said Karina Bagryan, co-president of Women in Engineering Concordia. "Based on other universities' experiences, if we bring chemical engineering it's going to help the issue."
During November 2016's Board of Governor's meeting it was unanimously decided that a Chemical and Materials Engineering Department would be created at Concordia.
Amir Asif, Dean of the Faculty of Engineering and Computer Science, pushed for the new department. "Within Chemical Engineering there is an opportunity," he said. "Since female students feel connected to Chemical Engineering."
This is because many female science students want to be doctors, explained Fariha Kamal, the other co-president of Women in Engineering Concordia. She said that, for example, in CEGEP they might enjoy chemistry, organic chemistry and biology, but when they finish their time they realize they don't want to be a doctor.
"They look into their options," she said, and chemical engineering has a lot in common to health sciences in regards to its inclusion of organic chemistry and biology.
The new department follows a broad push in Concordia's Engineering Faculty to attract more female students. The school introduced 25 new scholarships last year. "Eighteen are open to anyone," said Asif, "but seven are for female students only."
Of course, other factors influenced the approval of the new department.
"There is a major distinction between chemistry and chemical engineering that has started in the last 15 to 20 years," said Asif. This distinction is rooted in the growth of hands-on applications of chemical research. "We thought it is a great time for us to offer a chemical engineering program."
The new department, said Asif, will consolidate teaching and research across Concordia. And according to the proposal that was presented to the Board of Governors in December, the department is expected to operate at a $2.5 to $3.5 million surplus within three to four years.
The successful proposal for the new department describes the space needed for the new department. It points to two areas that will need to be renovated in the Hall building, on the 10th and 14th floors. According to the document, the 14th floor space is around 160 square metres and renovations will cost $1.3 million. For the 10th floor, the expected cost of renovating 500 square metres is $4 million.
"There's a discussion with the federal and provincial government in terms of new space," says Asif.
Until the renovations are complete, the department will be housed in the Science Complex at Loyola for the 2018-2019 academic year, when classes in the department are set to begin.
Asif said the department should start functioning soon, but that there isn't yet a firm date. So far, they have hired one professor, Dr. Alex De Visscher, whose focus is in chemical and petroleum engineering. He hopes that the department will be up and running before this summer semester.
Source: https://thelinknewspaper.ca/article/chemic...
← Editorial: The Curse of the Short-Term Contract The Concordia "Bomb Hoax" Was a Form of Terrorism → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,973 |
Yes, one Yes, one of note would be Boonah. There is a rec flying school there, gliding club and numerous hangers. Nigel Arnot also has his facilities there.
Hi Marwil, Have you had any luck in getting info about your Shadow. Give me a call on 0402107927 and I will assist with what I can. Regards dave.
Hi Marwil, I am the previous owner of 19-0917, and can enlighten you on most of the details of the Shadow. I have a second one that I am currently rebuilding. I have a feeling that I stillbhave some copies of the relevent paperwork of your aircraft. The log books should have been supplied with the aircraft upon sale, and are required for continued complience for registration. If you would like some assistance, visit "tp21.net" for contact details. I would be happy to help. Regards Dave.
Hi Marwil, My appologies, but haven't been on for several months. Yes, I did own 19.0917. You can contact me at 0402107927. Regards Dave.
Hi dtanner, I believe I bought your former Streak Shadow 19-917 from a guy in Bundy. I didn't have a clue about them (or RAA) or how they flew, their faults, how to fly one, I have no license, not affiliated with any club but it just it just looked so good. If you have time I would like to hear how you came to own this CFM and any stories, mods, recommendations that you have. Thanks. Mark.
is there a finer flying machine than this?
Hey Huggy, Maybe we should'nt have sold you that bird, I'm just wondering what is going to happen when you get your ticket. We won't see you for clouds. Will have to call you Biggles mate. Just havin fun. Will call you on the weekend about the panel and delivery. Cheers. Dave.
hi all. new Streak Shadow owner.
Hi all, I am the previous owner and I can tell you categorically that this Jabiru does not run hot at all and never has. The amount of cooling available ensures ample cooling on hot days even in a good climb, and can be too efficient in cold weather. I have two high density plugs for the intakes for winter. The temps marked are easily abided by. Regards Dave.
Hi yogasri, havent been for months, work and life in general. I own two streak shadows, and know a little about them, so if you have any questions, let me know and I will try to answer them. As for the landing gear, one of our Shadows was built about 1994, and still has the original landing gear. It has over 750 hours on it and a couple thousand landings, and still ok. The other has a glass gear fitted, and stands much higher than original. This gives excellent ground clearance and stance as it is 2 mitres wide, and from a local manufacturer (Qld). One shadow can be seen at "tp21.net" members market. I will try to answer any questions you have. Cheers. Dave. Happy Flying.
Hey Guys, Where do you start new forum threads? Thanks Dave.
Hey Gerry, If I could put in my two bobs worth. I have no problems with transmission or receiving signal, and as we both have the same type suggest you either mount aerial either side of tail boom on metal ground plane under skin, in wing root center section, or at 45* downward facing forward from nose cone allowing enough clearance for turf under nose. I have used both and found works well. I now have a UHF antenna in the under nose position as it seems to work best for UHF as most stations I communicate with are on the ground. With the aerial standing vertical for VHF works well and whether its because it bends in flight or just the ground plane doing it's thing, I'm not sure. Anyway hope this helps. Dave. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,660 |
The 1993 Norwegian Football Cup Final was the final match of the 1993 Norwegian Football Cup, the 88th season of the Norwegian Football Cup, the premier Norwegian football cup competition organized by the Football Association of Norway (NFF). The match was played on 24 October 1993 at the Ullevaal Stadion in Oslo, and was contested between the Tippeligaen side Bodø/Glimt and the First Division side Strømsgodset. Bodø/Glimt defeated Strømsgodset 2–0 to claim the Norwegian Cup for a second time in their history.
Match
Details
References
1993
Football Cup
FK Bodø/Glimt matches
Strømsgodset Toppfotball matches
Sports competitions in Oslo
1990s in Oslo
October 1993 sports events in Europe | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,825 |
Q: Finding Lattice points on a Cubic I want to study the rational points on a cubic. Eventually I found Nagell's algorithm from http://webs.ucm.es/BUCM/mat/doc8354.pdf, but I cannot immediately apply it because I don't know a rational point on the curve. I tried to use the point at infinity and skip to step 2 of the algorithm, but this causes $e_2$ and $e_3$ to both be $0$ even though the curve doesn't factor. The examples in the book practically skip over finding an initial rational point, but after some thought I'm almost certain that using the point at infinity doesn't work and have some ideas about finding an initial rational point. The main problem is that my cubic is a trivariate polynomial.
The cubic equation I'm studying is
$$F(a_x, b_x, r)=40(a_x+b_x)a_x b_x - (4r+100)(a_x^2 + b_x^2) - (4r + 300)a_x b_x + (20r + 500)(a_x + b_x) + (3r + 25)(r - 25).$$
Clearly this polynomial is symmetric in $a_x$ and $b_x$, cubic overall, and quadratic in each variable.
I think the problem with using a point at infinity is that the tangents to the graph at the point at infinity are the asymptotes. For this curve, the asymptotes are $a_x=\frac{4r+100}{40}$, $b_x=\frac{4r+100}{40}$, and $a_x+b_x=\frac{50-2r}{20}$. The horizontal and vertical asymptotes actually do intersect the graph at finite points, so I'm not sure if further abusing Nagell's algorithm by using one of these asymptotes as the tangent line and one of these finite intersections as $Q$ in the algorithm has any merit, but I will look into that. However, the oblique asymptote does not intersect the curve except at the critical values of $r$ ($0$, $25$, and $\frac{125}{2}$). (These are where the graph transitions from having an ellipse like "island" and three narrow hyperbola like "branches", to having one narrow "branch" and two wide "branches" which bend in the region where the "island" would go. At these critical values, the graph has self intersections instead. There are two other critical values near $0.3$ and $18.1$ between which the "island" disappears, but these are not relevant.)
So it seems that using the point at infinity as the initial rational point in Nagell's algorithm is "cheating" and will not be rewarded. For any finite rational point, the discriminant of $F$ as a polynomial in $a_x$, $b_x$, and $r$ must be a perfect square over the rationals. As stated, these are all quadratic. This is pretty much where I've been directing my attention, but the discriminants are
$$16(a_x^4 + b_x^4) + 48a_x^2b_x^2 + 32(a_x^2 + b_x^2)a_x b_x - 160(a_x^3 + b_x^3) - 800(a_x + b_x)a_x b_x + 2000(a_x^2 + b_x^2) + 4800a_x b_x - 8000(a_x + b_x) + 10000$$
for $r$ and
$$1600b_x^4 + 320b_x^3 r - 8000b_x^3 - 48b_x^2 r^2 - 2400b_x^2 r + 10000b_x^2 - 320b_x r^2 + 8000b_x r + 48r^3 + 800r^2 - 10000r$$
for $a_x$ (obviously the former is symmetric in $a_x$ and $b_x$ and the discriminant for $b_x$ is the same as the one for $a_x$ but with $a_x$ replaced with $b_x$).
The first one is symmetric and so it's simpler so that's the one I've been focusing on, however it is not a perfect square, cannot be written as the sum or difference of the squares of two symmetric polynomials in $a_x$ and $b_x$, and cannot be written as the sum of the squares of two symmetric polynomials in $a_x$ and $b_x$ plus the square of an integer. I stopped checking at that point so perhaps it is the sum of three or more squares of such polynomials or two or more plus the square of an integer, but after two that's really of rapidly diminishing usefulness. (I checked this by writing coefficient matching equations and searching for solutions using groebner bases, eg to show the discriminant cannot be the sum of two squares we take the expression $A(a_x^2+b_x^2)+Ba_x b_x + C(a_x+b_x) + D$, square it, add to the same expression but with $E$, $F$, $G$, and $H$, and coefficient match.)
Is my understanding of why Nagell's algorithm fails if we use the point at infinity correct?
Is there a good way to find when the discriminants mentioned are perfect squares of rational numbers, or if not, is there a good way to find and initial rational point on the cubic? I don't want an explanation of how to find all the rational points, just an initial one.
A: Introduction
I'm still not sure if there's a general finite rational point for the $a_x$ $b_x$ curve for fixed $r$, but I did find a general rational point for the $b_x$ $r$ curve for fixed $a_x$, and I've come to a better understanding of why using the point at infinity didn't work.
Why Nagell's Algorithm requires a finite rational point
In Nagell's algorithm, we try to convert $f(u, v)$ into
$$F(U, V, W) = F_3(U, V) + F_2(U, V)W + F_1(U, V)W^2,$$
where each $F_i$ is a homogeneous polynomial of degree $i$. In fact, each $F_i$ is the components of $f(u, v)$ with degree $i$ after the substitution $u=\frac{U}{W}$ and $v=\frac{V}{W}$.
We have to shift $f$ so that it has a rational point at the origin $(u,v)=(0,0)$. This means when we convert to projective coordinates, $F_1(U, V)$ is tangent to the curve at $(0,0)$
However, if we try to use a point at infinity instead, $F_1$ is zero and corresponds to the line at infinity, rather than a useful tangent line that we can find an intersection with the curve for. (We can't simply shift a point at infinity to the origin, instead we have to convert to projective coordinates and then shift and permute the coordinates so the point at infinity is mapped to the origin, but this isn't really relevant.)
So that's why using a point at infinity doesn't work.
Converting our cubic to an elliptic curve
As I mentioned in a comment, if we look at the discriminant of the curve as a quadratic in $r$, it's pretty inscrutable:
$$16(a_x^4 + b_x^4) + 48a_x^2b_x^2 + 32(a_x^2 + b_x^2)a_x b_x - 160(a_x^3 + b_x^3) - 800(a_x + b_x)a_x b_x + 2000(a_x^2 + b_x^2) + 4800a_x b_x - 8000(a_x + b_x) + 10000.$$
However, if we substitute $b_x=a_x$, which is one of the top three substitutions to try, we see something nice happens: the discriminant becomes:
$$144a_x^4 - 1920a_x^3 + 8800a_x^2 - 16000a_x + 10000$$
which factors as
$$16(a_x - 5)^2(3a_x - 5)^2.$$
This is very nice, we can use this to find a finite rational point on the curve for any fixed $a_x$, in particular $(a_x, b_x, r)=(a_x, a_x, \frac{5}{3}(4a_x - 5))$ or $(a_x, a_x, (2a_x - 5)^2)$
Let's proceed with the first point. We can define $f(u, v) = 3G_9(a_x, u + a_x, v + \frac{5}{3}(4a_x - 5))$ where $G_9(a_x, b_x, r)$ is simply our cubic. This $f$ is clearly a bivariate cubic in $u$, $v$ with a root at the origin and the $v$ coefficient nonzero, so we can proceed further into Nagell's algorithm. We substitute $u=\frac{U}{W}$ and $v=\frac{V}{W}$ and get
$$F=F_3+F_2 W+F_1 W^2=0$$
$$F_3=(-12)U^2 V$$
$$F_2=40(a_x - 5)U^2 - 12(3a_x - 5)UV + 9V^2$$
$$F_1=40(3a_x - 5)(a_x - 5)U - 12(a_x - 5)(3a_x - 5)V$$
Do note that this has a small hole where $a_x = 5$, since in that case both the $U$ and $V$ coefficients are zero, so the equation factors. We'll just assume $a_x \ne 5$ for the rest of this analysis. It's simple to analyze that case since it doesn't even require converting the cubic to an elliptic curve, it just simplifies. $a_x = \frac{5}{3}$ would also be problematic, but I actually only care about integral points not rational points, so $5$ is the only problematic value.
The next step of Nagell's algorithm effectively computes the point where the tangent ($F_1$) at the origin intersects the curve. The important parts of this though are two numbers $e_2$ and $e_3$ where $e_i=F_i(s9, -s8)$. $s8$ and $s9$ are the $U$ and $V$ coefficients. We get $e_2=2880(5-4a_x)(a_x-5)^2(3a_x-5)^2$ and $e_3=0$.
This is where the algorithm actually failed when I tried to use the point at infinity: both $e_2$ and $e_3$ were zero but $G_9$ did not factor. But now, for integral $a_x\ne 5$, $e_2$ is nonzero. $e_3$ is zero however, meaning the intersection point (the intersection between the tangent at the origin and the curve, NOT the rational point which was shifted to the origin) is a point at infinity. We proceed by using the substitution $U=U'+12(a_x-5)(3a_x-5)W'$, $V=V'+40(a_x-5)(3a_x-5)W'$, and $W=U'$. Then we convert back to cartesian coordinates $u'=\frac{U'}{W'}$ and $v'=\frac{V'}{W'}$. This gives us
$$f' = f_3' + f_2' + f_1'$$
$$f_1' = -2880(a_x-5)^2(3a_x-5)^2(4a_x-1)u' - 1728(a_x-5)^2(3a_x-5)^2v'$$
$$f_2' = -48u'(10(a_x-5)(a_x+6)(3a_x-5)u' + 3(a_x-5)(3a_x-8)(3a_x-5)v')$$
$$f_3' = u'(40(a_x-5)(3a_x-4)u'^2 - 12(13a_x^2 - 17a_x + 21)u'v' + 9v'^2)$$
(Hey look, it's a special guest appearance of the special elliptic curve number $1728$!)
The final change of variables is the most complicated yet, and occurs in a few steps. First, we let $t=\frac{v'}{u'}$ so that we can rewrite $f' = 0$ as
$$u'^2 f_3'(1,t) + u'f_2'(1,t) + f_1'(1,t) = 0.$$
Then we have a few more substitutions: $u'=\frac{-\phi_2\pm\sqrt{\delta}}{2\phi_3}$, $v'=tu'$, where $\phi_i = f_i'(1,t)$ and $\delta = \phi_2^2-4\phi_1\phi_3$. Then we substitute $t$ again: define $t_0=\frac{-s_8}{s_9}$ and write $t=t_0+\frac{1}{\tau}$. If we apply all these definitions and substitutions to $\rho=\tau^3\delta$, we get $\rho$ is a cubic in $\tau$, in particular $\rho=c\tau^3+d\tau^2+e\tau+k$. (This is the one inconsistency I still found with the algorithm: the book I linked in the question states that $\tau^4\delta$ is a cubic in $\tau$, but I found that only $\tau^3$ was needed to cancel the denominators and yield a cubic.)
The elliptic curve we get is then based on the coefficients from $\rho$:
$$y^2=x^3+dx^2+cex+c^2k.$$
$x$ and $y$ are linked to our existing variables as $\tau=\frac{x}{c}$, $\rho=\frac{y^2}{c^2}$, $t=t_0+\frac{c}{x}$, and $\delta=\frac{c^2 y^2}{x^4}$.
Continuing with our particular curve, we get
$$\phi_1=-576(a_x-5)^2(3a_x-5)^2(20a_x+3t-5)$$
$$\phi_2=-48(a_x-5)(3a_x-5)(9a_x t+10a_x-24t+60)$$
$$\phi_3=9t^2-12(3a_x^2-17a_x+21)t+40(a_x-5)(3a_x-4)$$
$$\delta=2304(a_x-5)^2(3a_x-5)^2((20a_x+3t-5)(-36a_x^2 t+120a_x^2+204a_x t-760a_x+9t^2-252t+800)+(9a_x t+10a_x-24t+60)^2).$$
Then for $\rho$ we get
$$\rho=2304(a_x-5)^2(3a_x-5)^2(-100(8a_x-7)\tau^3 -60(12a_x^3-77a_x^2+89a_x-3)\tau^2 -9(3a_x^2-40a_x-5)\tau+27),$$
so we've successfully converted the original cubic to an elliptic curve
$$y^2=x^3-2304\cdot 60(a_x-5)^2(3a_x-5)^2(12a_x^3-77a_x^2+89a_x-3)x^2 + 2304^2\cdot 100\cdot 9(a_x-5)^4(3a_x-5)^4(8a_x-7)(3a_x^2-40a_x-5)x + 2304^3\cdot 100^2\cdot 27(a_x-5)^6(3a_x-5)^6(8a_x-7)^2!$$
(The exclamation point denotes excitement, not factorial.)
This was a very long and tedious computation, I did it mostly in Sympy to avoid making mistakes, but I still have not completely verified it checks out. I tried to explain how I followed Nagell's algorithm in pretty high detail, but I also didn't literally write down every single step. The book (http://webs.ucm.es/BUCM/mat/doc8354.pdf, in chapter 1.4) explains each step of the algorithm, but worked through examples skip over a lot of things, so I hope that this more detailed walkthrough will be helpful to anyone else who, for example, also tries to use a point at infinity and runs into a similar problem. Even this answer is not as detailed as it could be, in particular I didn't explain how I actually used Sympy to do the computations (it's mostly factor, subs, expand, simplify, collect, and coeff, although coeff doesn't always work nicely with multivariate polynomials so take care).
| {
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the size of a loaf of bread.
encased in a rattling car.
Is it a goiter? Infection?
went three times around my waist.
And now you lay your hand on mine.
That was your first gift.
Science and art. The pencil and the notebook.
The carbon and its isotope.
scattered like shards on the night streets of Vienna.
No. Not us together. Only you.
lord of the bones of our hearts.
Dina is a former award-winning English teacher. She has written for the New York Times, London Times Online, Orion, and Full Grown People, and serves as a poetry reviewer for The Journal of Interdisciplinary Studies in Literature and Environment. This is her first published poem. | {
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{"url":"https:\/\/indico.nucleares.unam.mx\/event\/1541\/session\/29\/contribution\/179","text":"# 19th International Conference on Hadron Spectroscopy and Structure in memoriam Simon Eidelman\n\n26-31 July 2021\nMexico City\nMexico\/General timezone\nHADRON 2021 is over. Thanks for making it a success!\nHome > Timetable > Session details > Contribution details\n\n# Contribution Parallel\n\nMexico City -\nBaryon Spectroscopy\n\n# Novel pentaquark picture for Roper-like heavy baryons from chiral symmetry\n\n## Speakers\n\n\u2022 Dr. Daiki SUENAGA\n\n## Abstract\n\nWe propose a new type of structure for singly heavy baryons of $Qqq\\bar{q}q$ in addition to the conventional one of $Qqq$. Based on chiral symmetry of the light quarks, we find $\\Lambda_c(2765)$ and $\\Xi_c(2967)$ are mostly $Qqq\\bar{q}q$ while $\\Lambda_c(2286)$ and $\\Xi_c(2470)$ are mostly $Qqq$. The masses of negative-parity baryons are predicted. We also derive a mass formula and the extended Goldberger-Treiman relation that the masses of the baryons satisfy.","date":"2022-10-03 14:39:25","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\": 0.2564491033554077, \"perplexity\": 4111.981892052354}, \"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-2022-40\/segments\/1664030337421.33\/warc\/CC-MAIN-20221003133425-20221003163425-00263.warc.gz\"}"} | null | null |
Prometheus 2 Script In The Works, Says Star Noomi Rapace
By Nick Venable | 8 years ago
I'm not sure which move is the correct one for director Ridley Scott. Follow in the dumb-decision-making path of the characters in Prometheus and make another beautiful but vapid thorn in the Alien mythos? Or hedge his Michael Fassbenders and move on to more original ideas? Sure, there's always Option C, where he makes a sequel and it's highly successful and sensible, but we're not living in one of Scott's movies, readers. This is reality.
Prometheus star Noomi Rapace was promoting her latest film, Dead Man Down, in an interview with The Playlist when the conversation naturally turned toward that most notorious of recent science fiction blockbusters. Rapace shared that things are moving forward on the sequel's screenplay, which will be gloriously free of Damon Lindelof's involvement.
"They're working on the script," Rapace says. "I met Ridley in London a couple of weeks ago. I would love to work with him again and I know that he would like to do another one. It's just like we need to find the right story. I hope we will."
I'm all about them finding the right story as well. The first one had the basic framework for a modern epic, but then spacesuit bongs and Idris Elba's Southern accent dampened everything down. Perhaps an endless amount of criticism will fuel a more focused approach.
"And it's interesting because people, most people I've talked to who see the movie, see things that are quite different," she continued. "Some people who see the movie many times and discover new things. There are all these religious aspects and there are very interesting conversations. And for me, if we do a second one, there are a lot of things to explore in there and to continue."
When prodded for information about a screenwriter being hired, she responded, "You would have to ask someone at the studio about who the writer is." So I guess we can take that to mean there is a writer. And not a room full of monkeys and word processors.
Raised By Wolves Review: Ridley Scott's Sci-Fi Show Is Worth Your Time
Harrison Ford and Mark Hamill Returning To Star Wars?
Buck Rogers Movie Is Happening
Darth Vader TV Series Happening? | {
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} | 131 |
\section{The Cucker-Smale model}
{\LARGE I}\hspace*{-0.051cm}n 1998, Craig Reynolds won a Scientific and Engineering Award of the Academy of Motion Picture Arts and Sciences for ``\emph{pioneering contributions to the development of three dimensional computer animation for motion picture production}'', \cite{Rey98}. Reynolds was recognized for his work on realistic simulations of \emph{flocking}, \cite{Rey87}, proposing a collective dynamics of `bird-like objects' (or `boids') which are driven by pairwise interactions acting in three zones of \emph{repulsion}, \emph{alignment} and \emph{attraction}.
A similar 3Zone protocol is found in a broad spectrum of models for collective dynamics in different contexts: in modeling swarming dynamics in \emph{ecology} --- from fish, birds and sheep to bacteria locust and insects,
\cite{Aok82,Par82,Rey87,BCE91,KL93,VCBCS95,TT98,EKWG98,BDT99,WB01,
EK01,CKJRF02,PVG02,Ben03,CF03,TB04,GR05,HCK05,CKFL05,DCBC06,
OAGM06,CS07a,CS07b,Bal08,HH08,YBTBG08,BEBS09,LX10,JK10,
Cav10,SASBJ11,CCGPS12,Bia12,GWBL12,TDOEKB12,Bia14,
AA15,Gin15,Fon16,Jia17,PT17,Ari18,Cal18,USTB19,Liu20}; modeling social dynamics of \emph{human interactions} --- from pedestrians, exchange of opinions and ratings to markets and marketing, \cite{DeG74,Axe84,Axe97,Kra00,Hel01,HK02,WDA05,BHOT05,BeN05,
Wei06,Lor07,BHT09,CFL09,Hel09,FG10,Hel10,PT11,BT15,RDW18,BCD19}; and in modeling the dynamics of \emph{sensor-based networks} --- ranging from macro-molecules and metallic rods to control and mobile robot networks, \cite{Kur75,OK91,JLM03,OSM04,CMB06,OS06,JE07,ZP07,ZEP11,Rin12,BV13,MT14b}.
The common theme of the different models is crowd dynamics dictated by pairwise interactions between members of the crowd which are viewed as \emph{agents}.
A main question of interest is to understand how the small scale pairwise interactions within the crowd, are self-organized into a large scale patterns of the whole crowd, so that ``the whole is greater than the sum of its parts''. One then refers to the \emph{emergent behavior} of the crowd, where the larger patterns are realized by a crowd forming a flock, reaching a consensus, admitting a synchronized state, aggregate into one or more clusters, etc.
\smallskip\noindent
\paragraph{{\bf The class Cucker-Smale alignment models}} Pairwise attraction and repulsion are familiar from particle physics, for example, particle dynamics driven Coulomb and other singular potentials, \cite{ST97,Ser17,LS17,Ser18,LS18}. Here, we focus our attention on \emph{alignment} dynamics, driven by pairwise interactions in which agents steer towards \emph{average heading}. We consider the agent-based system in which $N$ agents, identified with (position, velocity) pairs $({\mathbf x}_i(t),{\mathbf v}_i(t)): {\mathbb R}_+ \mapsto (\Omega,{\mathbb R}^d)$ and subject to prescribed initial conditions, $({\mathbf x}_i(0),{\mathbf v}_i(0))=({\mathbf x}_{i0},{\mathbf v}_{i0})\in (\Omega,{\mathbb R}^d)$, are driven by
\begin{equation}\label{eq:CS}
\left\{ \ \ \begin{split}
{{\mathbf x}}_i(t+\step)&={\mathbf x}_i(t)+\step{\mathbf v}_i(t)\\
{{\mathbf v}}_i(t+\step)&={\mathbf v}_i(t) + \step\sum_{j\in {\mathcal N}_i} m_j\phi_{ij}(t)({\mathbf v}_j(t)-{\mathbf v}_i(t)).
\end{split}\right.
\end{equation}
The dynamics is dictated by a symmetric \emph{communication kernel},
\[
\phi({\mathbf x},\bx'} %{{\mathbf y})=\phi(\bx'} %{{\mathbf y},{\mathbf x})\geq 0.
\]
Its dynamic values, $\phi_{ij}(t)=\phi({\mathbf x}_i(t),{\mathbf x}_j(t))$,
encode the `rule of engagement' between agents, and in particular the neighborhood ${\mathcal N}_i=\{j : \phi_{ij}(t)>0\}$, which contributes to the steering of a `boid' positioned at ${\mathbf x}_i$.
The spatial domain $\Omega$ is either ${\mathbb T}^d$ or ${\mathbb R}^d$, so that boundaries are avoided, and $\step>0$ is a small, possibly variable time-step, $\step=\step(t)$. Different agents, $({\mathbf x}_i,{\mathbf v}_i)$, are assumed to have different masses, $m_i$, or other constant traits attributed to an agent positioned at ${\mathbf x}_i$.\newline
We refer to \eqref{eq:CS} as the class of Cucker-Smale (C-S) models for alignment dynamics. Different models are attached to different $\phi$'s and different $m_i$'s.
The original model of Cucker \& Smale (C-S) \cite{CS07a,CS07b} is \emph{the} canonical model for the class of alignment dynamics \eqref{eq:CS} with $\phi({\mathbf x},{\mathbf x}')\sim (1+|{\mathbf x}-{\mathbf x}'|)^{-\beta}, \ \beta>0$, which assumes a uniform mass distribution $m_i\equiv \nicefrac{1}{N}$,
\begin{equation}\label{eq:equiCS}
{{\mathbf v}}_i(t+\step)={\mathbf v}_i(t) + \frac{\step}{N}\sum_{j\in {\mathcal N}_i} \phi_{ij}(t)({\mathbf v}_j(t)-{\mathbf v}_i(t)).
\end{equation}
The work of Cucker \& Smale attracted a considerable attention in the literature and motivated the study of many variants of the C-S alignment models; we refer to \cite{She07,PKH10,Pes15,BDT17-19,Tad17,CHL17, CHJK19,Shv21} and the references therein.
In particular, a more general alignment model based on the formation of `blobs' or multi-flocks of agents with different masses was derived in \cite{ST21b}. In other models, different $m_i$'s can be identified with different intrinsic `traits' of different agents, such as degree, temperature, \cite{MT11, HKR18, Jin18, CHJK19, Has21}. We further elaborate on one example.
\smallskip\noindent
\paragraph{{\bf The Motsch-Tadmor model}}
If each of the terms contributing to the C-S alignment on the right of \eqref{eq:equiCS}, $ \sum_j \phi_{ij}({\mathbf v}_j-{\mathbf v}_i)$, is of the same ${\mathscr O}(1)$-order, then its total action of order ${\mathscr O} (N)$ will peak at time $t = {\mathscr O} (\nicefrac{1}{N})$. Thus, as noted in \cite[\S2] {ST21b}, the pre-factor $\nicefrac{1}{N}$ is C-S model \eqref{eq:equiCS} is in fact a \emph{scaling} factor, so that the dynamics peaks at the desired time $t \sim {\mathscr O} (1)$.\newline
In \cite{MT11} we advocated a more realistic scaling which is adapted to
spatial variability in the intensity of different alignment terms,
\begin{equation}\label{eq:MT}
{{\mathbf v}}_i(t+\step)={\mathbf v}_i(t) + \frac{\step}{\sum \limits_{k\in {\mathcal N}_i} \phi_{ik}(t)}\sum_{j\in {\mathcal N}_i} \phi_{ij}(t)({\mathbf v}_j(t)-{\mathbf v}_i(t)).
\end{equation}
Here the scaling depends on the \emph{degree} of different agents,
\[
deg_i:=\sum_{k\in {\mathcal N}_i} \phi_{ik}(t).
\]
It should be emphasized that the communication array in M-T model, $\ds \Big\{\frac{1}{deg_i}\phi_{ij}\Big\}$ is not symmetric. Nevertheless, it does fit the general symmetric framework of C-S class \eqref{eq:CS} with a proper choice of `masses'
$\displaystyle m_i =\frac{1}{L}deg_i$, and \emph{symmetric} interactions
$\ds \widetilde{\phi}_{ij}= L\phi_{ij}\frac{1}{deg_i}\frac{1}{deg_j}$, recovering \eqref{eq:MT},
\begin{equation}\label{eq:MTS}
{{\mathbf v}}_i(t+\step) = {\mathbf v}_i(t) + \step\sum_{j\in {\mathcal N}_i} m_j\widetilde{\phi}_{ij}(t)({\mathbf v}_j(t)-{\mathbf v}_i(t)), \quad m_i=\frac{1}{L}deg_i, \quad \widetilde{\phi}_{ij}= \frac{1}{L}\phi_{ij}\frac{1}{m_i}\frac{1}{m_j}
\end{equation}
The scaling parameter $L$ has no effect on the alignment and was introduced here in order to re-scale the total mass\footnote{\label{foot:MT}For example,, in the case of long range all-to-all communication where $\phi_{ij}={\mathcal O}(1)$, then $deg_j={\mathcal O}(N)$ and we set $L=N^2$ so that $M=\frac{1}{L}\sum_jdeg_j ={\mathcal O}(1)$.} so that $M:=\sum_i m_i ={\mathcal O}(1)$.
In this case, however, the degrees vary in time, $\ds m_i=\frac{1}{L}deg_i(t)$ and the discussion below needs to be modified to include time-dependent masses. This will be further explored in section \ref{sec:MT} below.\newline
As another example, we mention a similar situation that arises in the context of \emph{thermodynamic} C-S model \cite{HKR18,CHJK19}, where $m_i$'s can be identified with the different temperatures $m_i=\theta_i(t)$ of agents with re-scaled velocities
$\frac{1}{\theta_i}{\mathbf v}_i$. Again, one needs to address the time-dependence of the temperatures which are dictated by a separate dynamics.
\section{Communication kernels}
The dynamics of \eqref{eq:CS} is dictated by a symmetric \emph{communication kernel},
$\phi(\cdot,\cdot)\geq 0$.
Where do these communication kernels come from? they arise from a combination of empirical and phenomenological considerations.
A sample of the large literature can be found in
\cite{Kur75,TT98,Kra00,WB01,CF03,JLM03,CKFL05,GR05,WDA05,Wei06,ZP07,Jac10,ZEP11,SASBJ11,VZ12,Gin15} and the references therein.
We mention several primary examples.\newline
A large part of current literature is devoted to the generic class of \emph{metric-based} kernels,
\[
\phi({\mathbf x},\bx'} %{{\mathbf y})=\varphi(|{\mathbf x}-\bx'} %{{\mathbf y}|).
\]
The choice of metric kernels $\varphi(r)=\mathds{1}_{[0,R_0]}$ and $\varphi(r)=(1+ r)^{-\beta}, \ 0<\beta<1$ are found in the seminal works of Vicsek et. al. \cite{VCBCS95} and respectively Cucker \& Smale \cite{CS07a}. They are motivated by a \emph{phenomenological} reasoning that the strength of pairwise interactions is short-range or at least decreasing with the relative distance, ``\emph{birds of feather flock together}'' \cite{MSLC01}; this should be contrasted with an opposite heterophilous protocol, \cite{MT14a}, based on tendency to attract diverse groups so that $\varphi(r)$ is \emph{increasing} over its compact support.
A particular sub-class of such metric-based protocols are the \emph{singular} kernels, $\varphi(r)=r^{-\beta}, 0<\beta <d+2$,
which emphasize near-by neighbors, $r\ll 1$, over those farther away,
\cite{Pes15,CCMP17,ST17a,PS17,ST18,DKRT18,MMPZ19}. The case of non-summable kerenls, $\beta=d+2s, s\in (0,1)$ correspond to Riesz kernels and could be properly interpreted as principle values of summation in the commutator form \cite{ST17a}
\[
\sum_j \phi_{ij}m_j({\mathbf v}_j-{\mathbf v}_i)
= \sum_j \frac{m_j{\mathbf v}_j-m_i{\mathbf v}_i}{|{\mathbf x}_j-{\mathbf x}_i|^{d+2s}}
-\sum_j \frac{m_j-m_i\hspace*{0.6cm}}{|{\mathbf x}_j-{\mathbf x}_i|^{d+2s}}{\mathbf v}_i.
\]
An important source for communication kernels are detailed observations. As a prime example we mention the class of \emph{topologically-based} kernels, dictated by the \emph{size} of the crowd in between agents positioned at ${\mathbf x}$ and $\bx'} %{{\mathbf y}$
\begin{equation}\label{eq:whatismu}
\phi({\mathbf x},\bx'} %{{\mathbf y})=\varphi(\mu({\mathbf x},\bx'} %{{\mathbf y})), \quad \mu({\mathbf x},\bx'} %{{\mathbf y}):=\frac{1}{N}\#\{k:\, {\mathbf x}_k\in {\mathcal C}({\mathbf x},\bx'} %{{\mathbf y})\}.
\end{equation}
Here, ${\mathcal C}({\mathbf x},\bx'} %{{\mathbf y})$ is a pre-determined communication region enclosed between ${\mathbf x}$ and $\bx'} %{{\mathbf y}$.
In particular, if ${\mathcal C}$ is shifted to $R$-ball centered at ${\mathbf x}$, one ends up with the \emph{non-symmetric} topological kernel \cite{MT11}
$\displaystyle \phi({\mathbf x},\bx'} %{{\mathbf y})=\frac{\varphi(|{\mathbf x}-\bx'} %{{\mathbf y}|)}{\mu(B_R({\mathbf x}))}$.
Topologically-based communication was observed in starflag project reported in
\cite{Bal08,Cav08,Cav10,CCGPS12},
where birds react to the number of closest neighbors rather than their metric distance, and in pedestrian dynamics \cite{RDW18},
where communication is decreasing in more crowded regions, and was analyzed in \cite{Bal08,BD16,ST20b}.\newline
More on topologically-based kernels can be found in \cite{OSM04,Li08,CCGPS12,Has13,BD17}\newline
As a third example, we mention \emph{random-based} communication protocols found in chemo- and photo-tactic dynamics, \cite{HL09a},
the Elo rating system, \cite{JJ15,DTW19},
voter and related opinion-based models, \cite{BeN05},
or a random-batch method and consensus-based optimization.
\cite{DAWF02,CFL09,GWBL12,PTTM17,JLL20}.
Another class of communication kernels are those learned from the data,
\cite{BPK16,LZTM19,MLK19}.
Finally, we mention communication kernels which are derived from `higher order' principles; for example, a minimum entropy principle \cite{Bia12,Bia14},
and the paradigm of anticipation \cite{GTLW17}.
\section{Long time dynamics}
A key aspect in the long time behavior of \eqref{eq:CS} is the decay in time of the fluctuations of velocities $\{{\mathbf v}_i-{\mathbf v}_j\}$. Velocity fluctuations can be measured in a weighted-$\ell^2$ average sense quantifying \emph{energy fluctuations}, or in a uniform sense quantifying the $\ell^\infty$-\emph{diameter} of the discrete crowd of velocities.
\subsection{Energy fluctuations}\label{sec:energy}
We let ${\delta \mathscr E}(t)$ denote the \emph{energy fluctuations}, scaled by the total mass\footnote{Here and below,
$|\cdot|$ denotes an arbitrary vector norm on ${\mathbb R}^d$.}
\[
{\delta \mathscr E}(t):=\frac{1}{2M^2}\sum_{i,j}|{\mathbf v}_i(t)-{\mathbf v}_j(t)|^2m_im_j, \qquad M=\sum_i m_j.
\]
Thus, ${\delta \mathscr E}(t)$ is the weighted $\ell^2$-diameter of the set of velocities $\{{\mathbf v}_i\}_{i=1}^N$ at time $t$. Equivalently, we can express it as fluctuations around the mean velocity $\overline{{\mathbf v}}$
\begin{equation}\label{eq:mean}
{\delta \mathscr E}(t)=\frac{1}{M}\sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}(t)|^2m_i, \qquad \overline{{\mathbf v}}:=\frac{1}{\sum_i m_i}\sum_i m_i{\mathbf v}_i(t)
\end{equation}
The energy balance encoded in \eqref{eq:CS}${}_2$ implies (for simplicity we suppress the time dependence on $t$ on the right-hand side)
\begin{equation}\label{eq:energy}
\left\{\ \begin{split}
\frac{1}{M}\sum_i m_i&|{\mathbf v}_i(t+\step)|^2 -
\frac{1}{M}\sum_i m_i|{\mathbf v}_i(t)|^2 \\
& = \frac{2\step}{M}\sum_i \big\langle m_i{\mathbf v}_i, \sum_j m_j\phi_{ij}({\mathbf v}_j-{\mathbf v}_i)\big\rangle + \frac{\step^2}{M}\sum_im_i\big|\sum_j m_j\phi_{ij}({\mathbf v}_j-{\mathbf v}_i)\big|^2.
\end{split}\right.
\end{equation}
Since the communication kernel is symmetric, $\phi_{ij}=\phi_{ji}$, the total momentum is conserved
\begin{equation}\label{eq:momentum}
{\mathscr M}(t+\step)-{\mathscr M}(t) =\frac{\step}{M}\sum_{i,j}m_im_j\phi_{ij}({\mathbf v}_j-{\mathbf v}_i)=0, \qquad {\mathscr M}(t):=\frac{1}{M}\sum_i m_i{\mathbf v}_i(t).
\end{equation}
This implies that the incremental change in energy of the left of \eqref{eq:energy}
is the same as the incremental change of energy fluctuations. Indeed,
\[
\frac{1}{M}\sum_i m_i|{\mathbf v}_i(t)|^2 \equiv \frac{1}{2M^2}\sum_{i,j}|{\mathbf v}_i(t)-{\mathbf v}_j(t)|^2m_im_j + \frac{1}{M^2}\big|\sum_i m_i{\mathbf v}_i(t)\big|^2= {\delta \mathscr E}(t)+\big|{\mathscr M}(t)\big|^2,
\]
and the same applies at $t+\step$,
\[
\frac{1}{M}\sum_i m_i|{\mathbf v}_i(t+\step)|^2 \equiv {\delta \mathscr E}(t+\step)+\big|{\mathscr M}(t+\step)\big|^2,
\]
and since the squared terms on the right of the last two equalities are the same, we find
\begin{subequations}\label{eqs:abc}
\begin{equation}\label{eq:abc}
\frac{1}{M}\sum_i m_i |{\mathbf v}_i(t+\step)|^2 -\frac{1}{M}\sum_i m_i|{\mathbf v}_i(t)|^2 ={\delta \mathscr E}(t+\step) - {\delta \mathscr E}(t).
\end{equation}
We now come to the main point, namely, that the alignment operator of CS dynamics is coercive in the sense that
\begin{equation}\label{xyz}
\frac{2}{M}\sum_i\big\langle m_i{\mathbf v}_i, \sum_j m_j\phi_{ij}({\mathbf v}_j-{\mathbf v}_i)\big\rangle = - \frac{1}{M}\sum_{i,j}\phi_{ij}|{\mathbf v}_i-{\mathbf v}_j|^2m_im_j.
\end{equation}
The \emph{weighted} fluctuations on the right is identified as the \emph{enstrophy}.
We can bound the last squared term on the right of \eqref{eq:energy} in terms of the enstrophy and the maximal \emph{weighted degree},
$deg_+(t):=\max_i\sum_j \phi_{ij}(t)m_j$
\begin{equation}\label{rst}
\frac{1}{M}\sum_i m_i\big|\sum_j m_j\phi_{ij}({\mathbf v}_j-{\mathbf v}_i)\big|^2 \leq
deg_+(t)\frac{1}{M}\sum_{i,j} \phi_{ij}|{\mathbf v}_j-{\mathbf v}_i\big|^2m_im_j;
\end{equation}
\end{subequations}
Inserting \eqref{eqs:abc} back into the energy balance \eqref{eq:energy} we find
\begin{equation}\label{eq:final}
{\delta \mathscr E}(t+\step) - {\delta \mathscr E}(t) \leq -\step(t) \big(1-deg_+(t)\cdot\step(t)\big)\frac{1}{M}\sum_{i,j} \phi_{ij}(t)|{\mathbf v}_j(t)-{\mathbf v}_i(t)\big|^2m_im_j.
\end{equation}
Observe that we now pay attention to the time dependence on the right; in particular, the possibly variable time step, $\step=\step(t)$, and the time-dependent communication weights, $\phi_{ij}(t)=\phi({\mathbf x}_i(t),{\mathbf x}_j(t))$.\newline
We let ${\mathbb A}(t)$ denote the $N\times N$ \emph{adjacency matrix} ${\mathbb A}(t)=\{\phi_{ij}(t)\}$ encoding the edges of communication at time $t$, and define $\Delta_{{\mathbf m}}{\mathbb A}(t)$ is the \emph{weighted graph Laplacian}
\begin{equation}\label{eq:weighted}
\left(\Delta_{{\mathbf m}}{\mathbb A}\right)_{\alpha\beta}=\left\{\begin{array}{ll}
-\ds \phi_{\alpha\beta}\sqrt{m_\alpha m_\beta}, & \alpha\neq \beta\\ \\
\ds \sum_{\gamma\neq \alpha}\phi_{\alpha\gamma}m_{\gamma} & \alpha=\beta.
\end{array}\right.
\end{equation}
The weighted graph Laplacian, weighted by the masses ${\mathbf m}=(m_1,\ldots, m_N)$, has real eigenvalues, $\lambda_1=0 \leq \lambda_2 \leq \ldots \lambda_N$. This generalizes the usual notion of graph Laplacian, e.g.,\cite{Mer94,Chu97}, corresponding to the case of uniform weight, $m_i={\mathcal O}(\nicefrac{1}{N})$. We now summarize the computations above, quantifying the decay of energy fluctuations in terms of the spectral gap, $\lambda_2\big(\Delta_{\mathbf m}{\mathbb A}(t)\big)$.
\begin{theorem}[{\bf Decay of energy fluctuations}]\label{thm:main1} Consider the C-S dynamics \eqref{eq:CS} with time-steps small enough such that
\begin{equation}\label{eq:CFL}
\step(t) \cdot \max_i\sum_j \phi_{ij}(t)m_j \leq \frac{1}{2}.
\end{equation}
Then the following bound of energy fluctuations holds
\begin{equation}\label{eq:result}
{\delta \mathscr E}(t_n) \leq exp\Big\{-\sum_{k=0}^{n-1} \lambda_2(t_k)\step(t_k)\Big\}{\delta \mathscr E}_0, \quad \lambda_2(t)=\lambda_2\big(\Delta_{\mathbf m}{\mathbb A}(t)\big), \ \ t_{k+1}=t_k+\step(t_k).
\end{equation}
\end{theorem}
\begin{proof}
We return to the energy fluctuations bound \eqref{eq:final}. It remains to relate the enstrophy on the right of \eqref{eq:final} to the energy fluctuations on the left.
To this end, we use the following sharp lower bound on the enstrophy \cite[\S3]{HT21}, expressed in terms of its spectral gap $\lambda_2(t)=\lambda_2\big(\Delta_{{\mathbf m}}{\mathbb A}(t)\big)$,
\begin{equation}\label{eq:laplacian}
\sum_{i,j} \phi_{ij}(t)|{\mathbf v}_j(t)-{\mathbf v}_i(t)\big|^2m_im_j \geq \frac{\lambda_2(t)}{M} \sum_{i,j} |{\mathbf v}_j(t)-{\mathbf v}_i(t)\big|^2m_im_j, \quad \lambda_2= \lambda_2(\Delta_{{\mathbf m}}{\mathbb A}).
\end{equation}
Inserted into \eqref{eq:final}, the time-step restriction \eqref{eq:CFL} and \eqref{eq:laplacian} yield
\[
\begin{split}
{\delta \mathscr E}(t+\step) & \leq {\delta \mathscr E}(t) - \frac{\step(t)}{2}\lambda_2(t)\frac{1}{M^2}\sum_{i,j} |{\mathbf v}_j(t)-{\mathbf v}_i(t)\big|^2m_im_j \\
& \ \ = \Big(1-\step(t) \lambda_2(t)\Big){\delta \mathscr E}(t) \leq e^{-\step(t)\lambda_2(t)}{\delta \mathscr E}(t),
\end{split}
\]
and \eqref{eq:result} follows.
\end{proof}
\noindent
\begin{remark}[{\bf Graph connectivity}]
The weighted graph Laplacian $\Delta_{{\mathbf m}}{\mathbb A}$ is symmetrizable, with real eigenvalues $\lambda_1=0\leq \lambda_2 \leq \ldots \leq \lambda_N$. The weighted Poinacr\'{e} inequality \eqref{eq:laplacian} provides a sharp lower bound on the enstrophy in terms of the spectral gap
$\lambda_2(\Delta_{{\mathbf m}}{\mathbb A})>0$, which reflects the connectivity of the weighted graph $({\sf V},{\sf E})$, where vertices of ${\sf V}$ tag the positions $\{{\mathbf x}_i\}$ and the edges ${\sf E}$ quantify the connections $\{\phi_{ij}\}$. The intricate aspect here is the interplay between the graph which is time dependent, $({\sf V}(t),{\sf E}(t))$, hence its various properties are dictated by the alignment dynamics on the graph, and at the same time, as we observe in theorem \ref{thm:main1}, the fluctuations of alignment dynamics are dictated by the connectivity of the underlying graph.\newline
The spectral gap, $\lambda_2(\Delta_{\mathbf m}{\mathbb A})$, generalizes the usual notions of graph connectivity in terms of the \emph{Fiedler number} in case of uniform weights $m_i\equiv \nicefrac{1}{N}$. In particular, we point out that the weighted Poincar\'{e} bound \eqref{eq:laplacian} depends only on the total mass $M$ but otherwise is independent of the condition number, $\ds \frac{\max_i m_i}{\min_i m_i}$.
\end{remark}
Theorem \ref{thm:main1} describes the long time behavior of a fully-discrete C-S dynamics \eqref{eq:CS} under a general setup based on symmetric communication kernel, $\phi_{ij}=\phi_{ji}$, which involves variable spatial weights $m_i$ and variable time stepping, $\step=\step(t_k)$ satisfying a CFL-like time-step restriction \eqref{eq:CFL}. In particular, letting $\max_k \step(t_k) \rightarrow 0$ we recover the semi-discrete CS model
\begin{equation}\label{eq:SDCS}
\left\{ \ \ \begin{split}
\frac{\textnormal{d}}{\textnormal{d}t}{{\mathbf x}}_i(t)&={\mathbf v}_i(t)\\
\frac{\textnormal{d}}{\textnormal{d}t}{{\mathbf v}}_i(t)&=\sum_{j\in {\mathcal N}_i} m_j(t)\phi_{ij}(t)({\mathbf v}_j(t)-{\mathbf v}_i(t)).
\end{split}\right.
\end{equation}
and theorem \ref{thm:main1} tells that
\begin{equation}\label{eq:SDresult}
{\delta \mathscr E}(t) \leq exp\Big\{-\int_0^t \lambda_2(s)\textnormal{d}s\Big\}{\delta \mathscr E}_0.
\end{equation}
\subsection{Fluctuations revisited--- $\ell^\infty$-diameter of fluctuations}
We measure the fluctuations of velocities in terms of the the $\ell^\infty$-diameter of the collection of velocities $\{{\mathbf v}_i\}$
\[
{\delta \mathscr V}(t):=\max_{i,j}|{\mathbf v}_i(t)-{\mathbf v}_j(t)|.
\]
It will be convenient to trace the scalar components which form this diameter. To this end we fix an arbitrary unit vector\footnote{The vector norm $|\cdot|$ is assumed to have it dual $|{\boldsymbol \omega}|_*=\sup_{|{\mathbf v}|=1}\langle {\mathbf v},{\boldsymbol \omega}\rangle$.}, $|{\boldsymbol \omega}|_*=1$. Since
$|{\mathbf v}|=\max_{|{\boldsymbol \omega}|_*=1} \langle {\mathbf v}, {\boldsymbol \omega}\rangle$ then
\[
{\delta \mathscr V}(t):=\max_{|{\boldsymbol \omega}|_*=1}\max_{p,q}\big(v_p(t)-v_q(t)\big), \qquad v_p(t):=\langle {\mathbf v}_p(t),{\boldsymbol \omega}\rangle.
\]
We now trace the decay of these scalar components of velocity fluctuations, considering an arbitrary $(p,q)$ pair, $v_p(t)-v_q(t)$, where as before we suppress the dependence on time $t$ on the right,
\[
\begin{split}
v_p(t+\step)&-v_q(t+\step) \\
& =v_p-v_q + \step \sum_j m_j\phi_{pj}(v_j-v_p)-\step \sum_j m_j\phi_{qj}(v_j-v_q) \\
& = \big(1-\step \sum_j m_j\phi_{pj}\big)v_p -\big(1-\step \sum_j m_j\phi_{qj}\big)v_q
+ \step\sum_j m_j\phi_{pj}v_j - \step \sum_j m_j\phi_{qj}v_j\\
& = \big(1-\step \sum_j m_j\phi_{pj}\big)v_p -\big(1-\step \sum_j m_j\phi_{qj}\big)v_q \\
& \hspace*{2cm}+ \step\sum_j m_j(\phi_{pj}-c_j)v_j - \step \sum_j m_j(\phi_{qj}-c_j)v_j.
\end{split}
\]
In the last step we introduced arbitrary scalars $c_j$'s --- their contribution to the last two terms on the right cancel out. By the CFL condition \eqref{eq:CFL}, the first two parenthesis on the right are positive. We now set
$c_j:=\min_{p,q}\{\phi_{pj},\phi_{qj}\}$ --- with this choice the last two parenthesis on the right are also non-negative. Hence, if we let $v_+$ and $v_-$ denote the extreme values $ v_+ :=\max_p v_p$ and $v_-:=\min_q v_q$ we conclude
\[
\begin{split}
v_p(t+\step)&-v_q(t+\step) \\
& \leq \big(1-\step \sum_j m_j\phi_{pj}\big)v_+ -\big(1-\step \sum_j m_j\phi_{qj}\big)v_- \\
& \hspace*{2cm} + \step\sum_j m_j(\phi_{pj}-c_j)v_+ - \step \sum_j m_j(\phi_{qj}-c_j)v_-\\
& = v_+ - v_- -\step \sum_j m_jc_j(v_+-v_-)
=\Big(1-\step \sum_j m_jc_j\Big)\Big(\max_p v_p-\min_q v_q\Big) \\
& =\big(1-\step \erg({\mathbb A})\big) \max_{p,q}\big(v_p-v_q), \qquad \erg({\mathbb A}):= \sum_j m_j\min_{p,q}\{\phi_{pj},\phi_{qj}\}.
\end{split}
\]
Since $(p,q)$ is an arbitrary pair we conclude
\begin{equation}\label{eq:ergodicity}
{\delta \mathscr V}(t+\step)= \max_{|{\boldsymbol \omega}|_*=1}\max_{p,q}\big(v_p(t+\step)-v_q(t+\step)\big)\leq \big(1-\step \erg({\mathbb A}(t) \big){\delta \mathscr V}(t).
\end{equation}
\begin{theorem}[{\bf Decay of uniform fluctuations}]\label{thm:main2} Consider the C-S dynamics \eqref{eq:CS} with time-steps small enough such that \eqref{eq:CFL} holds
\
\step(t) \cdot \max_i\sum_j \phi_{ij}(t)m_j(t) \leq \frac{1}{2}.
\
Then the following bound of the diameter of fluctuations holds
\begin{equation}\label{eq:etaresult}
{\delta \mathscr V}(t_n) \leq exp\Big\{-\sum_{k=0}^{n-1} \erg\big({\mathbb A}(t_k)\big)\step(t_k)\Big\}{\delta \mathscr V}_{\!\!0}, \quad \erg\big({\mathbb A}(t)\big)=\sum_j m_j(t) \min_{p,q}\{\phi_{pj}(t),\phi_{qj}(t)\}.
\end{equation}
\end{theorem}
We emphasize that the bound \eqref{eq:etaresult} applies to C-S dynamics with general communication, $\{m_j(t)\phi_{ij}\}$, which need \emph{not} be symmetric, as it allows for time-dependent masses.
In particular, it applies to both the Cucker-Smale alignment model with symmetric interactions, \eqref{eq:equiCS}, $m_j=\nicefrac{1}{N}$ and the Motsch-Tadmor alignment model with non-symmetric, time-dependent interactions \eqref{eq:MT}, $m_i(t)=\frac{1}{L}deg_i(t)$.
\begin{remark}[{\bf Spectral gap vs. coefficient of ergodicity}]
The role of spectral gap in the present context of connectivity of graph goes back to Fiedler \cite{Fie73,Fie89}. In the case of equal weights, the so-called Fiedler number $\lambda_2(\Delta{\mathbb A})>0$ quantifies the algebraic connectivity of the graph $({\sf V},{\sf E})$ supported at vertices ${\sf V}=\{i : \, {\mathbf x}_i\}$ with weighted edges ${\sf E}=\{(i,j) : \, \phi_{ij}>0\}$.\newline
The inequality \eqref{eq:laplacian} is sharp in the sense that
\[
\frac{1}{M}\lambda_2(\Delta_{\mathbf m}{\mathbb A})
=\min \frac{\sum_{i,j} \phi_{ij}|{\mathbf v}_i-{\mathbf v}_j|^2 m_im_j}{\sum_{i,j} |{\mathbf v}_i-{\mathbf v}_j|^2m_im_j}.
\]
The obvious bound that follows,
\begin{equation}\label{eq:minphi}
\lambda_2(\Delta_{\mathbf m}{\mathbb A})
\geq M\min_{i,j}\phi_{ij},
\end{equation}
shows that $\phi_{ij}(t)>0 \ \leadsto \ \lambda_2(\Delta_{\mathbf m}{\mathbb A})(t)>0$. This is the scenario of a global, all-to-all connectivity between every pair of agents. The bound \eqref{eq:minphi} is not sharp: we may have certain edges vanish while still maintaining a connected graph, that is, the strict inequality $\lambda_2 > M\min_{ij}\phi_{ij}=0$ holds. A positive coefficient of ergodicity allows more general scenarios, in which pairs of agents, positioned at say ${\mathbf x}_p$ and ${\mathbf x}_q$, may lack direct communication, $\phi_{pq}=0$, but they still communicate through an intermediate agent positioned at ${\mathbf x}_k$. That is, for each $(p,q)$ there exists (at least) one agent positioned at ${\mathbf x}_k, \ k = k(p,q)$, which is the 'go between' agent so that\footnote{Of course the special case $k=p$ recovers the direct pairwise communication.} $\min\{\phi_{pk}, \phi_{qk}\} > 0$. This one-layer of communication is captured by the refined lower-bound
\[
\lambda_2(\Delta_{\mathbf m}{\mathbb A})
\geq \erg({\mathbb A})=\sum_j m_j \min_{p,q}\{\phi_{pj},\phi_{qj}\} \geq M\min_{i,j}\phi_{ij}.
\]
\end{remark}
The estimate \eqref{eq:ergodicity} in its $\ell^1$-dual form for goes back to Dobrushin \cite{Dob56}, quantifying the contractivity of column-stochastic matrices in terms of the so-called coefficient of ergodicity, denoted here $\erg({\mathbb A})$, \cite{IS11}. It was revisited in many follow-up works, e.g., its used to quantify
the relative entropy in discrete Markov processes \cite{CDZ93,CIR93} scrambling in models of opinion dynamics \cite{Kra00,Kra15} and flocking dynamics \cite[\S2.1]{MT14a}.
\subsection{Energy fluctuations revisited --- time dependent masses}\label{sec:MT}
The study of long time behavior based on $\ell^\infty$-diameter of velocity fluctuations enjoyed the advantage of addressing time-dependent messes. In contrast, our study of energy fluctuations in section \ref{sec:energy} was restricted to constant masses. Here we observe that the proof of theorem \ref{thm:main1} can be adapted to include the case of time-dependent masses, $m_i=m_i(t)$. Indeed, the time variability of the masses enters at precisely in two places: the time invariant total momentum in \eqref{eq:momentum}
\begin{subequations}\label{eqs:equalities}
\begin{equation}\label{eq:equa}
{\mathscr M}(t+\step)={\mathscr M}(t), \qquad {\mathscr M}(t):=\sum_i m_i{\mathbf v}_i(t),
\end{equation}
and the evaluation of the incremental energy fluctuations \eqref{eq:abc}
\begin{equation}\label{eq:equb}
{\delta \mathscr E}(t+\step) - {\delta \mathscr E}(t) = \sum_i m_i |{\mathbf v}_i(t+\step)|^2 -\sum_i m_i|{\mathbf v}_i(t)|^2.
\end{equation}
\end{subequations}
To pursue our line of proof when $m_i=m_i(t)$, the momentum ${\mathscr M}(t+\step)$ and energy fluctuations ${\delta \mathscr E}(t+\step)$ need to be weighted by $m_i(t+\step)$ rather than $m_i(t)$. Thus, the two qualities above admit the additional terms $\sum_i |m_i(t+\step)-m_i(t)|\times |{\mathbf v}_i(t+\step)|$ and, respectively, $\sum_i |m_i(t+\step)-m_i(t)|\times |{\mathbf v}_i(t+\step)|^2$, so one needs to control the incremental changes $\sum_i |m_i(t+\step)-m_i(t)|$.
Consider the example of the M-T model \eqref{eq:MT} with metric kernel
$\phi({\mathbf x},{\mathbf x}')=\varphi(|{\mathbf x}-{\mathbf x}'|)$ where the time-dependent masses are then given by the degrees
\[
m_i(t)=\frac{1}{L}deg_i(t)=\sum_j \varphi(|{\mathbf x}_i(t)-{\mathbf x}_j(t)|).
\]
Assume that $\varphi$ is a \emph{smooth} communication kernel such that
$|\varphi'| \leq C|\varphi|$. Then
\[
\sum_i |m_i(t+\step)-m_i(t)|\times |{\mathbf v}_i(t+\step)| \\
\leq
C_1\tau \sum_{i,j} |{\mathbf v}_i(t)-{\mathbf v}_j(t)| \phi_{ij}(t) \times |{\mathbf v}_i(t+\step)|.
\]
Now, using a uniform bound on the velocities $\max_i |{\mathbf v}_i(t+\step)|\leq C_2$ and the exponential decay of ${\delta \mathscr V}(t)$, \eqref{eq:etaresult}, and recalling $\phi_{ij}= L\widetilde{\phi}_{ij}m_im_j$ \eqref{eq:MTS}, we find
\[
\begin{split}
\sum_i |m_i(t+\step)&-m_i(t)|\times |{\mathbf v}_i(t+\step)| \\
& \leq C_1C_2 \tau \cdot {\delta \mathscr V}(t) \sum_j \widetilde{\phi}_{ij}(t) m_i(t) m_j(t)
\\
& \leq C' \tau e^{-\eta t}, \qquad C':=C_1C_2M^2\max |\phi|.
\end{split}
\]
Hence, the equalities \eqref{eqs:equalities} in the case of constant masses are now replaced by the corresponding
\begin{subequations}
\begin{equation}
\left|{\mathscr M}(t+\step)-{\mathscr M}(t)\right|\leq C'\tau e^{-\eta t},
\end{equation}
and, respectively,
\begin{equation}
\Big|\Big({\delta \mathscr E}(t+\step) -\sum_i m_i(t+\step)|{\mathbf v}_i(t+\step)|^2 \Big)- \Big( {\delta \mathscr E}(t) - \sum_i m_i(t)|{\mathbf v}_i(t)|^2\Big) \Big| \leq C'\tau e^{-\eta t}.
\end{equation}
\end{subequations}
Thus, presence of smoothly varying time-dependent masses, accounts for additional terms which have a bounded accumulated effect.
One can then study the long time behavior based on energy fluctuations in the presence of time-dependent masses, similar to our discussion in the next section, of flocking/swarming phenomena with constant masses.
\section{Flocking and Swarming}
The phenomena of \emph{flocking} or \emph{swarming} require the emergence of coordinated long time behavior of velocities, while the crowd of agents remains contained within finite diameter
\begin{equation}\label{eq:diameter}
D(t):=\max_{i,j}|{\mathbf x}_i(t)-{\mathbf x}_j(t)| \leq D_+ <\infty.
\end{equation}
The emerging behavior of velocities in intimately linked to the decay bounds of energy fluctuations.
Indeed, \eqref{eq:result} and its corresponding semi-discrete \eqref{eq:SDresult} imply that \emph{if} the weighted graph of communication remains sufficiently strongly connected in the sense that $\lambda_2(t)$ has diverging tail, then by \eqref{eq:mean}
\begin{equation}\label{eq:heavy}
\begin{split}
\int^\infty \lambda_2(s)\textnormal{d}s=0 \
\ \leadsto \ \ \sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}(t)|^2m_i \leq exp\Big\{{\small -\int_0^t \lambda_2(s)\textnormal{d}s}\Big\}{\delta \mathscr E}_0\stackrel{t\rightarrow \infty}{\longrightarrow}0.
\end{split}
\end{equation}
In particular, since the mean velocity is an invariant of the flow,
\[
\overline{{\mathbf v}}(t):=\frac{1}{M}\sum_j m_j{\mathbf v}_j(t)=\overline{{\mathbf v}}_0,
\]
\eqref{eq:heavy} tells us that a heavy-tailed $\lambda_2(t)$ implies the long time behavior of the velocities that align along the initial mean, ${\mathbf v}_i(t) \stackrel{t\rightarrow \infty}{\longrightarrow} \overline{{\mathbf v}}_0$
\begin{remark}[{\bf Emerging velocity in presence of time-dependent masses}] In case of constant masses, the mean velocity $\overline{{\mathbf v}}_(t)$ remains invariant in times, and the decay of velocity fluctuations implies the emergence of $\overline{{\mathbf v}}_0$ as the limiting velocity. The presence of time-dependent masses, however, leaves open the question of what is the emerging velocity. Thus, for example, in case of the M-T dynamics \eqref{eq:MT}, we expect that velocities will align along the corresponding mean $\overline{{\mathbf v}}$
\[
|{\mathbf v}_i(t) -\overline{{\mathbf v}}(t)| \stackrel{t\rightarrow \infty}{\longrightarrow}0,
\qquad \overline{{\mathbf v}}(t):= \frac{1}{\sum_j deg_j(t)}\sum_j deg_j(t){\mathbf v}_j(t).
\]
The question is if and when the emerging \emph{limiting} velocity,
$\ds \lim \limits_{t\rightarrow \infty} \frac{1}{\sum_j deg_j(t)}\sum_j deg_j(t){\mathbf v}_j(t)$, exists.
\end{remark}
\subsection{Long-range interactions}
But when does $\lambda_2(t)$ satisfy the `heavy-tail' condition sought in \eqref{eq:heavy}? this is a central question for tracing the phenomenon of flocking. It was addressed in many references, starting with the original \cite{CS07a,CS07b} followed by \cite{HT08,HL09b}; see \cite{CFTV10,Shv21} and the references therein. A definitive answer is provided in case of \emph{long-range kernels},
\begin{equation}\label{eq:long}
\phi({\mathbf x},{\mathbf x}')\gtrsim \frac{1}{(1+|{\mathbf x}-{\mathbf x}'|)^\beta}, \quad \beta>0.
\end{equation}
In this case we bound the tail of the spectral gap,
$\lambda_2(t)=\lambda_2\big(\Delta_{\mathbf m}{\mathbb A}(t)\big)$,
\begin{equation}\label{eq:connect}
\lambda_2\big(\Delta_{\mathbf m}{\mathbb A}(t)\big)
\geq M\min \phi_{ij}(t) \gtrsim \frac{M}{\big(1+D(t)\big)^\beta}\geq
\frac{M}{(1+D_0 + {\delta \mathscr V}_{\!\!0}\cdot t)^\beta}.
\end{equation}
The first inequality on the right follows from \eqref{eq:minphi}, the second follows from \eqref{eq:long} and the third follows from a uniform bound on the diameter of velocities\footnote{\label{foot:max}The result follows without appealing to the bound on diameter of velocities \eqref{eq:ergodicity}. Instead, a simpler maximum principle argument follows from the CFL condition \eqref{eq:CFL},
\[
|{\mathbf v}_i(t+\step)|\leq \Big(1-\step\sum_j\phi_{ij}m_j\Big)|{\mathbf v}_i(t)|+\step\sum_j \phi_{ij}m_j|{\mathbf v}_j(t)| \leq \max_j|{\mathbf v}_j(t)| \leq \ldots \leq v_+(0), \quad v_+(0)=\max_i |{\mathbf v}_i(0)|,
\]
and integration of \eqref{eq:CS}${}_1$, and likewise, \eqref{eq:SDCS}${}_1$ in the semi-discrete case, imply $D(t) \leq D_0+2v_+(0)\cdot t$.},
\[
D(t)\leq D_0 + \int^t {\delta \mathscr V}(s)\textnormal{d}s \leq D_0+{\delta \mathscr V}_{\!\!0}\cdot t,
\]
and hence the heavy-tailed bound, $\lambda_2(t)\gtrsim (1+t)^{-\beta}$ for $\beta\leq 1$. We conclude that the C-S dynamics \eqref{eq:CS} with long-range communication \eqref{eq:long}, $\beta\leq 1$, admits \emph{unconditional flocking}
\[
\sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}_0|^2m_i \lesssim
\left\{\begin{array}{ll}
exp\big\{-\frac{M}{(1-\beta){\delta \mathscr V}_{\!\!0}}\big(1+D_0+{\delta \mathscr V}_{\!\!0}\cdot t\big)^{1-\beta}\big\} & \beta<1\\ \\
\big(1+D_0+{\delta \mathscr V}_{\!\!0}\cdot t\big)^{\ds -\nicefrac{M}{{\delta \mathscr V}_{\!\!0}}} & \beta=1
\end{array}\right\} \stackrel{t\rightarrow \infty}{\longrightarrow}0.
\]
We can now use a bootstrap argument --- the fractional exponential decay of the fluctuations of order $1-\beta>0$ implies that the diameter remains uniformly bounded and hence uniform bounded connectivity
\[
D(t)\leq D_0 + \int^t {\delta \mathscr V}(s)\textnormal{d}s \leq D_+ \ \ \leadsto \ \ \lambda_2(t) \geq \eta:=\frac{1}{(1+D_+)^\beta}.
\]
Revising \eqref{eq:connect} with a finite diameter $\leq D_+$, yields the improved exponential bound
$\lesssim e^{-\eta t}$.
A similar argument applies in the borderline case of $\beta=1$: clearly, if ${\delta \mathscr V}_{\!\!0}<M$ then the finite tail of $\lesssim (1+t)^{-\nicefrac{M}{{\delta \mathscr V}_{\!\!0}}}$ will lead to a finite diameter; and indeed, since ${\delta \mathscr V}(t)$ is decaying, we will eventually reach the threshold ${\delta \mathscr V}(t_c)<M$ and exponential decay follows thereafter. We summarize.
\begin{theorem}[{\bf Flocking/swarming with long range kernels}]\label{thm:flocking1}
Consider the C-S dynamics \eqref{eq:CS} driven by long-range kernel \eqref{eq:long}, $\beta\leq1$. Then the crowd of agents has finite support $D_+$ and there is exponential decay
of fluctuations around the mean velocity,
\begin{equation}\label{eq:exp}
\sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}_0|^2m_i \lesssim
e^{-\eta t} {\delta \mathscr E}_0, \qquad \eta=\frac{1}{(1+D_+)^\beta}.
\end{equation}
\end{theorem}
The precise exponential bound, $\eta=\eta(\beta,D_0,v_+)$, was captured in \cite{HL09b} using an elegant argument based on a proper Liapunov functional
for C-S with metric kernel and uniform masses.\newline
\begin{remark}[{\bf No uniform bound}] We distinguish between two types of bounds on velocity fluctuations --- the $\ell^2$-based energy fluctuations, theorem \ref{thm:main1}, and the $\ell^\infty$ bounds, theorem \ref{thm:main2} or at least the uniform bound on velocities, see footnote \ref{foot:max}. Suppose we try to pursue a purely $\ell^2$-based argument
for flocking behavior. The energy bound \eqref{eq:result} implies
the uniform-in-time bound
$\max_i|{\mathbf v}_i(t)-\overline{{\mathbf v}}_0| \leq C\sqrt{N}$ which in turn yields a bound on the diameter $D(t) \leq D_0+2C\sqrt{N}t$. We now use the same bootstrap argument as before to find a uniform-in-time bound on the diameter $D(t) \lesssim D_+(N):=N^{\frac{\beta}{2(1-\beta)}}$. We conclude an exponential flocking of rate
\[
\sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}_0|^2m_i \lesssim D_+(N)
e^{-D_+(N)t}, \qquad D_+(N)=N^{\frac{\beta}{2(1-\beta)}}.
\]
As expected, there fluctuations bound grow with $N$. However, the point to note here is that the exponential decay in time enforces exponential alignment bound, uniform in $t$ \emph{and} $N$ when $t\gg N^{\frac{\beta}{2(1-\beta)}}$. For example, $\beta=\nicefrac{1}{4}$ requires a moderate time of $t\gg N^{\nicefrac{1}{6}}$ before exponential decay takes place.
\end{remark}
Theorem \ref{thm:flocking1} was derived based on considerations of energy fluctuations. Similarly, we can proceed using the $\ell^\infty$-diameter fluctuations of theorem \ref{thm:main2}. Its semi-discrete limit $\max_k\tau(t_k)\rightarrow 0$ reads
\[
\max_{i,j}|{\mathbf v}_i(t)-{\mathbf v}_j(t)|\leq exp\Big\{-\int_0^t \sum_j m_j(s)\min_{p,q}\{\phi_{pj}(s),\phi_{qj}(s)\}\textnormal{d}s\Big\}{\delta \mathscr V}_{\!\!0}.
\]
Here, we generalize theorem \ref{thm:flocking1} to the case of time-dependent masses. Using a bootstrap argument as before we end up with
\begin{theorem}[{\bf Flocking/swarming with long range kernels --- time-dependent masses}]\label{thm:flocking2}
Consider the C-S dynamics \eqref{eq:CS} with possibly time-dependent masses, $m_i=m_i(t)$, driven by long-range kernel \eqref{eq:long}, $\beta\leq1$. Then the crowd of agents has finite support $D_+$ and there is exponential decay
of fluctuations of velocities,
\begin{equation}\label{eq:exp2}
\max_{i,j} |{\mathbf v}_i(t)-{\mathbf v}_j(t)| \lesssim
exp\Big\{-\ds \eta \int_0^t \sum_j m_j(s) \textnormal{d}s\Big\} {\delta \mathscr V}_{\!\!0}, \qquad \eta=\frac{1}{(1+D_+)^\beta}.
\end{equation}
\end{theorem}
Observe that in the example of M-T model \eqref{eq:MT} the scaling alluded in footnote \ref{foot:MT}, $M={\mathcal O}(1)$, implies $\ds \int_0^t \sum_j m_j(s) \textnormal{d}s\geq Ct$ and hence we end up with an exponential decay
$e^{-\eta C t}$.
\smallskip
The arguments that led to theorem \ref{thm:flocking1} and the new theorem \ref{thm:flocking2} demonstrate a rather general methodology for studying flocking, swarming and more general emerging phenomena in alignment based dynamics. It consists of two main ingredients:
\begin{itemize}
\item Decay of energy fluctuations. This is tied to spectral analysis of the dynamic graph $\big({\sf V}(t),{\sf E}(t)\big)$. In typical cases, the dynamics is equipped with an intrinsic `energy' and energy fluctuations.
\item Bound on the velocities --- either a uniform bound on velocities or on $\ell^\infty$-diameter of velocities fluctuations. In either case, the purpose is to trace the size of the spatial diameter and show that the crowd does not disperse,
$D(t) \leq D_+$. In general, this is the more intricate bound to prove.
\end{itemize}
\noindent
As an example we mention alignment dynamics with external forcing \cite{ST20a}. Other examples include C-S dynamics with \emph{matrix} communication kernels, and C-S dynamics in which both, alignment and attraction, take place. We continue with this example in the context of \emph{anticipation dynamics}.
\subsection{From anticipation to Cucker-Smale dynamics}
Particles are driven by the external forces induced by the environment and/or by other particles. The dynamics of social particles, on the other hand, is driven by \emph{probing} the environment --- living organisms, human interactions and sensor-based agents have \emph{senses and sensors}, with which they actively probe the environment (and hence they are commonly viewed as `active matter' \cite{BDT17-19}). A distinctive feature of active particles in probing the environment is \emph{anticipation} --- the dynamics is not driven instantaneously, but reacts to positions ${\mathbf x}^\step(t):={\mathbf x}(t)+\step {\mathbf v}(t)$, \emph{anticipated} at $t+\step$, where $\step>0$ is an anticipation time increment. A general framework for anticipation dynamics, driven by pairwise interactions induced by radial potential $U=U(r)$, reads
\vspace*{-0.2cm}
\[
\left\{\quad
\begin{split}
\frac{\textnormal{d}}{\textnormal{d}t}{{\mathbf x}}_i(t)&={\mathbf v}_i(t)\\
\frac{\textnormal{d}}{\textnormal{d}t}{{\mathbf v}}_i(t)&= -\frac{1}{N}\sum_{j=1}^N\nabla U(|{\mathbf x}^\step_i(t)-{\mathbf x}^\step_j(t)|), \qquad {\mathbf x}^\tau_k(t):={\mathbf x}_k(t)+\tau {\mathbf v}_k(t).
\end{split}
\right.
\]
Our starting point is to expand the RHS in (the assumed small) $\step$, obtaining, \cite{ST21a},
\begin{equation}\label{eq:system-anticipation}
\frac{\textnormal{d}}{\textnormal{d}t} {\mathbf v}_i(t)=\overbrace{-\frac{1}{N}\sum_{j} \nabla U(|{\mathbf x}_j-{\mathbf x}_i|)}^{\text{repulsion+attrcation}} + \overbrace{\frac{\tau}{N}\sum_{j\in {\mathcal N}_i} \bbphi_{ij}({\mathbf v}_j-{\mathbf v}_i)}^{\text{alignment}}.
\end{equation}
\vspace*{0.2cm}
\begin{wrapfigure}{l}{0.35\textwidth}
\vspace*{-0.6cm}
\begin{center}
\includegraphics[width=0.22\textwidth]{figure_Lennard-Jones_v2.pdf
\end{center}
\vspace*{0.0cm}
\caption{Potential $U(r)$}\label{fig:Lennard-Jones}
\end{wrapfigure}
\vspace*{-0.0cm}\noindent
Thus, we \emph{derive} a general class of 3Zone models \eqref{eq:system-anticipation}, where the first terms on the right account for repulsion/attraction classified by their scalar amplitudes $U'_{ij}:=U'(|{\mathbf x}_i-{\mathbf x}_j|)<0 $ and respectively $U'_{ij}>0$, and the second term of alignment with \emph{matrix} coefficients, ${\bbphi}_{ij}= D^2U(|{\mathbf x}_i-{\mathbf x}_j|)$, see figure \ref{fig:Lennard-Jones}.
\vspace*{1.0cm}
\noindent
This leads us to consider an even larger class of 3Zone models with repulsion/attraction induced by potential $U$ and alignment term induced by a separate scalar symmetric kernel, $\phi_{ij}=\phi({\mathbf x}_i,{\mathbf x}_j)$ (independent of $U$),
\begin{equation}\label{eq:anticipation}
\frac{\textnormal{d}}{\textnormal{d}t} {\mathbf v}_i(t)=-\frac{1}{N}\sum_{j} \nabla U(|{\mathbf x}_j-{\mathbf x}_i|) + \frac{\tau}{N}\sum_{j: \phi_{ij}>0} \phi({\mathbf x}_i,{\mathbf x}_j)({\mathbf v}_j-{\mathbf v}_i).
\end{equation}
The special case of metric-based kernel $\phi_{ij}=\varphi(|{\mathbf x}_i-{\mathbf x}_j|)$ recovers the C-S dynamics \eqref{eq:equiCS}, $m_i\equiv\nicefrac{1}{N}$. For the special case of anticipation \eqref{eq:system-anticipation} we have $\bbphi_{ij}\geq U''(|{\mathbf x}_i-{\mathbf x}_j|){\mathbb I}$.
The energy fluctuations associated with \eqref{eq:anticipation}
\[
{{\delta \mathscr E}}(t):=\frac{1}{2N}\sum_i |{\mathbf v}_i(t)-\overline{{\mathbf v}}|^2 + \frac{1}{2N^2}\sum_{i,j}U(|{\mathbf x}_i(t)-{\mathbf x}_j(t)|),
\]
are dissipated due to alignment at a precise rate dictated by local velocity fluctuations,
\[
\frac{\textnormal{d}}{\textnormal{d}t} {{\delta \mathscr E}}(t) =-\frac{\step}{2N^2}\sum_{i,j}\phi_{ij}(t) |{\mathbf v}_{i}-{\mathbf v}_j|^2.
\]
We assume a smooth radial potential so that $U'(0)=U(0)=0$. It follows that the class of convex potentials and `fat-tailed' kernels such that
\begin{equation}\label{eq:cond}
U''(|{\mathbf x}_i-{\mathbf x}_j|)+\phi_{ij}\gtrsim \langle |{\mathbf x}_i-{\mathbf x}_j|\rangle^{-\gamma}, \quad \gamma<\nicefrac{4}{5},
\end{equation}
guarantee decay of energy fluctuations, $\ds {{\delta \mathscr E}}(t) \leq C_0 exp\{-t^{\frac{4-5\gamma}{4-3\gamma}}\}$, which in turn implies asymptotic flocking towards the average velocity, $\overline{{\mathbf v}}=\frac{1}{N}\sum_j {\mathbf v}_j$, \cite{ST21a}. Moreover, agents asymptotically congregate in space, forming a traveling wave dictated by the presence of an attractive potential $U$, e.g., a quadratic $U$ leads to a limiting harmonic oscillator.
\cite{ST20a}.
\smallskip
\paragraph{{\bf Open questions}} The arguments above exclude two important features in collective dynamics: since \eqref{eq:cond} implies $U$ is increasing, it does not address the role of \emph{repulsion} in shaping the emergent behavior. The large time behavior of 2Zone repulsion-attraction models were discussed in, e.g.,
\cite{CDMBC07,FHK11,FH13,DDMW15,CCP17,CFP17}.
The corresponding question for the full 3Zone model, in which attraction, alignment and repulsion co-exist, is mostly open.
Another key aspect is the long-range alignment sought by the 'heavy-tailed' kernels in \eqref{eq:cond} which does not address the local character of self-organized dynamics. The long time collective behavior based on \emph{short-range} protocols hinges on the \emph{graph connectivity} of the crowd, realized by the \emph{adjacency matrix} ${\mathbb A}(t):=\{\phi_{ij}(t)\}$. Short-range interactions may lead to instability. This can be traced by the \emph{graph Laplacian} $\Delta {\mathbb A}(t)$: while the initial configuration of the crowd is assumed to form one connected cluster expressed by the positivity of its spectral gap $\lambda_2(\Delta {\mathbb A}(0)>0$, it may break down into two or more disconnected clusters at a finite time when $\lambda_2(\Delta{\mathbb A}(t_c)=0$. Flocking analysis with short-range kerenls can be found in \cite{JE07,GPY16,Car17,MPT19,ST20b,DR21,Tad21}.
\section{Large crowd dynamics}
The question of instability for a fixed number of $N$ agents governed by short-range alignment is better addressed in the context of \emph{large crowd} dynamics of $N\gg 1$ agents. The latter is realized by the empirical distribution
\[
f_N(t,{\mathbf x},{\mathbf v}):=\frac{1}{M}\sum_i m_i\delta ({\mathbf x}-{\mathbf x}_i(t))\otimes \delta({\mathbf v}-{\mathbf v}_i(t)).
\]
The large crowd dynamics is captured by its first two ${\mathbf v}$ moments which are assumed to exist, \cite{Shv21}:
\[
\rho(t,{\mathbf x})=\lim_{N\rightarrow \infty} \int f_N(t,{\mathbf x},{\mathbf v})\d\bv, \quad \rho{\mathbf u}(t,{\mathbf x})=\lim_{N\rightarrow \infty} \int {\mathbf v} f_N(t,{\mathbf x},{\mathbf v})\d\bv
\]
These are the density and momentum which encode the macroscopic description of the agents based \eqref{eq:anticipation} (we abbreviate $\square=\square(t,{\mathbf x}), \square'=\square(t,\bx'} %{{\mathbf y})$)
\begin{equation}\label{eq:hydro}
\left\{\begin{split}
\rho_t +\nabla_{\mathbf x}\cdot (\rho {\mathbf u})&=0\\
(\rho{\mathbf u})_t + \nabla_{\mathbf x}\cdot (\rho{\mathbf u}\otimes {\mathbf u} +{\mathbb P})&=
\tau \int_{\bx'} %{{\mathbf y}\in \Omega} \phi({\mathbf x},\bx'} %{{\mathbf y})(\bu'-{\mathbf u})\rho\rho'\d\by
-\rho\nabla U * \rho(t,{\mathbf x})
\end{split}\right.
\end{equation}
There are several ingredients in the macroscopic description: the pressure (Reynolds stress) tensor, $\displaystyle {\mathbb P}(t,{\mathbf x}):=\lim_{N\rightarrow \infty} \int ({\mathbf v}-{\mathbf u})({\mathbf v}-{\mathbf u})^\top f_N(t,{\mathbf x},{\mathbf v})\d\bv$, encodes the second-order ${\mathbf v}$ moments of $f_N$. The closure of \eqref{eq:hydro} is imposed by assuming a limiting distribution
at thermal equilibrium -- a Maxwellian. But there is no generic closure in the present context of collective dynamics, since agents maintain their own detailed energy balance which is beyond the realm of collective motion. The two terms on the right capture scalar alignment and respectively attraction/repulsion induced by the potential $U$.
\noindent
\subsection{Short-range interactions}
For simplicity, we ignore the role of attraction/repulsion and conclude with three examples which trace the
flocking behavior of the purely alignment hydrodynamics \eqref{eq:hydro} with $U\equiv 0$.
\smallskip\noindent
{\bf Non-vacuous dynamics}.
In the first example, we consider the dynamics in the $2\pi$-torus driven by bounded short-range kernels, $\phi({\mathbf x},\bx'} %{{\mathbf y})$, localized along the diagonal
\begin{equation}\label{eq:bdd}
\frac{1}{\Lambda}\mathds{1}_{R_0}(|{\mathbf x}-\bx'} %{{\mathbf y}|) \leq \phi({\mathbf x},\bx'} %{{\mathbf y}) \leq {\Lambda}\mathds{1}_{2R_0}(|{\mathbf x}-\bx'} %{{\mathbf y}|), \quad R_0 \ll \pi.
\end{equation}
It follows that strong solutions with non-vacuous density
$\rho(t,\cdot)\gtrsim (1+t)^{-\nicefrac{1}{2}}$ flock around the limiting velocity $\overline{{\mathbf v}}$ due to the decay of energy fluctuations ${\delta \mathscr E}(t)\rightarrow 0$, \cite[Theorem 1.1]{ST20b}. As we noted in \cite[theorem 3.3]{Tad21}, the decay of energy fluctuations is independent of the specific closure of the pressure --- what really matters is the non-vanishing density, the connectivity of the $\textnormal{supp}\rho(t,\cdot)$ which enables to propagate information of alignment.
\smallskip\noindent
{\bf Topological interactions}.
The non-vacuous lower bound $(1+t)^{-\nicefrac{1}{2}}$ is not sharp. As a second example we mention a topologically-based \emph{singular} communication kernel,
corresponding to \eqref{eq:whatismu}
\begin{equation}\label{eq:topo}
\phi({\mathbf x},\bx'} %{{\mathbf y}) \sim \mathds{1}_{R_0}(|{\mathbf x}-\bx'} %{{\mathbf y}|)\times \frac{1}{\textnormal{dist}^d_\rho({\mathbf x},\bx'} %{{\mathbf y})},
\end{equation}
which involves the density weighted distance $\displaystyle \textnormal{dist}_\rho({\mathbf x},\bx'} %{{\mathbf y})=\Big(\int _{{\mathcal C}({\mathbf x},\bx'} %{{\mathbf y})}\hspace*{-0.3cm}{\textnormal{d}}\rho(t,{\mathbf z})\Big)^{1/d}$.
it follows that smooth solutions satisfying the relaxed non-vacuous condition, $\rho(t,\cdot)\gtrsim (1+t)^{-1}$, must flock, \cite{ST20b}. Again, no vacuum is a key aspect which
enables the propagation of information: as long as no vacuous islands are formed, alignment dictates flocking behavior.
\smallskip\noindent
{\bf Multi-species}. Our third example involves multi-species dynamics
\[
\left\{\begin{split}
(\rho_\alpha)_t +\nabla_{\mathbf x}\cdot (\rho_\alpha {\mathbf u}_\alpha)&=0\\
(\rho_\alpha{\mathbf u}_\alpha)_t + \nabla_{\mathbf x}\cdot (\rho_\alpha{\mathbf u}_\alpha\otimes {\mathbf u}_\alpha +{\mathbb P}_\alpha)&=
\tau \int_{\bx'} %{{\mathbf y}\in \Omega} \varphi_{\alpha\beta}(|{\mathbf x}-\bx'} %{{\mathbf y}|)(\bu'_\beta-{\mathbf u}_\alpha)\rho_\alpha\rho'_\beta\d\by.
\end{split}\right.
\]
In this case, different species tagged by the identifiers $\alpha,\beta\in{\mathcal I}$, are distinguished by their different protocol of communication with the environment of other species, $\phi_{\alpha\beta}$.
In \cite{HT21} it was shown that if the different species maintain non-vacuous densities $\rho_\alpha(t,\cdot)\gtrsim (1+t)^{-1}$ and if the communication array ${\mathbb A}(r):=\{\phi_{\alpha\beta}(r)\}$ forms a \emph{connected graph}, $\lambda_2 \big(\Delta{\mathbb A}(r)\big)\gtrsim (1+r)^{-\beta}$ with heavy-tail, $\beta<1$, then flocking follows, ${\mathbf u}_\alpha \stackrel{t\rightarrow \infty}{\longrightarrow} \overline{{\mathbf u}}:=\frac{1}{|{\mathcal I}|}\sum_{\alpha\in {\mathcal I}} {\mathbf u}_\alpha$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,055 |
A lha Fridtjof é uma ilha a a nordeste da Ilha Vásquez, no lado sudeste da Ilha Wiencke, no Arquipélago Palmer. Ela foi descoberta e nomeada pela Expedição Antártica Belga sob Adrien de Gerlache, 1897–99.
Ver também
Lista de ilhas antárticas e subantárticas
Referências
Ilhas do Arquipélago Palmer | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,110 |
/*
Authentication controller.
This file uses https://github.com/TheIronDeveloper/pokemon-wondertrade-analytics/blob/master/controllers/authentication.js as an example.
*/
var LocalStrategy = require("passport-local").Strategy,
User = require("../models/user.js");
var ensureAuthenticated = function(req, res, next) {
if (req.isAuthenticated()) {
return next();
}
res.redirect("/");
}
// TODO move this function to a user controller?
var find = function(req, res) {
res.render("profile", {
username: JSON.stringify(req.user.username),
email: JSON.stringify(req.user.email)
});
}
var login = function(req, res) {
res.render("login", {
message: req.flash("loginMessage")
});
}
var logout = function(req, res) {
req.logout();
res.redirect("/");
}
var register = function(req, res) {
res.render("register", {
message: req.flash("registerMessage")
});
}
module.exports = function(app, passport) {
passport.serializeUser(function(user, done) {
done(null, user.id);
});
passport.deserializeUser(function(id, done) {
User.findById(id, function(err, user) {
done(err, user);
});
});
passport.use("local-register", new LocalStrategy({
passReqToCallback: true
},
function(req, username, password, done) {
User.findOne({ username: username }, function (err, user) {
if (err) {
return done(err);
}
if (user) {
return done(null, false, req.flash("registerMessage", "Username already taken."));
}
var newUser = new User({
username: req.body.username,
// TODO don"t send password as plaintext
password: req.body.password,
email: req.body.email
});
// operate asynchronously, otherwise you might not redirect
newUser.save();
return done(null, newUser);
});
}
));
passport.use("local-login", new LocalStrategy({
passReqToCallback: true
},
function(req, username, password, done) {
req.logout();
User.findOne({ username: username }, function (err, user) {
if (err) {
return done(err);
}
if (!user) {
return done(null, false, req.flash("loginMessage", "Incorrect username."));
}
if (!user.validPassword(password)) {
return done(null, false, req.flash("loginMessage", "Incorrect password."));
}
return done(null, user);
});
}
));
// TODO move this to user controller?
app.get("/profile", ensureAuthenticated, find);
app.get("/login", login);
app.post("/login", passport.authenticate("local-login", {
successRedirect: "/profile",
failureRedirect: "/login",
failureFlash: true
}));
app.get("/logout", logout);
app.get("/register", register);
app.post("/register", passport.authenticate("local-register", {
successRedirect: "/profile",
failureRedirect: "/register",
failureFlash: true
}));
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,499 |
USS Cincinnati (LCS-20) is an of the United States Navy. She is the fifth ship to be named after Cincinnati, Ohio.
Design
In 2002, the United States Navy initiated a program to develop the first of a fleet of littoral combat ships. The Navy initially ordered two trimaran hulled ships from General Dynamics, which became known as the after the first ship of the class, . Even-numbered U.S. Navy littoral combat ships are built using the Independence-class trimaran design, while odd-numbered ships are based on a competing design, the conventional monohull . The initial order of littoral combat ships involved a total of four ships, including two of the Independence-class design. On 29 December 2010, the Navy announced that it was awarding Austal USA a contract to build ten additional Independence-class littoral combat ships.
Construction and career
Cincinnati was christened on 7 May 2018 by former Secretary of Commerce Penny Pritzker and commissioned on 5 October 2019. She has been assigned to Littoral Combat Ship Squadron One.
References
Independence-class littoral combat ships
2018 ships | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,706 |
Q: Instagram API Get max_tag_id I am an Instagram developer. I am trying to search images using the API with the hashtag.
I got the total count of the images but when I am trying to view these images it required "min_tag_id" and "max_tag_id".
Please tell me, any API which provides these two values.
A: try this
https://api.instagram.com/v1/tags/{tag-name}/media/recent?access_token=ACCESS-TOKEN
A: You won't have any post for that tag. That's what it shows like that.Then visit (https://api.instagram.com/v1/tags/{tag-name}/media/recent?access_token=ACCESS-TOKEN). It will work
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,011 |
Greasemonkey was a userscript manager for Firefox. With the release of version 4, it is now severely crippled and Greasemonkey recommends that you use some other engine.
Most scripts should now use the tampermonkey tag, even if they target Violentmonkey or versions of Greasemonkey prior to GM4. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,660 |
\section{Introduction} \label{section:intro}
For the last few years, political commentators have been indicating that we live in a \textit{post-truth} era \cite{davies2016age}, wherein the deluge of information available on the internet has made it extremely difficult to identify facts. As a result, individuals have developed a tendency to form their opinions based on the \textit{believability} of presented information rather than its truthfulness \cite{cone2019believability}. \textcolor{black}{This phenomenon is exacerbated by the business practices of social media platforms, which often seek to maximize the \textit{engagement} of their users at all costs. In fact, the algorithms developed by platforms for this purpose often promote conspiracy theories among their users \cite{tufekci2018youtube}.}
The sensitivity of users of social media platforms to conspiratorial ideas makes them an ideal terrain to conduct political misinformation campaigns \cite{kramer2014experimental, weedon2017information}. Such campaigns are especially effective tools to disrupt democratic institutions, because
\textcolor{black}{the functioning of stable democracies relies on \textit{common knowledge} about the political actors and the processes they can use to gain public support \cite{farrell2018common}. The trust held by the citizens of a democracy on common knowledge includes: (i) trust that all political actors act in good faith when contesting for power, (ii) trust that elections lead to a free and fair transfer of power between the political actors, and (iii) trust that democratic institutions ensure that elected officials wield their power in the best interest of the citizens.} In contrast, citizens of democracies often have a \textit{contested knowledge} regarding who should hold power and \textcolor{black}{how they should use it \cite{farrell2018common}.}
The introduction of \textit{alternative facts} can reduce the trust on common knowledge about democracy, \textcolor{black}{especially if they become accepted beliefs among the citizens.} Such disruptions on \textcolor{black}{the trust on common knowledge} can be found in the $2016$ U.S. elections \cite{allcott2017social} and Brexit Campaign in $2016$ \cite{oxford2018russia}, \textcolor{black}{where the spread of misinformation through social media platforms} resulted in a large number of citizens mistrusting the results of voting.
To tackle this growing phenomenon of misinformation, in this paper, we consider a finite group of social media platforms, \textcolor{black}{whose users represent the citizens in a democracy,} and a democratic government. Every post in the platforms is associated with a parameter that captures its informativeness, \textcolor{black}{which can take values between two extremes: (i) completely factual and (ii) complete misinformation.}
In our framework, posts that exhibit misinformation can lead to a decrease in trust on common knowledge among the users \textcolor{black}{\cite{bessi2015, brown2018,tucker2017, sternisko2020dark}.}
In addition, social media platforms are considered to have the technologies to \textit{filter}, or label, posts that intend to sacrifice trust on common knowledge. \textcolor{black}{Thus, the government seeks to incentivize the social media platforms to use these technologies and filter any misinformation included in the posts.}
Motivated by capitalistic values, \textcolor{black}{we induce a \textit{misinformation filtering game} to describe the interactions between the social media platforms and the government. In this game, each platform acts as strategic player seeking to maximize their advertisement revenue from the engagement of their users \cite{allcott2017social, jaakonmaki2017}.
User engagement is a metric that can be used to quantify the interaction of users with a platform, and subsequently, how much time they spend on the platform.
Recent efforts reported in the literature on misinformation in social media platforms have indicated that increasing filtering of misinformation leads to decreasing of user engagement \cite{candogan2020optimal}. There are many possible reasons for this phenomenon. First, filtering reduces the total number of posts propagating across the social network. Second, the users whose opinions are filtered may perceive this action as dictatorial censorship \cite{pew2020}, and as a result, they may chose to express their opinions in other platforms. Finally, misinformation tends to elicit stronger reactions, e.g., surprise, joy, sadness, as compared to factual posts \cite{vosoughi2018spread}, which may increase user engagement.
Thus, each platform is reluctant to filter misinformation.}
\textcolor{black}{In our framework, we consider that the government is also a strategic player, whose utility increases as the trust of the users of social media platforms on common knowledge increases. Consequently, increasing filtering of misinformation by the social media platforms increases the utility of the government. Thus the government is willing to make an investment to incentivize the social media platforms to filter misinformation. In our approach, we use mechanism design to distribute this investment among the platforms optimally, and in return, implement an optimal level of filtering.}
Mechanism design was developed for the implementation of system-wide optimal solutions to problems involving multiple rational players with conflicting interests, each with private information about preferences \cite{mas_colell1995}. Note that this approach is different from traditional approaches to decentralized control with private information \cite{2019Aditya_arXiv, mahajan2012, 17, Malikopoulos2018} because the players are not a part of the same time, but in fact, have private and competitive utilities.
The fact that Mechanism design optimizes the behaviour of competing players has led to broad applications spanning different fields including economics, politics, wireless networks, social networks, internet advertising, spectrum and bandwidth trading, logistics, supply chain, management, grid computing, and resource allocation problems in decentralized systems \cite{sharma2012local,sinha2013,kakhbod2011efficient, jain2010,zhang2019efficient,chremos2020_TCNS, chremos2019social}.
The contribution of this paper is as follows. We present an indirect mechanism to incentivize social media platforms to filter misleading information.
We show that our proposed mechanism is (i) feasible, (ii) budget balanced, (iii) individual rational, and (iv) strongly implementable at the equilibria of the induced game. We prove the existence of at least one generalized Nash equilibrium and show that our mechanism induces a Pareto efficient equilibrium.
The rest of the paper is organized as follows. In Section \ref{section:formulation}, we provide the modeling framework and problem formulation. In Section \ref{section:md_problem}, we present our mechanism, and in Section \ref{section:properties_of_mechanism}, we prove the associated properties of the mechanism. In Section V, we interpret the mechanism and present a descriptive example. Finally, in Section VI we conclude and present some directions for future research.
\section{Problem Formulation} \label{section:formulation}
We consider a democratic society consisting of a finite and nonempty set of social media platforms $\mathcal{I} = \{1, \dots, I\}$, $I \in \mathbb{N},$ and a government.
\color{black}
We refer to the social media platforms and the government collectively as the \textit{players}, and denote the set of all players by $\mathcal{J} = \mathcal{I} \cup \{0\}$, where the index $0$ corresponds to the government. The players strategically take actions in a \textit{misinformation filtering game} that is described in this section.
\subsection{Misinformation Filtering Game for Platforms}
\color{black}
Let the informativeness of a post on platform $i\in\mathcal{I}$ be denoted by $x_i \in [0,1]$, where $x_i = 0$ indicates that the post contains complete misinformation and $x_i = 1$ indicates that the post contains completely factual information. Our hypothesis, \textcolor{black}{inspired by \cite{farrell2018common, bessi2015, brown2018,tucker2017, sternisko2020dark},} states that the emergence of posts with many falsehoods and a low informativeness, i.e., $x_i \to 0$, leads to a decrease of trust of the users on common knowledge about democracy. \textcolor{black}{Recall that common knowledge about democracy refers to knowledge of political actors in a democratic society and the process they use to gain public support.}
Each social media platform $i \in \mathcal{I}$ has the technological {means} to detect and filter misinformation. \textcolor{black}{In the misinformation filtering game that we impose in our framework, the action $a_i$ of platform $i$ represents the level of filtering imposed by $i$ and takes values in a feasible set of actions $\mathcal{A} = [0,1]$. Each action $a_i$ minimizes the spread of a post that has informativeness $x_i < a_i$, while posts with informativeness $x_i \geq a_i$ are unaffected.} In practice, filtering of misinformation can be implemented in many ways.
The social media platform can place warnings on each post with $x_i < a_i$ to inform the users of their falsehood, or they can modify their algorithms to limit the propagation of such posts among users. Thus, the action $a_i$ represents the lower threshold on informativeness that is acceptable by platform $i$. \textcolor{black}{To this end, we refer to the action $a_i$ as the filter of platform $i$.}
\textcolor{black}{Each platform $i \in \mathcal{I}$ generates revenue by monetizing the \textit{engagement} of their users through advertisements \cite{jaakonmaki2017}.
By increasing filtering of misinformation there is a decrease in user engagement \cite{candogan2020optimal}.
This may be due to a perception of censorship among users \cite{pew2020}, and as a result, they may chose to express their opinions in other platforms.
Consider, for example, platform $l \in \mathcal{I}$ with a filter $a_l > a_i$. Some of the users of $l$, whose posts have been marked up by the filter, may migrate to platform $i$ which will lead to an increase in the engagement of platform $i$. This phenomenon motivates us to define a set of \textit{competing platforms}.}
\begin{definition}
For each platform $i \in \mathcal{I}$, the set $\mathcal{C}_i \subset \mathcal{I}$, with $i \in\mathcal{C}_i$, is the set of \textit{competing platforms} whose choice of filters has an impact on the engagement of platform $i$.
\end{definition}
To simplify the presentation of our results, we consider that for any two platforms $i,k \in \mathcal{I}$, if $i \in \mathcal{C}_k$, then $k \in \mathcal{C}_i$. However, our mechanism can easily be extended to the case of asymmetric competition among social media. Given the set of competing platforms $\mathcal{C}_i$, we can define a \textit{valuation function} of platform $i$.
\begin{definition}
The \textit{valuation function} of a social media platform $i \in \mathcal{I}$ is
$v_i\big(a_k : k \in \mathcal{C}_i \big) : \mathcal{A}^{|\mathcal{C}_i|} \to \mathbb{R}_{\geq 0}$. \textcolor{black}{It is a decreasing function with respect to $a_i$ and strictly increasing with respect to $a_l$ for all $l \in \mathcal{C}_{-i}$, where $\mathcal{C}_{-i} = \mathcal{C}_{i} \setminus \{i\}$.}
\end{definition}
\textcolor{black}{The valuation function $v_i\big(a_k : k \in \mathcal{C}_i \big)$ corresponds to the revenue generated by platform $i$ given the user engagement after all platforms have implemented their filters. A higher value of $a_i$ will result in decreasing user engagement in platform $i$, and thus their revenue. On the other hand, a higher value of $a_l$ of another competing platform $l \in \mathcal{C}_{-i}$ will result in increasing user engagement, and thus revenue, in platform $i$.}
\textcolor{black}{Next, recall from the discussion in the previous section that filtering of misinformation in a social media platform increases the trust of the users of this platform on common knowledge about democracy. Next, for each platform $i \in \mathcal{I}$, we define the average trust function on common knowledge.}
\begin{definition}
The \textit{average trust function} of the users of platform $i \in \mathcal{I}$ on common knowledge is $h_i(a_i) : \mathcal{A} \to [0, 1]$, \textcolor{black}{and it is a strictly increasing function with respect to $a_i$.}
\end{definition}
\textcolor{black}{The average trust function $h_i(a_i)$ captures the impact of filter $a_i$ on the trust on common knowledge across the users of platform $i$.
A low value of $h_i(a_i)$ implies that $a_i$ leads to low trust on common knowledge for the users of platform $i$, and vice versa.}
In practice, platform $i$ can measure the opinions expressed by their users \cite{ceron2014every} through surveys, and over time, use these measurements to estimate the impact of filter $a_i$ using the average trust function $h_i(a_i)$.
\color{black}
\subsection{Misinformation Filtering Game for the Government}
Recall that, in our framework, the government is considered the strategic player $0 \in \mathcal{J}$. The government's objective is to maximize the trust of the users of all social media platforms on common knowledge.
Therefore, the government selects an action $a_0\in\mathcal{A} = [0, 1]$ that designates a lower bound which must be satisfied by the aggregate average trust of all social media platforms in $\mathcal{I}$. To this end, we refer to the action $a_0$ as the government's lower bound on trust on common knowledge.
\color{black}
Let $N_i \in \mathbb{N}$ be the total number of users of the social media platform $i \in \mathcal{I}$. Then, the fraction of the number of users of $i$ with respect to the total number of users of all {platforms} is
\begin{equation}\label{eqn:n_i}
n_i = \frac{N_i}{\sum_{l \in \mathcal{I}} N_l}.
\end{equation}
The fraction $n_i$ represents the contribution of users in platform $i$ on the average trust on common knowledge about democracy. \textcolor{black}{Since $\sum_{i \in \mathcal{I}} n_i = 1$, the aggregate average trust on common knowledge is $\sum_{i \in \mathcal{I}} n_i \cdot h_i(a_i)$.
In our framework, the government's role is to select the lower-bound $a_0$ for the aggregate average trust.}
After the government decides on $a_0$, each {platform} $i \in \mathcal{I}$ who decides to participate in the game must select a filter $a_i$ that satisfies the following constraint:
\begin{equation}\label{eqn:a_0-constraint}
a_0 - \sum_{i \in \mathcal{I}} n_i \cdot h_i(a_i) \leq 0.
\end{equation}
Next, we define the government's valuation as a function of the lower bound on trust $a_0$.
\color{black}
\begin{definition}
The \textit{valuation function} of the government is $v_0(a_0) : [0, 1] \to \mathbb{R}_{\geq 0}$, and it is an increasing function with respect to the lower bound $a_0$.
\end{definition}
The government's valuation function $v_0(a_0)$ assigns a monetary value on the lower bound $a_0$. Recall that the government seeks to increase the trust on common knowledge among the users of all social media platforms. Thus, the government's valuation increases as the lower bound on aggregate average trust increases.
\color{black}
We also consider that the government might have limited resources available to invest in this problem, i.e., there exists a finite monetary budget $b_0 \in \mathbb{R}_{\geq 0}$
\color{black}
representing the maximum possible expenditure of the government for this problem.
\subsection{Information Structure}
\color{black}
\textcolor{black}{In this subsection, we specify the private and public information structure corresponding to each player in the imposed game.}
\textcolor{black}{\textit{1) Public information:}} The set of competing platforms $\mathcal{C}_i$ and fraction of users $n_i$ of each platform $i \in \mathcal{I}$ are known to all players in set $\mathcal{J}$. \textcolor{black}{Moreover, the set of feasible actions $\mathcal{A}$ is known to all players in the set $\mathcal{J}$.}
\textcolor{black}{\textit{2) Valuation functions:}} The valuation function $v_i(\cdot)$ of each social media platform $i \in \mathcal{I}$ is considered private information, and thus, it is known only to platform $i$. Similarly, the valuation function $v_0(\cdot)$ and the budget $b_0$ of the government are private information of the government.
\textcolor{black}{\textit{3) Average trust functions:}} The average trust function $h_i(\cdot)$ of social media platform $i \in \mathcal{I}$ is considered private information, and thus, it is known only to platform $i$ (it is not known to the government).
\color{black}
\subsection{Assumptions}
\color{black}
In the modeling framework presented above, we impose the following assumptions:
\begin{assumption} \label{assumption:cardinality}
For each platform $i \in \mathcal{I}$, \ $|\mathcal{C}_i| \geq 3$.
\end{assumption}
We impose this assumption to simplify the exposition of our mechanism. Assumption \ref{assumption:cardinality} implies that each user subscribes in multiple social media platforms.
\textcolor{black}{It has been shown in the literature that each user, on an average, subscribes to 8 social media platforms \cite{datareportal2020}.
Nevertheless,} we present an extension of our mechanism for $|\mathcal{C}_{i}| \geq 2$ in Appendix A.
\color{black}
\begin{assumption}\label{assumption:filter_compatibility}
The valuation function $v_i\big(a_k : k \in \mathcal{C}_i \big) : \mathcal{A}^{|\mathcal{C}_i|} \to \mathbb{R}_{\geq 0}$ of each social media platform $i \in \mathcal{I}$ is a concave and differentiable function with respect to $a_k$
\end{assumption}
\color{black}
The concavity of $v_i\big(a_k : k \in \mathcal{C}_i \big)$ captures the diminishing marginal change in engagement due to additional filtering. \textcolor{black}{Practically, the higher the value of $a_i$, the more users of platform $i$ will perceive the filter as censorship of their opinions.
Thus, for platform $i$, increasing a low-value filter may lead to a lesser loss in engagement as compared to increasing a filter whose value is already high.}
\textcolor{black}{Nevertheless, to ensure the robustness of our proposed mechanism,} we also present an analysis of our system by relaxing Assumption \ref{assumption:filter_compatibility} in Section IV-A.
\color{black}
\begin{assumption}\label{assumption:average_trust}
The average trust function $h_i(a_i) : \mathcal{A} \to [0, 1]$ of each social media platform $i \in \mathcal{I}$ is a concave and differentiable function with respect to $a_i$.
\end{assumption}
\color{black}
\textcolor{black}{The concavity of $h_i(a_i)$ implies that, for large values of $a_i$, a small incremental change of $a_i$ would not have a significant impact on the average trust of users on common knowledge. Practically, this implies low values of $a_i$ will have a major impact on the average trust function. Nevertheless, to ensure the robustness of our mechanism, we also present an analysis of our system by relaxing Assumption \ref{assumption:average_trust} in Section IV-A.}
\color{black}
\begin{assumption}\label{assumption:government_valuation}
The valuation function of the government $v_0(a_0) : [0, 1] \to \mathbb{R}_{\geq 0}$ is a concave and differentiable function with respect to the lower-bound $a_0$.
\end{assumption}
\color{black}
\textcolor{black}{Practically, for high values of $a_0$, the government might not be interested in investing additional resources to increase $a_0$ even more, as the impact on improving common knowledge would not be significant. Nevertheless,}
we also present an analysis of our system by relaxing Assumption \ref{assumption:government_valuation} in Section IV-A.
\begin{assumption} \label{assumption:assumed_knowledge_b}
The output of the function $h_i(a_i)$ can be monitored by any competing platform $l\in\mathcal{C}_{- i}$, and a violation of the condition \eqref{eqn:a_0-constraint} can be detected by the government.
\end{assumption}
\textcolor{black}{Assumption \ref{assumption:assumed_knowledge_b} helps us enforce the mechanism, which we present in Section \ref{section:md_problem}, in a static environment.
In the mechanism, each platform $i \in \mathcal{I}$ commits to a minimum value of the average trust function among their users which can be achieved by choosing an appropriate value for $a_i$. Consider that a platform $i$ selects a value $a_i$ that fails to satisfy this commitment. Practically, the government can detect a violation of \eqref{eqn:a_0-constraint} by gauging public opinion on the internet and through surveys. However, the government does not know the function $h_i(\cdot)$ of platform $i$, and thus, would penalize each platform in $\mathcal{I}$ equally for the violation of \eqref{eqn:a_0-constraint}. To avoid the penalty for the failure of platform $i$, a competing platform $l \in \mathcal{C}_{-i}$ can report the violation of $i$.
Thus, it is reasonable to consider that each platform $i \in \mathcal{I}$ monitors the output $h_l(a_l)$ of each competing platform $l \in \mathcal{C}_{-i}$ to maximize their own utility.}
We believe that using a dynamic mechanism, we could potentially relax Assumption \ref{assumption:assumed_knowledge_b} \cite{zhang2019efficient}. This would be a potential direction for future research.
\begin{assumption} \label{assumption:excludability}
The government ensures that any social media platform $i \in \mathcal{I}$ that does not participate in the mechanism receives no benefits from the filters of participating social media.
\end{assumption}
In static mechanisms, the ability to exclude a player from receiving benefits of some common resource is a necessary condition for voluntary participation of players without any monetary investment \cite{saijo2010fundamental}. This condition is often assumed implicitly in the literature \cite{sharma2012local,sinha2013,jain2010,kakhbod2011efficient}. In our mechanism, the government can make an investment up to the budget $b_0$. \textcolor{black}{Thus, we assume \textit{partial excludability} in Assumption \ref{assumption:excludability}, where a non-participating platform $i$ still receives the maximum valuation for selecting filter $a_i = 0$, but cannot receive benefits from the filters of any participating platforms.
In practice, the government can publicize that platform $i$ has chosen not to contribute in a collective endeavor to filter misinformation.
The resulting loss in credibility among the users of the platforms that participate will minimize their migration to platform $i$. This assumption may be relaxed using a dynamic mechanism, which could be another direction for future research \cite{farhadi2019}.}
\subsection{Problem Statement}
\textcolor{black}{Since there is a conflict of interest between the government and the social media platforms, we consider that the government} hires a social planner to design a mechanism \textcolor{black}{to impose the misinformation filtering game. The mechanism must serve two purposes: (i) incentivize all platforms to voluntarily participate in the game, and (ii) induce a selection of filters that maximizes the \textit{social welfare} of the system. The social welfare of the system is the sum of utilities of all players, formally defined next.
To meet these objectives, the social planner asks each player $i \in \mathcal{J}$ to send a message $m_i$ from a set of feasible messages $\mathcal{M}_i$. Based on the message profile $m = (m_0, m_1, \dots, m_{|\mathcal{I}|})$, the social planner assigns a tax $\textcolor{black}{\tau}_i(m) \in \mathbb{R}$ for each social media platform $i \in \mathcal{I}$, and an investment $\textcolor{black}{\tau}_0(m) \in \mathbb{R}_{\geq 0}$ for the government. The message and tax of each player is formally defined in Section III-B. By convention, a tax $\textcolor{black}{\tau}_i(m) > 0$ is a payment made by player $i \in \mathcal{J}$, and a tax $\textcolor{black}{\tau}_i(m) < 0$ is a subsidy given to player $i$.}
Thus, the taxes of the platforms can be either monetary payments or subsidies, whereas, the government may never collect a monetary subsidy from any platform.
\color{black}
Note that the social planner must not receive any profit, nor incur any losses, for designing and implementing the mechanism, which implies that the mechanism should be budget balanced, i.e., $\sum_{i \in \mathcal{J}} \textcolor{black}{\tau}_i(m) = 0$.
Next, we define the utilities of the players.
\begin{definition}
The \textit{utility} of each platform $i \in \mathcal{I}$ is given by $u_i\big(m , a_k: k \in \mathcal{C}_i\big) := v_i\big(a_k: k \in \mathcal{C}_i\big) - \textcolor{black}{\tau}_i(m),$
while the utility of the government is given by $u_0(m, a_0) := v_0(a_0) - \textcolor{black}{\tau}_0(m).$
\end{definition}
The social welfare is $u_0(m,a_0) + \sum_{i \in \mathcal{I}} u_i(m, a_k:k\in \mathcal{C}_i)$. The optimization problem for the social planner is to maximize the social welfare, and it is formulated as follows.
\begin{problem}\label{problem1}
\begin{align}
\max_{a, \textcolor{black}{\tau}(m)} \bigg(v_0(a_0) &- \textcolor{black}{\tau}_0(m) + \sum_{i \in \mathcal{I}} \Big(v_i\big(a_k:k \in \mathcal{C}_i\big) - \textcolor{black}{\tau}_i(m)\Big)\bigg), \label{eqn:objective_1} \\
\text{subject to:} \; & 0 \leq a_i \leq 1, \quad \forall i \in \mathcal{J} \label{eqn:constraint_1st}, \\
& a_0 - \sum_{i \in \mathcal{I}} n_i \cdot h_i(a_i) \leq 0, \label{eqn:constraint_2nd} \\
& 0 \leq \textcolor{black}{\tau}_0(m) \leq b_0, \label{eqn:constraint_tax_1st} \\
& \sum_{i \in \mathcal{J}} \textcolor{black}{\tau}_i(m) = 0, \label{eqn:constraint_tax_2nd}
\end{align}
where $a = \big(a_0, a_1 \dots, a_I\big)$ and $\textcolor{black}{\tau}(m) = \big(\textcolor{black}{\tau}_0(m), \textcolor{black}{\tau}_1(m),$ $\dots, \textcolor{black}{\tau}_{|\mathcal{I}|}(m)\big)$ denote the action and tax profiles of all players, respectively.
\end{problem}
In Problem \ref{problem1}, \eqref{eqn:constraint_2nd} ensures that the aggregate average trust of all users satisfies the government's lower bound $a_0$, \eqref{eqn:constraint_tax_1st}
restricts the government's investment $\textcolor{black}{\tau}_0(m)$ to be within the available budget, and \eqref{eqn:constraint_tax_2nd} ensures that the mechanism is budget balanced.
Note that, in Problem \ref{problem1}, the social planner does not have knowledge about the functional form of either the valuation function $v_i(\cdot)$ of any player $i \in \mathcal{J}$, or the average trust function $h_i(\cdot)$ of any platform $i \in \mathcal{I}$.
If the social planner knew these functions, then she could solve Problem \ref{problem1} using standard optimization methods to allocate the optimal filter $a_i$ and tax $\textcolor{black}{\tau}_i(m)$ to each platform $i \in \mathcal{I}$, and the optimal lower bound $a_0$ and investment $\textcolor{black}{\tau}_0(m)$ to the government. The objective function of Problem \ref{problem1} is differentiable and concave, and the set of feasible solutions is non-empty, convex, and compact. Thus, Problem \ref{problem1} is a convex optimization problem with a unique optimal solution \cite{boyd}.
However, this solution cannot be computed directly by the social planner because of the private information of the players. Note that if the social planner simply asks the players to report their private information, then the players may not be truthful. Thus, the social planner seeks to design the taxes $\textcolor{black}{\tau}_i(m)$ for each player $i \in \mathcal{J}$ to incentivize the players to be truthful while, at the same time, maximizing the social welfare.
\begin{remark}
The government has a no compelling reason to misreport to the social planner their budget $b_0$. Thus, we consider that the social planner has knowledge of $b_0$.
\end{remark}
\begin{remark}
By maximizing the social welfare $u_0(m,a_0) + \sum_{i \in \mathcal{I}}u_i(m,a_k:k\in \mathcal{C}_i)$ in Problem \ref{problem1}, the utility of each player is maximized. Hence, participation of the players in the mechanism is incentivized. Note that the government is not in the position to design the mechanism because they would seek to optimize only their own utility $u_0(m,a_0)$. Thus, the government hires the social planner to design and implement the mechanism described next.
\end{remark}
\color{black}
\section{Mechanism Design Approach} \label{section:md_problem}
\color{black}
In this section, we present a two-step mechanism to incentivize filtering of misinformation among social media platforms. The objective of the first step is to ensure that the social media platforms voluntarily agree to participate in the mechanism. The objectives of the second step are to: (i) extract truthful information from the participating platforms, (ii) derive the optimal level of investment for the government, and (iii) design appropriate taxes for the platforms to maximize the social welfare of the system.
\subsection{Step One - The Participation Step}
\color{black}
In step one of the mechanism, each social media platform $i \in \mathcal{I}$ must decide whether to participate in the mechanism, with complete knowledge of the rules of the second step of the mechanism described in the next subsection. Consider a platform $i \in \mathcal{I}$ that chooses not to participate in the mechanism. Thus, this platform neither pays taxes nor receives any subsidies from the government, i.e., $\textcolor{black}{\textcolor{black}{\tau}_i(m)} = 0$. Furthermore, platform $i$ is free to select the lowest value of $a_i = 0$ that maximizes the valuation $v_i\big(a_k:k\in\mathcal{C}_i\big)$. Meanwhile, another competing platform $l \in \mathcal{C}_{- i}$ may decide to participate in the mechanism and subsequently implement a non-zero filter $a_l$. \textcolor{black}{From Assumption \ref{assumption:excludability}, the government ensures that platform $l$ receives no utility as a result of filter $a_l$. Thus, the utility of the non-participating platform $i$ is given by $v_i\big(a_k = 0: k \in \mathcal{C}_i\big)$. We will use this utility for a non-participating platform in Theorem \ref{thm:ir} of Section IV to establish that all platforms decide to voluntarily participate in step one of the mechanism.}
\color{black}
\subsection{Step Two - The Bargaining Step}
\color{black}
\textcolor{black}{In step two, the social planner asks each player $i \in \mathcal{J}$ to broadcast a message $m_i$ from a set of feasible messages $\mathcal{M}_i$.} For each platform $i \in \mathcal{I}$, let $\mathcal{D}_{i} = \mathcal{C}_i \cup \{0\}$, and $\mathcal{D}_{-i} = \mathcal{D}_i \setminus \{i\}$.
The message of platform $i$ is defined as
\begin{equation}\label{eqn:defn_message}
m_i := (\textcolor{black}{\Tilde{h}_i}, \textcolor{black}{\Tilde{p}_i}, \Tilde{a}_i),
\end{equation}
where
$\textcolor{black}{\Tilde{h}_i} \in \mathbb{R}_{\geq 0}$ is the minimum average trust that platform $i$ proposes to achieve through filtering; $\textcolor{black}{\Tilde{p}_i} \in \mathbb{R}_{\geq 0} ^ {|\mathcal{D}_{-i}|}$ is the collection of prices that platform $i$ is willing to pay or receive per unit changes in the filters of other competing platforms \textcolor{black}{(except $i$)} and \textcolor{black}{the government's lower bound}, given by
\begin{equation}\label{eqn:prices}
\textcolor{black}{\Tilde{p}_i} := (\textcolor{black}{\Tilde{p}_l} ^ i : l \in \mathcal{D}_{-i});
\end{equation}
and $\Tilde{a}_i = (\Tilde{a}_k ^ i: k \in \mathcal{D}_i),$ $\Tilde{a}_i \in \mathbb{R} ^ {|\mathcal{D}_{i}|},$ is the profile of filters for all competing platforms \textcolor{black}{(including $i$)} and \textcolor{black}{government's lower bound proposed by platform $i$.}
\color{black}
\begin{remark}
Note that each platform proposes a filter for themselves, denoted by $\Tilde{a}_i^i$, in their message $m_i$. However, as it can be seen in \eqref{eqn:prices}, platform $i$ does not propose a price corresponding to $\Tilde{a}_i^i$. This is because we want to give every platform the ability to influence their filter, but not the ability to influence the price associated with their own filter.
\end{remark}
\color{black}
The message of the government is $m_0 := (\textcolor{black}{\Tilde{p}_0}, \Tilde{a}_0 ^ 0)$, where $\textcolor{black}{\Tilde{p}_0} \in \mathbb{R}_{\geq 0}$ is the price that the government is willing to pay or receive per unit change of the average trust, and $\Tilde{a}_0^0 \in \mathbb{R}$ is the \textcolor{black}{lower bound} proposed by the government. Note that our mechanism respects the privacy of each platform $i \in \mathcal{I}$ since she does not request either their valuation function $v_i\big(a_k : k \in \mathcal{C}_i \big)$ or their average trust function $h_i(a_i)$. Similarly, the government is not forced to publicly reveal the functional form of their valuation function $v_0(a_0)$. \textcolor{black}{Also each platform $i$ is free to select any feasible values for the components of the message $m_i$.}
\textcolor{black}{Based on the message profile $m := (m_0, m_1,$ $\dots, m_{|\mathcal{I}|})$ that the social planner receives, she allocates the following parameters to the players:}
\textcolor{black}{\textit{1)} The social planner allocates a filter to each platform $i \in \mathcal{I}$ and a lower bound to the government such that the constraints of Problem $1$ are satisfied.}
The filter allocated by the social planner to platform $i$ is $\textcolor{black}{\alpha}_i(m) := \sum_{k \in \mathcal{C}_i} \frac{\Tilde{a}_i ^ k}{|\mathcal{C}_i|}$, i.e., the average of the filters proposed by all competing platforms including $i$. The lower bound allocated by the social planner to the government is $\textcolor{black}{\alpha}_0(m) = \sum_{k \in \mathcal{J}} \frac{\Tilde{a}_0 ^ k}{|\mathcal{J}|}$, i.e., the average of the lower bounds proposed by all platforms and the government.
\textcolor{black}{\textit{2)}} The social planner allocates a minimum average trust {$\textcolor{black}{\eta_i}(m) \in [0,1]$} to each platform $i \in \mathcal{I}$, given by
\begin{equation}\label{eqn:defn_of_eta}
\textcolor{black}{\eta_i}(m) = \min\left\{ \frac{n_i \cdot \textcolor{black}{\Tilde{h}_i}}{\sum_{k \in \mathcal{I}} n_k \cdot \textcolor{black}{\Tilde{h}_k}} \cdot \textcolor{black}{\alpha}_0(m), \; 1\right\},
\end{equation}
where the social planner will not accept a message $m_i$ from a platform $i$ that might lead to a situation where $\sum_{k \in \mathcal{I}} n_k \cdot \textcolor{black}{\Tilde{h}_k} = 0$.
\textcolor{black}{The allocated minimum average trust, $\eta_i(m)$, is a lower bound on average trust that must be achieved by platform $i$.}
Let the filter implemented by platform $i$ be $a_i$. Then, platform $i$ must ensure that $n_i \cdot h_i(a_i)\geq \textcolor{black}{\eta_i}(m)$.
Recall \textcolor{black}{from the information structure} that a potential violation of this condition cannot be detected by the social planner since she does not have explicit knowledge of the function $h_i(\cdot)$. However, by Assumption \ref{assumption:assumed_knowledge_b}, the output of $h_i(a_i)$ can be monitored by any other competing platform $l \in \mathcal{C}_{- i}$.
Any violation of $n_i \cdot h_i(a_i)\geq \textcolor{black}{\eta_i}(m)$ will be reported by platform $l$ to the social planner, in order to ensure that platform $i$ implements the largest filter $a_i,$ and maximizes the utility $u_l(m, a_k:k\in \mathcal{C}_l)$. \textcolor{black}{This prevents platforms from violating the constraint imposed by the allocated minimum average trust $\eta_i(m)$.}
\textcolor{black}{\textit{3)}} The social planner allocates a price $\textcolor{black}{\pi_l} ^ i := \sum_{k \in \mathcal{C}_{- l} : k \neq i} \frac{\textcolor{black}{\Tilde{p}_l} ^ k}{|\mathcal{C}_{l}| - 2}$, $\textcolor{black}{\pi^i_l} \in \mathbb{R}_{\geq 0},$ to each platform $i \in \mathcal{I}$, corresponding to the allocated filter $\textcolor{black}{\alpha}_l(m)$ of every other competing platform $l \in \mathcal{C}_{- i}$. This price is derived as the average of prices proposed for the allocated filter $\textcolor{black}{\alpha}_l(m)$ by all competing platforms in $\mathcal{C}_{-l}$ except $i$. Thus, the allocated price $\textcolor{black}{\pi_l}^i$ is independent of the prices proposed by both platforms $i$ and $l$. \textcolor{black}{Similarly, the social planner allocates the price $\pi_0 = \sum_{i \in \mathcal{I}} \frac{\textcolor{black}{\Tilde{p}^i_0}}{|\mathcal{I}|}$ to the government.
Note that even though the prices allocated to each player
depend on the message profile $m$, we do not present them with the argument of $m$ to simplify our notation and improve the readability of the subsequent equations.}
\textcolor{black}{\textit{4)}} The social planner \textcolor{black}{allocates} the following tax to each social media platform $i \in \mathcal{I}$,
\begin{gather}
\textcolor{black}{\tau}_i(m) := - \textcolor{black}{\Tilde{p}_0} \cdot \textcolor{black}{\eta_i}(m) - \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_i} ^ l \cdot \textcolor{black}{\alpha}_{i}(m)
+ \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_l} ^ i \cdot \textcolor{black}{\alpha}_{l}(m) \nonumber \\
+ \sum_{l \in \mathcal{C}_{- i} \cup \{0\}} \textcolor{black}{\Tilde{p}_l} ^ i \cdot (\Tilde{a}_l ^ i - \Tilde{a}_l ^ {- i}) ^ 2, \label{eqn:payment_function}
\end{gather}
where $\Tilde{a}_l ^ {- i} = \sum_{k \in \mathcal{C}_{l}:k \neq i} \frac{\Tilde{a}_l ^ k}{|\mathcal{C}_{l}| - 1}$, for each $l \in \mathcal{C}_{- i}$,
is the average of the proposed filters for $l$ by all competing platforms except $i\in\mathcal{I}$, \textcolor{black}{and $\Tilde{a}_0 ^ {- i} = \sum_{k \in \mathcal{J}_{-i}} \frac{\Tilde{a}_0 ^ k}{|\mathcal{J}| - 1}$ is the average of lower bounds proposed by all players except $i$.} The tax $\textcolor{black}{\tau}_i(m)$ of platform $i$ in \eqref{eqn:payment_function} can be \textcolor{black}{interpreted as follows:
(i) the first term in \eqref{eqn:payment_function} represents a subsidy given by the government to platform $i$ for the increase in average trust among the users of platform $i$;
(ii) the second term in \eqref{eqn:payment_function} is a collection of subsidies given by each competing platform $l \in \mathcal{C}_{- i}$ to platform $i$ for the increase in valuation $v_l\big(a_k: k \in \mathcal{C}_l\big)$ due to the allocated filter $\textcolor{black}{\alpha}_i$; (iii) the third term in \eqref{eqn:payment_function} is a payment by platform $i$ for the increase in valuation $v_i\big(a_k:k \in \mathcal{C}_i\big)$ due to the allocated filter $\alpha_l$ of each competing platform $l \in \mathcal{C}_{- i}$; and
(iv) the fourth term in \eqref{eqn:payment_function} is a collection of penalties to platform $i$ \textcolor{black}{if either the filter proposed in message $m_i$ for any competing platform $l \in \mathcal{C}_{- i}$ is inconsistent} with the filters proposed by other platforms, \textcolor{black}{or if the lower bound proposed in $m_i$ is inconsistent with the lower bound proposed by other players.}} Note that the fourth term also penalizes platform $i$ for higher values of proposed prices $\textcolor{black}{\Tilde{p}_l} ^ i$
and thus, ensures that the platform $i$ proposes lower prices for the actions of other players.
Finally, the social planner \textcolor{black}{allocates the following investment to the government:}
\begin{equation}\label{eqn:payment_function_gov}
\textcolor{black}{\tau}_0(m) = \pi_0 \cdot \textcolor{black}{\alpha}_0(m) + (\textcolor{black}{\Tilde{p}_0} - \pi_0) ^ 2,
\end{equation}
\textcolor{black}{where} the first term is the total investment made by the government for the allocated low bound $\textcolor{black}{\alpha}_0(m)$, and the second term is a penalty when the price proposed by the government deviates from the price allocated to the government.
\color{black}
\begin{remark}
Note that in \eqref{eqn:payment_function}, for some filter $a_i>0$ of platform $i$, the social planner takes a payment from each competing platform $l \in \mathcal{C}_{-i}$ and allocates an equal subsidy to platform $i$. This subsidy serves a dual purpose: (i) it incentivizes platform $i$ to implement the filter $a_i$, and (ii) it eventually leads to a fair distribution of the government's investment among all platforms.
\end{remark}
\begin{remark}
We presented the step two of the mechanism under the implicit assumption that all social media platforms participate in the mechanism. This does not cause any implications, however, since, as we prove in Theorem \ref{thm:ir} next, all platforms eventually, indeed, participate in the mechanism in step one.
\end{remark}
\color{black}
\textcolor{black}{The step two of the mechanism is characterized by the tuple} $\langle \mathcal{M}, g(\cdot) \rangle$, where $\mathcal{M} = \mathcal{M}_0 \times \mathcal{M}_1 \times \dots \times \mathcal{M}_{|\mathcal{I}|}$ is the complete message space of all \textcolor{black}{players}, and $g(\cdot) : \mathcal{M} \to \mathcal{O}$ is the outcome function that maps each message profile to a set of outcomes $\mathcal{O}$. The set of outcomes is in the form
\begin{multline}
\mathcal{O} := \Big\{\big(\textcolor{black}{\alpha}_0(m), \textcolor{black}{\alpha}_1(m), \dots, \textcolor{black}{\alpha}_{|\mathcal{I}|}(m)\big), \big(\textcolor{black}{\tau}_0(m), \textcolor{black}{\tau}_1(m), \\
\dots, \textcolor{black}{\tau}_{|\mathcal{I}|}(m)\big) : \textcolor{black}{\alpha}_i(m) \in \mathcal{A}, \; \textcolor{black}{\tau}_i(m) \in \mathbb{R}, \; i\in\mathcal{J}\Big\},
\end{multline}
and the outcome function $g(m)$ determines the outcome of any given message profile $m = (m_0, m_1, \dots, m_I) \in \mathcal{M}$.
\subsection{Generalized Nash Equilibrium and the Induced Game}
Formally, a mechanism $\langle \mathcal{M}, g(\cdot) \rangle$ together with the utility functions $(u_i)_{i \in \mathcal{I}}$ induces a game in which \textcolor{black}{the social planner allocates the filters $(\textcolor{black}{\alpha}_1(m), \dots, \textcolor{black}{\alpha}_i(m))$ to the platforms and the lower bound $\textcolor{black}{\alpha}_0(m)$ to the government.} Each platform $i \in \mathcal{I}$ that participates in the mechanism must implement the filter $a_i = \textcolor{black}{\alpha}_i(m)$, and the government must select the lower bound $a_0 = \textcolor{black}{\alpha}_0(m)$.
\textcolor{black}{Note that platform $i$ can influence their allocated filter $\alpha_i(m)$ with their message $m_i$.} Thus, the strategy of platform $i$ in the induced game is given by the message $m_i \in \mathcal{M}_i$ \cite{mas_colell1995}, with a constraint that $\textcolor{black}{\alpha}_i(m) \in \mathcal{S}_i(m)$, where
\begin{equation}
\mathcal{S}_i(m) = \{a_i \in \mathcal{A} : n_i \cdot h_i(a_i) \geq \textcolor{black}{\eta_i}(m)\}.
\end{equation}
Thus, the set of feasible allocations $\mathcal{S}_i(m)$ for $i\in \mathcal{I}$ is a function of the messages of all social media in $\mathcal{I}$ and the government. The strategy of the government is denoted by the message $m_0$ and the set of feasible strategies is given by $\mathcal{M}_0$. For such a game, we select the solution concept of the generalized Nash equilibrium (GNE) \cite{facchinei2010generalized}. Let $m_{- i} = (m_0, \dots, m_{i - 1}, m_{i + 1}, \dots, m_{I})$.
A message profile $m ^ * = (m_i ^ * : i \in \mathcal{J})$ is the GNE of the induced game, if (i) for each $i \in \mathcal{I}$,
\color{black}
\begin{multline} \label{eqn:GNE_defn}
u_i\big((m_i ^ *, m_{- i} ^ *), \textcolor{black}{\alpha}_k(m_i ^ *, m_{- i} ^ *) : k \in \mathcal{C}_i\big) \\
\geq u_i\big((m_i, m_{- i} ^ *), \textcolor{black}{\alpha}_k(m_i, m_{- i} ^ *) : k \in \mathcal{C}_i\big),
\end{multline}
\color{black}
for all $m_i \in \mathcal{M}_i$ and $\textcolor{black}{\alpha}_i \in \mathcal{S}_i(m)$; and (ii) the message $m_0 ^ *$ of the government is such that $u_0\big((m_0 ^ *, m_{- 0} ^ *), \textcolor{black}{\alpha}_0(m_0 ^ *, m_{- 0} ^ *)\big) \geq u_0\big((m_0, m_{- 0} ^ *), \textcolor{black}{\alpha}_0(m_0, m_{- 0} ^ *)\big),$ for all $m_0 \in \mathcal{M}_0$.
\color{black}
To simplify the notation, in the remaining of the paper, we denote the utility of platform $i \in \mathcal{I}$ by $u_i(m_i,m_{-i})$ and the utility of the government by $u_0(m_0, m_{-0})$.
\begin{remark}
In general, the GNE solution concept is defined for a game with complete information. However, we adopt this solution in our induced game despite the fact that the valuation function $v_i\big(a_k : k \in \mathcal{C}_i\big)$ and the average trust function $h_i(a_i)$ are the private information of platform $i$. We resolve this discrepancy by considering that the induced game is played repeatedly over multiple iterations, and thus, the social media platforms can utilize an iterative learning process to find a GNE. This interpretation of a GNE is consistent with the theory of mechanism design \cite{groves_b}.
\end{remark}
\subsection{Summary of the Notation}
We summarize the variables introduced in Sections II and III in Table \ref{summary_table}. As a general guideline, we use lowercase letters of the English alphabet to denote variables and functions, lowercase letters with tilde to denote variables in a message, and lowercase letters of the Greek alphabet to indicate variables allocated to the players by the social planner. We use scripted letters to denote sets.
Furthermore, we use $\mathbb{R}$ to denote the set of real numbers, $\mathbb{R}_{\geq0}$ to denote the set of non-negative real numbers, and $\mathbb{N}$ to denote the set of natural numbers.
\begin{table}[t]
\color{black}
\caption{A summary of the key variables}
\label{summary_table}
\begin{tabular}{ p{30pt} p{195pt} }
\hline
\textbf{Symbol} & \multicolumn{1}{c}{\textbf{Explanation}}\\[3pt]
\hline
$m_i$ &The message broadcast by player $i \in \mathcal{J}$ \\ [3pt]
$a_i$ & The filter of platform $i \in \mathcal{I}$\\[3pt]
$\Tilde{a}^i_k$ & The filter proposed by platform $i \in \mathcal{I}$ for platform $k \in \mathcal{C}_i$ \\[3pt]
%
$\alpha_i(m)$ & The filter allocated to platform $i \in \mathcal{I}$\\[3pt]
$a_0$ & The government's lower bound on trust \\[3pt]
%
$\Tilde{a}_0$ & The lower bound proposed by the government \\[3pt]
\multirow{2}{*}{$\Tilde{a}^i_0$} & The lower bound proposed by platform $i \in \mathcal{I}$ for the government \\[3pt]
%
$\alpha_0(m)$ & The lower bound allocated to the government \\ [3pt]
%
$v_i(\cdot)$ & The valuation function of player $i \in \mathcal{J}$ \\ [3pt]
%
$h_i(\cdot)$ & The average trust function of platform $i \in \mathcal{I}$ \\ [3pt]
%
$\Tilde{h}_i$ & The proposed minimum average trust of platform $i \in \mathcal{I}$ \\ [3pt]
%
$\eta_i(m)$ & The allocated minimum average trust for platform $i \in \mathcal{I}$ \\ [3pt]
%
\multirow{2}{*}{$\Tilde{p}^i_l$} & The price proposed by platform $i \in \mathcal{I}$ corresponding to player $l \in \mathcal{D}_{-i}$ \\ [3pt]
%
\multirow{2}{*}{$\pi^i_l$} & The price allocated to platform $i \in \mathcal{I}$ corresponding to player $l \in \mathcal{D}_{-i}$ \\ [3pt]
$\Tilde{p}_0$ & The price proposed by the government \\ [3pt]
%
$\pi_0$ & The price allocated to the government \\ [3pt]
%
%
$\textcolor{black}{\tau}_i(m)$ &The tax allocated to player $i \in \mathcal{J}$\\
\hline
\end{tabular}
\color{black}
\end{table}
\color{black}
\section{Properties of the Mechanism} \label{section:properties_of_mechanism}
In this section, we show that our proposed mechanism has the following desirable properties: (i) budget balance at GNE, (ii) feasibility at GNE, (iii) strong implementation, (iv) existence of at least one GNE, and (v) individual rationality.
\textcolor{black}{Recall that each social media platform $i \in \mathcal{I}$ is a strategic player who seeks to maximize their utility $u_i(m_i, m_{-i})$ through the choice of message $m_i \in \mathcal{M}_i$. Thus, we can define the following optimization problem from the perspective of platform $i \in \mathcal{I}$ in the induced game.}
\begin{problem}\label{problem2}
The optimization problem for social media platform $i \in \mathcal{I}$ in the induced game is
\begin{align}
\max_{m_i \in \mathcal{M}_i} \; & v_i\big(\textcolor{black}{\alpha}_k(m) : k \in \mathcal{C}_{-i}\big) - \textcolor{black}{\tau}_i(m), \label{eqn:objective_nash} \\
\text{subject to: } & 0 \leq \textcolor{black}{\alpha}_i(m) \leq 1, \label{eqn:constraint_nash_1st} \\
& \textcolor{black}{\eta_i}(m) - n_i \cdot h_i\big(\textcolor{black}{\alpha}_i(m)\big) \leq 0, \label{eqn:constraint_nash_2nd}
\end{align}
\textcolor{black}{where the objective function in \eqref{eqn:objective_nash} is the utility $u_i(m_i, m_{-i})$ of platform $i$,}
\eqref{eqn:constraint_nash_1st} ensures that the allocated filter of platform $i$ is feasible,
and \eqref{eqn:constraint_nash_2nd} ensures that the fraction of average trust among users of platform $i$ is greater than the minimum average trust allocated by the social planner.
\end{problem}
Note that the social planner can ensure that \eqref{eqn:constraint_nash_1st} and \eqref{eqn:constraint_nash_2nd} are hard constraints by imposing a tax $\textcolor{black}{\tau}_i(m) \rightarrow \infty$ when they are violated. \textcolor{black}{Next, recall that the government is also a strategic player in the induced game who seeks to maximize their utility $u_0(m_0,m_{-0})$ through the choice of message $m_0 \in \mathcal{M}_0$.}
\begin{problem}\label{problem3}
The optimization problem for the government is
\begin{align}
\max_{m_0 \in \mathcal{M}_0} \; & v_0\big(\textcolor{black}{\alpha}_0(m)\big) - \textcolor{black}{\tau}_0(m), \label{eqn:objective_nash_gov} \\
\text{subject to: } & 0 \leq \textcolor{black}{\alpha}_0(m) \leq 1, \label{eqn:constraint_nash_1st_gov} \\
& \pi_0 \cdot \textcolor{black}{\alpha}_0(m) - b_0 \leq 0, \label{eqn:constraint_nash_2nd_gov}
\end{align}
where \textcolor{black}{the objective in \eqref{eqn:objective_nash_gov} is the utility $u_0(m_0, m_{-0})$ of the government,} \eqref{eqn:constraint_nash_1st_gov} ensures that the government's lower bound $a_0$ is feasible,
and \eqref{eqn:constraint_nash_2nd_gov} ensures that the total government's investment is less than their budget $b_0$.
\end{problem}
\color{black}
\begin{remark}
Consider an optimal solution $m_i^* \in \mathcal{M}_i$ of Problem \ref{problem2} for each platform $i \in \mathcal{I}$, and an optimal solution $m_0^* \in \mathcal{M}_0$ of Problem \ref{problem3} for the government. The message profile $m^* = \big(m_0^*, m_1^*, \dots, m_{|\mathcal{I}|}^*\big) \in \mathcal{M}$ satisfies \eqref{eqn:GNE_defn}, and thus, forms a GNE of the induced game.
\end{remark}
\color{black}
\textcolor{black}{Next, we establish some basic properties of the mechanism in Lemmas \ref{lemma:truthful_prices} and \ref{lemma:truthful_action_proposals} at any GNE, if one exists. In Lemma \ref{lemma:truthful_prices}, we refer to Problem \ref{problem3} to show} that the government's proposed price at any GNE of the induced game is equal to the average price proposed by all social media.
\begin{lemma}\label{lemma:truthful_prices}
Let the message profile $m ^ * \in \mathcal{M}$ be a GNE of the induced game. Then, $\textcolor{black}{\Tilde{p}_0} ^ * = \pi_0 ^ *$ for the government.
\end{lemma}
\begin{proof}
Since the objective function in Problem \ref{problem3} is concave with respect to the price $\textcolor{black}{\Tilde{p}_0}$, the price $\textcolor{black}{\Tilde{p}_0} ^ *$ at GNE can be using the equation $\frac{\partial u_0}{\partial \textcolor{black}{\Tilde{p}_0}} \big|_{\textcolor{black}{\Tilde{p}_0} ^ *} =$ $2 \cdot (\textcolor{black}{\Tilde{p}_0} ^ * - \pi_0 ^ *) = 0,$
which yields $\textcolor{black}{\Tilde{p}_0} ^ * = {\pi}_{- 0} ^ *$.
\end{proof}
\textcolor{black}{Similarly, in the next result (Lemma \ref{lemma:truthful_action_proposals}), we refer to Problem \ref{problem2} to establish that, at any GNE, the filters proposed by all social media platforms in $\mathcal{C}_i$ for platform $i$} are the equal, unless the corresponding price proposal is $0$. \textcolor{black}{Furthermore, at every GNE, if one exists, the lower bound proposed by all platforms is the same, unless the corresponding price proposal is $0$.}
\begin{lemma}\label{lemma:truthful_action_proposals}
Let the message profile $m ^ * \in \mathcal{M}$ be a GNE of the induced game. Then, for $\textcolor{black}{\Tilde{p}_k} ^ i \neq 0$, we have $\Tilde{a}_k ^ {i *} = \Tilde{a}_k ^ {- i *}$ for every social media platform $i \in \mathcal{I}$, for every $k \in \mathcal{D}_{-i}$.
\end{lemma}
\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma:truthful_prices}, and thus, \textcolor{black}{due to space limitations, it is omitted.}
\end{proof}
\textcolor{black}{Next, we use the properties established in Lemmas \ref{lemma:truthful_prices} and \ref{lemma:truthful_action_proposals} to show that our proposed mechanism is budget balanced at any GNE, if one exists, i.e., the social planner redistributes all the payments it collects from the players as subsidies to the players.
\begin{theorem}[\textbf{Budget Balance}] \label{budget_balance}
Consider any GNE $m ^ * \in \mathcal{M}$ of the induced game. Then, the proposed mechanism is budget balanced, i.e., $\sum_{i \in \mathcal{J}} \textcolor{black}{\tau}_i(m ^ *) = 0$.
\end{theorem}
\begin{proof}
From Lemmas \ref{lemma:truthful_prices} and \ref{lemma:truthful_action_proposals}, the tax $\textcolor{black}{\tau}_i ^ * = \textcolor{black}{\tau}_i(m ^ *)$ for social media platform $i$ at GNE is $\textcolor{black}{\tau}_i ^ * = - \textcolor{black}{\Tilde{p}_0} ^ * \cdot \textcolor{black}{\eta_i}(m ^ *) - \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_i} ^ l \cdot \textcolor{black}{\alpha}_{i}(m^*) + \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_l} ^ {i} \cdot \textcolor{black}{\alpha}_l(m^*).$ The tax $\textcolor{black}{\tau}_0 ^ *$ for the government at GNE is $\textcolor{black}{\tau}_0 ^ * = \textcolor{black}{\Tilde{p}_0} ^ * \cdot \textcolor{black}{\alpha}_0(m ^ *),$ where $\textcolor{black}{\Tilde{p}_0}^*$ is the price per unit change on average trust at GNE.
Since $\sum_{i \in \mathcal{I}} \textcolor{black}{\eta_i}(m) = \textcolor{black}{\alpha}_0(m),$ for all $m \in \mathcal{M}$, then at GNE we have
$
\sum_{i \in \mathcal{J}} \textcolor{black}{\tau}_i ^ * = \sum_{i \in \mathcal{I}} \Big[- \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_i} ^ l \cdot \textcolor{black}{\alpha}_i(m^*)
+ \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_l} ^ {i} \cdot \textcolor{black}{\alpha}_l(m^*)\Big]=0.
$
\end{proof}
\color{black}
In the next result (Lemma \ref{lemma:feasibility}), we establish that every GNE, $m^* \in \mathcal{M}$, if one exists, of the induced game leads to an allocation of filters for the platforms and a lower bound for the government that forms a feasible solution of Problem \ref{problem1}. In other words, every GNE of the induced game ensures that all constraints of Problem \ref{problem1} are satisfied.
\color{black}
\begin{lemma}[\textbf{Feasibility}] \label{lemma:feasibility}
Every GNE message profile $m ^ * \in \mathcal{M}$ leads to a filter profile $\big(\textcolor{black}{\alpha}_1(m ^ *), \dots, \textcolor{black}{\alpha}_{|\mathcal{I}|}(m ^ *)\big)$ and lower bound $\textcolor{black}{\alpha}_0(m ^ *)$, which is a feasible solution of Problem \ref{problem1}.
\end{lemma}
\begin{proof}
Every GNE message profile $m ^ *$ satisfies \eqref{eqn:constraint_nash_1st} - \eqref{eqn:constraint_nash_2nd} and \eqref{eqn:constraint_nash_1st_gov} - \eqref{eqn:constraint_nash_2nd_gov}. From Theorem \ref{budget_balance}, $\sum_{i \in \mathcal{J}} \textcolor{black}{\tau}_i(m ^ *) = 0$. For each $i \in \mathcal{I}$, $\textcolor{black}{\eta_i}(m) \leq n_i \cdot h_i(\textcolor{black}{\alpha}_i(m)),$ and $\sum_{i \in \mathcal{I}} \textcolor{black}{\eta_i}(m) = \textcolor{black}{\alpha}_0(m)$. Hence, $\sum_{i \in \mathcal{I}} h_i(\textcolor{black}{\alpha}_i(m)) \geq \textcolor{black}{\alpha}_0(m).$
\end{proof}
In the next result (Lemma \ref{lemma:achievability}), we establish that every social media platform $i\in \mathcal{I}$ can unilaterally deviate in the message $m_i \in \mathcal{M}_i$, to achieve any desired allocation of filters for every competing platform, including itself. \textcolor{black}{This property of our mechanism ensures that each platform $i \in \mathcal{I}$ can attain any filter $\hat{a}_i \in \mathcal{A}$, irrespective of the filters proposed by the competing platforms.}
\begin{lemma} \label{lemma:achievability}
Given the message profile $m_{- i} \in \mathcal{M}_{- i}$, the social media platform $i \in \mathcal{I}$ can unilaterally deviate in their message $m_i \in \mathcal{M}_i$ to attain any filter \textcolor{black}{$\hat{a}_k \in \mathcal{A}$ as the allocated filter $\textcolor{black}{\alpha}_k(m) \in \mathcal{S}_k(m)$, for all $k \in \mathcal{C}_{i}$.}
\end{lemma}
\begin{proof}
Let $m_{- i} = \big(m_0, \dots, m_{i-1}, m_{i+1}, \dots, m_{|\mathcal{I}|}\big)$ be the message profile of all players in $\mathcal{J}_{-i}$. Then, platform $i$ can propose a filter $\Tilde{a}^i_k = \hat{a}_k - \sum_{l \in \mathcal{C}_{k} : l \neq i} \frac{\Tilde{a}^l_k}{|\mathcal{C}_{k}| - 1}$,
\color{black}
to ensure that $\textcolor{black}{\alpha}_k(m) = \hat{a}_k$ for each $k \in \mathcal{C}_i$. Moreover, platform $i$ can propose a lower bound $\Tilde{a}^i_0 = - \sum_{l \in \mathcal{J}_{-i}}\Tilde{a}^l_0$ for the government,
to ensure that $\textcolor{black}{\alpha}_0(m) = 0$, and subsequently, $\textcolor{black}{\alpha}_k(m) = \hat{a}_k \in \mathcal{S}_k(m)$ for all $k \in \mathcal{C}_i$.
\end{proof}
Next, we establish that, at any GNE, \textcolor{black}{if one exists,} of the induced game the allocated filters for all platforms and the allocated lower bound for the government result in the optimal solution of Problem \ref{problem1}.
\begin{theorem}[\textbf{Strong Implementation}] \label{thm:implementation}
Consider any GNE $m ^ * \in \mathcal{M}$ of the induced game. Then, the allocated filter profile $\big(\textcolor{black}{\alpha}_1(m ^ *), \dots, \textcolor{black}{\alpha}_{|\mathcal{I}|}(m ^ *)\big)$ and the allocated lower bound $\textcolor{black}{\alpha}_0(m ^ *)$ at equilibrium is equal to the optimal solution $a ^ {*o}$ of Problem \ref{problem1}.
\end{theorem}
\begin{proof}
Let $\textcolor{black}{\alpha}(m ^ *) = \big(\textcolor{black}{\alpha}_1(m ^ *), \dots, \textcolor{black}{\alpha}_{|\mathcal{I}|}(m ^ *)\big)$. Then, the GNE message profile $m ^ *$ satisfies, for platform $i \in \mathcal{I}$, the following Kush-Kahn-Tucker (KKT) conditions for optimality:
\begin{align}
\frac{\partial v_i}{\partial \textcolor{black}{\alpha}_i} \Bigg|_{\textcolor{black}{\alpha}(m ^ *)} + \sum_{l \in \mathcal{I}_{- i}} \textcolor{black}{\pi^i_l} - \lambda_i ^ i + \mu_i ^ i + \nu_i ^ {i} & \cdot \frac{\partial h_i}{\partial \textcolor{black}{\alpha}_i} \Bigg|_{\textcolor{black}{\alpha}(m ^ *)} = 0, \label{eqn:implementation_1} \\
%
\frac{\partial v_i}{\partial \textcolor{black}{\alpha}_l} \Bigg|_{\textcolor{black}{\alpha}(m ^ *)} - \textcolor{black}{\pi_l}^{i} &= 0, \quad \forall l \in \mathcal{C}_{- i}, \label{eqn:implementation_1_2} \\
%
%
\textcolor{black}{\Tilde{p}_0} ^ * - \nu_i ^ i & = 0, \label{eqn:implementation_2} \\
%
\lambda_i ^ i \cdot \big(\textcolor{black}{\alpha}_i(m ^ *) - 1\big) & = 0, \label{eqn:implementation_3} \\
%
\mu_i ^ i \cdot \textcolor{black}{\alpha}_i(m ^ *) & = 0, \label{eqn:implementation_4} \\
%
\nu_i ^ i \cdot \big(\textcolor{black}{\eta_i}(m ^ *) - h_i(\textcolor{black}{\alpha}_i(m ^ *))\big) & = 0, \label{eqn:implementation_5} \\
%
\lambda_i ^ i, \mu_i ^ i, \nu_i ^ i & \geq 0, \label{eqn:implementation_6}
\end{align}
where \eqref{eqn:implementation_1} - \eqref{eqn:implementation_2} are the derivatives of the Lagrangian of platform $i$ with respect to $\textcolor{black}{\alpha}(m)$ and $\textcolor{black}{\eta_i}(m)$, for Problem \ref{problem2}, and \eqref{eqn:implementation_3} - \eqref{eqn:implementation_6} are constraints on the Lagrange multipliers $(\lambda_i^i,\mu_i^i,\nu_i^i)$. From \eqref{eqn:implementation_2}, $\nu_i ^ i = \textcolor{black}{\Tilde{p}_0} ^ *$ for all $i \in \mathcal{I}$.
%
Substituting \eqref{eqn:implementation_1_2} in \eqref{eqn:implementation_1}, we have
\begin{equation}
\sum_{k \in \mathcal{C}_{i}} \frac{\partial v_k}{\partial \textcolor{black}{\alpha}_i} \Bigg|_{\textcolor{black}{\alpha}(m ^ *)} - \lambda_i ^ i + \mu_i ^ i + \nu_i ^ i \cdot \frac{\partial h_i}{\partial \textcolor{black}{\alpha}_i} \Bigg|_{\textcolor{black}{\alpha}(m ^ *)} = 0,
\end{equation}
for all $i \in \mathcal{I}$. Similarly, the KKT conditions for Problem \ref{problem3} are:
\begin{align}
\frac{\partial v_0}{\partial \textcolor{black}{\alpha}_0} \Bigg|_{\textcolor{black}{\alpha}_0(m ^ *)} - \textcolor{black}{\Tilde{p}_0} ^ * - \lambda_0 ^ 0 + \mu_0 ^ 0 + \omega_0 ^ 0 \cdot \textcolor{black}{\Tilde{p}_0} ^ * & = 0, \label{imp_gov_1}\\
%
\lambda_0 ^ 0 \cdot \big(\textcolor{black}{\alpha}_0(m ^ *) - 1\big) & = 0, \label{imp_gov_2} \\
%
\mu_0 ^ 0 \cdot \textcolor{black}{\alpha}_0(m ^ *) & = 0, \label{imp_gov_3}\\
%
\omega_0 ^ 0 \cdot \big(\textcolor{black}{\Tilde{p}_0} ^ * \cdot \textcolor{black}{\alpha}_0(m ^ *) - b_0\big) & = 0, \label{im_gov_4}\\
%
\lambda_0 ^ 0, \mu_0 ^ 0, \omega_0 ^ 0 & \geq 0, \label{imp_gov_5}
\end{align}
where \eqref{imp_gov_1} is the derivative of the Lagrangian, and \eqref{imp_gov_2} - \eqref{imp_gov_5} are constraints on the Lagrange multipliers $(\lambda_0^0, \mu_0^0, \omega_0^0)$.
The optimal solution $a ^ {*o} = \big(a ^ {*o}_0, a ^ {*o}_1, \dots, a ^ {*o}_{|\mathcal{I}|} \big)$ of Problem \ref{problem1} satisfies the following KKT conditions:
\begin{align}
\sum_{k \in \mathcal{C}_{i}} \frac{\partial v_k}{\partial a_i} \Bigg|_{a_i ^ {*o}} - \lambda_i + \mu_i + & \nu \cdot \frac{\partial h_i}{\partial a_i} \Bigg|_{a_i ^ {*o}} = 0, \quad \forall i \in \mathcal{I}, \label{imp_central_1}\\
%
\frac{\partial v_0}{\partial a_0} \Bigg|_{a_0 ^ {*o}} - \lambda_0 + \mu_0 & - \nu - \omega \cdot \pi_0 = 0, \label{imp_central_2}\\
%
\lambda_i \cdot (a_i ^ {*o} - 1) & = 0, \qquad \forall i \in \mathcal{J}, \label{imp_central_3}\\
%
\mu_i \cdot a_i ^ {*o} & = 0, \qquad \forall i \in \mathcal{J}, \label{imp_central_4}\\
%
\nu \cdot \big(a_0 ^ {*o} - h_i(a_i ^ {*o})\big) & = 0, \label{imp_central_5}\\
%
\omega \cdot (\pi_0 \cdot a_0 ^ {*o} - b_0) & = 0, \label{imp_central_6}\\
%
\lambda_i, \mu_i, \omega, \nu & \geq 0, \qquad \forall i \in \mathcal{J}, \label{imp_central_7}
\end{align}
where \eqref{imp_central_1} - \eqref{imp_central_2} are the derivatives of the Lagrangian, and \eqref{imp_central_3} - \eqref{imp_central_4} are constraints on the Lagrange multipliers $(\lambda_i, \mu_i, \omega, \nu : i \in \mathcal{J})$.
By setting $\pi_0 = \textcolor{black}{\Tilde{p}_0} ^ *$, $\lambda_i = \lambda_i ^ i,$ $\mu_i = \mu_i ^ i,$ $\nu = \textcolor{black}{\Tilde{p}_0} ^ *,$ $\omega = \omega_0 ^ 0,$
$a_i ^ {*o} = \textcolor{black}{\alpha}_i (m ^ *)$, which implies that the efficient allocation of filters for all platforms and lower bound for the government is implemented by all GNE of the induced game.
\end{proof}
Next, we show that our mechanism guarantees the existence of at least one GNE \textcolor{black}{for the induced game. This ensures that the results of Lemmas \ref{lemma:truthful_prices} - \ref{lemma:feasibility} and Theorems \ref{budget_balance} - \ref{thm:implementation} are always valid for the induced game.}
\begin{theorem}[\textbf{GNE existence}] \label{thm:gne_existence}
Let $a ^ {*o} = \big(a ^ {*o}_0, a ^ {*o}_1, \dots, a ^ {*o}_{|\mathcal{I}|} \big)$ be the unique optimal solution of Problem \ref{problem1}. Then, there is a GNE message profile $m^* \in \mathcal{M}$ of the induced game that guarantees that the filter profile $\big( \textcolor{black}{\alpha}_1(m ^ *), \dots, \textcolor{black}{\alpha}_{|\mathcal{I}|}(m ^ *)\big)$ and lower bound $\textcolor{black}{\alpha}_0(m ^ *)$ at GNE satisfy $\textcolor{black}{\alpha}_i(m ^ *) = a_i ^ {*o}$, for all $i \in \mathcal{J}$.
\end{theorem}
\begin{proof}
Consider that the optimal solution $a ^ {*o}$ which satisfies the KKT conditions for Problem \ref{problem1} with the corresponding Lagrange multipliers $(\lambda_i, \mu_i, \nu, \omega : i \in \mathcal{J}).$ Taking similar steps to the proof of Theorem \ref{thm:implementation}, we can show that for $\textcolor{black}{\Tilde{p}_0} = \pi_0 = \nu$, the Lagrange multipliers of Problems \ref{problem2} and \ref{problem3} are $\lambda_i ^ i = \lambda_i, \; \mu_i ^ i = \mu_i, \; \nu_i ^ i = \nu, \; \omega_0^0 = \omega, \; i \in \mathcal{J},$ and the allocated prices are $\textcolor{black}{\pi_l}^{i} = \frac{\partial v_i}{\partial \textcolor{black}{\alpha}_l} \big|_{a^{*o}},$ for all $l \in \mathcal{C}_{- i}.$ This implies that the allocated filters at GNE are $\textcolor{black}{\alpha}_i(m ^ *) = a_i ^ {*o}$ for all platforms $i \in \mathcal{I}$, and \textcolor{black}{the allocated lower bound of the government is $\textcolor{black}{\alpha}_0(m ^ *) = a_0 ^ {*o}$.}
\end{proof}
\color{black}
Next, we consider the step one (the participation step) of our mechanism from Section III-A. We first note that the government always participates in the mechanism for the opportunity to incentivize misinformation filtering among the platforms. In the following result (Theorem \ref{thm:ir}), we invoke Assumption \ref{assumption:excludability} and the properties of our mechanism, to show that in step one, every social media platform voluntarily decides to participate in the mechanism. This property is also called individual rationality of the mechanism as it ensures voluntary participation of rational players without dictatorship.
\color{black}
\begin{theorem}[\textbf{Individually Rational}] \label{thm:ir}
The proposed mechanism is individually rational, i.e., each platform $i \in \mathcal{I}$ prefers the outcome of every GNE of the induced game to the outcome of not participating.
\end{theorem}
\begin{proof}
Consider any GNE message profile $m ^ *$. By Lemma \ref{lemma:achievability}, given profile $m_{- i} ^ *$, there exists a message $m_i \in \mathcal{M}_i$ for platform $i$ such that $\textcolor{black}{\alpha}_0(m_i, m_{- i} ^ *) = 0$. Furthermore, platform $i$ can unilaterally deviate in their message $m_i$ to ensure that for every platform $k \in \mathcal{C}_{i}$, the allocated filter is given by $\textcolor{black}{\alpha}_k(m_i, m_{- i} ^ *) = 0$. \textcolor{black}{Assumption \ref{assumption:excludability} implies that the utility of a non-participating platform $i \in \mathcal{I}$ is given by $v_i(a_k = 0: k \in \mathcal{C}_i)$.}
Consider the message $m_i = (\textcolor{black}{\Tilde{h}_i}, \textcolor{black}{\Tilde{p}_i}, \Tilde{a}_i)$ defined in \eqref{eqn:defn_message} with $\textcolor{black}{\Tilde{p}_l} ^ i = 0,$ for all $l \in \mathcal{C}_{- i} \cup \{0\}, $
$\Tilde{a}_k ^ i = - \sum_{l \in \mathcal{C}_{- i}} \Tilde{a}_k ^ l,$ for all $k \in \mathcal{C}_{- i},$ and $\Tilde{a}_0 ^ i = - \sum_{l \in \mathcal{J}_{- i}} \Tilde{a}_0 ^ i.$
Then, the allocation $\textcolor{black}{\alpha}_k(m_i, m_{- i} ^ *) = 0$ is feasible for every platform $k \in \mathcal{C}_{i}$ and the corresponding tax for social media platform $i$ is given by $\textcolor{black}{\tau}_i = 0$. The utility $u_i(m_i, m_{- i} ^ *)$ of social media platform $i$ is given by $u_i(m_i, m_{- i} ^ *) = v_i(0, \dots, 0) - 0.$ From the definition of the GNE in $\eqref{eqn:GNE_defn}$, we have $u_i(m ^ *) \geq u_i(m_i, m_{- i} ^ *)$. Hence, $u_i(m ^ *) \geq v_i(0, \dots, 0).$
We observe that the utility $u_i(m ^ *)$ at any GNE $m^* \in \mathcal{M}$ of a platform $i \in \mathcal{I}$, that decides to participate in the mechanism, is equal to or greater than their utility when not participating in the mechanism. Thus, in step one of the mechanism, the weakly dominant action of every social media platform $i \in \mathcal{I}$ is to participate in the mechanism.
\end{proof}
\subsection{Extension to Quasi-Concave Valuations}
In this subsection, we relax Assumptions \ref{assumption:filter_compatibility} - \ref{assumption:government_valuation}, and replace them with the following more general assumptions: (i) The valuation function \textcolor{black}{$v_i\big(a_k:k\in\mathcal{C}_i\big): \mathcal{A}^{|\mathcal{C}_i|} \to \mathbb{R}_{\geq0}$} of every platform $i \in \mathcal{I}$ is quasi-concave, differentiable, and have the same monotonic properties as before. (ii) The valuation function \textcolor{black}{$v_0(a_0): \mathcal{A} \to \mathbb{R}_{\geq0}$} of the government is quasi-concave, differentiable and increasing with respect to $a_0$. (iii) The average trust function \textcolor{black}{$h_i(a_i): \mathcal{A} \to [0,1]$} of any social media platform $i \in \mathcal{I}$ is a differentiable and increasing with respect to $a_i$. We cannot use the KKT conditions to prove the existence of a GNE and strong implementation under these relaxed assumptions. However, note that at any GNE, if one exists, the proposed mechanism is still budget balanced, feasible and individually rational. In addition, Lemmas \ref{lemma:truthful_prices}, \ref{lemma:truthful_action_proposals}, and \ref{lemma:achievability} also hold as they do not depend on the concavity of the valuation.
Next, we prove that for the relaxed assumptions, there exists a GNE and that it induces a Pareto efficient equilibrium in the game. Pareto efficiency refers to the condition where we cannot improve the utility of any player without decreasing the utility of another player in the induced game \cite{mas_colell1995}. Pareto efficiency is a weaker property in comparison to the strong implementation achieved by our mechanism for concave valuation functions.
\begin{theorem} \label{thm:quasi}
Let the valuation function $v_i(a_k:k\in\mathcal{C}_i)$ be quasi-concave and differentiable for all players $i \in \mathcal{J}$ and consider the game $\langle \mathcal{M}, g(\cdot), (u_i)_{i \in \mathcal{I}} \rangle$. Then, (i) there exists a GNE for the induced game, and (ii) every GNE of the induced game is Pareto efficient.
\end{theorem}
\begin{proof}
\textit{1) Existence:} Consider the social media platform $i \in \mathcal{I}$. Lemma 2 implies that at GNE, the message $m_i$ must lie in the set $\mathcal{M}_i' := \big\{m_i \in \mathcal{M}_i: \textcolor{black}{\Tilde{p}^i_l} \cdot (\Tilde{a}_l^i - a^{-i}_l) = 0, \; \forall l \in \mathcal{D}_{-i} \big\}$. For all $m_i \in \mathcal{M}_i'$, we can write the utility $u_i(m)$ as
\begin{align} \label{thm_5_1}
u_i(m) = \; &v_i(\textcolor{black}{\alpha}_k(m): k \in \mathcal{C}_i) + \textcolor{black}{\Tilde{p}_0} \cdot \textcolor{black}{\eta_i}(m) \nonumber \\
&+ \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_i} ^ l \cdot \textcolor{black}{\alpha}_{i}(m)
- \sum_{l \in \mathcal{C}_{- i}} \textcolor{black}{\pi_l} ^ i \cdot \textcolor{black}{\alpha}_{l}(m),
\end{align}
where the prices $\textcolor{black}{\Tilde{p}_0}$, $\textcolor{black}{\pi_i} ^ l$, and $\textcolor{black}{\pi_l} ^ i$ for any $l \in \mathcal{C}_{-i}$ are independent of message $m_i$. We observe that $u_i(m) = u_i(\textcolor{black}{\eta_i}, \textcolor{black}{\alpha}_k : \textcolor{black}{\alpha}_k \in \mathcal{D}_{i})$. Lemma 4 implies that given a message profile $m_{-i}$ of all platforms and the government in $\mathcal{J}_{-i}$, platform $i$ can unilaterally deviate in their message $m_i \in \mathcal{M}_i$ to receive any allocation $\textcolor{black}{\alpha}_k(m) \in \mathcal{A}$, for all $k \in \mathcal{D}_{i}$. Thus, instead of the message $m_i$, we equivalently consider that the action of platform $i$ is to select the tuple $\textcolor{black}{\beta}_i = \big(\textcolor{black}{\eta_i}, \textcolor{black}{\alpha}_k : k \in \mathcal{D}_{i} \big)$, that takes values in the set $\textcolor{black}{\mathcal{B}}_i = \big\{[0,1] \times \mathcal{A}^{|\mathcal{D}_i|} : n_i \cdot h_i(\textcolor{black}{\alpha}_i) - \textcolor{black}{\eta_i} \geq 0\big\}$. For the differentiable function $h_i(a_i)$, the set $\textcolor{black}{\mathcal{B}}_i$ is convex, compact, and independent of the message profile $m_{-i}$. Similarly, the action of the government $\textcolor{black}{\alpha}_0$ takes values in the set $\mathcal{A}$ that is compact, convex, and independent of the message profile $m_{-0}$.
Let the valuation $v_i\big(a_k:k\in\mathcal{C}_i\big)$ for every platform $i \in \mathcal{I}$ be quasi-concave and differentiable, and let \textcolor{black}{$\beta = \big(\beta_0, \beta_1, \dots, \beta_{|\mathcal{I}|}\big)$}. Then, for every $i \in \mathcal{I}$, the utility $u_i(\textcolor{black}{\beta})$ in \eqref{thm_5_1} is also quasi-concave and differentiable with respect to the action $\textcolor{black}{\beta}_i \in \textcolor{black}{\mathcal{B}}_i$. A similar argument implies that the government's utility $u_0(\textcolor{black}{\alpha}_0)$ is quasi concave and differentiable with respect to their action $\textcolor{black}{\alpha}_0$. Hence, it follows from Glicksberg's theorem that there exists a Nash Equilibrium (NE) for the induced game \cite{fudenberg1991}; since, by definition, any NE is also a GNE, it follows that there exists a GNE for the induced game.
\textit{2) Pareto efficiency:} It is sufficient in our case to show that the NE can be characterized by a Walrasian equilibrium as all Walrasian equilibria are Pareto efficient \cite{mas_colell1995}.
So, as in part 1, consider an arbitrary NE action profile $\textcolor{black}{\beta}^* = \big(\textcolor{black}{\alpha}_0^*, \textcolor{black}{\beta}_1^*, \dots, \textcolor{black}{\beta}_{|\mathcal{I}|}^*\big)$ that takes values in the set $\mathcal{A} \times \textcolor{black}{\mathcal{B}}_1 \times \cdots \times \textcolor{black}{\mathcal{B}}_{|\mathcal{I}|}$.
%
From the definition of the NE, for every platform $i \in \mathcal{I}$ it holds that
\begin{equation} \label{eqn:thm_NE}
u_i(\textcolor{black}{\beta}^*) \geq u_i(\textcolor{black}{\beta}_i, \textcolor{black}{\beta}_{-i}^*), \quad \forall \textcolor{black}{\beta}_i \in \textcolor{black}{\mathcal{B}}_i.
\end{equation}
{Note that the NE prices $\textcolor{black}{\Tilde{p}_0}^*$, $\textcolor{black}{\pi^{*i}_l}$, $\textcolor{black}{\pi_l}^{*i}$, for all $l \in \mathcal{I}_{-i}$ cannot be influenced by platform $i$, i.e., every social media platform is a price taker. Then, using the definition of the NE in \eqref{eqn:thm_NE} with the utility $u_i(m)$ in \eqref{thm_5_1}, we can write for platform $i$ that
\begin{multline}
\textcolor{black}{\beta}_i^* = \arg \max_{\textcolor{black}{\beta}_i \in \textcolor{black}{\mathcal{B}}_i} \Bigg\{ v_i(\textcolor{black}{\alpha}_k: k \in \mathcal{C}_i) + \textcolor{black}{\Tilde{p}_0}^* \cdot \textcolor{black}{\eta_i} \\
%
+ \sum_{l \in \mathcal{I}_{- i}} \textcolor{black}{\pi_i} ^ {*l} \cdot \textcolor{black}{\alpha}_{i} - \sum_{l \in \mathcal{I}_{- i}} \textcolor{black}{\pi_l} ^ {*i} \cdot \textcolor{black}{\alpha}_{l} \Bigg\}.
\end{multline}
Similarly, the government also behaves as a price taker because it cannot influence the NE price $\pi^*_{0}$. For the government at NE, we can write that
\begin{align}
\textcolor{black}{\alpha}_0^* = \arg \max_{\textcolor{black}{\alpha}_0 \in \mathcal{A}} \{v_0(\textcolor{black}{\alpha}_0) - \pi^*_{0} \cdot \textcolor{black}{\alpha}_0\}.
\end{align}
It follows immediately that the NE action profile $\textcolor{black}{\beta}^*$ constitutes a Walrasian equilibrium and thus, the NE for the induced game forms a Pareto efficient equilibrium \cite{mas_colell1995}. Since any NE is also a GNE by definition, it follows that every GNE of the induced game is Pareto efficient.}
\end{proof}
\color{black}
\begin{remark}
The GNE induced by our mechanism may not lead to allocated filters for platforms and lower bound for the government that form an optimal solution of Problem 1 using quasi-concave valuations. However, Theorem \ref{thm:quasi} establishes that a GNE still exists for such a system, and that it leads to a Pareto efficient allocation, where no player's utility can be improved without decreasing the utility of another player. Thus, from Theorem \ref{thm:ir}, we can conclude that for quasi-concave valuation functions, our mechanism incentivizes some misinformation filtering but may lead to suboptimal social welfare.
\end{remark}
\section{Discussion}\label{section:discussion}
\subsection{Interpretation of the Results}
In this subsection, we present an explanation of the mechanism presented in Section III and the main results derived in Section IV. The social planner seeks to design an efficient mechanism with the following two properties: (i) it should induce voluntary participation among all social media platforms, and (ii) it should maximize the social welfare, i.e., maximize the sum of utilities of all players.
Note that the social welfare increases as the valuation function $v_0(a_0)$ increases, which, in turn, increases with respect to the lower bound on aggregate average trust, $a_0$. A sufficiently high lower bound $a_0$ indirectly ensures that some platforms implement non-zero filters to raise the average trust of their users. Thus, a mechanism that satisfies properties (i) and (ii) also incentivizes platforms to implement filtering, conditional on the government's valuation $v_0(a_0)$ and budget $b_0$ being sufficiently large. The challenge faced by the social planner is to achieve these properties without knowledge of the valuation function $v_0(a_0)$ of the government, the valuation function $v_i\big(a_k:k\in\mathcal{C}_i\big)$ of any platform $i \in \mathcal{I}$, and the average trust function $h_i(a_i)$ of any social media platform $i \in \mathcal{I}$.
To meet this challenge, we present a two-step mechanism in Section III. In the step one (the participation step) of the mechanism, the social planner asks each social media platform to decide whether they wish to participate in the mechanism. This is an essential question because the government is not dictatorial, i.e., it cannot force platforms to participate in the mechanism. By refusing to participate in the mechanism, platform $i$ can select no filter and pay no tax. However, platform $i$ also receives no subsidy from the government, nor benefits from the filters of platforms that do participate. We prove in Theorem \ref{thm:ir} of Section IV that the utility of any platform $i \in \mathcal{I}$ after participating in the mechanism is greater than or equal to their utility when they do not participate. Thus, the weakly dominant action of every platform in step one is to participate in the mechanism, establishing property (i).
In the step two (the bargaining step) of the mechanism, the social planner asks each player $i \in \mathcal{J}$ to broadcast a message $m_i \in \mathcal{M}_i$. Based on the message profile $m = \big(m_0,m_1,\dots,m_{|\mathcal{I}|}\big)$, the social planner allocates a minimum average trust $\eta_i(m)$, a filter $\alpha_i(m)$, and a tax $\textcolor{black}{\tau}_i(m)$ to each platform $i \in \mathcal{I}$. Similarly, she allocates a lower bound $\alpha_0(m)$ and tax $\textcolor{black}{\tau}_0(m)$ to the government. By participating in the mechanism in step one, each player $i \in \mathcal{J}$ agrees to implement the allocated filters, and either pay or receive the allocated tax. The rules defined by the social planner induce a game among the players whose equilibrium is defined as a GNE. The structure of the messages, and various parameters allocated by the social planner lead to the properties of the mechanism in Section IV.
We derive most of the properties of the mechanism in Section IV for a state where the platforms and the government are at a GNE. Lemmas \ref{lemma:truthful_prices} and \ref{lemma:truthful_action_proposals} establish preliminary properties of the tax functions $\textcolor{black}{\tau}_i(m)$ of each player $i \in \mathcal{I}$. They show that at the GNE, each player $i$ has to be consistent in their message $m_i$ with respect to the messages of other players. This consistency check ensures that no player can benefit from a manipulation of the mechanism by proposing arbitrary prices, filters, or lower bounds.
Then, we use the results of Lemmas \ref{lemma:truthful_prices} and \ref{lemma:truthful_action_proposals} to derive Theorem \ref{budget_balance}, which proves that at any GNE the mechanism is budget balanced, i.e., the sum of all taxes is $0$. This is a desirable property for the mechanism because the social planner is now guaranteed to simply take the investment of the government $\textcolor{black}{\tau}_0(m)$ and redistribute it among the social media platforms, without worrying about leftover funds or insufficient funds.
Next, we show in Lemma \ref{lemma:feasibility} that every GNE of the induced game is a feasible solution to the problem of maximizing social welfare. Lemma \ref{lemma:achievability} proves that any social media platform $i \in \mathcal{I}$ can always achieve any desired filter in $\mathcal{A}$, including $0$, by selecting an appropriate message $m_i \in \mathcal{M}_i$. This property holds irrespective of the messages selected by the other players in $\mathcal{J}_{-i}$, and establishes that a participating platform has a free choice to control their allocated filter.
All preceding results allow us to prove in Theorem \ref{thm:implementation} that every GNE of the induced game maximizes the social welfare of the system. In Theorem \ref{thm:gne_existence} we prove that the induced game is guaranteed to have at least one GNE. Theorem \ref{thm:implementation} and Theorem \ref{thm:gne_existence}, together, imply that the mechanism maximizes the social welfare of the system, establishing property (ii). Thus, we have shown that our mechanism does, indeed, incentivize platforms to filter misinformation.
Finally, in Section IV-A we consider quasi-concave valuation functions for all players to relax some of our assumptions. In Theorem \ref{thm:quasi}, we establish that the induced game is still guaranteed to have a GNE, and that it is Pareto efficient. Thus, we observe that our mechanism still incentivizes some amount of filtering, but may lead to suboptimal social welfare.
\subsection{An Example}
In this subsection, we present a descriptive example of how our proposed mechanism may play out in a realistic setting. Consider three major social media platforms: Facebook, Twitter, and Reddit. These platforms allow users from different socioeconomic and political backgrounds to obtain the latest news. Typically, users access either Facebook, Twitter, or Reddit via their smartphone app and engage with them by scrolling down, liking, or sharing posts that feature news and personal opinions. The amount of time spent by all users on the platform and the number of actions taken by them collectively define the engagement generated by the platform \cite{allcott2017social,jaakonmaki2017}.
As user engagement is a primary driver of advertisement revenue, Facebook, Twitter, and Reddit regularly optimize their post recommendation algorithms to maximizing user engagement. Over time, these algorithms have evolved to promote posts with a high chance of generating engagement among users, without accounting for their impact on the opinions of the users \cite{tufekci2018youtube}. This has led to the formation of echo chambers, or opinion bubbles among many users, where they repeatedly interact only with posts that align with their own biases on any topic. For many users, their prior biases lead to a repeated exposure to misinformation and conspiracy theories \cite{margetts2018}. This causes
uncertainty among them regarding the integrity democratic institutions \cite{bessi2015, brown2018,tucker2017, sternisko2020dark}. For example, misinformation during elections reduces people's faith in the fairness of the election results \cite{farrell2018common}, and misinformation about precautions during a pandemic reduces people's trust in public health experts \cite{motta2020right}.
A democratic government can observe the trust of the country's citizens from the opinions expressed by them on various social media platforms. When the government realizes the impact of misinformation on the trust of the citizens, they seek to implement policies to minimize the spread of misinformation.
In practice, each social media platform can filter misinformation by either flagging posts with inaccurate information, following them up with truthful posts, or simply not recommending them to users.
However, filtering misinformation is an expensive undertaking for platforms because of (i) the high investment required to identify inaccurate information \cite{graves2018understanding}, and (ii) potential decrease in engagement of users who are censored \cite{candogan2020optimal}. Thus, the government decides to allocate a fixed budget for the problem, and appoints an independent agency to design appropriate incentives for Facebook, Twitter, and Reddit, while staying within the budget.
The agency presents the rules of our mechanism to the government, and confirms the government's participation. Then, the agency reveals the rules of the mechanism to the platforms, and announces that platforms who choose not to participate in this collaborative effort will be labelled as non-cooperative. Furthermore, the agency assures the three platforms that they need not reveal private information and that they can choose to avoid filtering misinformation even after participating in the mechanism (Lemma \ref{lemma:achievability}). These factors ensure that each platform participates voluntarily in the mechanism (Theorem \ref{thm:ir}).
Then, the agency asks each of Facebook, Twitter, and Reddit to propose a minimum threshold to which they will raise the trust of their users in democratic institutions. The tax incentives given to each platform will be proportional to this threshold. Simultaneously, the agency asks the government to propose a minimum acceptable level for the average of all platforms' thresholds. The government's investment will be proportional to this minimum average.
The agency also asks each platform to propose various filters and prices they are willing to pay or receive for the proposed filters. Similarly, the government proposes a price for their proposed minimum average.
The agency then publicly reveals all proposals and transparently uses the rules of the mechanism to assign a potential subsidy/payment, and potential filter to each platform. Similarly, she assigns a potential amount of investment and minimum average to the government. These assignments become binding only if all stakeholders, Facebook, Twitter, Reddit, and the government, accept the assignments. If any stakeholder is dissatisfied, the agency asks all of them to change their proposals and resubmit. This process is repeated until all the stakeholders reach a consensus.
The mechanism ensures that such a consensus exists (Theorem \ref{thm:gne_existence}) and that it is the best possible result for all stakeholders (Theorem \ref{thm:implementation}).
As long as the government is sufficiently committed to addressing the problem of misinformation, the mechanism ensures that at the consensus, the platforms will agree to implement misinformation filters. The allocations become binding on all stakeholders, and the independent agency collects the government's investment. This investment is paid out to each of Facebook, Twitter, and Reddit as a subsidy, only after they achieve the binding level of filtering.
\section{Conclusions and Future Work}\label{section:conclusion}
Our primary goal in this paper was to design a mechanism to induce a GNE solution in the misinformation filtering game, where (i) each platform agrees to participate voluntarily, and (ii) the collective utility of the government and the platforms is maximized. We designed a mechanism and proved that it satisfies these properties along with budget balance. We also presented an extension of the mechanism with weaker technical assumptions.
Ongoing work focuses on improving the valuation and average trust functions of the social media platforms based on data. We also consider incorporating uncertainty in a platform's estimates of the impact of their filter. These refinements of the modeling framework will allow us to make our mechanism more practical for use in the real world.
Future research should include extending the results of this paper to a dynamic setting in which the social media platforms react in real-time to the proposed taxes/subsidies. In particular, someone could develop an algorithm that the players can use to iteratively arrive at the Nash equilibrium. In such an algorithm, the social planner can receive additional information from the players while they iteratively learn the GNE. Then, she can use this information to change her allocations dynamically, allowing us to relax either Assumption \ref{assumption:assumed_knowledge_b} on monitoring of average trust, or Assumption \ref{assumption:excludability} on the excludability of the platforms.
\color{black}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 637 |
Q: Browsersync открывает сайт на пк но не открывает на телефон. Пут для телефона пишет External: http://false:3000 Будьте пожалуйста любезные, кто может подскажите пожалуйста в чем может быт проблема?
На папке где находится index.html запустил Browsersync в пк показивает сайт а на телефоне не показывает. Local: http://localhost:3000 для пк работает,
External: http://false:3000 для телефона не работает.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,798 |
\chapter{Introduction}
This thesis constructs and applies algorithms to carry out a wavelet analysis for data located in three-dimensional structures that are essentially ``fat surfaces'', i.e., are $\epsilon$-neighborhoods of possibly convoluted two-dimensional surfaces. This new approach was developed for the analysis of fMRI data.
This last paragraph contains several terms that we would like to explain already here, before going into the technical details of subsequent chapters.
\subsubsection{Wavelets}
A \emph{wavelet basis} is a collection of functions $(\psi_{j,m})$, which typically have two indices $j$ and $m$, denoting \emph{scale} and \emph{position}, respectively. In the classic setting, elements of a wavelet basis are translated and dilated versions of a prototype called the {``mother wavelet''} function. In other words, given a mother wavelet $\psi(x)$, the basis element $\psi_{j,m}$ is given by
$$
\psi_{j,m}(x) = C_j\psi(2^j(x-m)),
$$
where $j \in \mathbb Z$, $m \in 2^{-j}\mathbb Z$ and $C_j$ is the normalizing constant that ensures $\|\psi_{j,m}\|=1$.
It is generally expected that the mother wavelet has some degree of smoothness, and is either compactly supported, or has most of its energy contained in a compact domain.
With a good wavelet basis, only a small fraction of the coefficients in the wavelet expansion
\[
\label{wavelet_expansion}
f = \sum_{j,m}\gamma_{j,m}\psi_{j,m},
\]
suffices to approximate the function $f$ whenever it belongs to a signal class of interest, and a local change in the signal only affects a small number of corresponding coefficients. These properties make wavelets optimal bases in compression, denoising and estimation applications \cite{ donoho93, daubechies,sweldens_siam}.
\subsubsection{Wavelets in higher dimensions}
One dimensional wavelet designs can be translated into higher dimensional Euclidean spaces by means of tensor products, and these are called \emph{separable wavelet bases}. There are other means of obtaining wavelets directly in higher dimensions, typically leading to \emph{nonseparable wavelet bases}. The Fourier transform was used as an indispensable tool in most of the constructions in the earlier years of the development of wavelet theory: most of the classical wavelet bases for $\mathbb R^n$, whether separable or nonseparable, depend on the Fourier transform in their designs.
\subsubsection{The lifting scheme and the second generation wavelets}
Standard wavelet bases mentioned above have the entire vector space $\mathbb R^n$ as
their natural domain. It is fairly straightforward to adapt these to rectangular subdomains, and there also exist constructions for domains with more complex boundaries. However, those methods are ineffective when we deal with domains that
are very arbitrary and convoluted. A particular example of interest is the cortex of the
human brain, which is the place where all the cognitive activity takes place.
For such arbitrary subsets of $\mathbb R^n$, which do not possess a mathematical group structure, tools like the Fourier transform are not available to be utilized in the design. Fortunately however, very flexible frameworks such as the the lifting scheme \cite{sweldens96, sweldens_siam} are available to construct wavelet bases on these domains directly in the spatial domain. The wavelets obtained by such methods are no longer the translated and dilated copies of a prototype, and they are called the \emph{second generation wavelets}. The second generation wavelets possess many of the appealing features of the first generation wavelet transforms, such as leading to sparse and localized representations, and the availability of fast transform algorithms.
\subsubsection{Some prior applications of second generation wavelets}
One of the first constructions of biorthogonal wavelets on triangulated surfaces was by Launsbery et al. \cite{lounsbery94}. This scheme was later improved by means of the lifting scheme, which was published by Sweldens \cite{mallat09, sweldens96, sweldens_siam}.
Lifting-based wavelets on nonstandard grids have been used in numerous areas; an example is the construction in \cite{sweldens95}, of wavelet bases on the sphere, using an adaptive subdivision scheme. Spherical wavelets have applications in astronomical and geophysical data analysis.
Other examples of lifting-based wavelets have applications in computer graphics, where they can be used to efficiently represent meshes. In \cite{khodakovsky04}, they are used in compression of three dimensional meshes. In \cite{wei01}, an application in texture synthesis over arbitrary manifolds is given.
In \cite{cohen92, dahmen97, vasilyev00, griebel07} wavelets of this type are used for numerical solutions of partial differential equations.
An application in computational digital photography is given in \cite{fattal09}; in this paper a wavelet basis that is customized to a given image is constructed by the lifting scheme, which has wavelets and scaling functions that avoid edges, as much as possible. These wavelets are then successfully applied to problems of dynamic range compression, edge preserving smoothing, detail enhancement and image colorization.
A neuroimaging application is given in \cite{Yuetal07}, where it is applied to capturing shape variations of the cortex and in tracking cortical folding changes in newborns.
For a review of the theory of second generation wavelets and a survey of other applications, we refer to \cite{jansen2005second}.
\subsubsection{Functional brain imaging}
Functional brain imaging refers to acquiring time series of three-dimensional images of the human brain, which measure some correlate of the neural activity. This is different from anatomical (or structural) imaging in which the measured quantity is chosen to obtain as much contrast as possible to differentiate between different tissue types.
Functional magnetic resonance imaging (fMRI) has become the most widely used functional brain imaging
technology in neuroscience research. It has led to an explosion in papers aiming to
understand the organization of the human brain. Earlier
fMRI studies
mainly concentrated on determining the regions where the active component peaked, and were therefore not overly concerned about losing lower-amplitude information before
applying spatial low pass
filtering to the data to improve the statistical power \cite{beyond_mind}.
However, later studies have demonstrated that fMRI signal
contains much additional detail information. For instance, the work by J. Haxby and his collaborators \cite{haxby} showed, in an experiment measuring discrimination between different stimuli, that enough information was present, even after removal of the main activation area for each stimulus, to allow for successful stimulus classification. This
motivates the use of tools like wavelet analysis to detect
fine details that might be less apparent and that could be lost to thresholding in the spatial domain. However, the standard
wavelet transforms are mostly designed to have very special domains, like a rectangle in two dimensions or a cube in
three dimensions. The activity of interest in the brain, on the other hand, occurs on the cortex, which is an intricately convoluted set
in three dimensions. Using standard wavelets without any modification will have the obvious drawback of mixing signal from the cortex (gray matter) with the signal coming from the off-cortex regions. There would be an artificial edge at the boundary of the cortex, which would inflate the detail coefficients in the wavelet transform, reducing its ability to form a sparse representation of the data.
\subsubsection{Domain-adapted wavelets}
In this thesis, mainly motivated by the functional brain imaging problem, we use the lifting scheme to design wavelets on arbitrary two and three dimensional domains.
One of the key steps in the construction of the wavelets is to define a nested family of partitions on the domain. This
includes parent, child, neighbor and sibling relations. Our algorithm of defining these structures has an element of randomness in it, which results in getting multiple sets of wavelet bases, each coming from a different realization of the partitioning. This in turn allows us to repeat
the analysis with many different realizations of the wavelet bases and
averaging the results, a method that improves the power of the analysis.
An example of a nonstandard domain, a random partitioning on it and an example wavelet and scaling function (to be explained in the next chapter) is displayed in Figure \ref{fig:wavelet_and_scaling}.
\begin{figure}
\begin{center}
\subfigure[]{{\includegraphics[width=10cm]{./figures/chp1/random_parts.eps}}}\\
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp1/sf.eps}}}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp1/w.eps}}}
\end{center}
\caption{Illustration of partitioning and adapted wavelets on an annular shaped domain. (a) Sample random partitioning of two levels. (b) Scaling function. (c) Wavelet}
\label{fig:wavelet_and_scaling}
\end{figure}
\subsubsection{Wavelet-based statistical parametric mapping}
One of the basic objectives in an fMRI study is to detect whether a given location point (voxel) is responding to the given stimulus or not. The question can be answered by means of statistical tests that are applied to the corresponding time series. However, these statistical tests always have the chance of a positive result when there was no response (false positive), and a negative result when there actually was some response (false negative). In an fMRI image, there is a large number of voxels, and the statistical test needs to be performed for each of these voxels. This means that there is a large number of decisions to make about the absence or presence of response. Due to the noisy nature of the fMRI data, some spatial filtering is necessary in order to have control over the total number of false positives, while not losing the sensitivity of the test altogether. Statistical parametric mapping (SPM) \cite{hbf2}, which is one of the most widely used methods, uses a Fourier-based low pass filtering as a means of reducing noise. However this has the drawback of destroying fine-scale spatial details. An alternative is a wavelet-based framework is due to Van De Ville et al. \cite{surfing, vandeville04,
vandeville0406, vandeville07}, which includes theoretical bounds on the false positive rate, when the null hypothesis is true in the region of interest. This framework has no restrictions on the type of wavelet basis to be used, so it is suitable to use the domain adapted wavelets that are constructed so as to be spatially adapted to the cortex of the human brain.
\subsubsection{Organization of the thesis}
The organization of the thesis will be as follows. In Chapter 2 we will study the preliminaries about multiresolution analysis, wavelet transforms, and the lifting scheme. Chapter 3 will be about the methodology of construction of the adapted wavelets and will also give some numerical experiments. In Chapter 4 we will give results about the application of this type of wavelets in statistical analysis of fMRI data, in the wavelet-based statistical parametric mapping framework. Chapter 5 contains a summary and conclusion.
\chapter{Wavelets and the lifting scheme}
In this chapter, we review the basics of wavelet theory, as well as of the lifting scheme, a method that can be used to design wavelets on irregular domains, without relying on tools from Fourier analysis. It is due to Wim Sweldens \cite{sweldens_siam, sweldens96}.
Our presentation and notation will mostly follow \cite{sweldens_siam}. A generalized language for wavelets will be used, which also covers the classical (first generation) wavelets as a special case.
\section{Wavelet bases and wavelet transforms}
In a general setting, a wavelet basis is a collection of functions
\[
\label{wdef}
\left\{\psi_{j,m}(x)\colon j\in\mathcal J, m \in\mathcal M(j) \right\},
\]
where $\mathcal J\subset \mathbb Z$ is the index set for scale (frequency), and for each $j$, $\mathcal M(j)$ is the index set for location. The collection enables us to expand a given function $f$ as
\[
\label{wavelet_expansion}
f = \sum_{j\in\mathcal J,\ m \in\mathcal M(j)}\gamma_{j,m}\psi_{j,m},
\]
which induces the representation of the function $f$ with a sequence of numbers $(\gamma_{j,m})$
\begin{align*}
f \mapsto (\gamma_{j,m}),
\end{align*}
and this mapping is called a \emph{wavelet transform}.
In a good wavelet transform, only a small fraction of the coefficients suffices to approximate the function $f$ whenever it belongs to a signal class of interest, and a local change in the signal only affects a small number of corresponding coefficients. These make wavelets optimal bases in compression, denoising and estimation applications \cite{sweldens_siam, donoho93, mallat09}.
\subsection{First generation wavelets}
In the classic setting, a wavelet basis is obtained by translating and dilating (up to a normalizing constant) a single template function, which is called the \emph{mother wavelet}. These type of wavelets are referred as the \emph{first generation wavelets}. For the one-dimensional case, the domain is the infinite real line $\mathbb R$, the scale index set is $\mathcal J = \mathbb N$, and for $j \in\mathcal J$, the location index set is $\mathcal M(j) = 2^{-j}\mathbb Z$. Given a mother wavelet $\psi(x)$, the basis element $\psi_{j,m}$ is defined to be
$$
\psi_{j,m}(x) = C_j\psi(2^j(x-m)),
$$
where $C_j$ is the normalizing constant that ensures $\|\psi_{j,m}\|=1$.
In order to obtain a desirable wavelet basis, the mother wavelet should either be compactly supported, or have most of its energy be contained in a compact domain.
\subsection{Examples}
\begin{itemize}
\item The Haar basis is the earliest known example of a wavelet basis. It is generated by the Haar mother wavelet function
$$
\psi(x) = \left\{\begin{array}{rc}1 & \text{if } 0\leq x < 1/2 \\
-1 & \text{if } 1/2\leq x \leq 1\\
0 & \text{elsewhere}\end{array}\ \ \ . \right.
$$
Haar wavelets form an orthonormal basis and they are compactly supported. Their main drawback is that they are not smooth, which results in wavelet representations that are not very sparse, contrary to what was desired.
\item The Meyer wavelet is one of the first constructions of a smooth mother wavelet function that generates an \emph{orthogonal basis}, but it is not compactly supported, which is a drawback for computational purposes.
\item The Daubechies wavelet family comes next chronologically, it consists of wavelet bases that are both smooth, compactly supported and orthogonal.
\item There are wavelets that form a \emph{biorthogonal} basis (defined in Subsection \ref{subsec:bases_review}), one of which is illustrated in Figure \ref{wavelets}, along with the above examples.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=15 cm]{figures/chp2/wavelets.eps}
\caption{Examples of mother wavelets, belonging to some of the most commonly used wavelet bases: Haar, Meyer, Daubechies and Biorthogonal (average interpolating) wavelets. }
\label{wavelets}
\end{figure}
\section{Multiresolution analyses and fast wavelet transforms}
For the wavelet expansion
\[
\label{wavelet_expansion}
f = \sum_{j\in\mathcal J, \ m \in\mathcal M(j)}\gamma_{j,m}\psi_{j,m},
\]
orthogonal and biorthogonal wavelets have the relation
\[
\gamma_{j,m} = \langle f, \tilde \psi_{j,m} \rangle,
\]
that makes the computation of the coefficient functionals possible with a simple expression. However taking this inner product is computationally very slow.
Many of the wavelet bases, on the other hand, are implicitly tied to a structure formed by a multilayered sequence of subspaces, which is called a \emph{multiresolution analysis}.
A multiresolution analysis brings about a much faster way of computing the wavelet coefficients, which is called the \emph{fast wavelet transform} or the \emph{cascade algorithm}.
\subsection{Multiresolution analysis}
\label{subsec:multiresolution}
\begin{definition} A sequence of closed subspaces $\mbf M = \{V_j:j\in\mathcal J\subset \mathbb Z\}$ of $L^2(X,\Sigma,\mu)$ for some measurable space $X\subset\mathbb R^n$ is
called a \emph{multiresolution analysis} if it satisfies the following:
\begin{enumerate}
\item $V_j \subset V_{j+1}$
\item $\bigcap_{j}V_j = \{0\}$
\item $\overline{\bigcup_jV_j} = L^2(X,\Sigma,\mu)$
\item For each $j\in\mathcal J$, there exists a Riesz basis $\{\varphi_{j,k}:j\in\mathcal K(j)\}$ that spans $V_j$. The functions \((\varphi_{j,k})\) are called \emph{scaling functions}.
\end{enumerate}
\end{definition}
Here, $\mathcal K(j)$ is an index set indicating location, and in general it is assumed that $\mathcal K(j) \subset \mathcal(K(j+1))$. The scale index $\mathcal J$ is taken to be $\mathcal J = \mathbb Z$ when $X$ is an unbounded set of infinite measure, and $\mathcal J= \mathbb N$ when $X$ is bounded and of finite measure (which is the case for the focus of this thesis).
\subsubsection{Dual multiresolution analysis}
Let $\mbf M = \{V_j:j\in\mathcal J\subset \mathbb Z\}$ be a multiresolution analysis. Then, another multiresolution analysis
\[
\widetilde{\mbf M} = \{\widetilde V_j:j\in\mathcal J\subset \mathbb Z\}
\]
is called a \emph{dual multiresolution analysis} of $\mbf M$, if its scaling functions $(\tilde \varphi_{j,k})$ satisfy
\begin{equation}
\label{eqn:scaling_biorth}
\langle\varphi_{j,k},\tilde\varphi_{j,k'} \rangle = \delta_{k,k'} \ \ \ \text{ for } k,k'\in\mathcal K(j).
\end{equation}
For any given function $f\in V_j$, the coefficients in the expansion
\[
f = \sum_{k\in\mathcal K(j)}\lambda_{j,k} \varphi_{j,k},
\]
can be obtained by
\[
\lambda_{j,k} =\langle f , \tilde\varphi_{j,k} \rangle.
\]
\begin{remark}
Note that, according to this definition, a dual multiresolution is not unique. We also note that, since $(\varphi_{j,k})$ is a Riesz basis for $V_j$, it also has a dual basis \emph{within} $V_j$ that is biorthogonal to it, as mentioned in Subsection \ref{subsec:bases_review}. However a set of dual scaling functions that are denoted by $(\tilde\varphi_{j,k})$ here are not necessarily contained in $V_j$, and they do not necessarily coincide with the \emph{dual Riesz basis} of $(\varphi_{j,k})$ within $V_j$.
\end{remark}
\subsection{Refinement equation, finite filters and set of partitionings}
\label{subsec:ref_eqn}
Because of the inclusion $V_j \subset V_{j+1}$, $\varphi_{j,k}$ can be written as a linear combination of $(\varphi_{j+1,k})$'s:
\begin{equation}
\label{eqn:refinement}
\varphi_{j,k} = \sum_{l\in\mathcal K(j+1)}h_{j,k,l}\varphi_{j+1,l},
\end{equation}
for any $j\in\mathcal J$ and $k\in\mathcal K(j)$. This is a relation that must be satisfied in any multiresolution analysis. On the other hand, we shall see that a multiresolution analysis will be uniquely determined once the coefficients $(h_{j,k,l})$ are defined. In order for this to be possible, we need two definitions. One is the concept of a \emph{finite filter}, which puts some constraints on the coefficients $(h_{j,k,l})$ for them to result in a well defined multiresolution analysis. The other is the \emph{set of partitionings}, using which one will be able to pass from the finite filter to scaling functions in $L^2(X,\Sigma, \mu)$.
\subsubsection{Finite filters}
\begin{definition}
A set of real numbers $\{h_{j,k,l}\colon j\in\mathcal J,k\in\mathcal K(j), l\in\mathcal K(j+1)\}$ is called a \emph{finite filter} if the following are satisfied.
\begin{enumerate}
\item For each $j$ and $k$, $h_{j,k,l}$ is nonzero for only finitely many $l$'s. Hence, the set defined as
\begin{equation*}
\mathcal L(j,k) = \{l\in\mathcal K(j+1) |h_{j,k,l}\neq 0\}
\end{equation*}
is finite.
\item For each $j$ and $l$, $h_{j,k,l}$ is nonzero for only finitely many $k$'s. Hence, the set defined as
\[
\mathcal K(j,l) = \{k\in\mathcal K(j) |h_{j,k,l}\neq 0\}
\]
is finite.
\item
The sizes of sets $\mathcal L(j,k)$ and $\mathcal L(j,l)$ are uniformly bounded over all $j,k$ and $l$.
\end{enumerate}
\end{definition}
\subsubsection{Dual filter}
We note that, for the dual scaling functions, there also exists a refinement relation
\begin{equation}
\label{eqn:dual_refinement}
\tilde\varphi_{j,k} = \sum_{l\in\mathcal K(j+1)}\tilde h_{j,k,l}\tilde\varphi_{j+1,l},
\end{equation}
and using (\ref{eqn:scaling_biorth}), (\ref{eqn:refinement}) and (\ref{eqn:dual_refinement}), one obtains that the filter $(h_{j,k,l})$ and the dual filter $(\tilde h_{j,k,l})$ must satisfy the relation
\begin{equation}
\label{eq:h_biorth}
\sum_{l} h_{j,k,l}\tilde h_{j,k',l} = \delta_{k,k'}, \ \ \text{ for } j\in\mathcal J, \ \ k,k'\in\mathcal K(j).
\end{equation}
\subsubsection{Nested set of partitionings}
In order to be able to define scaling functions on $L^2(X,\Sigma,\mu)$, the next structure that is needed is a \emph{set of partitionings.} In the following definition, we assume that $\mathcal K(j) \subset \mathcal K(j+1)$.
\begin{definition}
\label{defn:partitionings}
A \emph{nested set of partitionings} is a collection $\{S_{j,k}\colon j\in\mathcal J, k\in\mathcal K\}$ of subsets of $X$ that satisfy
\begin{enumerate}
\item For all $j \in \mathcal J$, the collection $\{S_{j,k}\colon k\in \mathcal K(j)\}$ is disjoint and $\displaystyle X=\overline{\bigcup_{k\in\mathcal K(j)}S_{j,k}}$,
\item $S_{j+1,k}\subset S_{j,k}$,
\item For all $j\in \mc J$, and $k\in \mc K(j+1)$, there exists a $k' \in \mc K(j)$ such that $S_{j+1,k}\subset S_{j,k'}$,
\item For a fixed $k \in \mathcal K(j_0)$, the infinite intersection $\displaystyle\bigcap_{j>j_0}S_{j,k}$ is a single point set, whose unique element will be denoted by $x_k$.
\end{enumerate}
\end{definition}
\subsection{Synthesizing the scaling functions with the cascade algorithm}
\label{subsec:scaling_fun_cascade}
Having defined a set of partitionings $\{S_{j,k}\colon j\in\mathcal J, k\in\mathcal K(j)\}$, and a finite filter $\{h_{j,k,l}\colon j\in\mathcal J, k\in\mathcal K(j), l\in\mathcal K(j+1)\}$, one is now ready to synthesize the scaling function $\varphi_{j_0,k_0}$. Initially, setting $(\lambda_{j_0,k})$ to a Kronecker sequence as
\[
\lambda_{j_0,k} = \delta_{k,k_0}, \text{ for } k\in\mathcal K(j),
\]
we define the collection of sequences $(\lambda_{j,k})_{k\in \mathcal K(j)}$ for $j>j_0$, using the following recursive formula:
\[
\lambda_{j+1,l} = \sum_{k\in\mathcal K(j,l)}h_{j,k,l}\lambda_{j,k}.
\]
Then a sequence of functions $\left(f^j_{j_0,k_0}\right)_j$ for $j\geq j_0$ is defined as
\[
f^j_{j_0,k_0} = \sum_{k\in\mathcal K(j)}\lambda_{j,k}\chi_{S_{j,k}},
\]
which also satisfies
\begin{equation}
\label{f_refinement}
f^j_{j_0,k_0} = \sum_{l\in \mathcal K(j_0+1)}h_{j_0,k_0,l}f^j_{j_0+1,l}.
\end{equation}
Now the function $\varphi_{j_0,k_0}$ is defined to be
\[
\varphi_{j_0,k_0}=\lim_{j\to\infty} f^j_{j_0,k_0},
\]
assuming the limit exists in $L^2$.
\subsection{Wavelets}
\label{subsec:wavelets}
Given a multiresolution analysis $\mbf M = \{V_j:j\in\mathcal J\subset \mathbb Z\}$, and a dual $\widetilde{\mbf M} = \{\widetilde V_j:j\in\mathcal J\subset \mathbb Z\}$, the \emph{wavelet subspace} $W_j$ is the complement of $V_j$ in $V_{j+1}$ which is orthogonal to $\widetilde V_j.$ The \emph{wavelets} $\psi_{j,m}$ are the functions that span $W_j$.
\begin{definition}
\label{defn:wavelets}
Let $\mathcal M(j) = \mathcal K(j+1)\setminus \mathcal K(j)$ be the index set complementing $\mathcal K(j)$ in $\mathcal K(j+1)$, and let $\{\psi_{j,m}\colon j\in\mathcal J, m\in\mathcal M(j)\}\subset V_{j+1}$ be a collection of functions, and $W_j$ be the closure of its span. Then the collection $\{\psi_{j,m}\colon m\in\mathcal M(j) \}$ is a set of \emph{wavelet} functions if the following are satisfied.
\begin{enumerate}
\item $V_{j}\bigcap W_j = \{0\}$ and $W_{j} \perp \widetilde V_{j}$.
\item If $\mathcal J = \mathbb Z$, the set $\{\psi_{j,m}\colon j\in\mathcal J, m\in\mathcal M(j) \}$ is a Riesz basis for $L^2$.
If $\mathcal J = \mathbb N$, the set $\{\psi_{j,m}\colon j\in\mathcal J, m\in\mathcal M(j) \}\bigcup \{\varphi_{0,k}/{\|\varphi_{0,k}\|}\colon k\in\mathcal K(0)\}$ is a Riesz basis for $L^2$.
\end{enumerate}
\end{definition}
The Riesz bases mentioned above will be referred as \emph{wavelet bases} from now on.
\subsubsection{Dual wavelets}
Given a wavelet basis $(\psi_{j,m})$, there exists a corresponding dual Riesz basis (as explained in Subsection \ref{subsec:bases_review}), whose elements will be denoted by $(\tilde \psi_{j,m})$, and for a given $j$, the closed span of $(\tilde \psi_{j,m})$ will be denoted by $\widetilde W_j$. The dual wavelets satisfy
\[
\langle \psi_{j,m},\tilde \psi_{j',m'} \rangle = \delta_{m,m'}\delta_{j,j'},
\]
by their definition. The dual wavelet subspaces also satisfy $\widetilde W_j \bigcap \widetilde V_j = \{0\}$ and $\widetilde W_j \perp V_j$.
The dual wavelets can also be defined by reversing the roles of $\mbf M$ and $\widetilde{\mbf M}$, in Definition \ref{defn:wavelets}.
\subsection{Sets of biorthogonal filters}
\label{subsec:biorthogonality}
The finite filters $(h_{j,k,l})$ and $(\tilde h_{j,k,l})$ were defined in Subsection \ref{subsec:ref_eqn}. These filters take place in the refinement equations
\[
\varphi_{j,k} = \sum_{l\in\mathcal K(j+1)}h_{j,k,l}\varphi_{j+1,l},
\]
and
\[
\tilde\varphi_{j,k} = \sum_{l\in\mathcal K(j+1)}\tilde h_{j,k,l}\tilde \varphi_{j+1,l}.
\]
Now similarly, since $\psi_{j,m}\in V_{j+1}$ and $\tilde \psi_{j,m} \in \widetilde V_{j+1}$, we can define the filters $g_{j,m,l}$ and $\tilde g_{j,m,l}$ to be the filters that satisfy
\[
\psi_{j,m} = \sum_{l\in\mathcal K(j+1)}g_{j,m,l}\,\varphi_{j+1,l},
\]
and
\[
\tilde\psi_{j,m} = \sum_{l\in\mathcal K(j+1)}\tilde g_{j,k,l}\,\tilde \varphi_{j+1,l}.
\]
The functions $(\varphi_{j,k})$, $(\tilde \varphi_{j,k})$, $(\psi_{j,m})$ and $(\tilde\psi_{j,m})$ satisfy the biorthogonality relations
\begin{equation}
\label{eqn:func_biorth}
\begin{array}{lcl}
\langle \varphi_{j,k},\tilde \varphi_{j,k'} \rangle &=& \delta_{k,k'},\\
\langle \psi_{j,m},\tilde \psi_{j,m} \rangle &=& \delta_{m,m'},\\
\langle \varphi_{j,k},\tilde \psi_{j,m} \rangle &=& 0,\\
\langle \psi_{j,m},\tilde \varphi_{j,k} \rangle &=& 0.\\
\end{array}
\end{equation}
As a consequence of the above relations, the corresponding filters must satisfy
\begin{equation}
\label{eqn:biorthogonal_filters}
\begin{array}{rcl}
\disp \sum_{l\in\mathcal K(j+1)}\, h_{j,k,l}\tilde h_{j,k',l} & = & \delta_{k,k'}, \\
\disp \sum_{l\in\mathcal K(j+1)}\, g_{j,m,l}\tilde g_{j,m',l} &=& \delta_{m,m'} , \\
\disp \sum_{l\in\mathcal K(j+1)}\, g_{j,m,l}\tilde h_{j,k,l} &=& 0 ,\\
\disp \sum_{l\in\mathcal K(j+1)}\, h_{j,k,l}\tilde g_{j,m,l} &=& 0.
\end{array}
\end{equation}
\begin{definition}
A set of finite filters $\{h,\tilde h, g, \tilde g\}$ is called a set of {\rm\emph{biorthogonal filters}}, if the relations in {\rm(\ref{eqn:biorthogonal_filters})} are satisfied.
\end{definition}
\begin{remark}
The collection of wavelet and scaling functions that satisfy {\rm\eqref{eqn:func_biorth}} results in finite filters that satisfy {\rm(\ref{eqn:biorthogonal_filters})}. But also, thinking in the reverse direction, if one starts with a set of biorthogonal filters, one can obtain the wavelet and scaling functions using the cascade algorithm of Subsection {\rm\ref{subsec:scaling_fun_cascade}}, and they will have the desired biorthogonal relations of {\rm(\ref{eqn:func_biorth})}. The purpose of the lifting scheme is actually to design a set of biorthogonal filters and make succesive improvements on them without disturbing their biorthogonality.
\end{remark}
\subsection{Fast wavelet transforms}
The fast wavelet transform is an efficient algorithm that finds the expansion coefficients in the $N$-level wavelet expansion
\begin{equation}
\label{eqn:fast_wavelet}
f = \sum_{k\in\mathcal K(0)}\lambda_{0,k}\,\varphi_{0,k} + \sum_{j = 0}^{N-1}\sum_{m\in\mathcal M(j)} \gamma_{j,m}\, \psi_{j,m},
\end{equation}
for a positive integer $N$ and $f \in V_{N}$, by exploiting the refinement relations.
If we take a function $f$ in $V_N$, it can be written as a linear combination of the scaling functions of $V_N$ as
\begin{equation}
\label{eqn:finest_level_expansion}
f = \sum_{k\in\mathcal K(N)}\lambda_{N,k}\,\varphi_{N,k}.
\end{equation}
At the beginning of a fast wavelet transform, it is assumed that the scaling function coefficients $(\lambda_{N,k})_k$ at this finest level are given. Since $W_{N-1}$ is a complement of $V_{N-1}$ in $V_{N}$, $f$ can be written as $f = f_a + f_d$ with $f_a \in V_{N-1}$ and $f_d\in W_{N-1}$. If we expand $f_a$ and $f_d$ in terms of the scaling functions and wavelets, we get
\begin{align*}
f &= \sum_{k\in\mathcal K(N)}\lambda_{N,k}\,\varphi_{N,k}\\
&= \sum_{k\in\mathcal K(N-1)}\lambda_{N-1,k}\,\varphi_{N-1,k} + \sum_{m\in\mathcal M(N-1)}\gamma_{N-1,m}\,\psi_{N-1,m}.
\end{align*}
The key observation at this point is that the coefficient sequences $(\lambda_{N-1,k})_k$ and $(\gamma_{N-1,m})_m$ can be obtained by applying filters $\tilde h$ and $\tilde g$ to $(\lambda_{N,k})_k$, respectively. That is to say, the relations
\begin{align*}
\lambda_{N-1,k} = \sum_{l\in\mathcal K(N)}\tilde h_{N,k,l}\,\lambda_{N,l}
\end{align*}
and
\begin{align*}
\gamma_{N-1,k} = \sum_{l\in\mathcal K(N)}\tilde g_{N,k,l}\,\lambda_{N,l}
\end{align*}
can be obtained as a result of the biorthogonality relations. Recursive application of this relation to the $(\lambda_{N-i,k})$ after each step results in (\ref{eqn:fast_wavelet}).
The inverse transform of the above step is given by
\begin{align*}
\lambda_{N,l} = \sum_{k\in\mathcal K(N-1)} h_{N-1,k,l}\,\lambda_{N-1,k}
+
\sum_{m\in\mathcal M(N-1)} g_{N-1,m,l}\,\gamma_{N-1,m}.
\end{align*}
We recall that, in the present setting, it is assumed that all the summations in the above equations are finite, since the filters are assumed to be finite.
\section{The lifting scheme}
The lifting scheme is the framework through which one can obtain new biorthogonal filters from old ones, and tailor them according to the desired smoothness and vanishing moment properties of the corresponding scaling functions and wavelets.
\subsection{Operator notation}
Continuing to follow \cite{sweldens_siam}, we will introduce the operator notation corresponding to the filters of the previous section. The operator notation will allow a much compact expression for the lifting scheme.
First we will give a general definition for an operator $H$ and its adjoint $H^*$, corresponding to a filter $(h_{k,l})$. Then the definitions specific to present case will follow.
Let $\mathcal I_1$ and $\mathcal I_2$ be two index sets that are at most countable, and let $h = \{h_{k,l}\colon k\in\mathcal I_2, l\in\mathcal I_1\}$ be a filter with real coefficients. Let $\ell^2(\mathcal I_1)$ and $\ell^2(\mathcal I_2)$ be the spaces of the square summable, real valued discrete functions, defined on the index sets.
The operator $H$ corresponding to the filter $(h_{k,l})$ is be defined to be
\begin{align}
\label{eqn:operator}
H\colon & \ell^2(\mathcal I_1) \rightarrow \ell^2(\mathcal I_2) \notag
\\
& \left(a_{k}\right)_{m\in\mathcal I_1}
\mapsto \left(\sum_{l\in\mathcal I_1} h_{k,l}a_{l}\right)_{k\in\mathcal I_2}.
\end{align}
The adjoint operator of $H$, which is denoted by $H^*$, is an operator from $\ell^2(\mathcal I_2)$ to $\ell^2(\mathcal I_1)$ given by
\begin{align}
\label{eqn:operator_adjoint}
H^*\colon & \ell^2(\mathcal I_2) \rightarrow \ell^2(\mathcal I_1) \notag
\\
& \left(a_{k}\right)_{m\in\mathcal I_2}
\mapsto \left(\sum_{k\in\mathcal I_2} h_{k,l}a_{k}\right)_{l\in\mathcal I_1}.
\end{align}
Given an operator $H$, the adjoint operator is also the unique operator that satisfies
\[
\langle x, H y \rangle = \langle H^* x, y \rangle
\]
for all $x \in \ell(\mathcal I_1)$ and $y\in\ell(\mathcal I_2)$.
The above notation will be used for the rest of the section. Note that, in (\ref{eqn:operator}), the summation runs over the first index of $(h_{k,l})$, whereas in (\ref{eqn:operator_adjoint}), it runs over the second index. For the case of matrices, the adjoint operator corresponds to the transpose, and the order of indexes for the filters has been set accordingly.
Now, let us fix a level $j \in \mathcal J$. We have three index sets $\mathcal K(j+1)$, $\mathcal K(j)$ and $\mathcal M(j)$, which satisfy
\[
\mathcal K(j+1) = \mathcal K(j)\, \cup\, \mathcal M(j).
\]
Let us consider the spaces $\ell^2\left(\mathcal K(j+1)\right)$, $\ell^2\left(\mathcal K(j)\right)$ and $\ell^2\left(\mathcal M(j)\right)$. We define the operators $\tilde H_j$, $\tilde G_j$, $H_j^*$ and $G_j^*$ as follows:
\begin{align*}
\tilde H_j\colon &\ell^2(\mathcal K(j+1)) \rightarrow \ell^2(\mathcal K(j))
\\
&\left(a_k\right)_{k\in\mathcal K(j+1)} \mapsto \left(\sum_{l\in\mathcal K(j+1)}\tilde h_{j,k,l}\,a_l\right)_{k\in\mathcal K(j)},
\\
&\\
\tilde G_j\colon & \ell^2(\mathcal K(j+1)) \rightarrow \ell^2(\mathcal M(j))
\\
& \left(a_k\right)_{k\in\mathcal K(j+1)}
\mapsto
\left(
\sum_{l\in\mathcal K(j+1)}\tilde g_{j,m,l}a_{l}
\right)_{m\in\mathcal M(j)},
\end{align*}
and similarly,
\begin{align*}
H^*_j\colon & \ell^2(\mathcal K(j)) \rightarrow \ell^2(\mathcal K(j+1))
\\
& \left(a_{k}\right)_{k\in\mathcal K(j+1)}
\mapsto
\left(\sum_{l\in\mathcal K(j)} h_{j,k,l}a_{l}\right)_{k\in\mathcal K(j)}
,
\\
&\\
G^*_j\colon & \ell^2(\mathcal M(j)) \rightarrow \ell^2(\mathcal K(j+1))
\\
& \left(a_{m}\right)_{m\in\mathcal M(j+1)}
\mapsto \left(\sum_{l\in\mathcal K(j)} g_{j,k,l}a_{l}\right)_{k\in\mathcal K(j)}
.
\end{align*}
\subsection{Fast wavelet transform and biorthogonality with the operator notation}
Let us denote the sequence $(\lambda_{j,k})_{k\in\mathcal K(j)}$ simply by $\lambda_j$, and similarly $(\gamma_{j,m})_{m\in\mathcal M(j)}$ by $\gamma_j$. The fast wavelet transform is the process that starts with $\lambda_N$, and continues as
\begin{align*}
\lambda _N &\mapsto (\lambda_{N-1},\gamma_{N-1})
\\
&\mapsto (\lambda_{N-2},\gamma_{N-2},\gamma_{N-1})
\\
& \ \ \vdots\\
&\mapsto (\lambda_{0},\gamma_{0},\gamma_{1},\cdots,\gamma_{N-2},\gamma_{N-1}),
\end{align*}
where
\[
\lambda_{j} = \tilde H_{j}\lambda_{j+1} \ \ \text{ and } \ \ \ \gamma_{j} = \tilde G_{j}\gamma_{j+1},
\]
at each step.
The inverse of each step can be obtained by the operators $H^*$ and $G^*$ as
\[
\lambda_{j+1} = H_{j}^*\lambda_{j} + G_{j}^*\lambda_{j}.
\]
The biorthogonality relations (\ref{eqn:biorthogonal_filters}) in Subsection \ref{subsec:biorthogonality} can be written with the operator notation as
\begin{align}
\label{eqn:biorthogonality_op1}
\tilde G_j H^*_j = \tilde H_j G^*_j = 0 \\
\label{eqn:biorthogonality_op2}
\tilde H_j H^*_j = \tilde G_j G^*_j = I,
\end{align}
where $I$ is the identity operator. From these, it follows that
\[
\tilde H^*_j H_j + G^*_j \tilde G_j = I,
\]
provided the union of the ranges of the operators $H^*_j$ and $G^*_j$ span entire $\ell^2(\mathcal(K(j+1))$. In finite dimensions, it is enough that $|\mc K(j+1)| = |\mc K(j)| + |\mc M(j)|$, in order for this to be satisfied.
\subsection{Obtaining a new biorthogonal set of filters from old ones}
\subsubsection{Lifting}
The key point of the lifting scheme is to obtain new biorthogonal filters from old ones without losing the biorthogonality.
\begin{theorem}
\label{thm:lifting_primal}
Let ${H^{\text{\rm old}}, \tilde H^{\text{\rm old}}, G^{\text{\rm old}}, \tilde G^{\text{\rm old}}}$ be a set of biorthogonal filter operators. Then the following gives a new set of biorthogonal filter operators:
\begin{align*}
H_j &= H_j^{\,\text{\rm old}},\\
\tilde H_j &= \tilde H_j^{\text{\rm old}} + S \tilde G_j^{\text{\rm old}},\\
G_j &= G_j^{\text{\rm old}}- S^*H_j^{\text{\rm old}},\\
\tilde G_j &= \tilde G_j^{\text{\rm old}},
\end{align*}
where $S$ is any operator from $\ell^2(\mc M(j))$ to $\ell^2(\mc K(j))$.
\end{theorem}
\begin{proof}
For (\ref{eqn:biorthogonality_op1}) we have
\begin{align*}
\tilde G_j H^*_j &= \tilde G_j^{\,\text{old}} H_j^{\,\text{old}} = 0,\\
\tilde H_j G^*_j
&=
\left( \tilde H_j^{\,\text{old}} + S \tilde G_j^{\,\text{old}}\right) \Big(G_j^{\,\text{old}}- S^*H_j^{\,\text{old}}\Big)^*\\
&=
\left( \tilde H_j^{\,\text{old}} + S \tilde G_j^{\,\text{old}}\right) \Big(G_j^{* \, \text{old}}- H_j^{* \, \text{old}}S\Big)\\
& =
\tilde H_j^{\,\text{old}} G_j^{* \, \text{old}}
-\tilde H_j^{\,\text{old}} H_j^{* \, \text{old}}S
+S \tilde G_j^{\,\text{old}}G_j^{* \, \text{old}}
-S \tilde G_j^{\,\text{old}} H_j^{* \, \text{old}}S\\
&=0 -S + S +0= 0.
\end{align*}
Similarly, one can verify that (\ref{eqn:biorthogonality_op2}) also holds.
\end{proof}
The modification step to the original operators is called a \emph{lifting step}. This modification changes $\tilde H$ and $G^*$, but it does not change $H^*$ and $\tilde G$. This implies that, after such a modification, the scaling functions remain the same, but the dual scaling functions and the wavelets change. The dual wavelets also change, since the dual scaling functions change, although the coefficients $(\tilde g_{j,m})$ in their refinement relation stay the same.
\subsubsection{Dual lifting}
There also exists another version of Theorem \ref{thm:lifting_primal}, in which the filters $H^*$ and $\tilde G$ are modified and $\tilde H$ and $G^*$ remain unchanged, which we call as the \emph{dual lifting}.
\begin{theorem}
\label{thm:lifting_dual}
Let ${H^{\text{\rm old}}, \tilde H^{\text{\rm old}}, G^{\text{\rm old}}, \tilde G^{\text{\rm old}}}$ be a set of biorthogonal filter operators. Then the following gives a new set of biorthogonal filter operators:
\begin{align*}
H_j &= H_j^{\,\text{\rm old}} + R^*G_j^{\text{\rm old}},\\
\tilde H_j &= \tilde H_j^{\text{\rm old}},\\
G_j &= G_j^{\text{\rm old}},\\
\tilde G_j &= \tilde G_j^{\text{\rm old}} - R \tilde H_j^{\text{\rm old}},,
\end{align*}
where $R$ is any operator from $\ell^2(\mc K(j))$ to $\ell^2(\mc M(j))$.
\end{theorem}
\subsection{Heuristics for a good discrete wavelet transform}
\label{subsec:heuristics}
Now, we can focus on a {(single-level) discrete wavelet transform} $T_j$, which maps a given discrete function $\lambda_{j+1}\in\ell^2(\mathcal (K(j+1)))$ into two discrete functions $(\lambda_j, \gamma_j)$, where $\lambda_{j}\in\ell^2(\mathcal (K(j)))$ and $\gamma_{j}\in\ell^2(\mathcal (M(j)))$ as
\begin{align*}
T_j: \ell^2{\mc K(j+1)} &\rightarrow \ell^2{(\mc K(j))} \times \ell^2{(\mc M(j))}\\
\lambda_{j+1} &\mapsto (\lambda_j, \gamma_j).
\end{align*}
Here, the discrete functions $\lambda_j$ and $\gamma_j$ are interpreted as approximation and detail functions, respectively.
Now, we can list the properties generally expected from a wavelet transform. In the following, we will implicitly identify the discrete function
$(\lambda_j)$, with the corresponding function $f\in L^2(X,\Sigma,\mu)$ given by
\(
f = \sum_{k}\lambda_{j,k}\varphi_{j,k},
\)
when talking about concepts like \emph{smoothness}.
\begin{enumerate}[W1:]
\item \label{wt1} If $\lambda_{j+1}$ belongs to a class of smooth functions to be specified (most commonly, a class of polynomials up to a certain degree), then the detail function $\gamma_j$ is expected to be exactly zero.
\item \label{wtapp} $\lambda_{j}$ is expected to be an approximation of $\lambda_{j+1}$, in the sense that setting $\gamma_j =0 $ and taking the inverse of $(\lambda_j, 0)$ under $T_j$ should give an approximation of $\lambda_{j+1}$.
\item \label{wtcont} The transform is expected to be stable, i.e., a small change in $\lambda_{j+1}$ should correspond to small changes in $\lambda_j$ and $\gamma_j$, both in the transform and its inverse.
\item \label{wtlocal} The transform is expected to be local. Changes in a localized area of the function should only affect a corresponding localized region of the transformed functions.
\item \label{wtsparse}As a consequence of the above conditions, if $\lambda_{j+1}$ is locally well approximated by the class of smooth functions, then many entries of the detail output $\gamma_j$ will be close to zero. This is the reason for the \emph{sparsity} property of the wavelet transforms, a key point of their success.
\end{enumerate}
\subsection{Tailoring a wavelet transform in successive lifting steps}
In order to design a wavelet transform, one can start with a very simple transform, and in successive steps, improve it to meet the guidelines of Subsection \ref{subsec:heuristics}. In doing so, Theorems \ref{thm:lifting_primal} and \ref{thm:lifting_dual} will be utilized.
\subsubsection{The lazy wavelet transform}
The lazy wavelet transform simply splits the given signal $\lambda_{j+1}$ into two, i.e. $\lambda_j$ and $\gamma_j$ are just restrictions of $\lambda_{j+1}$ to $\mc K(j)$ and $\mc M(j)$ respectively,
\begin{gather*}
T_j^{\text{lazy}}: \ell^2{\mc K(j+1)} \rightarrow \ell^2{(\mc K(j))} \times \ell^2{(\mc M(j))} \\
\lambda_{j+1} \mapsto (\lambda_{j+1}\rst{\mc K(j)}, \ \lambda_{j+1}\rst{\mc M(j)}).
\end{gather*}
In other words, the corresponding filters are given by $h^{\text{lazy}}_{j,k,l} = \delta_{k,l}$ and $g^{\text{lazy}}_{j,m,l} = \delta_{m,l}$.
We will denote the corresponding filter operators as $\tilde H_j^\text{lazy}$ and $\tilde G_j^\text{lazy}$, and use the ordered pair notation for the operator as
\[
T_j^{\text{lazy}} = (\tilde H_j^\text{lazy} , \tilde G_j^\text{lazy}).
\]
It is easy to see that the lazy wavelet transform does not satisfy property W\ref{wt1}. There is no reason to expect that the signal would have values close to zero on the complementary grid $\mc M(j)$, which are typically uniformly scattered through the initial grid $\mc K(j+1)$. However, the lazy wavelet transform is very helpful as an initial step in the lifting framework, to be followed by a sequence of improvement steps.
\subsubsection{Prediction}
The first improvement to the lazy wavelet transform comes with the help of a prediction operator. Let $\lambda^{(0)}_{j} = \tilde H_j^\text{lazy}\lambda_{j+1}$ and $\gamma^{(0)}_j = \tilde G_j^\text{lazy}\lambda_{j+1}$ be the outputs of the lazy wavelet transform. The output of the prediction operator is a function of $\lambda^{(0)}_{j}$ and under the smoothness assumption, it approximates the signal $\gamma^{(0)}_{j}$
\begin{gather*}
P\colon \ell^2(\mc K(j)) \rightarrow \ell^2(\mc M(j))\\
P \lambda^{(0)}_{j} \approx \gamma^{(0)}_{j}.
\end{gather*}
The improved transform now becomes
\begin{gather*}
T^{(1)}_j\colon \ell^2 ({\mc K(j+1)}) \rightarrow \ell^2 ({\mc K(j)})\times \ell^2({\mc M(j)})\\
\lambda_{j+1} \mapsto (\lambda^{(0)}_{j}, \gamma^{(0)}_{j} - P\lambda^{(0)}_{j})
\end{gather*}
If the smoothness assumptions are satisfied, and the prediction successfully satisfies $P \lambda^{(0)}_{j} \approx \gamma^{(0)}_{j}$, then the desired property W\ref{wt1}, follows immediately, as intended. The new filter operators are $\tilde H^{(1)}_j = \tilde H^{lazy}_j$ and $\tilde G^{(1)}_j = \tilde G^{lazy}_j - P H^{lazy}_j$. The complete set of biorthogonal filters can be obtained by Theorem \ref{thm:lifting_dual}.
\subsubsection{Locality of the Prediction}
In order to have the property W\ref{wtlocal}, we must impose a condition on the prediction operator $P$: the $m$th entry of $P\lambda^{(0)}_{j}$ should depend only on the \emph{neighbors} of the $m$th element of the complimentary grid $\mc M(j)$, noting that $\mc K(j) \bigcup \mc M(j) = \mc K(j+1)$. Without this condition, the transform would not be local, and would not possess the advantages of wavelet transforms.
\subsubsection{The Update Step}
At this point, we have a transform $T^{(1)}$ that satisfies property W\ref{wt1}. However, there is something unsatisfying about the \emph{approximation output} $\lambda^{(1)}_{j}:= \tilde H^{(1)}_j \lambda_{j+1}$, which is still simply a restriction of $\lambda_{j+1}$ to the subgrid $\mc K(j)$. This constitutes a violation of property W\ref{wtapp}. In order to see this, one can take an extreme example for $\lambda_{j+1}$, such as
\[ \lambda_{j+1,k}
= \left\{\begin{array}{rcl}
0 &\text{if}& k\in \mc K(j) \\
1 &\text{if}& k\in\mc M(j)
\end{array}
,\right.
\]
whose reconstruction after setting the detail coefficients $\gamma_j$ to $0$ will result in the $0$ function, which is obviously not a good approximation for the initial $\lambda_{j+1}$.
As a result, a new step is required, which will ensure that the local averages of the function reconstructed with $\lambda_{j}$ will match those of $\lambda_{j+1}$. It can be accomplished by means of an operator $U$ that is local, and acts on the detail output $\gamma^{(1)}_{j}:= \tilde G^{(1)}_j \lambda_{j+1}$ of the previous step. The new operators will be $\tilde H^{(2)}_j : = \tilde H^{(1)}_j + U \tilde G^{(1)}_j$, and $\tilde G^{(2)}_j := \tilde G^{(1)}_j$. The condition that is desired to be satisfied by the output $\lambda^{(2)}_{j}$ of $\tilde H^{(2)}_j$ is to preserve the weighted sum of the coefficients as
\[
\sum_{k\in\mc K(j+1)} \lambda_{j+1,k}\,\mu(S_{j+1,k})=\sum_{k\in\mc K(j)} \lambda^{(2)}_{j,k}\,\mu(S_{j,k}).
\]
The modifications to get a complete set of biorthogonal filters is given by Theorem \ref{thm:lifting_primal}. Note also that the update step for the primal filters corresponds to a prediction step for the dual filters.
\vspace{.6cm}
\noindent There can be as many ``predict'' and ``update'' steps as one desires, to be added with similar rules. However, in general additional steps with improved predictions requires the use of filters with a higher number of nonzero coefficients. This, in turn, results in less localized wavelets and scaling functions.
\subsubsection{}
In this chapter, we gave an overview of wavelets and the lifting scheme, a flexible framework that enables designing wavelets on unstructured domains. This chapter was essentially a preparation for the next chapter, in which we give a concrete construction that works for arbitrary subsets of $\mathbb R^2$ and $\mathbb R^3$.
\section{Appendix: Orthogonal bases, Riesz bases and frames}
\label{subsec:bases_review}
\subsection{Riesz bases and orthogonal bases}
In a Hilbert space \(\mathcal H \), a basis $(x_n)$ is called a \emph{Riesz basis} if there exist two constants $A$ and $B$ such that
\begin{equation}
\label{eqn:riesz_basis}
A\sum{|a_n|^2}\leq \left\| \sum{a_n x_n}\right\|^2 \leq B\sum{|a_n|^2},
\end{equation}
for all scalar sequences $(a_n)$. $A$ and $B$ are called the Riesz basis constants.
An \emph{orthogonal basis} is a special Riesz basis in which $A=B=1$. In this case, the basis satisfies
\[
\langle x_n, x_{n'}\rangle = \delta_{n,n'},
\]
and for an $f\in\mathcal H$ , the scalar coefficients $(a_n)$ in the basis expansion
\[
f = \sum_{n}a_n x_n
\]
can be computed by taking inner products with the corresponding basis elements:
\[
\label{ortho_coef_functional}
a_n = \langle f,x_n\rangle.
\]
\subsection{Biorthogonal bases}
For each Riesz basis \( (x_n) \) that is not orthogonal, there exist another Riesz basis \( (\tilde x_n) \) that satisfies
\begin{equation*}
\label{eqn:biorthogonal}
\langle x_n, \tilde x_{n'}\rangle = \delta_{n,n'},
\end{equation*}
and therefore one has the relation
\[
a_n = \langle f,\tilde x_n\rangle,
\]
useful for the computation of the coefficients. The bases \( ( x_n) \) and \( (\tilde x_n) \) are called \emph{biorthogonal} bases. For the proof of existence of a biorthogonal basis for every Riesz basis, we refer to \cite{christensen08}.
\subsection{Frames}
A frame is a countable collection of vectors $(e_n)$ in a Hilbert space $\mathcal H$, for which there exists constants $A$ and $B$ such that
$$A\|f\|^2 \leq \sum_n{|\langle f,e_n \rangle |^2} \leq B\|f\|^2.$$
Frames are generally regarded as redundant families of vectors: a frame is not necessarily linearly independent, and a frame expansion of a function is not unique.
\chapter{Randomized domain-adapted wavelets}
In this chapter we give details about our construction of two and three dimensional wavelets on an arbitrary domain, and study their properties.
This construction constitutes one of our main contributions to the thesis.
\section{The structures needed for the construction}
In order to start defining and shaping filter operators within the lifting framework, we first need the embedded index sets (grids) $\mc K(j)$ for $j\in \mc J$, as defined in Chapter 2. The scale index set will be $\mc J = \mathbb N$, but in practice we will only consider the finite subset $\mc J = \{0,1,2, \cdots N\}$ for some $N \in \mathbb N$. After constructing $(\mc K(j))$, we can give the corresponding detail index sets $(\mc M(j))$ and the nested partitionings $\left(S_{j,k}\right)$. We will also need \emph{neighbor, sibling, parent} and \emph{child} relations on $(\mc K(j))$, to be used in the \emph{prediction} and \emph{update} steps of the lifting.
\subsection{The domain}
\label{subsec:domain}
The domain $\mc X$, which will be an input to the algorithm, is a subset of $\mathbb Z^2$ or $\mathbb Z^3$. For the case of $\mathbb Z^3$, the domain $\mc X$ actually corresponds to the intersection of a set $X\subset \mathbb R^3$ with the discrete grid $\{(n_x d_x, n_y d_y, n_z d_z)\colon (n_x,n_y,n_z)\in\mathbb Z^3\}$, where $d_x$, $d_y$ and $d_z$ are the resolution parameters in the corresponding directions. Each $n\in\mc X$ represents the rectangular region
\begin{align}
S_n = &[n_x d_x - d_x/2, n_x d_x + d_x/2]\notag\\
&\times [n_y d_y - d_y/2, n_y d_y + d_y/2]\notag\\
&\times [n_z d_z - d_z/2, n_z d_z + d_z/2]\label{eqn:defn_sn},
\end{align}
which is called a \emph{voxel}. A similar definition holds for two dimensions, in which the set $S_n$ is called a \emph{pixel}. An example of a discrete domain and the corresponding pixels is displayed in Figure \ref{fig:discrete_dom}.
\begin{figure}
\centering
\includegraphics[width=12cm]{./figures/chp3/the_discrete_domain.eps}
\caption{A two dimensional discrete domain $\mc X$, whose elements are displayed as dots, and the corresponding pixels $(S_n)$ are displayed as the squares surrounding the dots.}
\label{fig:discrete_dom}
\end{figure}
\subsubsection{The measure}
The measure of each set $S_n$, denoted by $\mu(S_n)$, is another required input. In the regular setting, one takes $\mu(S_n)=1$ for all $n\in\mc{X}$, but in some applications it may be more meaningful to weight each voxel according to some related measure.
Any set that we shall consider in the finite setting will be a union of the sets $(S_n)$, i.e sets of the form
\[
S = \bigcup_{n\in \mc{X'}}S_n
\]
for some $\mc{X'}\subset \mc X$, whose measure is given by
\[
\mu(S) = \sum_{n\in\mc{X'}}\mu(S_n).
\]
\subsubsection{Neighboring elements}
We define two elements $n, m \in \mc X\subset \mbb Z^3$ to be \emph{neighbors}, if
\[
|n_x-m_x|+|n_y-m_y| + |n_y-m_y| = 1,
\]
where $n = (n_x,n_y,n_z)$ and $m = (m_x, m_y, m_z)$. This is equivalent to the condition that the rectangular prisms bounding regions $S_n$ and $S_m$ share a face. A similar definition also holds for two dimensions.
\subsection{Embedded grids and the random merging algorithm}
We choose an $N\in \mathbb N$, which is the number of decomposition levels. We take the scale index set $\mc J$ to be $\{0,1,2,\cdots,N\}$, where $j=N$ represents the scale of highest resolution available.
We set $\mc K(N)$ to be equal to the discrete domain $\mc X$. For every $k\in\mc K(N)$, we denote the set of {neighbors} of $k$ with \(\operatorname{Nbr}(N,k)\subset \mc K(N) \), based on the neighborhood structure of $\mc X$. Also we set $S_{N,k} := S_k$, where $S_k$ is the set defined in (\ref{eqn:defn_sn}). After defining $\mc K(N)$, we will define $\mc K(N-1), \mc K(N-2), \cdots ,\mc K(0)$, and the corresponding neighborhood structure with a random merging algorithm, inductively.
For a given $j$, assume we have $\mc K(j)$, the sets $(S_{j,k})_{k\in\mc K(j)}$, and a collection of sets $\{\opnm{Nbr}(j,k)\colon k\in\mc K(j)\}$ of the neighbors for each element.
\begin{enumerate}
\item Start with $\mc K(j-1) = \mc M(j-1) = \emptyset$.
\item Compute the centroids of all sets $S_{j,k}$, which is defined as
\[
C(S_{j,k}) = \frac{1}{\mu(S_{j,k})}\sum_{S_{N,k}\subset S_{j,k}}(k_x d_x, k_y d_y, k_z dz)\mu(S_{N,k}),
\]
which gives a vector $C(S_{j,k})\in\mathbb R^3$.
\item \label{step:randperm} Declare all elements of $\mc K(j)$ as available, and put them in a linear order that is determined by a random permutation,
\item \label{step:togo}For the first available $k\in\mc K(j)$, select at most $p$ of the available neighbors, ${s_1,s_2,\cdots, s_q}\in\mc K(j)$, where $q\leq p$,
in a random way, with probability of being selected being inversely proportional to the distance between the centroids,
\item Add $k$ to $\mc K(j-1)$ and each of $s_1, s_2, \cdots, s_q$ to $\mc M(j-1)$,
\item
Mark each of ${k,s_1,s_2,\cdots, s_q}\in\mc K(j)$ to be \emph{unavailable}, and declare them to be \emph{siblings}, a relationship denoted as
\[
\opnm{Sib}(j,k) = \opnm{Sib}(j,s_1) = \cdots =\opnm{Sib}(j,s_q)= \{k,s_1,s_2,\cdots,s_q\},
\]
\item Define the set $\disp S_{j-1,k} := \bigcup_{l\in\opnm{Sib}(j,k)} S_{j,l},$
\item Go to Step \ref{step:togo}, until all elements of $\mc K(j)$ are marked as unavailable,
\item Declare two elements $k_1, k_2\in \mc K(j-1)$ to be neighbors, if there exists $s_1 \in \opnm{Sib}(j,k_1)$ and $s_2 \in \opnm{Sib}(j,k_2)$ such that $s_1$ and $s_2$ are neighbors in $\mc K(j)$, i.e., $s_1 \in \opnm{Nbr}(j,s_2)$.
\end{enumerate}
There are two sources of randomness in this algorithm: one is in the random permutation of $\mc K(j)$ in Step \ref{step:randperm} and the other is in the random choice of neighboring sets to be selected to be merged in Step \ref{step:togo}.
It can be verified that the set of partitionings $(S_{j,k})$ that is output by the algorithm, after being extended for levels $j = N+1,N+2,\ldots$ in some suitable way, satisfies Definition \ref{defn:partitionings}.
The parameter $p$ in Step \ref{step:togo} is mostly set to 3 in our experiments.
The formation of siblings is displayed in Figure \ref{fig:grid_siblings}, an example output of the algorithm in Figure \ref{fig:embedded_grid}, and the graphs indicating the neighbors is shown in \mbox{Figure \ref{fig:grid_neighbors}}.
\begin{figure}
\centering
\includegraphics[width=8cm]{./figures/chp3/grid_siblings.eps}
\caption{Illustration of merging and splitting of the grid $\mc K(j)$. The top row is the original grid $\mc K(j)$. In the second row, elements of the grid are randomly merged with neighbors, and they form groups consisting of one, two or three elements. In the last row, one element in each group, determined in random way, is displayed in a lighter color. The elements shown in light color make up $\mc K(j-1)$, and grid elements that remain dark-colored belong to the complimentary grid $\mc M(j-1)$. }
\label{fig:grid_siblings}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=15cm]{./figures/chp3/embedded_grid_and_regions.eps}
\caption{The embedded grid and the corresponding regions for a three level decomposition. In each figure, the dots represent $\mc K(3), \mc K(2), \mc K(1)$ and $\mc K(0)$ in the reading order. The shaded regions bounded by lines represent
$(S_{3,k})_{k\in\mc K(3)}, (S_{2,k})_{k\in\mc K(2)}, (S_{1,k})_{k\in\mc K(1)}$
and $(S_{0,k})_{k\in\mc K(0)}$.
}
\label{fig:embedded_grid}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=11cm]{./figures/chp3/grid_neighbors.eps}
\caption{Grid neighbors and regions. In the top each grid element $k\in\mc K(j)$ is shown insinde the region $S_{j,k}$ it represents. The bottom row displays the neighborhood graph, where the condition for two elements to be neighbors is equivalent to the regions they represent sharing a nontrivial boundary. Note that there are grid element pairs that are geographically very close but that nevertheless are not neighbors of each other.}
\label{fig:grid_neighbors}
\end{figure}
\section{The wavelet transform}
After obtaining the necessary structures, such as the nested index sets $(\mc K(j))_j$, their complements $(\mc M(j))_j$, and the neighborhood and sibling relations on them, we are now ready to define the filter operators that identify the wavelet transform. Our wavelet transform is basically a version of David Donoho's \emph{average interpolating wavelets} \cite{donoho}, which can be obtained with a three-stage lifting framework \cite{Sweldens_buildingyour}. Following the splitting step (lazy wavelet transform), it continues with a prediction, an update and another prediction step. The first two steps are similar to Haar decomposition, and result in wavelets similar to the ones called \emph{unbalanced Haar wavelets} \cite{unbalanced_haar}. These wavelets are orthogonal, and the transform can be considered to be a meaningful wavelet transform in its own right.
The second prediction step is a version of the average interpolation prediction, which converts the Haar-like wavelets of the previous step into smoother functions. With this step, we lose orthogonality, in return for smooth wavelet and scaling functions, which entails better sparsity properties for the output.
\subsection{The finest available resolution}
In the practical implementation, the scale index $\mc J = \{1,2,\cdots,N\}$ is finite, there is a highest available resolution, which is determined by the measuring device's capabilities.
The discrete function input to the transform $f_d\in \ell^2(\mc X)$ is assumed to come from a corresponding function $f\in L^2(X)$, with
\beqn
\label{eqn:finest_dual_scaling}
f_d(n) = \frac{1}{\mu(S_{n})} \int_{S_n}f\, d\mu ,\ \ \ \ \text{for } n\in \mc X
\eeqn
for some measure that agrees with the $(\mu(S_n))_n$ of Subsection \ref{subsec:domain}. As such, the dual scaling functions of the finest level $N$ are defined to be
\[
\tilde \varphi_{N,k} = \frac{1}{\mu(S_{N,k})}\chi(S_{N,k}),
\]
and the primal scaling functions $(\varphi_{N,k})_{k\in\mc K(N)}$ are
\[
\disp\varphi_{N,k} = \chi (S_{N,k}),
\]
where $\chi$ denotes the characteristic function, which takes $1$ inside the set and $0$ elsewhere. Accordingly, the finest resolution scaling function coefficients $\lambda_{N,j}$ are simply taken to be equal to the $f_d(n)$'s, and as a result
\[
\sum_{k\in\mc K(N)}\lambda_{N,k}\varphi_{N,k} = \sum_{n\in\mc X}f_d(n)\chi(S_n) \approx f,
\]
noting that $\mc K(N) = \mc X$ and $S_k = S_{N,k}$.
In synthesizing the scaling functions and the wavelets with the algorithm in Subsection \ref{subsec:scaling_fun_cascade}, we will have to stop at $j = N$. Hence, in a sense, the scaling functions $(\varphi_{N,k})_k$ will be the atomic building blocks for the wavelets and scaling functions that we consider, in the same sense that the sets $(S_{N,k})_k$ are the building blocks for all sets that we consider, in the construction.
\subsection{The lifting steps}
We assume an input $\lambda_{j+1} = (\lambda_{j+1,k})_{k\in\mc K(j)}$ is given. The analysis filters will initially perform the lazy wavelet
transform, which is a simple splitting. This initial pair of analysis filters will be denoted as $(\tilde H^{\text{lazy}}_j, \tilde
G^{\text{lazy}}_j)$ and their outputs as $(\lambda^{(0)}_j,\gamma^{(0)}_j)$. Three lifting steps (a prediction, an update and another prediction)
will modify the filters. The resulting analysis filter pairs will be denoted as $\left(\tilde H^{(1)}_j, \tilde G^{(1)}_j\right)$, $\left(\tilde
H^{(2)}_j, \tilde G^{(2)}_j\right)$ and
$\left(\tilde H^{(3)}_j, \tilde
G^{(3)}_j\right)$. The outputs of these filters will be denoted as $\left(\lambda^{(1)}_j,\gamma^{(1)}_j\right)$, $\left(\lambda^{(2)}_j
,\gamma^{(2)}_j\right)$ and $\left(\lambda^{(3)}_j,\gamma^{(3)}_j\right)$, respectively. The lifting operators will be denoted by $P_1$, $U$ and $P_2$, respectively, and the filters will satisfy
{\allowdisplaybreaks
\begin{align*}
\tilde H^{(1)}_j &= \tilde H^{\text{lazy}}_j
\\
\tilde G^{(1)}_j &= \tilde G^{\text{lazy}}_j - P_1 H^{\text{lazy}}_j
\\
\tilde H^{(2)}_j &= \tilde H^{(1)}_j + U \tilde G^{(1)}
\\
\tilde G^{(2)}_j &= \tilde G^{(1)}_j
\\
\intertext{and,}
\tilde H^{(3)}_j &= \tilde H^{(2)}_j
\\
\tilde G^{(3)}_j &= \tilde G^{(2)}_j - P_2 \tilde H^{(2)}_j.
\end{align*}
}In the implementation, one does not need to explicitly compute these filters. Starting with $\lambda_{j+1}$ and, one can obtain $\lambda_{j}^{(3)}$ and $\gamma_{j}^{(3)}$ in a sequence of operations
\begin{align*}
& \lambda_{j}^{(0)} = \tilde H^{\text{lazy}}_j \lambda_{j+1},
&
& \gamma_{j}^{(0)} = \tilde G^{\text{lazy}}_j \lambda_{j+1},
\\
& \lambda_{j}^{(1)} = \lambda_{j}^{(0)},
&
& \gamma_{j}^{(1)} = \gamma_{j}^{(0)} - P_1 \lambda_{j}^{(0)},
\\
& \lambda_{j}^{(2)} = \lambda_{j}^{(1)} + U \gamma_{j}^{(1)},
&
& \gamma_{j}^{(2)} = \gamma_{j}^{(1)},
\\
& \lambda_{j}^{(3)} = \lambda_{j}^{(2)},
&
& \gamma_{j}^{(3)} = \gamma_{j}^{(2)} - P_2 \lambda_{j}^{(2)},
\end{align*}
as displayed in Figure \ref{fig:lifting_diagram}. Similarly, the inverse transform is implemented by reversing the steps:
\begin{gather*}
\begin{aligned}
& \lambda_{j}^{(2)} = \lambda_{j}^{(3)}, \hspace{3cm}
&
& \gamma_{j}^{(2)} = \gamma_{j}^{(3)} + P_2 \lambda_{j}^{(2)},
\\
&\gamma_{j}^{(1)} = \gamma_{j}^{(2)},
&
& \lambda_{j}^{(1)} = \lambda_{j}^{(2)} - U \gamma_{j}^{(1)},
\\
& \lambda_{j}^{(0)} = \lambda_{j}^{(1)},
&
& \gamma_{j}^{(0)} = \gamma_{j}^{(1)} + P_1 \lambda_{j}^{(0)},
\end{aligned}
\\
\lambda_{j+1} = H^{*\, \text{lazy}}_j\lambda_{j}^{(0)} +
G^{*\, \text{lazy}}_j\gamma_{j}^{(0)}.
\end{gather*}
The filters $\left(\tilde
H^{(2)}_j, \tilde G^{(2)}_j\right):=\left(\tilde H^{\text{Haar}}_j, \tilde G^{\text{Haar}}_j\right)$ that come after the second lifting step
actually corresponds to the unbalanced Haar transform, which we will give in the next subsection. The third lifting step is based on \emph{average interpolation}, and the resulting filters after this step, $\left(\tilde
H^{(3)}_j, \tilde G^{(3)}_j\right):=\left(\tilde H^{\text{AI}}_j, \tilde G^{\text{AI}}_j\right),$ are called average interpolating filters, to be explained in Subsection \ref{subsec:average_interp}.
\begin{figure}
\centering
\begin{overpic}[scale=.53
{./figures/chp3/lifting_diagram.eps}
\put(1.5,21){$\lambda_{j+1}$}
\put(39.5,20.5){$P_1$}
\put(57.5,20.5){$U$}
\put(74.5,20.5){$P_2$}
\put(38,13){$-$}
\put(36.5,11.5){$+$}
\put(73,13){$-$}
\put(71.5,11.5){$+$}
\put(54.5,30.5){$+$}
\put(56,29){$+$}
\put(21,32){\footnotesize{$\tilde H_j^{\text{lazy}}$}}
\put(92,32.25){$\lambda^{(3)}_{j}$}
\put(32,34.75){$\lambda^{(0)}_{j}$}
\put(48,34.75){$\lambda^{(1)}_{j}$}
\put(66,34.75){$\lambda^{(2)}_{j}$}
\put(32,6.5){$\gamma^{(0)}_{j}$}
\put(48,6.5){$\gamma^{(1)}_{j}$}
\put(66,6.5){$\gamma^{(2)}_{j}$}
\put(57.75,32.25){\footnotesize{$\Sigma$}}
\put(40.25,9.5){\footnotesize{$\Sigma$}}
\put(75,9.5){\footnotesize{$\Sigma$}}
\put(21,9){\footnotesize{$\tilde G_j^{\text{lazy}}$}}
\put(92,9.75){$\gamma^{(3)}_{j}$}
\end{overpic}
\caption{Diagram illustrating the implementation of the three-stage lifting.}
\label{fig:lifting_diagram}
\end{figure}
\subsection{Unbalanced Haar wavelets}
\subsubsection{First prediction}
The first prediction operator $P_1$, is a function from $\ell^2(\mc K(j))$ to $\ell^2(\mc M(j))$. First we note that, for each $m\in \mc M(j)$ there exists a unique $k_m\in\mc K(j)$ such that $k_m \in \opnm{Sib}(j+1,m)$. If $a = (a_k)_{k\in\mc K(j)}$ is the input to $P_1$, the $m$th entry in the output of $P_1$ is simply defined to be equal to $a_{k_m}$, i.e.,
\begin{align*}
P_1\colon & \ell^2( \mc K(j)) \rightarrow \ell^2( \mc M(j)), \ \ \ P_1 a = b \ \text{means}
\\
&\hspace{2.5cm} b_{m} = a_{k_m},
\end{align*}
where $k_m$ is the unique element of $\mc K(j) \cap \opnm{Sib}(j+1,m)$.
The first prediction step is based on predicting the value of an entry $\lambda_{j+1, k}$ to be the same value as $\lambda_{j+1,k_m}$, for a specific neighbor $k_m$ in $\mc K(j+1)$. The idea is very similar to \emph{nearest neighbor interpolation}, except that the neighbor used in prediction is not necessarily the nearest one, but is determined by the structure of the embedded grids. For the example in Figure \ref{fig:grid_siblings}, the value at the dark colored spots in the lowest row is to be predicted with the value in the corresponding light colored spot.
We note that, after this first prediction step, one has
\begin{align}
\label{eqn:after_first_pred}
\gamma_{j,s_1}^{(1)} = &\lambda_{j+1,s_1} - \lambda_{j+1,k}\notag\\
& \vdots \\
\gamma_{j,s_q}^{(1)} = &\lambda_{j+1,s_q} - \lambda_{j+1,k}.\notag
\end{align}
where $k\in\mc K(j)$ and $\opnm{Sib}(k,j+1) = \{k, s_1, s_2, \cdots, s_q\}$, with $q$ being the number of siblings.
\subsubsection{Update}
The update operator $U$ is a function from $\ell^2(\mc M(j))$ to $\ell^2(\mc K(j))$ that modifies the output of the previous step as $\lambda^{(2)}_j = \lambda^{(1)}_j + U \gamma^{(1)}_j$, in order to guarantee that
\[
\sum_{k\in\mc K(j+1)} \lambda_{j+1,k}\,\mu(S_{j+1,k})=\sum_{k\in\mc K(j)} \lambda^{(2)}_{j,k}\,\mu(S_{j,k}).
\]
It is sufficient, and usually expected, that this condition is satisfied locally, by having
\begin{align}
\label{eqn:measure_preserve_loc}
\lambda_{j,k}^{(2)}\,\mu(S_{j,k})
=&
\lambda_{j+1,k}\,\mu({S_{j+1,k}})
+ \lambda_{j+1,s_1}\,\mu({S_{j+1,s_1}})
\notag\\& + \cdots + \lambda_{j+1,s_q}\,\mu({S_{j+1,s_q}}),
\end{align}
for all $k\in\mc K(j)$, and $s_1,s_2,\cdots,s_q\in \opnm{Sib}(k,j+1)$.
This can be accomplished with an update operator $U$ given by
\begin{align*}
U_j\colon & \ell^2( \mc M(j)) \rightarrow \ell^2( \mc K(j)), \ \ \ U a = b \ \text{means}
\\
& b_{k} =
\frac
{a_{s_1}\,\mu(S_{j+1,s_1})+\cdots+a_{s_q}\,\mu(S_{j+1,s_q})}
{\mu(S_{j+1,k})+\mu(S_{j+1,s_1})+\cdots+\mu(S_{j+1,s_q})},
\end{align*}
where $\{k,s_1,s_2,\cdots,s_q\} = \opnm{Sib}(j+1,k)$ for all $k \in \mc K(j)$. It can be verified that after this step, one has
\begin{equation}
\label{eqn:averaging_property}
\lambda_{j,k}^{(2)} =
\frac{\lambda_{j+1,k}^{(0)}\,\mu(S_{j+1,k})+\lambda_{j+1,s_1}^{(0)}\,\mu(S_{j+1,s_1})+\cdots+\lambda_{j+1,s_q}^{(0)}\,\mu(S_{j+1,s_q})}{\mu(S_{j+1,k})+\mu(S_{j+1,s_1})+\cdots+\mu(S_{j+1,s_q})},
\end{equation}
which is the same as (\ref{eqn:measure_preserve_loc}), noting that
\begin{equation*}
\mu(S_{j,k})= \mu(S_{j+1,k})+\mu(S_{j+1,s_1})+\cdots+\mu(S_{j+1,s_q}).
\end{equation*}
\subsubsection{Wavelets and scaling functions}
Using \eqref{eqn:finest_dual_scaling} and \eqref{eqn:averaging_property}, it can be shown inductively that the dual scaling functions $\tilde \varphi_{j,k}$ are given by
\[
\tilde \varphi_{j,k} = \frac{1}{\mu(S_{j,k})} \chi(S_{j,k}),
\]
and using \eqref{eqn:after_first_pred} one gets the dual wavelets $(\tilde \psi_{j,m})$ as
\[
\tilde \psi_{j+1,\,m} = \frac{1}{\mu(S_{j+1,\,k_m})}\chi(S_{j+1,\,k_m}) - \frac{1}{\mu(S_{j+1,m})}\chi(S_{j+1,m}),
\]
where $k_m$ is the unique element of $\opnm{Sib}(j+1,m)\cap \mc K(j)$.
It is easy to verify that the collection of dual scaling functions and wavelets are orthogonal to each other, which implies that the primal scaling functions and wavelets are the same as the dual ones, up to normalizing constants. These type of wavelets are studied in \cite{unbalanced_haar}, where they are named \emph{unbalanced Haar wavelets}.
\subsection{Average interpolating wavelets}
\label{subsec:average_interp} The unbalanced Haar wavelets have the advantage of being orthogonal, and they are relatively simple to design, but they have the drawback of not being
smooth, not even continuous. In a wavelet representation
$$
f =
\sum_{j,m}\gamma_{j,m}\psi_{j,m},
$$
we
usually seek \emph{sparse representations}, meaning that most of the coefficients
$\gamma_{j,m}$'s are zero or at least negligible. This is usually not achieved by
Haar
wavelets to the most pronounced degree, because there is a link between the
smoothness of wavelets and fast decay of wavelet coefficients. Intuitively, we
can say that Haar wavelets, not being smooth, find it difficult to
represent
smooth functions and require a larger number of basis elements than is needed by a more smooth basis. Therefore we
would
like to have a wavelet basis without sharp discontinuities and certain
smoothness properties.
By adding a second prediction operator $P_2$ to the lifting scheme, it is possible to obtain
smoother wavelets out of Haar wavelets, and that give smaller detail coefficients for smooth signals.
Or, when we think the other way around, if the prediction operator $P_2$ is designed to make the entries of the detail output smaller for smooth functions, then the corresponding wavelets will turn out to be smoother than Haar wavelets.
The first prediction step $P_1$ was based on a version of \emph{nearest neighbor interpolation}, while this second prediction step is based on the notion of \emph{average interpolation}, which we explain next. The use of average interpolation within the context of wavelets is due to Donoho \cite{donoho}.
\subsubsection{Average interpolation}
In the following, all sets considered are subsets of $\mathbb R^n$, and $\mu$ is a measure on $\mathbb R^n$. The average $ \frac{1}{\mu(S)}\int_S f d\mu$ is denoted shortly as $ \dashint_S f$, for a given $S\subset \mathbb R^n$.
Let $S_1,S_2,\cdots, S_r,{C_1},C_2,\cdots, {C_t}\subset \mathbb R^n$ be mutually disjoint sets that are geographically close, for some integers $r$ and $t$.
Let $f$ be an unknown function, which is assumed to be locally well approximated by polynomials.
It is assumed that the averages $\dashint_{S_1} f, \cdots, \dashint_{S_r} f$ are known, and the goal is to estimate $\dashint_{C_1} f, \cdots, \dashint_{C_t} f$.
The approach is to find a polynomial $p$ whose average values agree with $f$ on on $S_1,\cdots,S_r$, i.e.,
\begin{equation}
\label{}
\dashint_{S_i} f = \dashint_{S_i} p
\end{equation}
for $i = 1,2,\cdots,r.$ After such a polynomial is found, the estimates for $\dashint_{C_1} f, \cdots, \dashint_{C_t} f$ are given to be $\dashint_{C_1} p, \cdots, \dashint_{C_t} p$, respectively.
\vspace{.5cm}
After this brief description of average interpolation, we can define the second prediction operator that is used to obtain the average interpolating wavelets.
\subsubsection{Second prediction}
The second prediction operator $P_2$ maps $\ell^2(\mc K(j))$ to $\ell^2(\mc M(j))$, aiming to estimate $\gamma^{(2)}_j$ using $\lambda^{(2)}_j$. Here using \eqref{eqn:averaging_property} and assuming the finest level coefficients $\lambda_{N,j}$ are local averages of some input function $f$ as in \eqref{eqn:finest_dual_scaling}, we get
\[
\lambda_{j,k}^{(2)} = \dashint_{S_{j,k}} f.
\]
Also for $m\in\mc M(j)$, by \eqref{eqn:after_first_pred}, one has
\begin{align*}
\gamma_{j,m}^{(2)} = \gamma_{j,m}^{(1)} &= \lambda_{j+1,m} - \lambda_{j+1,k_m}\\
&= \dashint_{S_{j+1,m}}\!\!\!\!f\ \ \ \ -\ \ \ \ \dashint_{S_{j+1,k_m}}\!\!\!\!f,
\end{align*}
where $k_m$ is the unique element of $\mc K(j)\cap\opnm{Sib}(j+1,m)$. We aim to estimate this value, using $\lambda^{(2)}_{j,n_1},\cdots, \lambda^{(2)}_{j,n_r}$, where $\{n_1,n_2,\cdots,n_r\}=\opnm{Nbr}(j,k_m)$. This is actually the problem of estimating $\dashint_{S_{j+1,m}} f$ and
$\dashint_{S_{j+1,k_m}}f$ using the values $\dashint_{S_{j,n_1}}f, \cdots, \dashint_{S_{j,n_r}}f$, which is to be handled by average interpolation.
In our implementation, for each $k\in\mc K(j)$ we solve for the first degree polynomial
\begin{equation}
\label{eqn:polynomial}
p_{j,k}(x,y,z) = a_{j,k} + b_{j,k}\, x + c_{j,k}\, y + d_{j,k}\, z
\end{equation}
that minimizes
\[
\sum_{n\in\opnm{Nbr}(j,k)}\left| \lambda^{(2)}_{j,n} - \dashint_{S_{j,n}} p_{j,k}\right|^2.
\]
This is a linear problem, and each of the polynomial coefficients in \eqref{eqn:polynomial} is a linear combination of $\lambda^{(2)}_{j,n_1},\cdots, \lambda^{(2)}_{j,n_r}$. So we can define $P_2$ to be the linear operator such that the $m$th entry of $P_2 \lambda_j^{(2)}$ is given by
\[
\dashint_{S_{j+1,m}}\!\!\!\!p_{j,k_m}\ \ \ \ -\ \ \ \ \dashint_{S_{j+1,k_m}}\!\!\!\!p_{j,k_m},
\]
noting that this quantity is a also a linear combination of $\lambda^{(2)}_{j,n_1},\cdots, \lambda^{(2)}_{j,n_r}$. If the input $f$ to the algorithm is itself a first degree polynomial, then the fit would be perfect and the prediction will give exact quantities.
After this final prediction step, one can add a normalizing step to the transform that makes each of the wavelets and the dual wavelets of unit norm.
\section{Numerical experiments}
\subsection{The potential to reduce noise by sparse representation}
The underlying point in transform-based noise reduction methods is the sparse representation property. A good transform enables a representation that puts most of the information about a smooth function into a relatively small fraction of the coefficients, while spreading out a signal uniformly to all components if the signal is noise-like, having little spatial correlation. More explicitly if we denote the wavelet representation of a function with a simplified single indexed notation as
\[
f = \sum_{m\in\mc M}\gamma_m \psi_m,
\]
then we assume that for the $f$ of interest to us, there exists a set $\mc A \subset \mc M$ with $|\mc A|\ll |\mc M|$ that gives
\begin{equation}
\label{eqn:thresholded_approximation}
f_{\text{approx}}:= \sum_{m\in\mc A}\gamma_m \psi_m \approx f .
\end{equation}
For the case of orthogonal wavelets, the best choice for such sets are generally of the form
\begin{equation}
\label{eqn:thresholded_set}
\mc A = \{m\in\mc M \colon |\gamma_m|>\tau\},
\end{equation}
for some threshold $\tau$. These type of sets give the best approximation in the $L^2$ sense, among the sets of the same size. This way of choosing $\mc A$ usually still works even when the wavelet basis is not orthogonal for some signal subclasses, if the corresponding frame bounds are close to tight.
\subsubsection{Comparing adaptive wavelets with standard wavelets}
In this numerical experiment, we started with a domain $X$ consisting of concentric rings, contained in a rectangle $R$, as displayed in the top row of Figure \ref{fig:wav_sparsity_comp}. We generated a smooth function on $R$, which is a different random linear combination of two dimensional Gaussians on each connected component of the $X$. On $R\setminus X$ the domain also contains another smooth function of the same type. That is to say, if the domain can be written as $X = S_1\cup\cdots\cup S_n$, where each of $S_1,\cdots,S_n$ is a single, connected ring, and if $f_1,\cdots,f_{n+1}$ are smooth functions on $R$, then our input $f$ is taken to be
\[
\label{eqn:random_smooth}
f = f_1\,\chi_{S_1} + f_n\,\chi_{S_n} + f_{n+1}\,\chi_{R\setminus X},
\]
$\chi_S$ denoting the characteristic function of a set $S$.
This choice of $f$ is motivated by the fMRI problem, where the brain cortex, which is the natural domain of the measured data, has a convoluted structure, and geographically close parts of it may carry signals that are of different nature.
We computed the coefficients under different wavelet transforms, and computed reconstructed approximations in the form of \eqref{eqn:thresholded_approximation}, the coefficient sets $\mc A$ being selected with thresholding as in \eqref{eqn:thresholded_set}. For each different threshold, we computed the approximation error as
\[
\frac {\|f-f_{\text{approx}}\|}{\|f\|}.
\]
Although the standard wavelet transforms give reconstructions over the whole rectangle $R$, we compute the norms only over the domain $X$, i.e.,
\[
\|f\| = \sqrt{\int_X |f(x)|^2 dx}.
\]
We also generated multiple realizations of the noise $n$ over the whole square,
whose samples are i.i.d. standard Gaussian random variables. We transformed
each of the realizations of the noise with the same wavelets, and reconstructed it using only entries from
the coefficient set $\mc A$ that was determined by the function $f$, and
computed the expected norm $\text{E}[{\|n_{\text{rec}}\|}]$, of the
reconstructed noise. That is, we process the signal $f$ and the noise $n$
separately, and plot $\frac {\|f-f_{\text{approx}}\|}{\|f\|}$ versus
$\text{E}[{\|n_{\text{rec}}\|}]$, while the threshold parameter $\tau$ is being
reduced. This gives a measure of the {potential to reduce noise} of the related
transform on the domain $X$.
The results are plotted in Figure \ref{fig:wav_sparsity_comp}, where the
experiment is done with three standard wavelet bases that have the domain $R$,
and the domain-adapted wavelets that are constructed for $X$. For each choice of
wavelets, the experiment is repeated for the decomposition levels 1,2 and 3.
The results show that increasing the decomposition level gives a much more
pronounced positive effect with the domain adapted wavelets than the standard
wavelets.
\subsubsection{Comparing different domains}
We repeated the experiment while changing the thickness of each of the rings of the domain, or the gap between the rings corresponding to the off-domain regions. The results are given in Figure \ref{fig:four_domains}. This experiment shows that, in order for the adapted wavelets to have a clear advantage over the standard ones, the rings or the gaps between them must be sufficiently thin. For an example like the one given in the first row, the adapted wavelets do not have a considerable advantage over the wavelets that have the whole rectangle $R$ as their domain.
\begin{figure}
\centering
\includegraphics[width=13cm]{./figures/chp3/comparison_wavelets_levels_corrected.eps}
\caption{Comparing denoising potentials of different conventional wavelets, and the domain-adapted wavelets. Relative approximation error (vertical) versus expected noise norm (horizontal), as the threshold is reduced. All error norms are obtained from the surviving coefficients of the original image, and are computed only on the annular domain. A curve in the lower part of the plane implies a better performance. a) Domain, b) Image, c) Haar wavelets d) Daubechies-3 wavelets e) Biorthogonal 3.3 wavelets f) Domain-adapted wavelets. The vertical axis in figures c-g is the relative $L_2$ approximation error, which is
$\frac {\|f-f_{\text{approx}}\|}{\|f\|}$. Note that increasing the level of the transform has a stronger effect with the domain-adaptive wavelets.
}
\label{fig:wav_sparsity_comp}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=15cm]{./figures/chp3/four_domains.eps}
\caption{Comparison of the denoising potential of different wavelets on four different two-dimensional domains. Each of the wavelet transforms is computed at Level-3. Figures show the relative approximation error (horizontal) versus expected noise norm (vertical), as the threshold is reduced. As the domain gets thinner, or the gaps between different circles gets smaller, the domain-adaptive wavelets outperform the conventional wavelets. }
\label{fig:four_domains}
\end{figure}
\subsection{Improvement due to averaging}
The randomness in our wavelet algorithm allows us to repeat a signal processing task with multiple realization of the wavelets, and then take the average of the results. In this experiment, we work with a domain consisting of concentric circles and generate smooth signals on them, similar to the examples in the previous subsection. We add i.i.d. Gaussian noise, and perform a wavelet denoising. The signal-to-noise-ratio (SNR) versus number of realizations plot is given in Figure \ref{fig:denoising_with_averaging}. For chosen domain, with the given noise, the tensor product Daubechies-3 wavelet transform initially performs better denoising than the domain adapted wavelets. However, as we average over multiple realizations, the domain-adapted wavelets result in better SNR values.
\begin{figure}
\centering
\includegraphics[width=14cm]{./figures/chp3/denoising_with_averaging.eps}
\caption{Denoising with multiple wavelet realizations. The performance improves as we average the results over multiple realizations. The dashed line in the plot represent the performance of the Daubechies-3 wavelet, which performed the best among the standard wavelets we tested in the previous subsection. SNR values computed only over the domain. }
\label{fig:denoising_with_averaging}
\end{figure}
\subsection{The effect of the second prediction}
The second prediction step of the transform is designed to make the detail coefficients smaller, or the wavelets and scaling functions smoother, as explained in Subsection \ref{subsec:average_interp}. The wavelet transform before this step is the \emph{unbalanced Haar wavelet} transform, which is orthogonal. After the second prediction, the wavelets are \emph{average interpolating wavelets}, and they are smooth but not orthogonal.
We tested the effect of this step in noise reduction with an experiment similar to the one in the previous subsection. The SNR versus number of wavelet realizations is given in Figure \ref{fig:snr_haar_ai}. This result shows an improvement of nearly 2.5 dB as a result of the second prediction step. A fixed threshold is empirically determined, and used in all wavelet realizations for the same type. Since the Haar wavelets are orthogonal, the thresholded approximation gives always the best approximation that can be achieved with the same number of surviving coefficients. However this is not necessarily the case for the average interpolating wavelets, because they are not orthogonal. This explains the sudden drops in the SNR curve corresponding to some outlying wavelet realizations.
\begin{figure}
\centering
\includegraphics[width=12cm]{./figures/chp3/haar_ai_comparison_snr.eps}
\caption{Comparison of unbalanced Haar wavelets and average interpolating wavelets in a noise reduction experiment. Performance improves as the results obtained by newer wavelet realizations are included in the cumulative average. The average interpolating wavelets give better performance than the unbalanced Haar wavelets, but occasional drops in performance are observed when an outlying wavelet realization is encountered. }
\label{fig:snr_haar_ai}
\end{figure}
\subsection{Translation-invariant processing}
In a reliable signal processing algorithm, one would intuitively expect a
translation-invariance property. If we translate the signal without
distorting, process it and then translate it back, we would expect the
result to be the same as if it were processed without being translated. The simplest version of a multiresolution-based wavelet algorithm does not have this invariance, which is why Donoho and Coifman
propose the wavelet spin cycle algorithm in \cite{donoho_transinv}, which achieves translation invariant
denoising in one dimension. Translational invariance in higher dimensions can be achieved in the same way. In dimensions two and higher, one can similarly desire rotational invariance, which can be achieved by introducing redundancy in angular resolution as well, e.g., via steerable filters \cite{freeman91} or dual-tree wavelet transforms \cite{selesnick05}.
We tested the invariance property under translations and rotations of our randomized wavelet transform algorithm, when the results are averaged over multiple realizations, as follows. We took an image on a 64$\times$64 square domain, and we considered a 5-level randomized wavelet decomposition.
We chose a processing task of projecting an image onto the spaces $V_0, W_0, W_1, \cdots W_5$, which are described in Subsection \ref{subsec:multiresolution} and Subsection \ref{subsec:wavelets}. This corresponds to
transforming the image, and reconstructing from only a single level of
coefficients. We took averages of the results according to 100 different
wavelet realizations, and found that the realization-averaged transform is (almost) invariant under translations and rotations. This is illustrated in The result is shown in Figure \ref{fig:translation_invariance}, for rotations (which is the harder of the two tasks); and it is
seen that the process commutes with a rotation of 45 degrees, as desired.
\begin{figure}
\centering
\includegraphics[width=10.5cm]{./figures/chp3/translation_invariance.eps}
\caption{An image transformed and reconstructed back form a single level of wavelet coefficients, i.e, projected on to the corresponding subspaces for a 5-level wavelet decomposition.
The six rows correspond to the projection onto the spaces $V_0, W_0, W_1, \cdots W_5$, respectively. In the first and third
columns, images are processed without being translated whereas in the second and fourth columns they are first rotated 45 degrees, and then processed, and rotated back. The left two columns show the result of a single realization, and the right two columns show the result of computing the average of 100 realizations. We see that the processing
becomes much more powerful, and almost rotationally invariant when averaged over multiple realizations.}
\label{fig:translation_invariance}
\end{figure}
\chapter{Application to wavelet-based statistical analysis of fMRI data}
One of the most widely used and recognized methods for fMRI analysis is
Statistical parametric mapping (SPM) \cite{hbf2}. In SPM, a key step is
spatial prefiltering with a Gaussian window, as a means of reducing noise. However, Gaussian filtering has the drawback of destroying the fine spatial details. A wavelet-based alternative of SPM, which is called WSPM, is due to Van De Ville et al. \cite{surfing, vandeville04,
vandeville0406, vandeville07}. It replaces the Gaussian filtering step with
wavelet filtering, and it involves thresholding in the wavelet domain as a denoising step, followed
by a thresholding in the spatial domain. The threshold parameters in WSPM are selected so as to control the false positive rate, while minimizing the reconstruction error.
Standard wavelets used in this framework
are defined on rectangular domains, typically a square (in two dimensions) or a cube (in
three dimensions). On the other hand, the natural domain of the neural activity
is the brain cortex, which is an intricately convoluted three dimensional
domain.
This chapter essentially consist of the application of
the domain-adapted wavelets of the previous chapter to the WSPM framework. The wavelets are constructed so as to have the brain cortex as their
natural domain.
\section{Statistical testing of fMRI data}
\subsection{One sample t-test}
\label{one_sample_t_test}
One of the most basic tasks in fMRI data analysis is to determine whether there is any brain response to a given stimulus or a given train of stimuli, and if there is, to determine its location. In a simplified paradigm, we may assume the measurement we get from a given voxel is a random variable of the form
$$
v_n = \mu_n + e_n,
$$
where $e_n$ is a noise random variable with zero mean and unknown variance, and $\mu_n$ is the true activation in the voxel $n$, i.e. $\mu_n = 0$ when there is no activation and $\mu_n>0$ when there is activation. For each voxel $n \in V \subset \mathbb Z^3$ in the region of interest, one may consider using a standard one-sample t-test to decide whether that voxel is active or not. Given a series of measurements $v_n(1), v_n(2) \cdots, v_n(N)$ from voxel $n$, the goal is to decide whether $\mu_n$ is zero or strictly positive, i.e., to decide between the hypotheses
\begin{align*}
&\mathbf{H_0} \colon \mu_n = 0 \\
&\mathbf{H_1} \colon \mu_n > 0.
\end{align*}
To decide which of the hypotheses is true, one passes to another random variable, which is called as a $t$-statistic, and distributed with Student's $t$-distribution of $N$ degrees of freedom. In order to obtain a $t$-variable, the population mean $\hat \mu_n$ and an unbiased estimate of the variance $s^2_N$ are computed from the samples as follows:
$$
\hat \mu_n = \frac{1}{N}\sum_{k = 1}^N v_n(k)\hspace{2cm} s^2_N=\frac{1}{N-1}\sum_{k=1}^{N}\left(v_n(k) - \hat \mu_n\right)^2.
$$
Then the ratio $t = \displaystyle \frac{\hat \mu}{\sqrt{s^2_N/N}}$ has a Student's $t$-distribution with $N$ degrees of freedom. A preselected extremely unlikely region is used to reject the null hypothesis. In our case one would use the so called one-sided tail as in Figure \ref{ttest}, since the mean is assumed to be nonnegative. This region is chosen to have a small probability of occurrence, a typical value for the significance level $\alpha$ being $0.05$, which is also the false-positive rate, i.e., the chance that null hypothesis will be rejected incorrectly.
\begin{figure}
\begin{center}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp4/studentstdist.eps}}}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp4/reject_null.eps}}}
\end{center}
\caption{(a) The density functions of Student's $t$-distribution with $n=1, 2$ and 10 degrees of freedom, respectively, and the standard Gaussian distribution, which can also be viewed as a $t$-distribution with infinitely many degrees of freedom. (b) The 5 \% region to reject the null hypothesis, for number of degrees of freedom $n=5$. }
\label{ttest}
\end{figure}
\subsection{The general linear model}
Although the model that is described in subsection \ref{one_sample_t_test} may be helpful for illustration purposes, it was not realistic, since data from a real voxel is never silent, due to the highly active nature of the human brain. That is why one can observe only a slight increase in the activity of a voxel in response to stimuli. In Figure \ref{fig_patterns_and_regressors} data from a real experiment by Haxby et al. \cite{haxby} are plotted, in which signals from 577 voxels are averaged for the correlation to be visually detectable. In this setting, the question becomes whether there is a positive correlation between the stimulus and the response from the voxel and the decision is less straightforward. Fortunately, there exist straightforward extensions of the one sample $t$-test into this setting.
\begin{figure}
\centering
\includegraphics[width=14cm]{./figures/chp4/patterns_and_regressors.eps}
\caption{The stimuli and fMRI response. Data from a visual stimuli experiment by Haxby et.al \cite{haxby}. The dashed line represents the on/off timing of the visual stimuli, and the solid line is the fMRI response, averaged over 577 voxels from the VT cortex. The linear trend in the stimuli is also removed. The correlation between the stimuli and the response is clearly visible in the figure, although in the signal from a single voxel the correlation would be virtually impossible to see. }
\label{fig_patterns_and_regressors}
\end{figure}
Let $v_n(1), v_n(2), \cdots, v_n(N)$ be the time samples obtained from the voxel $n$ at times $t = 1,2,\cdots, N$, and let us represent these samples as an $N\times 1$ column vector,
$$
\boldsymbol v_n = \left[ \begin{array}{c} v_n(1) \\ v_n(2) \\ \vdots\\ v_n(N) \end{array} \right].
$$
In a typical setting the subject is presented
with stimuli during certain intervals for a certain amount of time.
Let the voxel location $n$ be fixed, and
consider the time series $v(t)$, denoted as simply $v$. We set up a model,
known as the \emph{general linear model}, or GLM, to analyze the time course of
this voxel:
\begin{equation*}
\mbf{v}_n = \mbf X \boldsymbol{\beta} + \mbf e_n
\end{equation*}
where $\mbf X$ is a matrix each column of which is the expected time course if the subject
had responded only to a particular type of stimulus, with different stimulus type time courses given by the different columns of $\mbf X$. After the linear regression,
each component of the estimate of $\beta$ will correspond to the level of
response to the corresponding category.
A simple example for a single, on-off type stimulus with only eight time points can be given as
follows: Let us present the subject with stimuli for two time points, and then
give a rest for the next two time points, and repeat this procedure one more time. Then a
simple choice for our model would be
\begin{equation}
\label{glm_example}
\mathbf v_n =
\left[ \begin {array}{rrr}
1&t_0&1\\\noalign{\smallskip}1&t_1&1\\\noalign{\smallskip}1&t_2&-1\\\noalign{\smallskip}
1&t_3&-1\\\noalign{\smallskip}1&t_4&1\\\noalign{\smallskip}1&t_5&1\\\noalign{\smallskip}
1&t_6&-1\\\noalign{\smallskip}1&t_7&-1\end {array} \right]
\left[ \begin {array}{c}
{\beta_{0,n}}\\\noalign{\medskip}{\beta_{1,n}}\\\noalign{\medskip}{\beta_{2,n}}\end {array} \right]
+ \mathbf e
.
\end{equation}
The constant vector in the first column accounts for the mean. The second
column is the time vector, and its coefficient $\beta_{1,n}$ accounts for the linear trend. The last column corresponds to the actual stimulus, having value 1 when the stimulus is on, and $-1$ when it is off. The only parameter of interest here would be
$\beta_{2,n}$, corresponding to this last column, which would be interpreted as the magnitude of response from the current
voxel to the stimulus train.
Then this analysis is repeated for each voxel, and the resulting map
$\left\{\boldsymbol{\beta}_{2,n}\colon n\in R \right\}$ is then the activation parameter map for the given stimulus, where $R$ is the region of interest. Significance of the response can be determined with the $t$-test, and the voxels passing the test are marked to be active. This is illustrated with sample time courses from a real experiment, in Figure \ref{glm_illustration}.
\begin{remark}
A more advanced approach would be to convolve the box functions in the columns of interest in $\mathbf X$ with what is called as the \emph{hemodynamic response function}, which
makes it resemble the actual expected time course more, taking account the
dynamics of the blood-oxygenation in the brain. See Figure {\rm\ref{hemodynamic_response}} for the plot of the convolved regressors and the hemodynamic response function, and \rm{\cite{sarty}} for more information.
\end{remark}
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{./figures/chp4/homodynamic_response.eps}
\caption{Original on-off stimulus (top), convolved with the homodynamic response function (middle), and the hemodynamic response function (bottom).}
\label{hemodynamic_response}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=15cm]{./figures/chp4/glm_illustration.eps}
\caption{Plots of time series from various voxels, from the experiment by Haxby et al. \cite{haxby}. Each voxel is fitted to the model, as a linear combination of a straight line and the box train function corresponding to the on-off stimuli. A $t$ variable is derived and corresponding $p$ values are computed. The smaller values of $p$ imply a stronger rejection of the null hypothesis. }
\label{glm_illustration}
\end{figure}
\subsubsection{Statistical testing}
The general linear model (GLM) for the time course of the voxel $n$ is
\begin{equation*}
\mbf{v}_n = \mbf X \boldsymbol{\beta} + \mbf e,
\end{equation*}
where $\boldsymbol{\beta}$ is an $N\times 1$ vector of unknown parameters, and $\mathbf e$ is an $N\times 1$ Gaussian random vector with zero mean and unknown variance. The matrix $\mathbf X$ is $N\times L$, which is known beforehand and is called the \emph{design matrix}.
Usually not all components of $\boldsymbol{\beta}$ are of interest. One usually needs to know whether the quantity $\mbf{c}^{\operatorname{T}}\boldsymbol{\beta}$ is zero or strictly positive, and $\mbf c$ is called the \emph{contrast vector}. For the example of (\ref{glm_example}), the contrast vector would be taken as $\mbf{c}^{\operatorname{T}} = [0\ 0\ 1]^{\operatorname{T}} $, indicating that one is interested in whether the last column of the design matrix has positive correlation with the data, after removing the effects of the first two columns. In another case when one is interested in whether column $i$ of the design matrix has more correlation with the data than column $j$, the contrast vector $\mbf c$ would be a vector of all zeros except for the $i$th and $j$th entries, which would be $1$ and $-1$, respectively.
We observe a realization of the random vector $\mbf v_n$, and would like to decide between the two hypothesis about $\mbf{\boldsymbol{\beta}}$:
\begin{align*}
&\mathbf{H_0} \colon \mbf{c}^{\operatorname{T}}\boldsymbol{\beta} = 0 \\
&\mathbf{H_1} \colon \mbf{c}^{\operatorname{T}}\boldsymbol{\beta} > 0.
\end{align*}
In order to do a $t$-test, we first need to obtain the $t$-variable. One starts with computing an unbiased estimate for $\boldsymbol{\beta}$ by an ordinary least squares formula
\[
\hat{\boldsymbol{\beta}} = (\tran{\mbf{X}} \mbf{X})^{-1}\mbf{X} \mbf{y},
\]
implicitly assuming the columns of $\mbf X$ are linearly independent. Then an estimate for the error would be
\[
\hat{\mbf{e}} = \mbf{y} - \mbf{X}\hat{\boldsymbol{\beta}}.
\]
Now if one defines
\begin{align}
g_n & = \tran{\mbf{c}}\hat{\boldsymbol{\beta}},\\
s^2_n &= {\hat{\mathbf e_n}}^{\operatorname T}{\hat{\mathbf
e_n}}\ \mathbf c ^{\operatorname T}(\mathbf X^{\operatorname T} \mathbf
X)^{-1} \mathbf c.
\end{align}
then the scalar random variable $g$ would have a Gaussian distribution with mean $\tran{\mbf{c}}{\boldsymbol{\beta}}$, and $s^2$ will follow a Chi-square distributions with $J = N - \operatorname{rank}(X)$ degrees of freedom, and they are independent \cite{vandeville0406}. Following this, one can obtain a $t$ variable by
$$
t_n = \frac{g_n}{\sqrt{s^2_n/J}}, \ \ \text{with} \ J =
N - \operatorname{rank}(\mathbf X),
$$
and one can then proceed similarly to the $t$-test of Subsection \ref{one_sample_t_test} for a statistical test of the significance of the activation.
\subsection{Multiple comparison problem}
When multiple statistical tests are to be performed based on the same data set, the problem of \emph{multiple comparison} needs to be dealt with. In the case of fMRI, a typical dataset has of the order of $10^4$ voxels. A significance level of $\alpha = 0.05$ would result in $10^2$ to $10^3$ false detections; this number may be as large as the total number of voxels in the entire region of activation! In general, for a dataset of $K$ voxels, one expects to get $\alpha K$ false positives. One solution offered to remedy this problem is to use the \emph{Bonferroni correction}, which is a conservative method to reduce the expected number of total false positives by reducing $\alpha$, and considering voxels in (sub)collections rather than individually. For a total number of $K$ voxels, the Bonferroni correction asks to reduce the significance level from $\alpha$ to $\alpha/K$, which in turn reduces the expected number of total false positives from $\alpha K$ to $\alpha$. By the \emph{union bound}, getting a single falsely detected voxel within the set of $K$ voxels would have a rate smaller than $\alpha$. For a set with, e.g., $10^2$ voxels and an initial $\alpha$ selected to be 0.05, Bonferroni correction would ask to reduce $\alpha$ from $5\times 10^{-2}$ to $5\times10^{-4}$. Although it can be improved by restricting the region of interest to a smaller set of voxels, this correction has the obvious drawback of extremely reduced sensitivity for fMRI datasets, almost to the level of detecting no activation \cite{surfing}: although the Bonferroni argument reduces the number of false positives, it \emph{increases} the number of false negatives. Such small choices of sensitivity rates make the tests useless, because fMRI data sets are very noisy and there are not enough samples to get detections at such a conservative specificity.
\section{Spatial filtering for improving statistical power}
\subsection{Statistical parametric mapping and wavelets}
The problem with the Bonferroni correction is that it does not make use of the spatial correlation between voxels \cite{surfing}. There are methods that remedy this situation by essentially performing a transformation, which practically maps the large number of noisy and highly correlated voxels into a small number of uncorrelated and less noisy transform domain coefficients. The most widely used approach in this category is the Statistical parametric mapping (SPM) package \cite{SPM}, in which the essential step is prefiltering by a Gaussian window, which corresponds to going to the Fourier domain, and throwing out the high frequency components and reconstructing. Note that a spatial convolution with a Gaussian window is equivalent to multiplying the Fourier transform of the data with the Fourier transform of the Gaussian window; since this is another Gaussian, this multiplication is a form of weighted thresholding. As depicted in Figure \ref{squarewave}, Gaussian and wavelet filtering can be viewed as two different forms of the same idea. In Gaussian filtering, the coefficients to be discarded are the high frequency coefficients that are predetermined, which makes it linear, while in wavelet filtering, coefficients are to be discarded are determined by thresholding, therefore it is a form of nonlinear filtering.
\begin{figure}
\begin{center}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp4/denoise_square_wave.eps}}}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp4/denoise_square_wave2.eps}}}
\subfigure[]{{\includegraphics[width=7.5cm]{./figures/chp4/denoise_square_wave3.eps}}}
\end{center}
\caption{Depiction of noise reduction by Gaussian and wavelet filtering. Spatially correlated functions have their transform domain concentrated to a few coefficients, while noise processes have their transforms spread out to all coefficients uniformly. (a) First Row: A square wave function, and a Gaussian which will be used in filtering, Second Row: Noisy (left) and filtered (right) versions of the square wave function. (b) First row: The square wave function and its Fourier transform, overlaid by the Fourier transform of the Gaussian window, Second Row: White noise and its Fourier transform. (c) First Row: The square wave function and its Wavelet transform, Second Row: White noise and its wavelet transform.}
\label{squarewave}
\end{figure}
\label{sec:theory}
\subsection{Gaussian versus wavelet filtering }
While Gaussian filtering has the advantages of being a well established linear noise reduction method, its main disadvantage is that it destroys all fine spatial details, which might be important in fMRI images. In Figure \ref{wavelet_filtering}, this phenomenon is illustrated with a single dimensional example. That is the main motivation of exploring wavelet based methods, which are the subject of the rest of this chapter.
\begin{figure}
\centering
\includegraphics[width = 15cm]{./figures/chp4/wavelet_denoising_1d.eps}
\caption{Wavelet denoising. A square wave signal is contaminated with noise. In the second row, it is denoised by Gaussian filtering and by wavelets, respectively. Although Gaussian filtering achieves a visually detectable noise reduction, it also destroys the fine spatial details, which corresponds, in this example, to the steepness of the edges; this steepness is much better preserved by wavelet denoising.}
\label{wavelet_filtering}
\end{figure}
\subsection{Classical wavelet-based analysis}
The classical wavelet-based methods \cite{ruttimann9801}, mainly transfer each spatial volume to the wavelet domain, perform a $t$-test on each wavelet coefficient, and reconstruct the volumes back by only using the wavelets whose coefficients survive the $t$-test.
Let us denote an fMRI
data set with $v_{\mathbf n}(t)$, where $\mathbf n \in \mathbb Z^3$ is the
spatial index, and $t\in\mathbb Z$ is the temporal index. With a simplified
notation, the wavelet transform corresponds to representing the data under the form
$$
v_\mathbf n (t) = \sum_\mathbf k w_\mathbf k(t) \psi_\mathbf k (\mathbf n),
$$
where $\{\psi_\mathbf k\}$ is the wavelet basis. The basis functions
are translated and dilated versions of some prototype in the case of standard wavelets, but they take more arbitrary forms in the adapted case.
Now let $\mathbf w _\mathbf k$ be the vector of wavelet coefficients
corresponding to $\psi_k$; i.e.,
$$\mathbf w_ \mathbf k = [w_\mathbf k (1)\
\cdots\ w_\mathbf k (N_t)]^T,$$ where $N_t$
is the total number of time samples. Due to linearity of the wavelet transform, we can write the same general linear model
as in the spatial domain
$$
\mathbf w_\mathbf k = \mathbf X \mathbf y
_\mathbf k + \mathbf e_\mathbf k,
$$
where $\mathbf X$ is the $N_t \times L$
design matrix, $\mathbf y_\mathbf k$ is the $N \times 1$ vector of unknown
parameters, and $\mathbf e_\mathbf k$ is the residual error. Assuming the noise to be independent and identically
distributed Gaussian, the unbiased estimate of $\mathbf y_\mathbf k$
is given by
$$
\hat{\mathbf y}_\mathbf k = (\mathbf X^{\operatorname T} \mathbf
X)^{-1}\mathbf X^{\operatorname T} \mathbf w_\mathbf k .$$
Corresponding to each index $\mathbf k$ of the wavelet basis, we obtain two
scalar values:
\begin{align*}
g_\mathbf k &= {\mathbf c}^{\operatorname T} \hat {\mathbf y}_\mathbf k,\\
s^2_\mathbf k &= {\hat{\mathbf e}_\mathbf k}^{\operatorname T}{\hat{\mathbf
e}_\mathbf k} \mathbf c ^{\operatorname T}(\mathbf X^{\operatorname T} \mathbf
X)^{-1} \mathbf c,
\end{align*}
where $\mathbf g_\mathbf k$ and $\mathbf s_\mathbf k$ follow a Gaussian and
Chi-squared distribution, respectively, and $\mbf c$ is the contrast vector. From these one can obtain a $t$ value
for each wavelet coefficient $\mathbf k$:
$$
t_\mathbf k = \frac{g_\mathbf k}{\sqrt{s^2_\mathbf k/J}}, \ \ \text{with} \ J =
N_t - \operatorname{rank}(\mathbf X),
$$
which can be tested against a threshold $\tau_w$, which is chosen in accordance
with the desired significance level. After testing, the
detected coefficients are reconstructed as
\begin{equation}
\label{eq:recon}
\mathbf r_\mathbf n = \sum_\mathbf k T_{\tau_w}(t_\mathbf k)g_\mathbf k
\psi_\mathbf k(\mathbf n),
\end{equation}
where $T$ is the thresholding function corresponding to the two
sided $t$ test; i.e., $T(t_\mathbf k) = 1$ if $|t_\mathbf k|\geq \tau_w$, and
zero otherwise. The volume $r_\mathbf n$ contains many nonzero voxels, each of
which is a function of many voxels from the original data. One must rely on heuristic thresholds on $r_\mathbf n$ to
obtain acceptable detection maps. Moreover, $r_\mathbf n$ does not have a direct
statistical interpretation. These disadvantages are overcome with the WSPM, as explained in the next section.
\subsection{WSPM: Joint spatio-wavelet statistical analysis}
Wavelet based Statistical Parametric Mapping (WSPM) is a method proposed by Van De Ville et al. \cite{surfing, vandeville04,
vandeville0406, vandeville07}, which is a modification of SPM where the denoising step is performed by thresholding in the spatial wavelet domain. This makes the typical advantage of wavelets, which is about reducing noise while keeping high frequency detail information, apparent in
the results. The underlying theorem guarantees control over the false-positive rate by a bound of the null-hypothesis rejection probability. Moreover, empirical results show similar sensitivity to that obtained by SPM with improved spatial detail. The resulting map of active voxels can be seen to align with the
cortex, as a demonstration of preserving the detail information.
The main idea in the joint spatio-wavelet statistical analysis is
to perform two consecutive thresholding operations: first in the wavelet domain
and then in the spatial domain. There are two corresponding threshold
parameters,
$\tau_w$ and
$\tau_s$, to be determined \cite{vandeville04, vandeville0406}. A spatial map
is obtained after the first thresholding as in (\ref{eq:recon}). Then $r_\mathbf
n$ is weighted by $1/\sum_\mathbf k \sigma_\mathbf k |\psi(\mathbf n)|$ and
thresholded by $\tau_s$. That is, the set of voxels that are declared to be
active would be
$$
\left\{\mathbf n \colon \frac{\left|\sum_\mathbf k T_{\tau_w}(t_\mathbf
k)g_\mathbf k
\psi_\mathbf k(\mathbf n) \right|}{\sum_\mathbf k \sigma_\mathbf k
|\psi(\mathbf n)|}\geq \tau_s\right\}.
$$
Given the desired significance level $\alpha$, an optimal choice for $\tau_s$
and
$\tau_w$, which
minimizes
the approximation error between the reconstruction from the fitted parameters
and two times
thresholded reconstruction, can be computed to be
$$
\tau_w =
\sqrt{-W_{-1}\left(-\frac{\alpha^2\pi}{2}\right)},\ \ \ \tau_s = 1/\tau_w,
$$
where $W_{-1}$ is the $-1$-branch of the Lambert-$W$ function
\cite{vandeville04}.
\begin{figure}
\centering
\includegraphics[width=14cm]{./figures/chp4/vandeville_figure.eps}
\caption{Diagram illustrating the wavelet-based approach proposed in \cite{vandeville0406}. a) Classical wavelet based method, and b) Integrated wavelet based method (WSPM).
In both methods, the wavalet transform of each volume is computed (DWT), the variables $g_\mathbf k$ and $s^2_\mathbf k$ are computed for each wavelet coefficient (LM), a $t$ variable is obtained for each wavelet coefficient and the $t$-test is performed, and the inverse wavelet transform computed after throwing out the coefficients failing the test (IDWT). In the classical method, one must rely on heuristic thresholds for the reconstructed volume. In the integrated framework, the thresholds $\tau_w$ and $\tau_s$ are determined together, as a function of the input sensitivity parameter $\alpha_B$. In the integrated method, one also needs to compute a reconstruction with the absolute value of the wavelets, before the spatial thresholding. Figure replicated from \cite{vandeville0406}.}
\label{fig:classical_vs_vspm}
\end{figure}
\subsection{Anatomically adapted wavelets}
In the methods mentioned above methods, the wavelet transforms are either performed slice by
slice as two dimensional wavelet transforms, or are applied to the whole
volume as a three dimensional wavelet transform. In either case, the domain of
the signals are assumed to be the
rectangular. However, in reality the activity takes place in a subset
corresponding to the brain cortex, which is a highly convoluted three
dimensional structure. This motivates us to obtain domain-adaptive wavelets for fMRI data, using the construction in Chapter 3. We input the segmented brain cortex to the algorithm, and use the resulting wavelets in the statistical framework for analyzing fMRI data. We refer to these type of wavelets as \emph{anatomically adapted wavelets}.
\section{Experimental results}
\label{sec:expresults}
\subsection{Simulated data}
We generated smooth functions over a domain consisting of concentric
circles, to represent the fine layered structure of the gray-matter cortical layer with different widths. We contaminated the data with white Gaussian noise, whose magnitude is decreasing from left to right, as shown in Figure
\ref{fig:res2}.
The adapted wavelets showed an improvement in the
sensitivity, while keeping the specificity within the theoretical limits. As the level of the wavelet decomposition increases, the sensitivity keeps
increasing when adapted wavelets are used, while it remains unchanged for
standard wavelets; this is illustrated by the comparison in Figure \ref{fig:res3} .
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\epsfig{figure=./figures/chp4/simdata.eps,width=12cm}}
\medskip
\end{minipage}
\caption{The detected map of active voxels in an example with simulated data. The domain consists of concentric rings. Gaussian white noise is superimposed on the data, with $\sigma$ kept constant within each ``column'', but as the ``row'' number $r$ increases, $\sigma$ decreases, with $\sigma r = \text{constant}$. For $r=2$, $\sigma$ is such that the noise overpowers the signal completely; for $r=64$, $\sigma$ has decreased to a level where the signal is detected without problems. The left column shows results with
standard orthogonal wavelets, and the right column shows the result with the
anatomically adapted wavelets, which detect the signal in some locations where standard wavelets don't. In the second row the level of wavelet
decomposition is increased, and the performance of adapted wavelets has
increased correspondingly while the performance of standard wavelets remain unchanged.}
\label{fig:res2}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\epsfig{figure=./figures/chp4/alpha_sens.eps,width=12cm}}
\end{minipage}
\caption{The ROC curves with standard (tensor-product orthogonal cubic B-spline
wavelets) and anatomically adapted wavelets. The $\alpha$-value represents the
type-I error control rate that is input to the algorithm.}
\label{fig:res3}
\end{figure}
\subsection{Real data}
We tested the proposed method with data obtained from a visual stimulation
experiment, with 16 slices of $128\times128$ voxels, the size of which is 1.8 mm $\times$ 1.8
mm $\times$ 5 mm. We performed segmentation with the SPM software package, and generated the adaptive
wavelets using the domain corresponding to the gray-matter layer. The thresholded binary domain is shown in Figure \ref{fig:cortical_mask}. Using the
standard
orthogonal wavelets, the analysis resulted in 1032 detected voxels, with
the adapted wavelets it resulted in 1214 active voxels. In both cases the
sensitivity parameter $\alpha$ is taken to be 0.001. This suggests an
improved sensitivity, with a detection of larger number of voxels, as shown in
Figure \ref{fig:res4}.
\begin{figure}
\centering
\includegraphics[width=14cm]{./figures/chp4/cortical_mask.eps}
\caption{The binary mask used in the algorithm, which identifies the brain cortex. The volume corresponding to the brain cortex is displayed slice by slice. This map is obtained by thresholding the output of SPM's brain segmentation algorithm. This domain information is input to the wavelet construction algorithm, as a binary volume. }
\label{fig:cortical_mask}
\end{figure}
\begin{figure}[htb]
\begin{minipage}[b]{1\linewidth}
\centering
\centerline{\epsfig{figure=./figures/chp4/realdata.eps,width=12cm}}
\end{minipage}
\caption{Slices from detected activation maps with real visual stimulation
data. The left image is obtained with adaptive wavelets, and the right image is obtained
with orthogonal cubic B-spline wavelets.}
\label{fig:res4}
\end{figure}
\nocite{sweldens_siam}
\nocite{surfing}
\nocite{vandeville0406} \nocite{vandeville04} \nocite{sarty}
\nocite{daubechies}
\nocite{ruttimann9801} \nocite{vandeville07}
\nocite{sweldens96}
\chapter{Summary and future work}
In this thesis, we constructed two and three dimensional wavelets for arbitrary domains, using the lifting scheme. When the original domain of the signals to be analyzed has a significant proportion of boundary pixels or voxels, our adapted wavelets demonstrate better sparsity properties and a superior performance in denoising, compared to standard wavelets that have rectangular domains.
In the construction, a nested grid structure is required to be defined on the domain. In defining the nested grid, we used a randomized algorithm. This randomization gives us the chance to have multiple sets of wavelet bases on the same domain, which allows one to process the data multiple times and then take the average of the results. This, in general, improves the results whenever it is possible to average the output of the wavelet application. Our test with two dimensional images showed that this algorithm is nearly translation and rotation invariant.
The new class of wavelets are then used in the brain imaging problem, after being adapted to the
anatomy of the brain cortex. In the wavelet-based Statistical Parametric Mapping
framework, we have observed an improved sensitivity, while
retaining the same amount of control over type-I errors, compared to wavelet transforms having rectangular domains. In simulated data, contrary
to what is observed with the standard wavelet transform, use of the adapted wavelets
shows clear improvement as the level of the wavelet decomposition increases.
As the high resolution fMRI scanners are becoming more widely available, spatial filtering tools like ours, which treat the brain cortex as an arbitrary three dimensional volume, may become an alternative to the well-established spatial filtering tools that are currently used in brain imaging analysis.
\subsubsection{Future work}
In utilizing multiple sets of wavelet bases, we processed the data separately with each individual non-redundant basis, and took the average of the results. However, the average may include results from some bad realizations of the basis, which may deteriorate the performance. As an alternative, we will explore considering of the union of the multiply generated bases as a single overcomplete basis, and use it together with sparsity-constrained risk minimization algorithms.
In another direction, we will test the performance of the anatomically adapted wavelets in multi-subject fMRI studies.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,260 |
La batalla de Pozoblanco fue un enfrentamiento ocurrido durante la guerra civil española que se llevó a cabo en la localidad de Pozoblanco, Provincia de Córdoba (España). Los combates duraron entre el 6 de marzo y el 16 de abril de 1937. Terminó en una victoria republicana, constituyendo una de las batallas más destacadas de los llamados frentes de Andalucía y Córdoba. Dado que sucedió al tiempo que estaba teniendo lugar la famosa Batalla de Guadalajara, pasó prácticamente desapercibida y no alcanzó la importancia que realmente tuvo.
Antecedentes
Desde el comienzo de la Guerra Civil, las Fuerzas sublevadas consolidaron su dominio sobre importantes áreas de la Provincia de Córdoba, especialmente las de la capital cordobesa, la zona industrial de Peñarroya-Pueblonuevo o la Subbética. Así, a finales de 1936 estos matenían bajo control buena parte de la provincia salvo la zona de los Pedroches.
A comienzos de 1937 los sublevados consolidaban sus dominios en la Andalucía Occidental, especialmente tras la conquista de Málaga. Después de la Ofensiva de la aceituna y la Batalla de Lopera, los avances de los sublevados en el Valle del Guadalquivir se encontraban estancados ante la creciente resistencia republicana. Más aún, esto ocurría al tiempo que estaba teniendo lugar el Asedio del Santuario de Nuestra Señora de la Cabeza, donde la resistencia de un grupo de guardias civiles sublevados se encontraba cada vez en una situación más complicada. Es entonces cuando se plantea en el Cuartel general del General Queipo de Llano la posibilidad de reactivar este sector.
Operaciones
Primeros ataques
El 6 de marzo diversas fuerzas del Ejército sublevado del Sur al mando de Queipo de Llano lanzan una serie de ataques en el sector de los Pedroches, con el objetivo de tomar Pozoblanco y Villanueva de Córdoba, para después avanzar hacia Andújar y allí liberar a los sitiados del Santuario de la Cabeza. El subsector de Pozoblanco, a cargo del comandante Francisco Blanco Pedraza, estaba defendido por las brigadas 73.ª y 74.ª; otra brigada, la 75.ª, se encontraba en Andújar situada en reserva.
Los franquistas pusieron en movimiento tres columnas, mandadas respectivamente por los comandantes Álvarez-Rementería, Gómez Cobián e Hidalgo. El ataque progresa inicialmente, llegando a avanzar unos 16 km en dirección a Villanueva de Córdoba; El coronel republicano Gabriel Morales reorganiza sus fuerzas y traslada a las brigadas mixtas 20.ª y 25.ª, logrando retrasar el avance sublevado. La 20.ª Brigada, junto a dos batallones y una agrupación de caballería de la 63.ª Brigada Mixta, atacó el flanco de las fuerzas de Queipo de Llano, logrando detener su avance en el cruce de las carreteras de Peñarroya y Villanueva del Duque con la de Belmez a Hinojosa.
La presión de los sublevados se dirigió entonces hacia Pozoblanco, llegando sus vanguardias a las orillas del río Cuzna.
Clímax de la batalla
El coronel Morales —comandante del Ejército del Sur— solicitó el envío de refuerzos. El Estado Mayor republicano de Valencia le envió la posteriormente denominada «Brigada Móvil de Pozoblanco», compuesta por dos batallones de carabineros del centro de instrucción de Requena, un batallón de Linares, el tercer batallón del regimiento valenciano «Pablo Iglesias» y una Batería de Almansa. Empieza a evidenciarse que las fuerzas sublevadas no tienen suficiente fuerza para hacer frente a las fuerzas del coronel Joaquín Pérez Salas, militar republicano que ya se había destacado en el Frente de Córdoba desde el comienzo de la contienda. Entre los días 9 y 12 la batalla alcanzó su momento clímax, ya que los ataques sublevados se intensificaron para intentar alcanzar Pozoblanco.
El día 9 sus vanguardias alcanzan Villanueva del Duque, donde se enfrentaron a la 25.ª Brigada Mixta. El día 10 las fuerzas de Queipo de Llano entraron en Alcaracejos, localidad que ocupan, al tiempo que otra columna que avanza hacia el norte intenta enlazar con las fuerzas de Alcaracejos, aunque la endurecida resistencia republicana lo impide. En este momento las bajas son muy numerosas por ambas partes, y aunque los republicanos mantienen la resistencia, Queipo de Llano insistió en continuar con el ataque durante los días 12 y 13. Pérez Salas logró asegurar Pozoblanco, al tiempo que reorganizaba sus fuerzas. Según el historiador Salas Larrazábal, en ese momento disponía de seis brigadas mixtas listas para intervenir. Las fuerzas franquistas debieron sostener fuertes combates en Villanueva del Duque, atacada por las brigadas republicanas 63.ª, 20.ª y 25.ª; la columna del coronel Manuel Baturone logró aliviar parcialmente esta presión.
En estas batallas la caballería todavía jugaría un papel activo: por ejemplo, el día 9 los republicanos sufrieron 9 muertos, de los que 7 lo fueron directamente tras una carga de caballería. El 13 de marzo el jefe de la 25.ª Brigada Mixta fue destituido debido al «mal rendimiento» que estaba teniendo la unidad desde el inicio de las operaciones, siendo entonces sustituido por el comandante García Moreno. En vista de que la situación no progresaba, Queipo de Llano ordenó la retirada paulatina a sus puntos de partida anteriores a la ofensiva. En los primeros momentos, los republicanos no fueron conscientes de esta retirada pero pronto pasarían a la acción.
El 18 de marzo las columnas de Baturone e Hidalgo presionaron fuertemente sobre Pozoblanco, que estuvo a punto de caer en manos franquistas. Sin embargo, la llegada de refuerzos —consistentes en los tanques soviéticos y algunos elementos de la 52.ª Brigada Mixta— lograron salvar la situación.
Contraataque republicano
A partir de ese momento los republicanos lanzaron todas sus fuerzas al contraataque aunque, según Salas Larrazábal, tras la retirada de las fuerzas de Queipo de Llano «no existía ninguna fuerza enemiga en frente de las unidades del Ejército Popular». Queipo de Llano, consciente de la amenaza que corrían sus desgastadas fuerzas, ordenó el 29 de marzo que las unidades bajo su mando se retiraran y regresasen a sus puntos de partida. Los republicanos llegan a mandar al frente a unidades de la Brigada de caballería, así como tanques y carros de combate. En los siguientes días las brigadas mixtas 20.ª y 25.ª, junto a la Columna «Andalucía-Extremadura», avanzaron sobre Villanueva del Duque, combatiendo también en Alcaracejos e Hinojosa, aunque solo se encontraron con la resistencia de pequeños grupos. En poco tiempo, los republicanos lograron restaurar las líneas defensivas anteriores al inicio de la ofensiva, e incluso avanzar hasta las cercanías de Peñarroya-Pueblonuevo.
El 22 de marzo llegaron nuevos refuerzos republicanos, consistentes en una compañía de tanques, varias baterías de artillería y el resto de la 52.ª Brigada Mixta —procedente del frente de Almería—. La 52.ª BM vino a sustituir a la 20.ª Brigada Mixta, que para entonces se encontraba muy desgastada.
La victoria del Ejército republicano animó a sus mandos a lanzar el 27 de marzo una ofensiva que pretendía la captura de Peñarroya-Pueblonuevo, y aunque no se logró tal objetivo, si consiguió algunas ganancias territoriales con la conquista de Valsequillo, La Granjuela, Los Blázquez. Después de alcanzar las inmediaciones de Peñarroya el 8 de abril, los avances republicanos quedaron detenidos y para el 13 de abril habían terminado todas las operaciones en el Frente de Córdoba.
Consecuencias
Así, la tentativa de Queipo de Llano terminó en un estrepitoso fracaso para el general sublevado, al tiempo que constituía un éxito del Ejército republicano del Sur y reforzaba su moral tras el Desastre de Málaga. Aun así, esta batalla ha pasado prácticamente desapercibida para la historiografía, dado que quedó ensombrecida por la victoria republicana en la Batalla de Guadalajara; La brillante actuación del Teniente Coronel Joaquín Pérez Salas hizo que posteriormente éste llegara a ser propuesto para la concesión de la Placa Laureada de Madrid. El resultado de este enfrentamiento terminaría decidiendo el resultado del Asedio del Santuario de Nuestra Señora de la Cabeza, que acabó siendo capturado el 1 de mayo por las fuerzas al mando del Teniente Coronel Martínez Cartón.
Referencias
Bibliografía
López Romero , Laura (2003); Joaquín Pérez Salas y la Batalla de Pozoblanco
Moreno Gómez, Francisco (1985); La Guerra Civil en Córdoba (1936-1939). Ed. Alpuerto.
Moreno Gómez, Francisco (1983); La República y la Guerra Civil en Córdoba (I). Ayuntamiento de Córdoba.
Salas Larrazábal, Ramón (2006); Historia del Ejército Popular de la República. La Esfera de los Libros S.L. ISBN 84-9734-465-0
Enlaces externos
Historia de Pozoblanco
Pozoblanco
Pozoblanco
Batallas en la provincia de Córdoba
Pozoblanco | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,145 |
{"url":"https:\/\/eprint.iacr.org\/2018\/344","text":"### Nothing Refreshes Like a RePSI: Reactive Private Set Intersection\n\nAndrea Cerulli, Emiliano De Cristofaro, and Claudio Soriente\n\n##### Abstract\n\nPrivate Set Intersection (PSI) is a popular cryptographic primitive that allows two parties, a client and a server, to compute the intersection of their private sets, so that the client only receives the output of the computation, while the server learns nothing besides the size of the client's set. A common limitation of PSI is that a dishonest client can progressively learn the server's set by enumerating it over different executions. Although these \"oracle attacks\" do not formally violate security according to traditional secure computation definitions, in practice, they often hamper real-life deployment of PSI instantiations, especially if the server's set does not change much over multiple interactions. In a first step to address this problem, this paper presents and studies the concept of Reactive PSI (RePSI). We model PSI as a reactive functionality, whereby the output depends on previous instances, and use it to limit the effectiveness of oracle attacks. We introduce a general security model for RePSI in the (augmented) semi-honest model and a construction which enables the server to control how many inputs have been used by the client across several executions. In the process, we also present the first practical construction of a Size-Hiding PSI (SHI-PSI) protocol in the standard model, which may be of independent interest.\n\nNote: Added reference to a paper by Chase et al. (Eurocrypt'15)\n\nAvailable format(s)\nPublication info\nPublished elsewhere. MINOR revision.ACNS 2018\nKeywords\nprivate set intersectionreactive functionalities\nContact author(s)\nme @ emilianodc com\nHistory\n2018-06-01: revised\nSee all versions\nShort URL\nhttps:\/\/ia.cr\/2018\/344\n\nCC BY\n\nBibTeX\n\n@misc{cryptoeprint:2018\/344,\nauthor = {Andrea Cerulli and Emiliano De Cristofaro and Claudio Soriente},\ntitle = {Nothing Refreshes Like a RePSI: Reactive Private Set Intersection},\nhowpublished = {Cryptology ePrint Archive, Paper 2018\/344},\nyear = {2018},\nnote = {\\url{https:\/\/eprint.iacr.org\/2018\/344}},\nurl = {https:\/\/eprint.iacr.org\/2018\/344}\n}\n\nNote: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.","date":"2023-02-06 02:35:46","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\": 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\": 0.26803168654441833, \"perplexity\": 4391.083971117232}, \"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-2023-06\/segments\/1674764500303.56\/warc\/CC-MAIN-20230206015710-20230206045710-00213.warc.gz\"}"} | null | null |
\section{Introduction}
The fifth generation of wireless communication (5G) must support novel traffic types for which low latency, high data rate, and ultra reliability are of interest. Particularly, in many applications
such as vehicle-to-vehicle and vehicle-to-infrastructure
communications for traffic efficiency/safety or real-time video
processing for augmented reality, the codewords are required to
be short (in the order of $\sim$ 100 channel uses) with stringent requirements on the latency and reliability \cite{tutor2}. Therefore, it is interesting to optimize the performance of wireless networks
in the presence of finite-length codewords.\par
In 2010, \cite{poly_main} presented accurate information-theoretic approximations for the achievable rates of finite blocklength codes. Using \cite{poly_main}, the performance of wireless networks with short packets has been studied in various papers, for the cases with cognitive radio \cite{cog_mak}, relay networks \cite{li2016throughput}, hybrid automatic repeat request technique \cite{harq}.\par
In this letter, we consider a wireless network with an access point (AP) serving multiple users. Using short packets, the AP transmits packets in the downlink to the users, which have different target error probability requirements. Particularly, using the recent results of \cite{poly_main}, we propose a joint sum rate and per-user error probability optimization problem and investigate the effect of the codeword length on the system performance. To solve the joint sum rate and per-user error probability optimization problem, we develop a low-complexity two-level algorithm based on the divide-and-conquer approach. Also, we derive a closed-form expression for the optimal per-user error probability (Theorem \ref{porp:er}). Finally, we find an efficient and close-to-optimal power allocation algorithm, in terms of sum rate and error probability, based on the augmented Lagrange method \cite{bert} (Algorithm 1).
\par
The simulation and analytical results show that 1) our proposed algorithm can reach (almost) the same performance as in the exhaustive search-based approach with considerably less implementation complexity (Figs. 1 and 2). Also, 2) the throughput is sensitive to the length of short packets while its sensitivity to the packet length decreases for long packets (Fig. 2b). Finally, 3) optimal error probability assignment with water-filling (WF) power allocation achieves higher throughput, compared to using the optimal power allocation with equal error probability assignment (Figs. 1, 2a).
\section{System Model}
We consider a downlink communication model with $N$ single-antenna users which are served by a single-antenna AP. It is assumed that each user is allocated an orthogonal channel to the AP, for instance they could be separated in the frequency or time domain. Let us denote the instantaneous channel gain between the AP and the $i$-th user, $i=1, \ldots, N,$ by $g_i$. The channel gain $g_i$ can be expressed as $g_i=\bar{g_i} \theta_i $
, where $\theta_i$ represents the small scale fading and $\bar{g_i}$
is the average channel gain, obtained by considering the path loss
effects and shadowing. Thus, with $i$-th user located at distance $d_i$ from the AP, we have $\bar{g_i}=\kappa_i d_i^{-\delta_i}$ where $\kappa_i$ is the signal power gain at distance 1 meter from the AP and $\delta_i$ is the path loss exponent. Moreover, the power allocated by the AP to the signal of user $i$ is denoted by $p_{i}$. Thus, the instantaneous signal-to-noise ratio (SNR) received by user $i$ is $\gamma_{i}=\frac{p_{i}g_{i}}{\sigma^2}$, where $\sigma^2$ is the noise power density.
We characterize the network performance when the AP employs packets of short length. Specifically, the user $i$'s message is encoded into the packets of length $L$ and transmitted with power $p_i$. In this way, the maximum achievable information rate in nats per channel use (npcu) for user $i$ which can be decoded with
block error probability no greater than $\epsilon_i$ is given by\cite[Thm. 1]{tan2015third}
\begin{equation}\label{eq:rate_fbl}
r_{i}=\log\left(1+\gamma_i p_i\right)-\sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_i p_i\right)^2}\right)}Q^{-1}\left(\epsilon_i\right) +\frac{\log\left(L\right)}{L} .
\end{equation}
In (\ref{eq:rate_fbl}), the achievable rate increases unboundedly as the error probability tends towards one. On the other hand, the rate decreases significantly in the cases with strict error probability requirements, i.e., small $\epsilon_i$'s. Also, the achievable rate increases with the signal length $L$ monotonically and letting $L\to\infty$, the achievable rate (\ref{eq:rate_fbl}) converges to Shannon's capacity formula in the cases with asymptotically long codewords.\par
Motivated by the tradeoff between the achievable rates and the error probability in (\ref{eq:rate_fbl}), we consider a joint sum rate maximization and per-user error probability minimization problem. Assuming perfect channel state information (CSI) at the AP, we study a multi-objective optimization problem
\begin{subequations}
\label{opt:main0}
\begin{align}
\underset{\bm{\epsilon},\bm{p}}{\text{maximize}}
& \quad \sum_{i=1}^{N}\log\left(1+\gamma_ip_i\right)- \sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} Q^{-1}\left(\epsilon_i\right) \label{cost_sr} \\
\underset{\bm{\epsilon},\bm{p}}{\text{minimize}}
& \quad \max\{\epsilon_1,\hdots,\epsilon_N\} \label{cost_er} \\
\text{subject to}
& \quad 0\leq\epsilon_i \leq \varepsilon_{\text{max},i} \quad \forall i\in \{1,\hdots,N\} ,\label{const_er}\\
& \quad 0\leq p_i \quad \forall i\in \{1,\hdots,N\}, \label{const_pow1}\\
& \quad \sum_{i=1}^{N} p_i \leq P_{\text{max}}, \label{const_pow2}
\end{align}
\end{subequations}
where $\bm{\epsilon}=[\epsilon_1,\hdots,\epsilon_N]$ and $\bm{p}=[p_1,\hdots,p_N]$. Also, in (\ref{cost_er}), the goal is to minimize the maximum error probability of the users. Then, in (\ref{const_er}), $\varepsilon_{\text{max},i}$ is the maximum error probability constraint of user $i$ which indicates the supporting quality of service (QoS) requirement of user $i$. Also, the total power constraint of the AP is denoted by $P_{\text{max}}$. In this way, (\ref{opt:main0}) is of interest in emerging applications of 5G calling for heterogeneous QoS requirements on data rate and reliability. For instance, massive machine-type communication and ultra-reliable and low latency communication scenarios demand short-length packet exchange with stringent requirement on reliability at moderately low rate\cite{tutor2}. The requirements of the aforementioned services illustrate how the
framework of (\ref{opt:main0}) can be utilized to balance conflicting performance objectives, namely, sum rate and maximum error probability. Moreover, as seen in the following, our discussions are well applicable to the cases when optimizing the network sum throughput, which is defined as the product of the rate and the successful message decoding probability. However, as opposed to throughput optimization, (\ref{opt:main0}) is flexible in optimizing the rates and error probabilities individually based on the QoS requirements. \par
Depending on the number of users, there may be no closed-form solution for (\ref{opt:main0}). Thus, we follow the same method as in \cite{yu2016tradeoff} to convert the problem into a single objective optimization using the weighted sum method while normalizing the objectives. Also, as seen in Section \ref{sec:sim}, our proposed sub-optimal approach can reach (almost) the same performance as in the optimal exhaustive search-based scheme. With no loss of generality, we assume $\varepsilon_{\text{max},1}\leq \varepsilon_{\text{max},2} \leq \hdots \leq \varepsilon_{\text{max},N} < \frac{1}{2}$. Also, to guarantee a consistent comparison between the objectives in (\ref{cost_sr}) and (\ref{cost_er}), we normalize them as
\begin{align}
&U_1\left(\bm{p},\bm{\epsilon}\right)=\frac{\left[\sum_{i=1}^{N}\log\left(1+\gamma_ip_i\right)- \sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} Q^{-1}\left(\epsilon_i\right)\right]}{\text{SR}_{\infty}}\\
&U_2\left(\bm{\epsilon}\right)=\frac{\varepsilon_{\text{max},N}-\max\{\epsilon_1,\hdots,\epsilon_N\}}{\varepsilon_{\text{max},N}},
\end{align}
where $\text{SR}_{\infty}$ is a normalization factor that can be found by plugging the water-filling power allocation expression into the Shannon's capacity formula which provides an upper bound for (\ref{cost_sr}). Then, we use the weighted sum method to rewrite (\ref{opt:main0}) as the single-objective optimization problem
\begin{subequations}
\label{opt:main_after}
\begin{align}
\underset{\bm{\epsilon},\bm{p}}{\text{maximize}}
& \quad \omega U_1\left(\bm{p},\bm{\epsilon}\right)+\left(1-\omega\right) U_2\left(\bm{\epsilon}\right) \\
\text{subject to}
&\quad (\text{\ref{const_er}})-(\text{\ref{const_pow2}}).
\end{align}
\end{subequations}
Here, $0 \leq \omega \leq 1$ is the weighting parameter. Note that, with $\omega$ ranging from $0$ to $1$, scenarios with strict rate requirements and relaxed error probability requirements to scenarios with low rate requirements and ultra-reliability are addressed.
\section{Proposed Algorithm}\label{sec:alg}
\vspace{-1mm}
The optimization problem (\ref{opt:main_after}) belongs to the class of non-convex problems which has a multi-modal objective function, so finding its global optimal solution is computationally infeasible. For this reason, we apply the primal decomposition approach \cite{palomar2006tutorial} to optimize $\bm{\epsilon}$ and $\bm{p}$ separately. In this way, to solve (\ref{opt:main_after}), we use the following iterative approach
\begin{equation*}
\begin{aligned}
\underbrace{\bm{p}[0]}_{\text{initialization}}&\rightarrow \bm{\epsilon}[1]\rightarrow\bm{p}[1]\rightarrow \hdots \underbrace{\bm{\epsilon}[T]\rightarrow\bm{p}[T]}_{\text{optimal solution}},
\end{aligned}
\end{equation*}
where $\bm{\epsilon}[t]$ and $\bm{p}[t]$ are the optimal error probability and power allocation vectors at iteration $t$, and $T$ is the maximum number of iterations considered by the network designer. The details of our proposed optimization approach are as follows.
\subsection{Error Probability Optimization For A Given Power Allocation}
Here, for a given power allocation $\bm{p}^{\star}\left[t-1\right]:= \bm{p}$, we find the optimal error probabilities of each user at iteration $t$ denoted by $\bm{\epsilon}^{\star}[t]$. Setting $z=\max\{\epsilon_1,\hdots,\epsilon_{N}\}$ and assuming a given power allocation, (\ref{opt:main_after}) is rephrased as
\begin{subequations}
\label{opt:sub_er}
\begin{align}
\underset{\bm{\epsilon},z}{\text{minimize}}
& \quad \frac{\omega}{\text{SR}_{\infty}} \sum_{i=1}^{N} \sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} Q^{-1}\left(\epsilon_i\right)+\frac{1-\omega}{\varepsilon_{\text{max},N}}z \label{subprob1:obj} \\
\text{subject to}
& \quad \epsilon_i\leq z \quad \quad \forall i\in \{1,\hdots,N\} \label{const:sub11} \\
& \quad 0\leq z \leq \epsilon_{\text{max},N} \label{const:sub12}\\
& \quad 0\leq\epsilon_i \leq \varepsilon_{\text{max},i} \quad \forall i\in \{1,\hdots,N\} \label{const:sub13}
\end{align}
\end{subequations}
Theorem \ref{porp:er} gives a closed-form expression for the optimal error probability assignment of each user in terms of (\ref{opt:sub_er}).
\begin{theorem}\label{porp:er}
The optimal error probabilities of the users are given by
\begin{equation} \label{eq:opt_err_nahaie}
\bm{\epsilon}^{\star}=\begin{cases}\vspace{1mm}
\Big[\varepsilon_{\text{max},1},\hdots,\varepsilon_{\text{max},k-1},\underbrace{\beta_{k},\hdots,\beta_{k}}_{N-k+1 \ \text{times}} \Big] & \beta_{k} \in \mathcal{I}_{k}\\
\left[\varepsilon_{\text{max},1},\hdots,\varepsilon_{\text{max},N}\right] & \text{otherwise}
\end{cases},
\end{equation}
for $k=1,\hdots,N$. Here, we define the intervals $\mathcal{I}_{k}=(\varepsilon_{\text{max},k-1},\varepsilon_{\text{max},k}]$ with $\mathcal{I}_{1}=(0,\varepsilon_{\text{max},1}]$, and \\ $\beta_{k}= Q\Bigg(\sqrt{\smash[b]{2\log\Big(\frac{\sqrt{L}\left(1-\omega\right) \text{SR}_{\infty}}{ \varepsilon_{\text{max},N} \omega\sqrt{2\pi}\sum_{i=k}^{N}\frac{\sqrt{\gamma_i^2 p_i^2+2\gamma_i p_i}}{1+\gamma_i p_i} }\Big)}}\Bigg)$.
\end{theorem}
\begin{proof}
Since the constraints in (\ref{const:sub11})-(\ref{const:sub13}) are affine functions in $\bm{\epsilon}$ and $z$, it is sufficient to prove that the objective function in (\ref{subprob1:obj}) is convex. The second derivative of $Q^{-1}\left(x\right)$ is given by $\frac{\text{d}^2 Q^{-1}\left(x\right)}{\text{d}x^2}=2\pi Q^{-1}\left(x\right)\exp\left(\left( Q^{-1}\left(x\right)\right)^2\right) > 0$ if $x < \frac{1}{2}$. Therefore, considering the fact that $\epsilon_i \leq \epsilon_{\text{max},i} \leq \frac{1}{2}$, the objective function in (\ref{subprob1:obj}) is a sum of convex functions and an affine function, i.e., $z$. Hence, (\ref{opt:sub_er}) is a convex optimization problem, and the optimal solution can be found by considering Karush-Kuhn-Tucker (KKT) conditions. Thus, we write the Lagrangian function of (\ref{opt:sub_er}) as
\begin{equation}\nonumber
\mathcal{L}\left(\bm{\epsilon},z,\bm{\lambda},\bm{\nu},\eta\right)= \frac{\omega}{\text{SR}_{\infty}} \sum_{i=1}^{N} \sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} Q^{-1}\left(\epsilon_i\right)+
\end{equation}
\begin{equation}
\frac{1-\omega}{\varepsilon_{\text{max},N}}z-\sum_{i=1}^{N}\lambda_i\left(z-\epsilon_i\right)-\sum_{i=1}^{N}\nu_i\left(\epsilon_{\text{max},i}-\epsilon_i\right)-\eta\left(\varepsilon_{\text{max},N}-z\right),
\end{equation}
where $\bm{\lambda}=\left[\lambda_1,\hdots,\lambda_N\right] \succeq 0$, $\eta\geq 0$, and $\bm{\nu}=\left[\nu_1,\hdots,\nu_N\right] \succeq 0$ are dual variables associated with constraints in (\ref{const:sub11}), (\ref{const:sub12}), and (\ref{const:sub13}), respectively. According to the KKT conditions, the optimal solution, which is denoted by $\bm{\epsilon}^{\star}$ and $z^{\star}$, should satisfy
\begin{subequations}
\label{KKT:COND}
\begin{align}
&\frac{\partial \mathcal{L}}{\partial \epsilon^{\star}_i}= -\frac{\omega}{\text{SR}_{\infty}}\sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} \sqrt{2\pi} \exp\left(\frac{\left(Q^{-1}\left(\epsilon^{\star}_i\right)\right)^2}{2}\right) \nonumber \\
&+\lambda^{\star}_i+\nu^{\star}_i=0, \label{KKT:derv1}\\
&\frac{\partial \mathcal{L}}{\partial z^{\star}} = \frac{1-\omega}{\varepsilon_{\text{max},N}}-\sum_{i=1}^{N}\lambda^{\star}_i+\eta^{\star}=0, \label{KKT:derv2} \\
&\lambda^{\star}_i\left(z^{\star}-\epsilon^{\star}_i\right)=0,\label{KKT:eps1}\\
&\nu^{\star}_i\left(\varepsilon_{\text{max},i}-\epsilon^{\star}_i\right)=0,\label{KKT:eps2}\\
&\eta^{\star}\left(\varepsilon_{\text{max},N}-z^{\star}\right)=0\label{KKT:eps5},\\
&\text{(\ref{const:sub11})-(\ref{const:sub13})}\label{KKT:eps4}.
\end{align}
\end{subequations}
In (\ref{KKT:derv1}), we have used $\frac{\text{d} Q^{-1}\left(x\right) }{\text{d} x}=-\sqrt{2\pi}\exp\left(\frac{\left(Q^{-1}\left(x\right)\right)^2}{2}\right)$. From (\ref{KKT:eps1}) and (\ref{KKT:eps2}), it can be verified that $\epsilon_i^{\star}$ is equal to either $z^{\star}$ or $\varepsilon_{\text{max},i}$; otherwise, $\lambda^{\star}_i$ and $\nu^{\star}_i$ must be equal to zero which contradict with (\ref{KKT:derv1}). Assume $z^{\star} < \varepsilon_{\text{max},N}$ and $z^{\star}\in \mathcal{I}_k$, so according to (\ref{KKT:eps5}), we have $\eta^{\star}=0$. Note that, for $ 1 \leq i \leq k-1$, $\epsilon_{i}^{\star}$ must be equal to $\varepsilon_{\text{max},i}$ since we have $z^{\star} > \varepsilon_{\text{max},i}$. Also, due to the fact that $z^{\star} < \varepsilon_{\text{max},i}$, it can be inferred that $\epsilon_{i}^{\star}=z^{\star}$ for $ k \leq i \leq N$. Thus, in the cases with $z^{\star}\in \mathcal{I}_k$, we have $\bm{\epsilon}^{\star}=\left[\varepsilon_{\text{max},1},\hdots,\varepsilon_{\text{max},k-1},\underbrace{z^{\star},\hdots,z^{\star}}_{\text{$N-k+1$ times}}\right]$. Then, from (\ref{KKT:eps1}) and (\ref{KKT:eps2}), it can be concluded that $\lambda^{\star}_{i}=0$ for $1 \leq i \leq k-1$ and $\nu^{\star}_i=0$ for $k \leq i \leq N$. In this way, (\ref{KKT:derv1}) can be expressed as
\begin{equation}\label{eq:lambda_megh}
\frac{\omega}{\text{SR}_{\infty}}\sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} \sqrt{2\pi} \exp\left(\frac{\left(Q^{-1}\left(z^{\star}\right)\right)^2}{2}\right)=\lambda^{\star}_i,
\end{equation}
for $k \leq i \leq N$. Also, from (\ref{KKT:derv2}) and (\ref{eq:lambda_megh}), we have
\begin{equation}\label{eq:find_b}
\begin{aligned}
&\frac{1-\omega}{\varepsilon_{\text{max},N}}=\sum_{i=k}^{N}\lambda^{\star}_i.
\end{aligned}
\end{equation}
Plugging (\ref{eq:lambda_megh}) into (\ref{eq:find_b}), the upper branch of (\ref{eq:opt_err_nahaie}) is found. In this way, depending on $z^{\star}$ being in each region $\mathcal{I}_{k}$, the closed-form solution for $\bm{\epsilon}^{\star}$ is provided. Then, given $z^{\star}=\varepsilon_{\text{max},N}$, it is straightforward
to show that the objective function in (\ref{subprob1:obj}) is a decreasing function in each $\epsilon_i$, so the lower branch of (\ref{eq:opt_err_nahaie}) provides the optimal solution. Note that because of the strict convexity of (\ref{subprob1:obj}), there is an optimal solution for $\bm{\epsilon}$ which is found by searching in $N+1$ branches of (\ref{eq:opt_err_nahaie}).
\end{proof}
\vspace{-4mm}
\subsection{Optimal Power Allocation For A Given Error Probability}
Consider a given $\bm{\epsilon}[t] := \bm{\epsilon}$. Then, (\ref{opt:main_after}) is relaxed to
\begin{subequations}
\label{opt:sub_pow}
\label{opt:subprolme2}
\begin{align}
\underset{\bm{p}}{\text{maximize}}
& \quad \omega U_1\left(\bm{p},\bm{\epsilon}\right) \label{subprob2:obj} \\
\text{subject to}
& \quad \text{(\ref{const_pow1})-(\ref{const_pow2})}.
\end{align}
\end{subequations}
Since the function in (\ref{subprob2:obj}) is non-concave in $\bm{p}$, problem (\ref{opt:sub_pow}) belongs to the class of non-convex optimization problems. In a non-convex problem, there is a nonzero duality gap between primal and dual problems. Here, we use the \textit{augmented Lagrange approach} \cite[sec 4]{bert} to deal with this non-convex optimization which reduces the duality gap by augmenting a penalty-like quadratic term added to the Lagrangian function. In
\cite[Sec 4.2]{bert}, it has been proved that the augmented Lagrangian is locally convex when the penalty parameter is sufficiently large. In contrast to the penalty functions approach, the
augmented Lagrangian function largely preserves smoothness
and does not require an asymptotically large penalty parameter for the method to converge, meaning that the penalization is exact. Augmented Lagrangian algorithms are based on successive maximization of the augmented Lagrangian function in which the multiplier estimates and penalty parameter are fixed in each iteration and then updated
between iterations. Applying the augmented Lagrangian method on (\ref{opt:sub_pow}), which eliminates the constraints and adds them to the objective function, gives the augmented Lagrangian function
\begin{equation}
\begin{aligned}
&\mathcal{L}_{\mu,\zeta}(\bm{p})=
\frac{\omega}{\text{SR}_{\infty}}\left[\sum_{i=1}^{N}\log\left(1+\gamma_ip_i\right)- \sqrt{\frac{1}{L}\left(1-\frac{1}{\left(1+\gamma_ip_i\right)^2}\right)} Q^{-1}\left(\epsilon_i\right)\right] \\
&-\frac{1}{2\mu} \left[\left(\max\left\lbrace 0,\zeta-\mu\left(P_{\text{max}}-\sum_{i=1}^{N}p_i\right)\right\rbrace \right)^2-\zeta^2\right],
\end{aligned}
\end{equation}
where $\mu$ is a positive coefficient denoting the penalty parameter and
$\zeta$ is the Lagrangian dual variable associated with (\ref{const_pow2}). Then, at stage $l$ of the power allocation problem, we solve
\begin{equation}
\underset{\bm{p}}{\text{maximize}}
\quad \mathcal{L}_{\mu^{(l)},\zeta^{(l)}}\left(\bm{p}\right), \label{ag:}
\end{equation}
which approximates (\ref{opt:sub_pow}) to find the power allocation at iteration $ l+1 $ denoted by $\bm{p}^{(l+1)}$. Moreover, variables $\zeta^{(l)}$ and $\mu^{(l)}$ are updated according to
\begin{equation}
\begin{aligned}
\zeta^{(l+1)}&=\max\left\{ 0,\zeta^{(l)}-\mu^{(l)}\left( P_{\max}-\sum_{i=1}^{N} p_{i}^{(l)}
\right) \right\}, \\
\mu^{(l+1)}&=2\mu^{(l)}, \label{up:zeta_eta}%
\end{aligned}
\end{equation}
respectively. In this way, as $\mu^{(l)}$ increases, the
violations introduced by constraints are penalized more severely so that the maximizer of the penalty function in (\ref{ag:}) gives the results closer to the feasible region. In \cite[Sec 4.2]{bert}, it has been shown that while the constraints are nonlinear, the convergence rate of the augmented Lagrangian method is linear.
\par
The iterative joint error probabilities and power allocation
algorithm is summarized in \textbf{Algorithm \ref{CHalgorithm}}. In order to analyze the complexity order of the proposed algorithm, we note that the optimal error probabilities can be found by (\ref{eq:opt_err_nahaie}) with the complexity $\mathcal{O}\left(N\right)$. Also, the complexity of the power allocation at each iteration is $\mathcal{O}\left(N^2\right)$. Thus, the complexity of Algorithm \ref{CHalgorithm} is $\mathcal{O}\left(N^2\right)$ + $\mathcal{O}\left(N\right)= \mathcal{O}\left(N^2\right).$
\begin{algorithm}
\caption{Error Probabilities Assignment and Power Allocation}
\label{CHalgorithm}
\begin{algorithmic}[1]
\State For every given $\omega$, $P_{\text{max}}$, $\{\varepsilon_{\text{max},1},\hdots,\varepsilon_{\text{max},N}\}$, $\mu^{(0)}$, and $\zeta^{(0)}$.
\State Initialize: $\bm{p}[0]$ and $t=0$.
\While{$\{\epsilon^{\star}[t] \}$ converges}
\State Calculate $\bm{\epsilon}^{\star}[t+1]$ via (\ref{eq:opt_err_nahaie}) with given $\bm{p}^{\star}[t]$.
\State Initialize: $l=0$.
\While{$\{p^{(l)} \}$ converges}
\State Calculate $\bm{p}^{(l+1)}$ via (\ref{ag:}) with given $\bm{\epsilon}^{\star}[t+1]$, $\zeta^{(l)}$, and $\mu^{(l)}$.
\State Update $\zeta^{(l+1)}$ and $\mu^{(l+1)}$ via (\ref{up:zeta_eta}).
\State $l=l+1$.
\EndWhile
\State $\bm{p}^{\star}[t+1]=\bm{p}^{(l)}$.
\State $t=t+1$
\EndWhile
\end{algorithmic}
\end{algorithm}
\section{Numerical Results and Conclusion}\label{sec:sim}
Here, we study the trade-off between the sum rate and maximum error probability. We set the noise power $\sigma^2 = 1$, the number of users $N=4$, and the error probability constraints
$\bm{\varepsilon_{\text{max}}}=[10^{-5},5\times 10^{-5},10^{-4},5\times 10^{-4}]$ which are often assumed for vehicular-to-vehicular communications\cite{tutor2}. Also, it is assumed that the users are equidistant from the AP. Also, we consider
Rayleigh fading with mean 1. Finally, we set $\mu^{(0)}=1$ and $\zeta^{(0)}=0.15$ in Algorithm 1.
For the numerical results, we consider the cases with $ L \geq 100$ channel uses, for which the
approximation (\ref{eq:rate_fbl}) is tight enough \cite{poly_main}. Also, we compare our method with three baseline algorithms: 1) WF-based power allocation with the
error probabilities of all users set to the minimum of the required error probabilities, called minmax error
probability assignment, i.e., $\bm{\epsilon}=[\varepsilon_{\text{max},1},\hdots,\varepsilon_{\text{max},1}]$, 2) the proposed method for power allocation with the minmax error probability assignment, and 3) equal power allocation with the proposed method for the error probabilities assignment. Finally, the results are obtained by averaging over $10^4$ different channel realizations.
\par
Figure 1 shows the the tradeoff between the sum rate and the error probability for different algorithms with $P_{\text{max}}=6$ dB and $L=200$ channel uses. As a performance metric, we define the sum throughput as
\begin{equation}
\mathcal{T}=\sum_{i=1}^{N}r_i\left(1-\epsilon_i\right),
\end{equation}
where the user $i$'s codeword rate and error probability are given by $r_i$ and $\epsilon_i$, respectively.
Then, Fig. 2a demonstrates the sum throughput versus the AP's
total power constraint $P_{\text{max}}$ by setting $L=100$ channel uses and $\omega=0.9$. Finally, Fig. 2b evaluates the effect of the codeword length on the sum throughput. The results lead to the following conclusions:\\
\begin{figure}
\centering
\includegraphics[width=5in, height=3in]{tradeoff.eps}
\caption{Sum rate vs maximum error probability. }
\vspace{-5mm}
\end{figure}
$\bullet$ The scheme with the WF power allocation and the error probability assignment based on Theorem 1 achieves the tradeoff region close to the proposed method optimizing both the error probability and the power allocation (Fig. 1).\\
$\bullet$ For short codewords, the throughput is remarkably affected by the length of the codeword. However, the effect of increasing the codeword length decreases for long codewords (Fig. 2b). Also, optimal error probability assignment with WF power allocation achieves higher throughput compared to optimal power allocation with equal error probability assignment. Moreover, the performance of WF with minmax error probability assignment is close to that of the scheme with the proposed power allocation with minmax error probability assignment (Figs. 2a and 2b). \\
$\bullet$ For short codeword (say,
$L\leq 1000$ channel uses),
the proposed algorithm leads to considerable throughput improvement in comparison with other schemes. For instance, when $L=100$ cu, the performance of the proposed method has $66 \%$ of improvement. However, the performance difference of the schemes decreases in the cases with long codewords.\\
$\bullet$ Finally, as observed in Figs. 1, 2a, and 2b, the gap between the developed algorithm and the exhaustive search-based algorithm diminishes by increasing $P_{\text{max}}$ or $L$. Thus, our proposed algorithm can be effectively applied to jointly optimize the sum rate and the error probability of multi-user networks in delay-constrained applications.
\begin{figure}
\centering
\includegraphics[width=5in, height=3in]{thr_pow.eps}
\includegraphics[width=5.18in, height=3in]{throughput.eps}
\caption{Sum throughput of the considered algorithms. Subplot (a): Sum throughput vs the total power constraint of the AP. Subplot (b): Sum throughput vs the codeword length.}
\vspace{-5mm}
\end{figure}
\begin{comment}
\section{Conclusion}\label{sec:conc}
In this letter, we investigated a tradeoff between sum-rate and
error probability in downlink of cellular network, and proposed an
efficient algorithm for joint optimization of maximal error probabilities assignment and power allocation when codewords are
of finite length. Simulation results show that, using the optimal error probability assignment with a fixed power allocation outperforms the system with a given error probability assignment with a proposed power allocation. Finally, as illustrated, the proposed algorithm reaches to the same performance as the exhaustive search method
\end{comment}
\footnotesize
\vspace{-3mm}
\bibliographystyle{IEEEtran}
\vspace{-1mm}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Ansible: make output from a command become a key-value item/variable for the next command I want to use this output (from a previous command) as an array of key-values or as an inventory for the next command in the same playbook
stdout:
hot-01: 10.100.0.101
hot-02: 10.100.0.102
hot-03: 10.100.0.103
....
hot-32: 10.100.0.132
like this:
- shell: "echo {{ item.key }} has value {{ item.value }}"
with_items: "{{ output.stdout_lines }}"
or:
- add_host: name={{ item.key }} ansible_ssh_host={{ item.value }}
with_items: "{{ output.stdout_lines }}"
Desired output of the echo command:
hot-01 has value 10.100.0.101
I also tried with with_dict: "{{ output.stdout }}" but still no luck
"fatal: [ANSIBLE] => with_dict expects a dict"
A: AFAIK there are no Jinja2 filters to convert strings to dictionaries.
But in your specific case, you can use the python's split string function to separate the key from the value:
- shell: "echo {{ item.split(': ')[0] }} has value {{ item.split(': ')[1] }}"
with_items: "{{ output.stdout_lines }}"
I know, having to use split twice is a bit sloppy.
As in this case your output is a valid YAML, you can also do the following:
- shell: "echo {{ item.key }} has value {{ item.value }}"
with_dict: "{{ output.stdout | from_yaml }}"
As a last resort, you can also create your own ansible module to create a Jinja2 filter to cover your case. There is an split module filter that you can use as inspiration here: https://github.com/timraasveld/ansible-string-split-filter
| {
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} | 8 |
Bolangir: Additional District and Sessions Court of Bolangir, on Friday, rejected the bail application of Punjilal Meher, the prime accused in Patnagarh wedding gift blast case in which two persons including newly-wed Soumya Sekhar Sahu was killed on February 23.
The Court had yesterday reserved the verdict after completion of the hearing on Punjilal's bail plea.
Official sources said, Punjilal's bail petition was rejected on several grounds including concrete evidences collected by the Crime Branch Special Investigation Team (SIT) and confession statements of the accused leading to discovery of crucial evidences.
The Crime Branch had earlier arrested Punjilal on April 25 for masterminding and executing the parcel bomb explosion.
According to the investigating agency, Punjilal had planned the crime and executed it almost single handedly. He himself made the bomb with his own technique and went to Raipur carrying it in the shape of a parcel and booked it at a courier office near a railway station.
He had conspired to kill the entire family of Soumya Sekhar to avenge humiliation that he underwent following his removal from the post of principal of Bhainsa college.
The post was later occupied by deceased Soumya Sekhar's mother. | {
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from __future__ import unicode_literals
from __future__ import absolute_import
from setuptools import setup, find_packages
import re
import os
import codecs
def read(*parts):
path = os.path.join(os.path.dirname(__file__), *parts)
with codecs.open(path, encoding='utf-8') as fobj:
return fobj.read()
def find_version(*file_paths):
version_file = read(*file_paths)
version_match = re.search(r"^__version__ = ['\"]([^'\"]*)['\"]",
version_file, re.M)
if version_match:
return version_match.group(1)
raise RuntimeError("Unable to find version string.")
with open('requirements.txt') as f:
install_requires = f.read().splitlines()
setup(
name='koopa',
version=find_version("koopa", "__init__.py"),
description=('Drake for Luigi'),
author='Cirruspath, Inc.',
author_email='jhorey@cirruspath.com',
license='Apache License 2.0',
packages=find_packages(),
include_package_data=True,
install_requires=install_requires,
entry_points="""
[console_scripts]
koopa=koopa.client.cli:main
"""
)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,485 |
Países Baixos - autoridade competente (Art. 13)
For the Kingdom in Europe:
In proceedings at first instance the district courts and the presidents of the district courts, in appeal proceedings the courts of appeal, and in cassation proceedings the supreme court are competent to render decisions relating to maintenance obligations in the Netherlands.In proceedings at first instance the district courts, in appeal proceedings the courts of appeal, and in cassation proceedings the supreme court are competent to enforce foreign judicial decisions relating to maintenance.
For Aruba:
The Joint Court of Justice of Aruba, Curaçao and Sint Maarten is competent to render decisions relating to maintenance, in first instance and in appeal.
For Curaçao:
For Sint Maarten:
The Courts of first instance and the Joint Court of Justice of Aruba, Curaçao and Sint Maarten is competent to render decisions relating to maintenance.
For the Caribbean part of the Netherlands (the islands of Bonaire, Sint Eustatius and Saba):
The authorities of the European part of the Netherlands are also competent for the Caribbean part of the Netherlands. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: How to detect if Mac if rebooting after a power failure using terminal? As I asked in the title, how can I check if a Mac is rebooting after experiencing a power failure. It would be preferable to check this using terminal or some other script, as I would like to test this automatically in a script upon system startup.
A: You can check for the "Shutdown Cause" in the log.
*
*5 is a normal shutdown
*3 is pressing the power button
*0 is loss of power
In El Capitan and earlier:
cat /var/log/system.log | grep -i "shutdown cause"
In Sierra and newer:
log show --predicate "process == kernel" | grep -i "shutdown cause"
Or....just using awk to get the code:
log show --predicate "process == kernel" | awk '/shutdown cause/ {print $12}'
In Mojave and Catalina, you can issue the command to get a more fine grained approach to the log:
log show --predicate '(process = "kernel") && (eventMessage CONTAINS "shutdown cause")' --last 48h --style compact
This will process only the last 48 hours and output in a nice "compact" layout.
A: In High Sierra, 10.13.5 I used System Information (alt About this Mac) to examine the Power Management log, under Software/Logs.
This clearly showed I'd had a power cut at or shortly after 02:35:14 am. I originally spotted it when the clock on my microwave showed 00:00. Below you cans see the power off and reboot times.
2018-07-07 02:35:14 +0100 Assertions PID 39(UserEventAgent) Released BackgroundTask "com.apple.backupd-auto" 00:00:34 id:0x0xb00009108 [System: BGTask]
Time stamp Domain Message Duration Delay
========== ====== ======= ======== =====
UUID: (null)
2018-07-07 10:25:13 +0100 Start powerd process is started
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,444 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/301505\/coding-schemes-for-analysing-multinomial-data-using-lme4-binary-logistic-regress","text":"# Coding schemes for analysing multinomial data using lme4 binary logistic regression\n\nData: I have data where each trial is a categorical response from a four-way un-ordered forced choice task (A,B,C,D) by participants. I am interested in the factors that influence the choice. The data set is unbalanced (incomplete crossover of participants and stimuli)\n\nMethods: I am analysing it using mixed effect binary logistic modelling using the lme4 library, since there are no easy-to-use multinomial mixed effect models implemented in R at the moment. I have two ideas that involve using lme4.\n\nQuestions: I am particularly curious about idea 2 and whether it is kosher or not.\n\nIdea 1: Analyse it using four separate binary logistic models, each with the response variable being one of the four categories vs rest (e.g. A vs non-A, B vs non-B etc.)\n\nIdea 2: Analyse it using one single binary logistic model. I would recode each trial as four fake trials with the four target responses (A,B,C,D) respectively and the actual response is True or False; say the original trial has the response A; the fake trial with the target response A would get the value True, the other three would get the value False.\n\ne.g. Say the raw data is: Trial 1: A, Trial 2: C, Trial 3: B ... I would code each trial as four fake trials, I would have a column \"Target Response\" which is always A,B,C and D (one each), and an additional column \"Choice\" with TRUE\/FALSE. \"Choice\" would get the TRUE value if it's the actual one being picked\", else FALSE. \"Trial ID\", \"Choice\", \"Target Response\" 1, TRUE , A 1, FALSE, B 1, FALSE, C 1, FALSE, D 2, FALSE , A 2, FALSE, B 2, TRUE, C 2, FALSE, D 3, FALSE , A 3, TRUE, B 3, FALSE, C 3, FALSE, D\n\n\u2022 Is this method kosher at all? I am thinking this might not be kosher because it is not possible for each of the \"Trial ID\" to have more than one TRUE in reality but the coding scheme here allows for that impossibility\n\n\u2022 To mediate this faking of trials, should I group each of these four fake trials with a code and include this code as a random effect? That is the inclusion of \"Trial ID\" as a random intercept might help (or does it?).\n\n\u2022 Should I also include \"target response (A,B,C,D)\" as a random effect? Would this be able to model the inherent preference for each of the four choices?\n\n\u2022 Besides the ability to model the choice of picking A,B,C and D in a single model, I like idea 2 because it seems to allow me to model why people didn't pick the other 3 alternatives for each trial (that is the picking of \"A\" is influenced by why they do not want to pick \"B\", \"C\" and \"D\"). If I do not recode the data at all, I don't seem to be able to model this.\n\n## migrated from stackoverflow.comSep 5 '17 at 15:22\n\nThis question came from our site for professional and enthusiast programmers.\n\n\u2022 I think you should look into doing this using MCMCglmm instead. Regardless, this would be better suited at Cross Validate, and I'm voting to migrate. \u2013\u00a0Axeman Sep 5 '17 at 14:46\n\u2022 AFAIK, you can also fit this in brms (via family = categorical). A reproducible example would be useful, though. \u2013\u00a0alexforrence Sep 5 '17 at 17:57","date":"2019-07-23 18:10:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.441084623336792, \"perplexity\": 1768.572996257224}, \"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-2019-30\/segments\/1563195529481.73\/warc\/CC-MAIN-20190723172209-20190723194209-00349.warc.gz\"}"} | null | null |
Гай Кальпу́рний Пизо́н (; родился около 110 года до н. э. — умер вскоре после 61 года до н. э.) — римский политический деятель из плебейского рода Кальпурниев, консул 67 года до н. э. Во время консулата возглавлял борьбу против Габиниева закона о наделении Гнея Помпея Великого чрезвычайными полномочиями для борьбы с пиратами. Управлял Нарбонской Галлией (66—65 годы до н. э.), враждовал с Гаем Юлием Цезарем.
Происхождение
Гай Кальпурний принадлежал к плебейскому роду Кальпурниев, происходившему, согласно поздним генеалогиям, от Кальпа — мифического сына второго царя Рима Нумы Помпилия (к Нуме возводили свои родословные также Пинарии, Помпонии и Эмилии). Ветвь Пизонов была наиболее влиятельной: её представители регулярно занимали высшие должности в последний век существования Римской республики.
Биография
Гай Кальпурний родился приблизительно в 110 году до н. э. Первое упоминание о нём в сохранившихся источниках, предположительно, относится к 76 году до н. э.: Пизон мог быть одним из судей, принимавших решение по делу актёра Квинта Росция Галла. В 69 году до н. э. он защищал интересы Секста Эбуция в его тяжбе с Авлом Цециной из-за наследства (интересы Цецины представлял Марк Туллий Цицерон). По мнению Фридриха Мюнцера, именно в 69 году Гай Кальпурний мог быть претором; но Роберт Броутон отмечает, что, согласно закону Виллия, самая поздняя из возможных датировок — 70 год до н. э.
В 67 году до н. э. Гай Кальпурний получил консулат на двоих с ещё одним плебеем — Манием Ацилием Глабрионом. Вместе консулы добились принятия закона против предвыборных махинаций (de ambitu): осуждённый по такому закону должен был выплатить крупный штраф, исключался из сената и на всю жизнь терял пассивное избирательное право. Некоторые источники упоминают в связи с этой инициативой только Пизона, и в историю закон вошёл как Lex Calpurnia.
Пизон возглавил оптиматов, пытавшихся противодействовать возвышению Гней Помпея Великого. В частности, он выступил против предложения Авла Габиния о наделении Помпея чрезвычайными полномочиями для борьбы с пиратами. В пылу полемики Гай Кальпурний заявил, что, «если Помпей желает подражать Ромулу, то ему не избежать участи последнего» (имелась в виду версия об убийстве Ромула сенаторами); после этих слов консула едва не растерзала толпа. Когда инициатива Габиния стала законом, Пизон, движимый, по словам Плутарха, завистью и ненавистью, приказал распустить экипажи уже собранного флота. Но к тому времени корабли уже отплыли из Брундизия. Габиний подготовил законопроект об отрешении Гая Кальпурния от должности, который не был внесён в народное собрание только благодаря протесту Помпея.
Пизон помешал популяру Марку Лоллию Паликану стать одним из консулов на следующий год и продемонстрировал при этом, по словам Валерия Максима, «замечательную душевную стойкость». По истечении полномочий он стал на два года (66—65 годы до н. э.) проконсулом Нарбонской Галлии. Во время наместничества Гай Кальпурний подавил волнения в землях аллоброгов. По возвращении в Рим он был обвинён Гаем Юлием Цезарем в вымогательстве и казни без суда жителя Транспаданской Галлии, но Цицерон, участвовавший в судебном процессе в качестве защитника, добился оправдательного приговора.
В конце 63 года до н. э., когда был раскрыт заговор Катилины, Пизон приложил серьёзные усилия, чтобы участником заговора признали и Цезаря. Вместе с Квинтом Лутацием Катулом Капитолином он пытался убедить Цицерона, консула, возглавившего борьбу с Катилиной, чтобы тот выдвинул против Цезаря ложное обвинение. Не добившись этого, Гай Кальпурний, по словам Саллюстия, начал распространять явную клевету о причастности Гая Юлия к планам Катилины; поверив ему, некоторые всадники даже угрожали Цезарю смертью. Пизон свидетельствовал перед сенаторами против Гая Корнелия Цетега и участвовал в дебатах 5 декабря 63 года до н. э., в ходе которых было решено казнить заговорщиков без суда. Позже Цицерон называл его в числе нобилей, одобрявших итоги его консульства.
Последние упоминания о Гае Кальпурнии относятся к декабрю 61 года до н. э. Цицерон с неудовольствием пишет Аттику, что при обсуждении одного вопроса в сенате первым предоставили слово Пизону, а не ему; а претендовавший на консулат 59 года до н. э. Марк Кальпурний Бибул рассчитывал через Пизона заключить союз с ещё одним соискателем — Луцием Лукцеем. Поскольку более поздних упоминаний нет, историки полагают, что Гай Кальпурний умер вскоре после этого.
Интеллектуальные занятия
Цицерон называет Пизона в своём трактате «Брут», перечисляя ораторов, бывших современниками Квинта Гортензия Гортала. По его словам, Гай Кальпурний был «оратор спокойный, словоохотливый, сообразительный, однако выражение лица он обычно принимал такое, что казался куда умнее, чем был на самом деле».
Примечания
Источники и литература
Источники
Литература
Ссылки
Кальпурнии Пизоны
Википедия:Персоналии, не категоризованные по месту рождения
Преторы
Проконсулы Нарбонской Галлии | {
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Deunte Raymon Heath (né le à Atlanta, Géorgie, États-Unis) est un lanceur droitier des Ligues majeures de baseball qui a joué avec les White Sox de Chicago entre 2012 et 2013.
Carrière
Alors qu'il joue au collège, Deunte Heath est drafté trois fois par une équipe du baseball majeur sans signer de contrat avec un club. Les Mets de New York ( en 2003), les Rays de Tampa Bay ( en 2004) et les Angels de Los Angeles ( en 2005) le choisissent tour à tour, mais Heath choisit plutôt de s'engager avec l'Université du Tennessee. En 2006, il est réclamé au de sélection par les Braves d'Atlanta, l'équipe de sa ville natale, et il commence sa carrière en ligues mineures avec un club-école de la franchise en 2007.
Le , Heath est accusé en Floride pour avoir présumément répondu à une annonce en ligne offrant des services sexuels et proposé de payer 75 dollars pour l'acte. Il est suspendu par son équipe puis libéré par les Braves le 8 avril. Heath signe un contrat avec les White Sox de Chicago le 22 avril suivant.
Deunte Heath atteint le baseball majeur à l'âge de 27 ans le 1er septembre 2012 alors qu'il lance en relève pour les White Sox.
Notes et références
Liens externes
Naissance en août 1985
Naissance à Atlanta
Joueur américain de baseball
Joueur des Barons de Birmingham
Joueur des Knights de Charlotte
Joueur des White Sox de Chicago
Joueur des Hiroshima Toyo Carp
Joueur des Saitama Seibu Lions
Lanceur des ligues majeures de baseball | {
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At 24/7 Drainage UK, we are veterans in our field. That sets apart from the rest. Our certified experts are all qualified professionals and we carry insurance to operate in the field. Our broad customer base is a direct result of serving Denton with distinction for so long. Anytime, day or night, count on us always to fix all your drainage trouble, from blocked sinks to blocked drains.
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We support all our blocked sink services in Denton with only top quality products from reliable brands and manufacturers. Keeping your family safe, we make sure that all the products we are using are safest possible, and in the event that strong chemicals are required, we advise you on all required protective measures.
In Denton, we remain the most trusted name in the industry as far as block drains and sinks are concerned; 24/7 Drainage UK is your best bet when you are confronted with kitchen, bathroom and toilet blockages. We have thousands of happy customers behind us, and so you can trust us to get the job done right first time. We are the best blocked sink service in the area, so get in touch today! | {
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Der Fußball in London war an der Entwicklung des Fußballsports in England maßgeblich beteiligt und zudem stellt die Hauptstadt Englands seit Jahrzehnten die Stadt mit den meisten Vereinen in den vier Profiligen des Landes.
Geschichte
Londoner Vereine im FA Cup
Obwohl der heute älteste Profiverein Londons, FC Fulham, erst 1879 gegründet wurde und somit um mehr als ein Jahrzehnt jünger ist als die in Nottingham beheimateten ältesten Profivereine Notts County und Nottingham Forest, die 1862 und 1865 entstanden sind, wurde bereits 1859 im Londoner Stadtteil Battersea mit dem Wanderers FC der erste "Seriensieger" des Landes gegründet. Der Wanderers FC gehörte 1863 zu den Gründungsmitgliedern der Football Association und war 1871/72 Sieger des erstmals ausgetragenen Football Association Challenge Cups. Bei der folgenden Austragung konnte er seinen Titel auf Anhieb verteidigen und erzielte ferner zwischen 1876 und 1878 einen Titelhattrick. Doch in den folgenden Jahren des ausgehenden 19. Jahrhunderts konnte kein Londoner Verein mehr den Pokalwettbewerb gewinnen und so dauerte es bis 1901, ehe der Tottenham Hotspur FC den FA Cup erstmals wieder in die Hauptstadt holte. Anschließend dauerte es zwanzig Jahre, ehe 1921 erneut die Spurs erfolgreich waren. In den 1930er Jahren war es ausgerechnet der zum Erzrivalen der Spurs erkorene FC Arsenal, der ebenfalls zweimal den FA Cup in die Hauptstadt holte. Zum Ende der Saison 2015/16 ist Arsenal mit 12 gewonnenen Trophäen der erfolgreichste Londoner Verein im FA Cup und gemeinsam mit Manchester United Rekordpokalsieger Englands. An dritter Stelle folgt Tottenham Hotspur mit acht Titeln und dahinter liegen gleichauf mit sieben Titeln der FC Chelsea, der FC Liverpool und Aston Villa. Ansonsten konnten lediglich West Ham United (dreimal) und Charlton Athletic (einmal) den FA Cup in die englische Hauptstadt holen.
Londoner Vereine in der ersten Liga
Als die Football League als damals höchste Spielklasse im englischen Fußball in der Saison 1888/89 ihren Spielbetrieb mit 12 Mannschaften aufnahm, war kein einziger Verein aus London vertreten. Erstmals stieg am Ende der Saison 1903/04 mit dem Zweitliga-Vizemeister Woolwich Arsenal ein Hauptstadtverein in die höchste Spielklasse auf. Nach Arsenals Erstligadebüt in der Saison 1904/05 folgten schon bald der FC Chelsea (1907/08) und der Tottenham Hotspur FC (1909/10).
Mit 13 Titeln stellt der FC Arsenal, der seit der Saison 1919/20 dauerhaft in der ersten englischen Liga vertreten ist, die erfolgreichste Londoner Mannschaft und die dritterfolgreichste in England nach Manchester United und FC Liverpool, die 20 bzw. 19 Meisterschaften zu ihren Gunsten entscheiden konnten. Von allen anderen Londoner Vereinen konnten lediglich der FC Chelsea (sechsmal) sowie der Tottenham Hotspur FC (zweimal) die Meisterschaft gewinnen.
Londoner Vereine im Ligapokal
Den erst 1960 eingeführten League Cup gewann von den Londoner Vereinen der FC Chelsea mit fünf Titeln am häufigsten und liegt somit gleichauf mit Aston Villa hinter dem Rekordsieger FC Liverpool, der achtmal erfolgreich war. Weitere Londoner Vereine, die den Ligapokal gewinnen konnten, sind Tottenham (4 Titel), Arsenal (2 Titel) und die Queens Park Rangers, die 1967 erfolgreich waren.
Fußball in den Regionen
Aufgrund der Größe der englischen Hauptstadt und der Vielzahl der in ihr ansässigen Traditionsvereine, die stets einen bestimmten Bezirk repräsentieren, bestehen Rivalitäten vorwiegend zwischen Vereinen bzw. ihren Anhängerschaften, die in derselben Region beheimatet sind. Daher werden im folgenden Text die Vereine in die vier Himmelsrichtungen unterteilt, in denen sie – vom Stadtzentrum aus betrachtet – beheimatet sind. Der FC Millwall und Charlton Athletic, die beide im Südosten residieren, werden bei solchen Einteilungen häufig dem Süden zugeschlagen und nur gelegentlich dem Osten. Weil die heftigste Rivalität in dieser Region jedoch zwischen Millwall und dem im Osten ansässigen West Ham United besteht – eine Feindschaft, die zu den brutalsten Derbys weltweit zählt –, wurden Millwall und Charlton dem Osten zugeschlagen.
Rivalität im Norden Londons
Der Tottenham Hotspur FC wurde 1882 von jungen Männern aus dem Cricket-Verein Hotspur und dem örtlichen Gymnasium im Nordlondoner Stadtteil Tottenham gegründet. Sein späterer Erzrivale wurde 1886 von Arbeitern der Waffenfabrik Royal Arsenal ins Leben gerufen. Die Fabrik befand sich im Südostlondoner Stadtteil Woolwich, der ein Teil der Docklands ist. Der noch junge Verein geriet jedoch schon bald in finanzielle Schwierigkeiten, die auch auf die geographische Lage des Klubs zurückzuführen waren; denn weil die Region in Plumstead relativ einwohnerschwach war, litt man auch unter einem geringen Zuschaueraufkommen. Daher verließ der FC Arsenal 1913 seine angestammte Heimat und verzog nach Highbury, wo er das Highbury-Stadion errichtete, das sich in unmittelbarer Nähe zur White Hart Lane, der Heimat des Hotspur FC, befand. Durch die erzwungene Nähe wurde Arsenal von Hotspur als lästiger "Eindringling ins eigene Revier" empfunden, wodurch eine der erbittertsten Rivalitäten des englischen Fußballs entstand. Bereits wenige Jahre nach dem Umzug von Arsenal wurde die Intensität ihrer Rivalität noch verstärkt. Als 1919 bei der Wiederaufnahme des Spielbetriebs nach dem Ende des Ersten Weltkriegs die erste Liga von 20 auf 22 Mannschaften erweitert wurde, mussten die Spurs den Abstieg hinnehmen, den sie als Tabellenletzter der Saison 1914/15 sportlich erlitten hatten, während der FC Arsenal als Fünftplatzierter der Zweitliga-Saison 1914/15 in die höchste Spielklasse aufgenommen wurden, obwohl die Mannschaft hierfür nicht die sportlichen Voraussetzungen geschaffen hatte. Und so halten sich auf Seiten der Spurs bis zum heutigen Tag hartnäckig Gerüchte, dass der seinerzeitige Arsenal-Boss Sir Henry Norris bei dieser Entscheidung seine Finger maßgeblich und auf unlautere Weise im Spiel gehabt habe.
Vereine im Osten Londons
War der Umzug für die Spurs weniger erfreulich, entwickelte er sich für den 1905 im Arbeiterviertel Charlton entstandenen Charlton Athletic FC zum Glücksfall. Denn der Wegzug des FC Arsenal hinterließ in der Region eine Lücke, die die Verantwortlichen von Charlton Athletic inspirierte, den Schritt in den Profifußball zu wagen. Auch wenn die Addicks zwischen 1936 und 1957 dauerhaft in der ersten Liga vertreten waren (sowie später noch einmal für mehrere Spielzeiten in den 1980er Jahren und zu Beginn des 21. Jahrhunderts), 1937 Vizemeister wurden und 1947 den FA Cup gewannen, wurden sie von ihren beiden örtlichen Rivalen West Ham United (zwischen denen keine nennenswerte Abneigung besteht) und FC Millwall (zwischen denen das Verhältnis angespannter ist) nie als auf einer Stufe stehend anerkannt. Denn deren äußerst gehässige Rivalität stellt alles in den Schatten und gilt als das gefährlichste Derby Englands, wenn nicht überhaupt. Wann immer die beiden Vereine in der Vergangenheit aufeinander trafen, brachen zwischen den rivalisierenden Fans bereits am frühen Morgen heftige Schlägereien aus, die sich bis in die folgende Nacht hinein fortsetzten. Ihre intensive gegenseitige Abneigung geht bis auf ihre Gründungszeit zurück, als ihre Sportplätze keine fünf Kilometer voneinander entfernt lagen und ein Großteil ihrer Anhänger für konkurrierende Unternehmen der Schifffahrtsbranche arbeiteten. Als 1926 ein Streik ausgerufen wurde, dem sich die Arbeiter aus dem Umfeld von West Ham anschlossen, vom Umfeld des FC Millwall aber nicht befolgt wurde, war das Klima endgültig vergiftet. Seither schmähten die Fans von West Ham die streik brechenden Millwall-Anhänger als Verräter.
Ebenfalls im Osten der Stadt ist der 1881 gegründete Leyton Orient FC beheimatet, dessen Name sich von der Orient Steam Navigation Company ableitet, bei dem einer ihrer Gründer arbeitete. Obwohl der Verein sich in unmittelbarer Nachbarschaft von West Ham United befindet und somit auch in einer problematischen Gegend, läuft im Umfeld ihrer Begegnungen alles friedlich ab. Immerhin spielte Leyton seit 1982 nicht mehr höher als drittklassig und außerdem fehlt dem zweitältesten Profiverein Londons das "gewisse Etwas".
Vereine im Westen Londons
Im Londoner Westen sind die Rivalitäten zwischen den Vereinen bzw. ihren Fangruppen unterschiedlich ausgeprägt, aber insgesamt weniger brisant als im Norden und Osten. Der älteste der vier Profivereine aus dieser Region ist der FC Fulham, der bereits 1879 gegründet wurde und somit der älteste aller Londoner Profifußballvereine ist. 1885 folgte die Gründung der Queens Park Rangers und 1889 des FC Brentford, ehe erst 1905 der FC Chelsea ins Leben gerufen wurde. Der erfolgreichste Verein im Londoner Westen wurde nur gegründet, um in dem bereits bestehenden Stadion an der Stamford Bridge zu spielen, das bisher von keiner Mannschaft genutzt wurde. Ursprünglich sollte der bereits seit einem Vierteljahrhundert bestehende FC Fulham in dem Stadion spielen, was dieser wegen der aus seiner Sicht zu hohen Pacht ablehnte. So kam es, dass der Stadioneigentümer Gus Mears mit dem FC Chelsea schließlich seinen eigenen Verein gründete, der sich bis 1982 im Besitz der Familie Mears befand. Anschließend befand der Verein sich im Besitz des Fußballfunktionärs Ken Bates, bevor er 2003 von dem russischen Oligarchen Roman Abramowitsch erworben wurde.
Obwohl Chelsea von seinen unmittelbaren Nachbarn Fulham und QPR angefeindet wird, nahm die Anhängerschaft der Blues üblicherweise kaum Notiz von den beiden Nachbarvereinen, so dass in diesem Teil der englischen Hauptstadt keine gewachsene Rivalität besteht.
Während eines Teils der 1970er und 1980er Jahre waren die Queens Park Rangers der populärste Verein im Londoner Westen, weshalb es für die eingefleischten Fans heute schwer zu ertragen ist, dass Chelsea ihnen inzwischen deutlich den Rang abgelaufen hat. QPR wiederum gilt als Hauptfeind der Anhänger des FC Brentford, die es den Rangers nie verziehen haben, dass diese in den 1960er Jahren einen Versuch unternommen hatten, ihren Verein zu schlucken. Einen ähnlichen Versuch hatte 1987 Marler Estates als Eigentümer des FC Fulham bei QPR unternommen, um den fusionierten Verein fortan unter der Bezeichnung Fulham Park Rangers ins Rennen zu schicken. Massive Fanproteste aus beiden Lagern verhinderten dieses Vorhaben ebenso wie eine ablehnende Haltung der Fußball-Liga.
Vereine im Süden Londons
Wie bereits eingangs erwähnt, gehören die südlich der Themse beheimateten FC Millwall und Charlton Athletic geografisch eigentlich auch in diese Region und ihre Duelle mit dem weiter südlich ansässigen Crystal Palace F.C. sind durchaus von gegenseitiger Abneigung geprägt. Doch während beispielsweise Millwalls größte Abneigung West Ham United gilt, ist Brighton & Hove Albion aus dem südenglischen Seebad Brighton Hauptfeind der Anhängerschaft von Crystal Palace; und nur diese Begegnung von Palace hat das Potenzial zu gewalttätigen Auseinandersetzungen zwischen den Fanlagern.
Der Crystal Palace F.C., der 1905 im gleichnamigen Gebäude gegründet wurde, trägt seine Heimspiele seit 1924 im vereinseigenen Selhurst Park aus, den man zwischen 1986 und 1991 mit Charlton Athletic und anschließend von 1991 bis 2003 mit dem ebenfalls im Süden beheimateten FC Wimbledon teilte. Obwohl bereits 1889 gegründet, stieg der FC Wimbledon erst 1964 in den Profifußball ein und wurde in die semiprofessionelle Southern Football League aufgenommen, die er zwischen 1975 und 1977 dreimal in Folge gewann. Von nun an begann sein rasanter Aufstieg; für die Saison 1977/78 wurde er in die viertklassige Football League Third Division aufgenommen und schaffte innerhalb von acht Jahren den Sprung in die erste Liga. Zwei Jahre später folgte der Gewinn des FA Cup. 1991 verließ der Verein sein seit 1912 genutztes Stadion an der Plough Lane und trug seine Heimspiele fortan im Selhurst Park aus, wo er ein höheres Besucheraufkommen erzielen konnte. Dennoch blieb die Unterstützung für den Verein auf eher niedrigem Niveau und so folgte dem Abstieg aus der ersten Liga im Sommer 2000 der baldige Konkurs. Mit dem AFC Wimbledon wurde bereits 2002 ein Nachfolgeverein gegründet, der seine Heimspiele in Kingsmeadow im Stadtteil Kingston upon Thames austrägt.
Vereinsstatistiken
Beste Ergebnisse in den nationalen Wettbewerben
In der nachfolgenden Tabelle werden alle Londoner Vereine aufgeführt, die mindestens eine Saison in der höchsten Spielklasse Englands (von 1888 bis 1892 The Football League, die nächsten einhundert Jahre Football League First Division und seit 1992 Premier League) vertreten waren. Genannt wird ihre jeweilige Zugehörigkeit zur ersten Liga, ihre beste Platzierung darin sowie ihr bestes Ergebnis im FA Cup und im League Cup. Die Übersicht befindet sich auf dem Stand Saisonende 2015/16.
Eckdaten und Europapokalbilanzen
Die nachfolgende Übersicht informiert über das jeweilige Gründungsjahr sowie den Sitz der in der oberen Tabelle genannten Vereine. Ebenfalls eingebaut sind ihre größten Erfolge im Europapokal (sofern vorhanden), weil diese Information hier aus Platzgründen besser unterzubringen war als in der vorherigen Tabelle.
Sonstiges
Bis auf wenige Ausnahmen wurden die Finalspiele um den FA Cup stets in London ausgetragen. Zwischen 1872 und 1892 fanden die Begegnungen in 19 von 21 Fällen im Kennington Oval statt, zwischen 1895 und 1914 regelmäßig im Crystal Palace National Sports Centre und von 1923 bis 2000 im "alten" Wembley-Stadion, ehe seit 2007 das neue Wembley-Stadion die permanente Austragungsstätte ist. Auch die englische Nationalmannschaft trägt ihre Heimspiele meistens in London aus.
Überregional bekannt sind die im Nordosten der Stadt gelegenen Hackney Marshes mit ihren ursprünglich 88 Fußballfeldern, auf denen Weltstars wie Bobby Moore und David Beckham in jungen Jahren gegen den Ball getreten haben.
Einzelnachweise
London
Sport (London)
Fußballgeschichte
London | {
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Author James W.P. Campbell , is a fellow and director of studies in architecture and history of art at Queens'...
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Among the Headhunters: An Extraordinary World War II Story of Survival in the Burmese Jungle
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I am not an expert seamstress, but I sure have fun figuring things out! If you're not an expert either, you'll enjoy my step-by-step instructions. Happy sewing! You'll find all of my sewing projects here!
Turn boring socks into cool pumpkin Halloween decorations!
Who knew that this annoying space could turn into such a fun bunk bed fort?
Thank you sharing such interesting stuffs.I liked 4th number image in first raw and I am going to create similar one for my nephew.
Great stuff you posted here. I especially like the Felt Ice Cream it looks very real. Thanks.
Thanks, Stephanie. It was a really fun project.
Thanks Stephanie for sharing this awesome sewing project.I believe that it will help lots of people and kids.
Your sewing projects are really good. Who says that you are not an expert seamstress. You have a creative mind. That chair back cover with pocket is really great. I already made one.
Thanks, Kelly. I hope you find some great projects.
thanks for sharing , you have compiled very good list of project i am naive in sewing but still i will start with easy one and let's see what i can do with it. | {
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{"url":"https:\/\/jakubmarian.com\/integral-of-cos2x\/","text":"# Integral of $\\cos^2(x)$\n\nby Jakub Marian\n\nTip: See my list of the Most Common Mistakes in English. It will teach you how to avoid mis\u00adtakes with com\u00admas, pre\u00adpos\u00adi\u00adtions, ir\u00adreg\u00adu\u00adlar verbs, and much more.\n\nThe easiest way to calculate this integral is to use a simple trick. First, we write $\\cos^2(x) = \\cos(x)\\cos(x)$ and apply integration by parts:\n\n$$\u222b \\cos(x)\\cos(x)\\,dx = \\cos(x)\\sin(x)-\u222b(-\\sin(x))(\\sin(x))\\,dx$$\n\nIf we apply integration by parts to the rightmost expression again, we will get $\u222b\\cos^2(x)dx = \u222b\\cos^2(x)dx$, which is not very useful. The trick is to rewrite the $\\sin^2(x)$ in the second step as $1-\\cos^2(x)$. Then we get\n\n\\begin{align*} \u222b \\cos^2(x)\\,dx &= \\cos(x)\\sin(x)+\u222b(1-\\cos^2(x))\\,dx \\\\ &= \\cos(x)\\sin(x)+x-\u222b\\cos^2(x)\\,dx \\end{align*}\n\nNow, all we have to do is to get $\u222b\\cos^2(x)\\,dx$ from the right-hand side to the left-hand side of the equation:\n\n$$2\u222b \\cos^2(x)\\,dx = \\cos(x)\\sin(x)+x$$ $$\u222b \\cos^2(x)\\,dx = \u00f7{1}{2}(\\cos(x)\\sin(x)+x)+c$$\n\nOne last question remains: where the hell did the $+c$ come from? I will answer that in a separate article in the future.\n\nBy the way, I have written several educational ebooks. If you get a copy, you can learn new things and support this website at the same time\u2014why don\u2019t you check them out?","date":"2019-02-20 07:36:01","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\": 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\": 0.9943912625312805, \"perplexity\": 248.048785578909}, \"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-2019-09\/segments\/1550247494485.54\/warc\/CC-MAIN-20190220065052-20190220091052-00106.warc.gz\"}"} | null | null |
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To describe about my family, we are a joint family with strong traditional values. I belong to the Dheevara caste and am looking for a match from the same community.
I am a middle class family.my father passed away 2015..Me and amma settled in kollam .I have one sister and brother ..
Myself is Sindhu working in Sabarigiri School as a Teacher. My Parent are late. One sister is living in Kochuveli.
My sister is employed in government sector as a teacher with a bachelor's degree. She currently lives in Kollam. We belong to a middle class, nuclear family with traditional values.
I'm successfully employed as a Teaching / Academician in a private firm. I belong to the SC caste and willing to marry a suitable prospect from other communities also. To describe about my family, we are a nuclear family with strong traditional values. I reside in the beautiful city of Kollam.
I'm successfully running a business. I belong to the SC caste and am looking for a match from the same community. To describe about my family, we are a nuclear family with strong traditional values. I reside in the beautiful city of Kollam.
This is for my sister. She completed her master's degree and is now working as an officer in government sector. We belong to a middle class, nuclear family with liberal values, currently settled in Kollam. | {
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The Luo Question: A Kenyan Problem that Refuses to Go Away
21st September 2017 No Comments |
I remember feeling like a non-Kenyan for the first time in my life on that dark night Uhuru Kenyatta was declared the re-elected president, a decision that has since been nullified in a historic Supreme Court ruling. The court found the Independent Electoral and Boundaries Commission (IEBC), the body charged with conducting elections, guilty of committing what it termed "illegalities and irregularities" that tainted the entire electoral process.
I recall clambering onto the roof of our apartment building, transfixed by the turn of events, painfully watching the twinkling stars, listening to Jubilee Alliance Party's frenzied supporters screaming and beating drums and yelling – it was like the chasing away of nyawawa among my Luo folk. I picked up my phone and called a few of my friends, to calm my anguish at the wickedness of democracy.
Like a lot of African countries, elections are turning into charades. Elections are conducted as a mere formality to create a semblance of democratic progress. While significant gains have been achieved since African countries attained independence more than 50 years ago – like the shift from manual transmission of votes (which was usually marred with stuffing malpractices) to technology-driven transmission, a lot remains to be done. The vexing matter of free, fair, and credible elections remains the greatest challenge since it's a matter of public opinion and not the prerogative of institutions per se. Another setback that complicates the democratization process is when political elites advocate for ethnic hegemony, as witnessed with the Kikuyu-Kalenjin alliance beginning to entrench itself as a national norm.
According to the CIA World Fact Book as of July 2017, Kenya has a population of 47.6 million people with Asian-Indians recently getting official recognition. This scenario implies that in our quest for the nation-state ideal, other communities deserve a chance at the highest office to evoke a sense of political equality. Nobody should fool us that the institution of the presidency is 'neutral' when it comes to the distribution of resources. Our history is replete with shocking accounts of how Jomo Kenyatta, a celebrated nationalist, turned the presidency into a personal turf only beholden to his Kikuyu community. And that's what brings me to the emotive issue of the Luo question that remains a constant talking point in the country.
The announcement of Kenyatta's re-election as president in the 2017 General elections by the IEBC chair Wafula Chebukati triggered an avalanche of mixed feelings across Kenya. Kenyatta "won" by 54.27 percent against Odinga's 44.74 percent of all the votes cast. It is crucial to understand the "win" was an increase from the 50.51 percent for Kenyatta in 2013. In a functioning democracy, if indeed the win was valid, that should have heralded mostly positively feelings and emotions. But ng'o. It was not. Kisumu, a bastion of opposition politics since the Jomo Kenyatta-Oginga Odinga fallout in 1966, and the soul of the Luo people, was swiftly cordoned by the state machinery – as is the custom to preempt protests.
The kind of bloodletting that followed in the name of "law and order" and safeguarding property was not something out of the blue for those who understand the historical victimization of the Luo. Since independence, the victimhood of the Luo community has been cleverly distorted as an act of their own making. A section of the Kikuyu community whose elites have suppressed the Luo through economic stagnation and marginalization now want young Luo men and women to explain why they are still economically impoverished and anti-state and anti-police. They are asked to explain why their fellow community members are anti-development when the rest of Kenyans have accepted and moved on after the announcement of the presidential results.
The public reaction following reports that the police were murdering Luos – I call it murder, for that is what it has been – was that of further dehumanization and demonization of the Luo body. In other words, the life of a Luo is something the state can easily dispense with (at the pull of a trigger) because Luos have been persistently stereotyped as being 'rowdy, violent, anti-government, anti-police, idle…etc., etc.
But the deliberate failure by the state led by the presidency is the refusal to allow the Luos and other communities to express their frustrations – of systemic victimization and exclusion for questioning the government of the day. Tolerating such an act has the potential of opening a Pandora's Box. And none from the past Kalenjin and Kikuyu-led governments would consent to actions amounting to "confessions" from any of those ethnic groups that continue to be sidelined from the political center.
However, the Luo question has been compounded by the gradual buying of the falsehood by other equally ostracized communities. This came into sharp focus when the state agents were killing Luos in informal settlements and other perceived National Super Alliance (NASA) strongholds. I would say it was a pointer of a lost chance to understand that the continued persecution of any dissenting ethnic community is just a sinister scheme to suppress peripheral voices and banish them away into political wilderness. And that is common sense. When the state has subdued voices that demand accountability and transparency in terms of good governance and proper management of the economy, then, surely, we are sliding into a farce.
A farce is when economic development is constantly parroted even at the expense of the underlying politics that influence economic policies. A good example is the historical injustices inflicted against such suffering communities who are forbidden from questioning the reasons for their continued suffering. Questioning attracts state terror as wenye wanakula nyama tukimeza mate cheer on.
A glimpse into volume three of the Truth, Justice and Reconciliation (TJRC) Commission Report titled National Unity, Healing and Reconciliation reveals explicitly what needs to be done. Among other pertinent issues that are highlighted include the need for formal recognition and right to identity for such communities. Others are violations of right to development which the Luo and other historically excluded communities have experienced firsthand since independence, but subsequent regimes turn around to accuse them of imagining those tales.
Lastly, for the country to enjoy unity and reconciliation, the report further recommends that:
The President issues an official, public and unconditional apology to minority and indigenous communities in Kenya for the State's systematic discrimination against these groups and communities. This should be done within six months of the issuance of the report.
Now one understands why the Jubilee Alliance is averse to the full implementation of the report. The Deputy President, William Ruto echoed the stand two months ago while addressing Kilifi residents on a campaign trail:
Our opponent says the solution to [the] land problem is to rely on the TJRC report where each community will be questioned. I'm asking you Kilifi residents, if we resort to this path, will we divide Kenyans or not?
By posing that rhetorical question, Ruto is cleverly insinuating to the marginalized communities to avoid raising historical grievances. Now, as we hold our breaths waiting for a fresh presidential election as ordered by the Supreme Court, we can only hope the incoming government will promote inclusivity for all Kenyans so that we can soar higher again.
About the Writer:
Amol Awuor is a trained high school teacher practising journalism. He is a former editorial intern with The Standard newspaper.
Amol Awuor
"Free, Fair & Credible," au Kura si Kesi
Waganga, Wauguzi, etc
A Sense of Where We Are, an introduction | {
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Q: php HP cloud Creating Signatures for FormPOST python to php translation I am currently trying out the HP cloud object storage API but all their examples are in Python and I'm working in PHP.
It's saying my signature is invalid, if anyone can help see where I'm going wrong, I've attached the python example https://docs.hpcloud.com/api/object-storage/#formpost as well as my attempt in PHP.
import hmac
from hashlib import sha1
from time import time
path = '/v1/12345678912345/container/object_prefix'
redirect = 'https://myserver.com/some-page'
max_file_size = 104857600
max_file_count = 10
expires = int(time() + 600)
tenant_id = '12345678912345'
access_key_id = 'GP54NNRN2TKBVWH449AG'
secret_key = 'EHLzysK9S1QRWkwvVpVHsGZyM715OH4S2kJ'
hmac_body = '%s\n%s\n%s\n%s\n%s' % (path, redirect,
max_file_size, max_file_count, expires)
signature = tenant_id + ':' + access_key_id + ':' + hmac.new(secret_key, hmac_body, sha1).hexdigest()
And this is my attempt...
<?php
$expires = time()+600;
$hmac_body = 'https://region-a.geo-1.objects.hpcloudsvc.com/v1/xxx/'.'http://www.test.com/test.php'.'41943040'.'1'.$expires;
$signature = 'tenant_id:access_key:'.hash_hmac(sha1,$hmac_body,'secret_key', FALSE);
?>
<form action="<?php echo 'https://region-a.geo-1.objects.hpcloudsvc.com/v1/xxx/';?>" method="POST" enctype="multipart/form-data">
<input type="hidden" name="redirect" value="http://www.test.com/test.php" />
<input type="hidden" name="max_file_size" value="41943040" />
<input type="hidden" name="max_file_count" value="1" />
<input type="hidden" name="expires" value="<?php echo $expires;?>" />
<input type="hidden" name="signature" value="<?php echo $signature;?>" />
<input type="file" name="testupload" />
<input type="submit" />
</form>
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\section{Introduction}
\label{introduction}
In recent years, a considerable number of studies
have been made on
the $SU(N)$ spin Calogero-Sutherland (CS) model
\cite{HaHaldane,MP2}.
This model describes $n$ particles system on a circle of length
$L$ interacting with the inverse-square type potential.
Each particle is labelled by its coordinate $x_j$ and spin
with $N(\geq 1)$ possible values.
(When $N=1$, this is the CS model \cite{Cal,Suth}.)
The Hamiltonian of the model is given by
\begin{equation}
\label{CShamiltonian}
H
=
-\frac{1}{2}\sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}
+
\left(\frac{\pi}{L}\right)^2
\sum_{1\leq i<j\leq n}
\frac{\beta(\beta+P_{ij})}
{\sin^2\frac{\pi}{L}(x_i-x_j)},
\end{equation}
where $\beta$ is a coupling parameter
and $P_{ij}$ is the spin exchange operator
for particles $i$ and $j$.
In this paper,
we take $N=2$ and $\beta$
to be a positive integer.
A lot of intriguing results have been obtained
in connection with the spin CS model.
In particular,
the eigenfunctions of the spin CS model
have been explicitly constructed and then,
using these properties,
the dynamical correlation functions of
this model were computed.
For $\beta=1$ which is the simplest nontrivial case,
the hole propergator of the $SU(2)$ spin CS model
(with finite $n$ and in the thermodynamic limit)
has been calculated by Kato \cite{Green}.
He also gave a conjectural formula for the arbitrary
integer coupling case.
Using the Jack polynomials with prescribed symmetry
\cite{BF1,Dunkl3},
this conjecture was recently confirmed by
Kato and one of the authors \cite{KaYa}.
On the other hand,
introducing the new class of orthogonal polynomials
\cite{TU,Uglov},
exact results have been obtained by Uglov \cite{Uglov}.
He computed the dynamical density and spin-density
two-point correlation functions of
the $SU(2)$ spin CS model with finite $n$.
In \cite{KYA},
Kato and authors have studied the construction for
the dynamical correlation functions of
the $SU(N)$ spin CS model
in the thermodynamic limit.
We gave the formula for
the density two-point correlation function
in the thermodynamic limit
and checked the consistency with the predictions
from conformal field theory \cite{Kawakami}.
However, the relation between Uglov's work and ours
is missing.
In this paper, from Uglov's formulae
for the dynamical correlation function
of the spin CS model,
we take the thermodynamic limit
which is technically nontrivial.
For the density two-point correlation function,
we prove our previous result in
the thermodynamic limit microscopically.
Here,
we give our main results together with those of
the work in ref. \cite{Green,KaYa}.
We introduce the following notations:
for non-negative integers $a$, $b$ and $c$,
\begin{eqnarray}
\label{ene-td}
{\cal E}(u,v,w;a,b,c)
&=&
\sum_{i=1}^a\epsilon_{\rm p}(u_i)
+\sum_{j=1}^b\epsilon_{\rm h}(v_j)
+\sum_{k=1}^c\epsilon_{\rm h}(w_k),
\\
\label{mome-td}
{\cal P}(u,v,w;a,b,c)
&=&
\frac{\pi\rho_0}{2}
\Big[
-(2\beta+1)\sum_{i=1}^a u_i
+\sum_{j=1}^b v_j
+\sum_{k=1}^c w_k
\Big],
\\
\label{int}
I(a,b,c)[\ast]
&=&
\prod_{i=1}^{a}
\int_1^\infty du_i
\prod_{j=1}^{b}
\int_{-1}^1dv_j
\prod_{k=1}^{c}
\int_{-1}^1dw_k
(\ast)
|F_\beta(u,v,w;a,b,c)|^2,
\end{eqnarray}
where
$\rho_0$ is the density of particles,
variables
$u=(u_1,\cdots,u_a)$,
$v=(v_1,\cdots,v_b)$ and $w=(w_1,\cdots,w_c)$ represent
the normalized momenta of quasiparticle with spin $\sigma$,
quasiholes with spin $-\sigma$ and $\sigma$,
respectively ($\sigma=\pm 1/2$).
The quasiparticle and quasihole dispersions
are introduced by
\begin{eqnarray}
\label{particle-dis}
&&
\epsilon_{\rm p}(y)
=
(2\beta+1)^2
\frac{1}{2}
\left(\frac{\pi\rho_0}{2}\right)^2(y^2-1),
\\
\label{hole-dis}
&&
\epsilon_{\rm h}(y)
=
(2\beta+1)
\frac{1}{2}
\left(\frac{\pi\rho_0}{2}\right)^2(1-y^2),
\end{eqnarray}
respectively.
The function $F_\beta$ is defined by
\begin{eqnarray}
\label{form-factor}
&&F_\beta(u,v,w;a,b,c)
\nonumber
\\
&=&
\frac{
\prod_{1\leq i < j\leq b}
(v_i - v_j )^{g_{\rm h}^{\rm d}}
\prod_{1\leq i < j\leq c}
(w_i - w_j )^{g_{\rm h}^{\rm d}}
\prod_{i=1}^{b}\prod_{j=1}^{c}
(v_i - w_j)^{g_{\rm h}^{\rm o}}
}
{
\prod_{i=1}^{a}\prod_{j=1}^{b}
(u_i - v_j)
\prod_{i=1}^{a}
(u_i^2-1)^{(1-g_{\rm p}^{\rm d})/2}
\prod_{j=1}^{b}
(1-v_j^2)^{(1-g_{\rm h}^{\rm d})/2}
\prod_{k=1}^{c}
(1-w_k^2)^{(1-g_{\rm h}^{\rm d})/2}
},
\end{eqnarray}
where
\begin{equation}
\label{stat-mat}
g_{\rm p}^{\rm d}=\beta+1,\
g_{\rm h}^{\rm d}=(\beta+1)/(2\beta+1),\
g_{\rm h}^{\rm o}=-\beta/(2\beta+1).
\end{equation}
The retarded Green function \cite{Green,KaYa},
density and spin-density two-point correlation functions
can respectively be written as the following form:
\begin{eqnarray}
\label{rGreen-td}
\langle \psi(x,t)\psi(0,0)\rangle
&=&
A(\beta)I(0,\beta+1,\beta)
[(\pi\rho_0/2)e^{{\rm i}({\cal P}(0,\beta+1,\beta)x
-({\cal E}(0,\beta+1,\beta)-\zeta)t)}],
\\
\label{dd-td}
\langle \rho(x,t)\rho(0,0)\rangle
&=&
B(\beta)I(1,\beta+1,\beta)
[{\cal P}(1,\beta+1,\beta)^2
\cos({\cal P}(1,\beta+1,\beta)x)
e^{-{\rm i}{\cal E}(1,\beta+1,\beta)t}],
\\
\label{ss-td}
\langle s(x,t)s(0,0)\rangle
&=&
C_{\mbox{\scriptsize I}}(\beta)I(1,\beta,\beta+1)
[(\pi\rho_0/2)^2\cos({\cal P}(1,\beta,\beta+1)x)
e^{-{\rm i}{\cal E}(1,\beta,\beta+1)t}]
\nonumber
\\
&+&
C_{\mbox{\scriptsize II}}(\beta)I(1,\beta+2,\beta-1)
[(\pi\rho_0/2)^2\cos({\cal P}(1,\beta+2,\beta-1)x)
e^{-{\rm i}{\cal E}(1,\beta+2,\beta-1)t}],
\end{eqnarray}
where
$\zeta=(\pi(2\beta+1)\rho_0/2)^2$ is
the chemical potential and
we use the convention
$\prod_{i=1}^0(\ast)$=1.
Constant factors in the above formulae are defined by
\begin{eqnarray}
A(\beta)&=&
\frac{1}{\pi(2\beta+1)^\beta}D(\beta),
\\
B(\beta)&=&
\frac{1}{\pi^2(2\beta+1)^{\beta+1}}D(\beta),
\\
C_{\mbox{\scriptsize I}}(\beta)&=&
\frac{1}{4\pi^2(2\beta+1)^{\beta-1}}D(\beta),
\\
C_{\mbox{\scriptsize II}}(\beta)&=&
\frac{1}{4\pi^2(2\beta+1)^{\beta-1}}\frac{\beta}{\beta+2}D(\beta),
\end{eqnarray}
where, using the gamma function ${\mit \Gamma}(z)$,
the constant $D(\beta)$ is given by
\begin{equation}
D(\beta)
=
\frac{1}{{\mit \Gamma}(\beta+2)}
\prod_{j=1}^{2\beta+1}
\frac{{\mit \Gamma}\big((\beta+1)/(2\beta+1)\big)}
{{\mit \Gamma}^{2}\big(j/(2\beta+1)\big)}.
\end{equation}
The paper is organized as follows.
In section \ref{Uglov-work},
we recall Uglov's results about
the dynamical correlation functions
of the $SU(2)$ spin CS model.
In section \ref{thermodynamic-limit},
firstly, we examine the excitation contents
of the intermediate states of
the dynamical correlation functions.
Secondly,
taking the thermodynamic limit,
we derive the formulae (\ref{dd-td}) and (\ref{ss-td}).
The conclusion is presented in section \ref{conclusion}.
Appendix A contains
proof of the statement in the subsection
\ref{intermediate-states}.
In appendix B,
we give the examples of the explicit
formulae of the building blocks
for the dynamical correlation functions.
\section{Uglov's formulae
for the dynamical correlation functions}
\label{Uglov-work}
In this section, we fix notations and then
recall Uglov's exact results
for the dynamical correlation functions \cite{Uglov}.
We only give the final results of Uglov's paper.
For details, see ref. \cite{Uglov}.
\subsection{Notations}
\label{notations}
In Uglov's formulation,
the states of the (transformed) Hamiltonian
are labeled by the colored partitions.
A brief mathematical preliminary here may be in order.
We fix notations which will be
to the fore in this paper (see refs. \cite{Macd,Uglov}).
For a fixed non-negative integer $n$, let $\Lambda_n
=\{\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)\in(\mbox{{\bf Z}}_{\geq 0})^n\,
|\,\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\}$
be the set of all partitions with length less or equal to $n.$
The weight of a partition $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)$
is defined by
$|\lambda |=\sum_{i=1}^n\lambda_i$.
A partition can be represented by a Young diagram.
For example, the partition $\lambda=(4,3,1)$ is expressed as
$$
\lambda=
\Yvcentermath1
\yng(4,3,1)
$$
When there is a square in the $i$th row and $j$th
column of $\lambda$, we write $(i,j)\in\lambda$.
The conjugate of a partition $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)$
is the partition
$\lambda'=(\lambda'_1,\lambda'_2,\cdots,\lambda'_{\lambda_1})$ whose diagram is the
transpose of the diagram $\lambda$.
For instance, if $\lambda=(4,3,1)$, then $\lambda'=(3,2,2,1)$:
$$
\lambda'=
\Yvcentermath1
\yng(3,2,2,1)
$$
Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)$ be a partition.
For a square $s=(i,j)\in\lambda$, the numbers
\begin{equation}
\begin{array}{ll}
\label{arm-leg}
a(s)=\lambda_i-j, \quad & a'(s)=j-1, \\
l(s)=\lambda'_j-i, \quad & l'(s)=i-1,
\end{array}
\end{equation}
are called arm-length, coarm-length, leg-length, and coleg-length,
respectively:
\newcommand{\stackrel{{\tiny a}}{-}}{\stackrel{{\tiny a}}{-}}
\newcommand{a}{a}
\newcommand{a'}{a'}
\newcommand{l}{l}
\newcommand{l'}{l'}
$$
\lambda=
\Yvcentermath1
\young(%
\hfil\hfil\hfil\uparrow\hfil\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfill'\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\downarrow\hfil\hfil\hfil\hfil\hfil\hfil,%
\leftarrowa'\rightarrow s\leftarrow-a-\rightarrow,%
\hfil\hfil\hfil\uparrow\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil |\hfil\hfil\hfil,%
\hfil\hfil\hfill\hfil\hfil\hfil,%
\hfil\hfil\hfil |\hfil\hfil,%
\hfil\hfil\hfil\downarrow,%
\hfil\hfil
)
$$
Also the numbers
\begin{eqnarray}
\label{content}
&& c(s)=a'(s)-l'(s)=j-i, \\
\label{hooklength}
&& h(s)=a(s)+l(s)+1=\lambda_i+\lambda'_j-i-j+1,
\end{eqnarray}
are called content and hook-length, respectively.
For $\alpha\in\mbox{{\bf C}}$, their refinements are defined by
\begin{eqnarray}
\label{modi-content}
&& c(s;\alpha)
=a'(s)-\alpha l'(s)=j-1-\alpha(i-1), \\
\label{upper-hook-length}
&& h_\lambda^*(s;\alpha)
=a(s)+1+\alpha l(s)=\lambda_i-j+1+\alpha(\lambda'_j-i), \\
\label{lower-hook-length}
&& h^\lambda_*(s;\alpha)
=a(s)+\alpha(l(s)+1)=\lambda_i-j+\alpha(\lambda'_j-i+1).
\end{eqnarray}
Moreover we define the following numbers:
\begin{eqnarray}
&&
d(s;\alpha)
=h_\lambda^*(s;\alpha)h^\lambda_*(s;\alpha),\\
&&
e(s;\alpha)
=\frac{a'(s)+\alpha(n-l'(s))}{a'(s)+1+\alpha(n-l'(s)-1)}
=\frac{j-1+\alpha(n-i+1)}{j+\alpha(n-i)}.
\end{eqnarray}
We recall a coloring scheme of diagrams.
(Here we only need a coloring by two colors, white and black,
since we consider the case with $N=2$.)
For a partition $\lambda$, we define two subsets of $\lambda$ by
$W_\lambda=\{s\in\lambda\,|\,c(s)\equiv 0\ \mbox{mod}\, 2\}$
and
$B_\lambda=\{s\in\lambda\,|\,c(s)\equiv 1\ \mbox{mod}\, 2\}$.
We call the color of $s\in\lambda$ white (black)
if $s\in W_\lambda$ ($\in B_\lambda$),
and call $\lambda=W_\lambda\sqcup B_\lambda$ the colored partition.
(Notice that $(1,1)\in W_\lambda$ (if $\lambda\ne\O$).)
For example, if $\lambda=(4,3,1)$, then
$W_\lambda=\{\Yvcentermath1\young(\hfill)\ \mbox{in the following diagram}\}$,
and
$B_\lambda=\{\Yvcentermath1\young(\bullet)\ \mbox{in the following
diagram}\}$:
$$
\lambda=
\Yvcentermath1
\young(
\hfill\bullet\hfil\bullet,%
\bullet\hfil\bullet,%
\hfil
)
$$
We define an another subset of $\lambda$ by
$H_2(\lambda)=\{s\in\lambda\,|\,h(s)\equiv 0\ \mbox{mod}\, 2\}$.
For example, if $\lambda=(4,3,1)$, then
$H_2(\lambda)=\{\Yvcentermath1\young(\star)\ \mbox{in the following
diagram}\}$:
$$
\lambda=
\Yvcentermath1
\young(
\star\star\hfil\hfil,%
\star\star\hfil,%
\hfil
)
$$
\subsection{Dynamical correlation functions}
\label{dcf-finite}
We now recall Uglov's formulae
for the dynamical correlation functions \cite{Uglov}.
We take the number of particles $n$
to be an even number such that $n/2$ is odd \cite{Uglov}.
In Uglov's formalism,
the states of the CS Hamiltonian
are labeled by the colored partitions.
First of all, the total energy
with respect to the transformed Hamiltonian,
total momentum, and
total $z$-component of spin for the colored partition
$\lambda$ are respectively given by\footnote{%
The formula for $E_\lambda$ in ref. \cite{Uglov}
has typographical error.}
\begin{eqnarray}
\label{ene-finite}
&&
E_\lambda
=
\frac{1}{2}\left(\frac{2\pi}{L}\right)^2
\left[
n_w(\lambda')-\gamma n_w(\lambda)
+
\frac{1}{2}((n-1)\gamma+1)|W_\lambda|
\right], \\
\label{mome-finite}
&&
P_\lambda
=
\frac{2\pi}{L}|W_\lambda|, \\
\label{spin-finite}
&&
S_\lambda
=
|W_\lambda|-|B_\lambda|,
\end{eqnarray}
where $\gamma=2\beta+1(\in\mbox{{\bf Z}}_{>0})$, and
\begin{eqnarray}
&&
n_w(\lambda)=\sum_{s\in W_\lambda}l'(s), \\
&&
n_w(\lambda')=\sum_{s\in W_\lambda}a'(s).
\end{eqnarray}
Here, for any subset $\mu\subset\lambda$, we denote by $|\mu|$
the number of squares in $\mu$.
Next, the building blocks for the main factors of
the dynamical correlation functions are defined as follows:
for a colored partition $\lambda\in\Lambda_n$,
\begin{eqnarray}
&&
X_\lambda
=\prod_{s\in W_\lambda\setminus\{(1,1)\}} c(s;\gamma)^2,
\\
&&
Y_\lambda
=\prod_{s\in H_2(\lambda)} d(s;\gamma),
\\
&&
Z_\lambda
=\prod_{s\in W_\lambda} e(s;\gamma).
\end{eqnarray}
For the system with Hamiltonian (\ref{CShamiltonian}),
we denote the ground state expectation value
of the operator ${\cal O}$ by $\langle{\cal O}\rangle_n$.
Then,
the (ground state) dynamical density
and spin-density two point correlation functions
are respectively given by \cite{Uglov}
\begin{eqnarray}
\label{dd-finite}
\langle\rho(x,t)\rho(0,0)\rangle_n
&=&
\frac{4}{\pi^2}
\sum_{{{{{\footnotesize \lambda\in\Lambda_n:\,
\mbox{{\scriptsize colored partition}}}}
\atop
{\footnotesize |\lambda|:\,\mbox{{\scriptsize even}}}}
\atop
{\footnotesize S_\lambda=0}}
\atop
{\footnotesize |W_\lambda|=|H_2(\lambda)|}}
|P_\lambda|^2 X_\lambda Y_\lambda^{-1} Z_\lambda
e^{-itE_\lambda}\cos(xP_\lambda),
\\
\label{ss-finite}
\langle s(x,t)s(0,0)\rangle_n
&=&
\frac{1}{2L^2}
\sum_{{{{{\footnotesize \lambda\in\Lambda_n:\,
\mbox{{\scriptsize colored partition}}}}
\atop
{\footnotesize |\lambda|:\, \mbox{{\scriptsize odd}}}}
\atop
{\footnotesize S_\lambda=\pm 1}}
\atop
{\footnotesize |W_\lambda|=|H_2(\lambda)|+1}}
X_\lambda Y_\lambda^{-1} Z_\lambda
e^{-itE_\lambda}\cos(xP_\lambda),
\end{eqnarray}
where $\rho(x,t)$ and $s(x,t)$
are the Heisenberg representations of
the reduced density operator
$\rho(x)=\sum_{i=1}^n\delta(x-x_i)-n/L$
and the $z$-component of spin-density operator
$s(x)=\sum_{i=1}^n\delta(x-x_i)\sigma_i^z/2$,
respectively.
Here $\sigma_i^z$ is the $z$-component of Pauli matrices.
\section{Thermodynamic limit of
the dynamical correlation functions}
\label{thermodynamic-limit}
In this section,
we take the thermodynamic limit of Uglov's exact
formulae (\ref{dd-finite}) and (\ref{ss-finite}).
Firstly,
we determine the excitation contents
of the intermediate states
for the dynamical correlation functions
(\ref{dd-finite}) and (\ref{ss-finite}).
Secondly,
we rewrite the formulae (\ref{dd-finite}) and (\ref{ss-finite})
in terms of parameters which
correspond to the elementary excitations.
Finally, we take the thermodynamic limit.
\subsection{Intermediate states}
\label{intermediate-states}
In order to take the thermodynamic limit,
we must determine the excitation contents
of the intermediate states
for the dynamical correlation functions
(\ref{dd-finite}) and (\ref{ss-finite}).
Except for the factor $X_\lambda$
which comes from the matrix element
of the local operators,
the factors in the sums of
the right hand side of (\ref{dd-finite}) and (\ref{ss-finite})
are non-zero for all $\lambda$.
Therefore, a summand in the sums of
the right hand side of (\ref{dd-finite}) and (\ref{ss-finite})
is non-zero if and only if $X_\lambda\ne 0$.
From the definition, it is easy to see that
$X_\lambda\ne 0\Leftrightarrow (2,\gamma+1)\notin\lambda$ \cite{Uglov}.
(Notice that $\gamma=2\beta+1\in\mbox{{\bf Z}}_{>0}$.)
As is the spinless case \cite{Ha,LPS},
this condition has the following physical interpretation:
The intermediate states contributing to
the dynamical correlation functions
(\ref{dd-finite}) and (\ref{ss-finite})
have one quasiparticle and $\gamma$ quasihole excitations.
We will see that these elementary excitations
have the $SU(2)$ spin degrees of freedom.
The above condition together with the conditions
on the sums in the formulae
(\ref{dd-finite}) and (\ref{ss-finite})
determine the intermediate states.
We parametrize the intermediate states
of the correlation functions
(\ref{dd-finite}) and (\ref{ss-finite}).
For this purpose,
we introduce some notations.
In consideration of above observation,
we define the subset of $\Lambda_n$ by
$\Lambda_n^{(\gamma)}
=\{\lambda\in\Lambda_n\,|\,(2,\gamma+1)\notin\lambda\}$.
That is, a partition $\lambda\in\Lambda_n^{(\gamma)}$
has $\gamma$ columns and one `arm'.
For a partition
$\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)\in\Lambda_n^{(\gamma)}$,
we introduce the notation
$
\lambda
=
\langle
\lambda'_1,\lambda'_2,\cdots,\lambda'_\gamma;r
\rangle,
$
which consists of $\gamma$ columns and one `arm'.
Here $r=\lambda_1-\gamma$. (If $\lambda_1<\gamma$, then $r=0$.)
For example,
$\lambda=(13,5,5,5,4,4,4,2,2,1)=\langle 10,9,7,7,4;8\rangle\in\Lambda_n^{(5)}$:
$$
\lambda=
\Yvcentermath1
\yng(13,5,5,5,4,4,4,2,2,1)
$$
Using these notations, we can state that
the intermediate states
for the density two-point
correlation function (\ref{dd-finite})
are colored partitions $\lambda\in\Lambda_n^{(\gamma)}$
with even weight, $S_\lambda=0$ and $|W_\lambda|=|H_2(\lambda)|$.
We call these colored partitions the d-d admissible.
Similarly,
the intermediate states
for the spin-density two-point
correlation function (\ref{ss-finite})
are colored partitions $\lambda\in\Lambda_n^{(\gamma)}$
with odd weight, $S_\lambda=\pm 1$ and $|W_\lambda|=|H_2(\lambda)|+1$.
We call these colored partitions the s-s admissible.
The above conditions on the intermediate states
of dynamical correlation functions
are rather complicated.
Technical difficulty in taking
the thermodynamic limit
comes form these complicated conditions.
Then, in the following, we simplify the conditions
for d-d and s-s admissible colored partitions.
For this purpose, we introduce more notations.
For
$\nu
=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}$,
we define two subsets $I_1(\nu)$ and $I_2(\nu)$
of $I=\{1,\cdots,\gamma\}$ by
\begin{eqnarray}
I_1(\nu)
&=&
\left\{j\in\{1,\cdots,\gamma\} \,\left|\,
\nu_j
=
\left\{
\begin{array}{ll}
(1+(-1)^j)/2,\quad & \mbox{if } \nu_{\gamma+1}=0 \\
(1+(-1)^{j-1})/2,\quad & \mbox{if } \nu_{\gamma+1}=1
\end{array}
\right.
\right.
\right\},
\\
I_2(\nu)
&=&
I\setminus I_1(\nu).
\end{eqnarray}
For example,
if
$\nu
=(\overbrace{1,1,\cdots,1}^{\gamma+1\ \mbox{{\footnotesize times}}})$,
then
$I_1(\nu)=\{j\in I \,|\, j : \mbox{odd}\}$
and
$I_2(\nu)=\{j\in I \,|\, j : \mbox{even}\}$.
We introduce the function
$
\label{parity}
\rho:\ \Lambda_n^{(\gamma)}
\longrightarrow\{0,1\}^{\times(\gamma+1)}
$
by
$$
\rho(\lambda)
=(\sigma(\lambda'_1),\cdots,\sigma(\lambda'_\gamma),\sigma(r)),\,
\mbox{if}\,\,
\lambda
=
\langle
\lambda'_1,\cdots,\lambda'_\gamma;r
\rangle,
$$
where
$\sigma(a)=0\ (1)$ if $a$ is even (odd).
We call $\rho(\lambda)$ the parity of $\lambda\in\Lambda_n^{(\gamma)}$.
It can be easy to show that,
for a colored partition
$\lambda\in\Lambda_n^{(\gamma)}$ with parity
$\rho(\lambda)=(\nu_1,\cdots,\nu_{\gamma+1})$,
\begin{eqnarray}
&&
\label{w-b}
S_\lambda
=|W_\lambda|-|B_\lambda|=\sum_{j=1}^{\gamma+1}(-1)^{j-1}\nu_j.
\end{eqnarray}
Using these formulae, we have
\begin{equation}
\label{cond1}
S_\lambda
=
\left\{
\begin{array}{rr}
0 \\
\pm 1
\end{array}
\right.
\Leftrightarrow
\#I_1(\rho(\lambda))
=
\left\{
\begin{array}{ll}
\beta+1 \\
\beta\,\mbox{or}\,\beta+2.
\end{array}
\right.
\end{equation}
Here, for a set $A$, $\#A$ denotes the number of elements.
Moreover we can show that,
for $\lambda\in\Lambda_n$ with even (resp. odd) weight,
\begin{equation}
\label{cond2}
S_\lambda=0\quad(\mbox{resp. } \pm 1)
\Leftrightarrow
|W_\lambda|=|H_{2}(\lambda)|\quad
(\mbox{resp. } |H_{2}(\lambda)|+1).
\end{equation}
The proof of the statement (\ref{cond2})
is given in Appendix A.
The statements (\ref{cond1}) and (\ref{cond2})
are essential to taking the thermodynamic limit.
From above statements,
we see that a colored partition
$\lambda\in\Lambda_n^{(\gamma)}$
is the d-d admissible
if and only if
$|\lambda|$ is even and $\#I_1(\rho(\lambda))=\beta+1$.
Similarly,
a colored partition
$\lambda\in\Lambda_n^{(\gamma)}$
is the s-s admissible
if and only if $|\lambda|$ is odd and
$\#I_1(\rho(\lambda))=\beta$ or $\beta+2$.
The s-s admissible colored partitions are divided into
two types which are characterized by
$\#I_1(\rho(\lambda))=\beta$ or $\beta+2$.
We call former type I and latter type II.
The $SU(2)$ spin degrees of freedom of the
elementary excitations
are assigned as follows.
For an admissible colored partition
$\lambda=
\langle
\lambda'_1,\cdots,\lambda'_\gamma;r
\rangle\in\Lambda_n^{(\gamma)}$ with
parity $\rho(\lambda)=(\nu_1,\cdots,\nu_{\gamma+1})$,
the spin of quasiparticle is $1/2$ (resp. $-1/2$) if
$\nu_{\gamma+1}=0$ (resp. $1$).
On the other hand,
the spin of quasihole corresponding to $\lambda'_j$
is $1/2$ (resp. $-1/2$)
if $\nu_{\gamma+1}=0$ and $j\in I_2(\rho(\lambda))$
or $\nu_{\gamma+1}=1$ and $j\in I_1(\rho(\lambda))$
(resp.
$\nu_{\gamma+1}=0$ and $j\in I_1(\rho(\lambda))$
or $\nu_{\gamma+1}=1$ and $j\in I_2(\rho(\lambda))$
).
Now, we can determine the excitation contents
of the intermediate states
for the dynamical correlation functions.
The excitation contents of the intermediate states
for the dynamical density two-point correlation function
is given by the following set of
quasiparticle and quasiholes:
\begin{equation}
\label{minimal-bubble-dd}
\left\{
\begin{array}{l}
\mbox{one quasiparticle with spin $\sigma$}
\\
\mbox{$\beta+1$ quasiholes with spin $-\sigma$}
\\
\mbox{$\beta$ quasiholes with spin $\sigma$},
\end{array}
\right.
\end{equation}
where $\sigma=\pm 1/2$.
This is consistent with the result in ref. \cite{KYA}.
Similarly,
the excitation contents of the intermediate states
for the dynamical spin-density two-point correlation function
is given by the following sets of
quasiparticle and quasiholes:
\begin{equation}
\label{minimal-bubble-ss1}
\left\{
\begin{array}{l}
\mbox{one quasiparticle with spin $\sigma$}
\\
\mbox{$\beta$ quasiholes with spin $-\sigma$}
\\
\mbox{$\beta+1$ quasiholes with spin $\sigma$},
\end{array}
\right.
\end{equation}
and
\begin{equation}
\label{minimal-bubble-ss2}
\left\{
\begin{array}{l}
\mbox{one quasiparticle with spin $\sigma$}
\\
\mbox{$\beta+2$ quasiholes with spin $-\sigma$}
\\
\mbox{$\beta-1$ quasiholes with spin $\sigma$}.
\end{array}
\right.
\end{equation}
It is remarkable that
two types of the set
of the elementary excitations contribute to
the dynamical spin-density two-point correlation function.
For later convenience, using (\ref{cond1}) and (\ref{cond2}),
we rewrite (\ref{dd-finite}) and (\ref{ss-finite}) as
\begin{eqnarray}
\label{dd-finite-parity}
\langle\rho(x,t)\rho(0,0)\rangle_n
&=&
\frac{4}{\pi^2}
\sum_{\nu=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}
\atop
{\scriptscriptstyle \#I_1(\nu)=\beta+1}}
\sum_{{{{{\footnotesize \lambda\in\Lambda_n^{(\gamma)}:\,
\mbox{{\scriptsize colored partition}}}}
\atop
{\footnotesize |\lambda|:\,\mbox{{\scriptsize even}}}}
\atop
{\footnotesize \rho(\lambda)=\nu}}
}
|P_\lambda|^2 X_\lambda Y_\lambda^{-1} Z_\lambda
e^{-itE_\lambda}\cos(xP_\lambda),
\\
\label{ss-finite-parity}
\langle s(x,t)s(0,0)\rangle_n
&=&
\frac{1}{2L^2}
\sum_{\nu=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}
\atop
{\scriptscriptstyle \#I_1(\nu)=\beta}}
\sum_{{{{{\footnotesize \lambda\in\Lambda_n^{(\gamma)}:\,
\mbox{{\scriptsize colored partition}}}}
\atop
{\footnotesize |\lambda|:\,\mbox{{\scriptsize odd}}}}
\atop
{\footnotesize \rho(\lambda)=\nu}}
}
X_\lambda Y_\lambda^{-1} Z_\lambda
e^{-itE_\lambda}\cos(xP_\lambda)
\nonumber
\\
&+&
\frac{1}{2L^2}
\sum_{\nu=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}
\atop
{\scriptscriptstyle \#I_1(\nu)=\beta+2}}
\sum_{{{{{\footnotesize \lambda\in\Lambda_n^{(\gamma)}:\,
\mbox{{\scriptsize colored partition}}}}
\atop
{\footnotesize |\lambda|:\,\mbox{{\scriptsize odd}}}}
\atop
{\footnotesize \rho(\lambda)=\nu}}
}
X_\lambda Y_\lambda^{-1} Z_\lambda
e^{-itE_\lambda}\cos(xP_\lambda).
\end{eqnarray}
The first summation in the right hand side
of (\ref{dd-finite-parity}) is
taken over $2{}_\gamma C_{(\gamma+1)/2}$ different parities, since
\begin{equation}
2{}_\gamma C_{(\gamma+1)/2}
=
\#\{
\nu
=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}\,|\,
\#I_1(\nu)=\beta+1
\}.
\end{equation}
Similarly, the first summations of the first and second line
in the right hand side of (\ref{ss-finite-parity}) are
respectively taken over $2{}_\gamma C_{(\gamma-1)/2}$ and
$2{}_\gamma C_{(\gamma+3)/2}$ different parities.
\subsection{Quasiparticle and quasihole description
of the dynamical correlation functions}
\label{quasiparticle-description}
To take the thermodynamic limit,
next our task is to rewrite the formulae
(\ref{dd-finite-parity}) and (\ref{ss-finite-parity})
in terms of parameters which
correspond to the momenta
of quasiholes and quasiparticle
(see \cite{Ha,LPS}).
We have already introduced such parameters,
{\it i.e.},
$
\lambda
=
\langle
\lambda'_1,\lambda'_2,\cdots,\lambda'_\gamma;r
\rangle.
$
The quantities $\lambda'_1,\lambda'_2,\cdots,\lambda'_\gamma$ and $r$
are respectively related to the momenta
of quasiholes and quasiparticle.
Although it can be possible to proceed the calculation
by using the parameters
$\lambda'_1,\lambda'_2,\cdots,\lambda'_\gamma$ and $r$,
it is appropriate to introduce new parameters as follows.
We define the following numbers \cite{Uglov}:
\begin{eqnarray}
\label{row-w}
&&w_i(\lambda)
=
\#\{s\in i\mbox{th row of }\lambda\,|\,s: \mbox{white}\},
\\
\label{column-w}
&&w_j(\lambda')
=
\#\{s\in j\mbox{th column of }\lambda\,|\,s: \mbox{white}\}.
\end{eqnarray}
We note that,
using these numbers, we have
$|W_\lambda|=\sum_{i=1}^n w_i(\lambda)
=\sum_{j=1}^{\lambda_1} w_j(\lambda')$.
Then,
instead of
$\lambda'_1,\lambda'_2,\cdots,\lambda'_\gamma$ and $r$,
we adopt
$w_1(\lambda'),\cdots,w_\gamma(\lambda')$, and $p=w_1(\lambda)-\gamma$
as the parameters.
These two sets of parameters are related by the formulae
\begin{equation}
\lambda'_j
=
\left\{
\begin{array}{ll}
2w_j(\lambda')-1,\quad & \mbox{if $j$: odd, $\lambda'_j$: odd}, \\
2w_j(\lambda'), \quad & \mbox{if $\lambda'_j$: even},\\
2w_j(\lambda')+1,\quad & \mbox{if $j$: even, $\lambda'_j$: odd},
\end{array}
\right.
\end{equation}
and
\begin{equation}
r
=
\left\{
\begin{array}{ll}
2p+1,\quad & \mbox{if $r$: odd}, \\
2p, \quad & \mbox{if $r$: even}.
\end{array}
\right.
\end{equation}
Let us rewrite the formulae for the dynamical
correlation functions (\ref{dd-finite-parity}) and
(\ref{ss-finite-parity})
by using
$w_1(\lambda'),\cdots,w_\gamma(\lambda')$, and $p$.
First of all,
for $\lambda\in\Lambda_n^{(\gamma)}$,
we have
\begin{eqnarray}
&&
|W_\lambda|
=\sum_{j=1}^{\lambda_1} w_j(\lambda')
=\sum_{j=1}^{\gamma} w_j(\lambda')+p, \\
&&
n_w(\lambda)=\sum_{s\in W_\lambda}l'(s)
=\sum_{{\scriptstyle 1\leq j\leq\gamma}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
(w_j(\lambda')^2-w_j(\lambda'))
+\sum_{{\scriptstyle 2\leq j\leq\gamma-1}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}}}}
w_j(\lambda')^2, \\
&&
n_w(\lambda')=\sum_{s\in W_\lambda}a'(s)
=
\sum_{j=1}^{\gamma}(j-1)w_j(\lambda')+p(p+\gamma).
\end{eqnarray}
Then, from the definitions
(\ref{ene-finite}) and (\ref{mome-finite}),
we obtain the formulae for $E_\lambda$ and $P_\lambda$.
Next we rewrite $X_\lambda$, $Y_\lambda$ and $Z_\lambda$.
For this purpose, following Ha \cite{Ha},
we decompose a partition
$\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)\in\Lambda_n^{(\gamma)}$
into three sub-diagrams
$\lambda={\cal A}_\lambda\sqcup{\cal B}_\lambda\sqcup{\cal C}_\lambda$ where
\begin{eqnarray}
&&
{\cal A}_\lambda=\{(1,j)\in\Lambda_n^{(\gamma)}\,|\,1\leq j\leq\gamma\},\\
&&
{\cal B}_\lambda=\{(i,j)\in\Lambda_n^{(\gamma)}\,|\,1\leq j\leq\gamma, 2\leq
i\leq\lambda'_j\},\\
&&
{\cal C}_\lambda=\{(1,j)\in\Lambda_n^{(\gamma)}\,|\,\gamma+1\leq j\leq\lambda_1\}.
\end{eqnarray}
For example, if
$\lambda=(13,5,5,5,4,4,4,2,2,1)=\langle 10,9,7,7,4;8\rangle\in\Lambda_n^{(5)}$,
then
${\cal A}_\lambda=$
sub-diagram which contains
$\Yvcentermath1\young(\heartsuit)$
in the following diagram,
${\cal B}_\lambda=$
sub-diagram which contains
$\Yvcentermath1\young(\spadesuit)$,
and
${\cal C}_\lambda=$
sub-diagram which contains
$\Yvcentermath1\young(\clubsuit)$:
$$
\lambda=
\Yvcentermath1
\young(
\heartsuit\heartsuit\heartsuit\heartsuit\heartsuit
\clubsuit\clubsuit\clubsuit\clubsuit\clubsuit\clubsuit\clubsuit\clubsuit,%
\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit\spadesuit\spadesuit,%
\spadesuit\spadesuit,%
\spadesuit\spadesuit,%
\spadesuit
)
$$
We denote
$W_{{\cal D}_\lambda}=W_\lambda\cap{\cal D}_\lambda$,
$B_{{\cal D}_\lambda}=B_\lambda\cap{\cal D}_\lambda$
and
$H_{2, {\cal D}_\lambda}=H_2(\lambda)\cap{\cal D}_\lambda$ for ${\cal D}={\cal A},{\cal B},{\cal C}$.
(Notice that $(1,1)\in W_{{\cal A}_\lambda}$ (if $\lambda\ne\O$).)
For a colored partition $\lambda\in\Lambda_n^{(\gamma)}$,
we denote
$X_{{\cal D}_\lambda}
=\prod_{s\in W_{{\cal D}_\lambda}\setminus\{(1,1)\}} c(s;\gamma)^2$ and
$Y_{{\cal D}_\lambda}
=\prod_{s\in H_{2, {\cal D}_\lambda}} d(s;\gamma)$
for ${\cal D}={\cal A},{\cal B},{\cal C}$.
It is obvious that
$X_\lambda=X_{{\cal A}_\lambda} X_{{\cal B}_\lambda} X_{{\cal C}_\lambda}$, $Y_\lambda=Y_{{\cal A}_\lambda}
Y_{{\cal B}_\lambda} Y_{{\cal C}_\lambda}$ and
$Z_\lambda=Z_{{\cal A}_\lambda} Z_{{\cal B}_\lambda} Z_{{\cal C}_\lambda}$.
Then, it is easy to show that, for $\lambda\in\Lambda_n^{(\gamma)}$,
\begin{eqnarray}
&&
X_{{\cal A}_\lambda}
=
2^{\gamma-1}{\mit \Gamma}^2\big((\gamma+1)/2\big),\\
&&
X_{{\cal B}_\lambda}
=
\xi^{\gamma+1}
\prod_{j=1}^\gamma
\xi^{-2w_j(\lambda')}
{\mit \Gamma}^{-2}(j/\gamma)
\nonumber\\
&&
\times
\prod_{{\scriptstyle 1\leq j\leq\gamma}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
{\mit \Gamma}^2\big(w_j(\lambda')-\xi(j-1)\big)
\prod_{{\scriptstyle 2\leq j\leq\gamma-1}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}}}}
{\mit \Gamma}^2\big(w_j(\lambda')+1/2-\xi(j-1)\big),\\
&&
X_{{\cal C}_\lambda}
=
2^{2p}
\frac{{\mit \Gamma}^2\big(p+(\gamma+1)/2\big)}
{{\mit \Gamma}^2\big((\gamma+1)/2\big)}, \\
&&
Y_{{\cal C}_\lambda}
=
2^{2p}
{\mit \Gamma}\big(p+1\big)
\frac{{\mit \Gamma}\big(p+(\gamma+1)/2\big)}
{{\mit \Gamma}\big((\gamma+1)/2\big)},\\
&&
Z_{{\cal A}_\lambda}
=
\prod_{{\scriptstyle 1\leq j\leq\gamma}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
\frac{\gamma n+j-1}{\gamma n+j-\gamma},\\
&&
Z_{{\cal B}_\lambda}
=
\prod_{{\scriptstyle 1\leq j\leq\gamma}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
\frac{{\mit \Gamma}\big(n/2+\xi j-\xi\big){\mit \Gamma}\big(n/2-w_j(\lambda')+\xi j+1/2\big)}
{{\mit \Gamma}\big(n/2+\xi j-1/2\big){\mit \Gamma}\big(n/2-w_j(\lambda')+\xi j-\xi+1\big)}
\nonumber\\
&&\ \ \ \
\times
\prod_{{\scriptstyle 2\leq j\leq\gamma-1}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}}}}
\frac{{\mit \Gamma}\big(n/2+\xi j-\xi+1/2\big){\mit \Gamma}\big(n/2-w_j(\lambda')+\xi j\big)}
{{\mit \Gamma}\big(n/2+\xi j\big){\mit \Gamma}\big(n/2-w_j(\lambda')+\xi j-\xi+1/2\big)},\\
&&
Z_{{\cal C}_\lambda}
=
\frac{{\mit \Gamma}\big(\gamma n/2+1\big){\mit \Gamma}\big(\gamma n/2+p+(\gamma+1)/2\big)}
{{\mit \Gamma}\big(\gamma n/2+(\gamma+1)/2\big){\mit \Gamma}\big(\gamma n/2+p+1\big)},
\end{eqnarray}
where $\xi=(2\gamma)^{-1}$.
Notice that we can derive
all above formulae without fixing
the parity $\rho(\lambda)$.
On the other hand, to derive
the explicit forms of $Y_{{\cal A}_\lambda}$ and $Y_{{\cal B}_\lambda}$
for $\lambda\in\Lambda_n^{(\gamma)}$, we must fix
the parity $\rho(\lambda)$.
In fact, to write down the explicit forms
of $Y_{{\cal A}_\lambda}$ and $Y_{{\cal B}_\lambda}$,
we need more complicated notations.
However, for the purpose of taking the thermodynamic limit,
the necessary information are the sets $I_1(\rho(\lambda))$,
$I_2(\rho(\lambda))$, and the quantities of order
${\cal O}(n)$.
We see that, after replacing
the elements of sets $I_1$ and $I_2$ appropriately,
the thermodynamic limit of $Y_{{\cal D}_\lambda}$ and $Y_{{\cal D}_{\lambda'}}$
with $\rho(\lambda)\ne\rho(\lambda')$
coincide with each other $({\cal D}={\cal A}, {\cal B})$.
We do not give the explicit forms of
$Y_{{\cal A}_\lambda}$ and $Y_{{\cal B}_\lambda}$ for
the general admissible colored partition $\lambda$.
In Appendix B, we give the examples
for some admissible colored partitions.
The thermodynamic limit of $Y_{{\cal A}_\lambda}$ and
$Y_{{\cal B}_\lambda}$ for general admissible colored partitions
are easily obtained from those examples.
Finally, we change the summation indices for
the sums in the dynamical correlation functions.
For example, we rewrite the sum
in the density two-point
correlation function (\ref{dd-finite-parity})
as
\begin{equation}
\label{sum-dd}
\sum_{\nu=(\nu_1,\cdots,\nu_{\gamma+1})\in \{0,1\}^{\times(\gamma+1)}
\atop
{\scriptscriptstyle \#I_1(\nu)=\beta+1}}
\quad
\sum_{p\geq 0}
\quad
\sum_{n/2\geq w_{j_1}(\lambda')\geq\cdots\geq w_{j_{\beta+1}}(\lambda')\geq 0}
\quad
\sum_{n/2\geq w_{k_1}(\lambda')\geq\cdots\geq w_{k_{\beta}}(\lambda')\geq 0},
\end{equation}
where $\{j_l\}_{l=1}^{\beta+1}=I_1(\rho(\lambda)=\nu)$
such that
$j_1\geq\cdots\geq j_{\beta+1}$
and
$\{k_l\}_{l=1}^{\beta}=I_2(\rho(\lambda)=\nu)$
with
$k_1\geq\cdots\geq k_{\beta}$.
\subsection{Thermodynamic limit}
\label{limit}
In this subsection,
we take the thermodynamic limit, {\it i.e.},
$n\rightarrow\infty$, $L\rightarrow\infty$
with $\rho_0=n/L$ fixed.
Let us introduce the momenta
$u$ and $v_j$ for $j=1,\cdots,\gamma$ of
the quasiparticle and quasiholes, respectively,
by the formulae,
\begin{eqnarray}
\label{particle-velocity}
\frac{1}{\gamma}\frac{p}{n}
&\longrightarrow&
-\frac{u+1}{4},
\\
\label{hole-velocity}
\frac{w_j(\lambda')}{n}
&\longrightarrow&
\frac{v_j+1}{4}.
\end{eqnarray}
Then we have the thermodynamic limit
of the energy and total momentum,
\begin{eqnarray}
\label{ene-td'}
&&
E_\lambda
\longrightarrow
{\cal E}
=
\sum_{j=1}^\gamma\epsilon_{\rm h}(v_j)+\epsilon_{\rm p}(u),
\\
\label{mome-td'}
&&
P_\lambda
\longrightarrow
{\cal P}
=
\frac{\pi\rho_0}{2}
\Big[
\sum_{j=1}^\gamma v_j-\gamma u
\Big],
\end{eqnarray}
where
\begin{eqnarray}
\label{hole-dis'}
&&
\epsilon_{\rm h}(y)
=
\gamma
\frac{1}{2}
\left(\frac{\pi\rho_0}{2}\right)^2(1-y^2), \\
\label{particle-dis'}
&&
\epsilon_{\rm p}(y)
=
\gamma^2
\frac{1}{2}
\left(\frac{\pi\rho_0}{2}\right)^2(y^2-1).
\end{eqnarray}
We have adopted the normalization
(\ref{particle-velocity}) and (\ref{hole-velocity})
of $u$ and $v_j$ so that
the Fermi points coincides with $\{\pm 1\}$.
Also, using the formula
$\lim_{|a|\rightarrow \infty}{\mit \Gamma}(a+z)/{\mit \Gamma}(a)=a^z$,
we can obtain the thermodynamic limit of
$X_{\lambda}Y_{\lambda}^{-1}Z_{\lambda}$.
In the following,
we consider the case of density two-point
correlation function.
In this case, we have
\begin{eqnarray}
&&
X_{\lambda}Y_{\lambda}^{-1}Z_{\lambda}
\longrightarrow
L^{-(\gamma+1)}2^{2\gamma}(\gamma\rho_0)^{-(\gamma+1)}
{\mit \Gamma}\big((\gamma+1)/2\big)
\prod_{j=1}^\gamma
{\mit \Gamma}\big(\xi+1/2\big)
{\mit \Gamma}^{-2}\big(j/\gamma\big)
\nonumber
\\
&&
\qquad
\times
(u^2-1)^{(\gamma-1)/2}
\prod_{j=1}^\gamma
(1-v_j^2)^{\xi-1/2}
\prod_{j\in I_1(\rho(\lambda))}
(u-v_j)^{-2}
\nonumber
\\
&&
\qquad
\times
\prod_{s=1,2}
\prod_{{\scriptstyle j, k\in I_s(\rho(\lambda))}
\atop
{\scriptstyle j<k}}
(v_j-v_k)^{-2(\xi+1/2)}
\prod_{j\in I_1(\rho(\lambda))}
\prod_{k\in I_2(\rho(\lambda))}
(v_j-v_k)^{-2(\xi-1/2)}
\end{eqnarray}
for a d-d admissible colored partition
$\lambda\in\Lambda_n^{(\gamma)}$ with the fixed parity
$\rho(\lambda)$.
For each d-d admissible colored partition
$\lambda\in\Lambda_n^{(\gamma)}$ with the fixed parity
$\rho(\lambda)$,
we replace $\{v_j\}_{j\in I_1(\rho(\lambda))}$
and $\{v_j\}_{j\in I_2(\rho(\lambda))}$
by $\{v_j\}_{j=1}^{\beta+1}$
such that $v_1\geq \cdots\geq v_{\beta+1}$
and $\{w_j\}_{j=1}^\beta$
such that $w_1\geq \cdots\geq w_{\beta}$,
respectively.
In the thermodynamic limit,
we rewrite the sums as integrals
\begin{eqnarray}
\label{sum2int-hole}
&&
\sum_{n/2\geq w_{j_1}(\lambda')\geq\cdots\geq w_{j_{\beta+1}}(\lambda')\geq 0}
\quad
\sum_{n/2\geq w_{k_1}(\lambda')\geq\cdots\geq w_{k_{\beta}}(\lambda')\geq 0}
\nonumber
\\
&&
\longrightarrow
L^\gam2^{-2\gamma}\rho_0^\gamma
\int_{1\geq v_{1}\geq v_{2}\geq\cdots\geq v_{\beta+1}\geq -1}
dv_1dv_2\cdots dv_{\beta+1}
\int_{1\geq w_{1}\geq w_{2}\geq\cdots\geq w_{\beta}\geq -1}
dw_1dw_2\cdots dw_{\beta},
\\
\label{sum2int-particle}
&&
\sum_{p\geq 0}
\longrightarrow
-
L2^{-2}\gamma\rho_0\int_{-\infty}^{-1}du.
\end{eqnarray}
Notice that,
for each d-d admissible colored partition
$\lambda\in\Lambda_n^{(\gamma)}$ with the fixed parity
$\rho(\lambda)$,
the energy ${\cal E}$, total momentum ${\cal P}$ and
thermodynamic limit of the quantity $X_\lambda Y_\lambda^{-1}Z_\lambda$
are invariant under the exchange $v_i\leftrightarrow v_j$
and/or $w_k\leftrightarrow w_l$.
Then, finally, after removing the order on momenta,
we arrive at the formula (\ref{dd-td}).
This formula coincides with our previous result
in \cite{KYA} up to the constant factor.
The formula (\ref{ss-td}) can be derived in the same way.
Essential part of
the formulae (\ref{rGreen-td}), (\ref{dd-td}) and (\ref{ss-td})
can be described by the function $F_\beta$.
As is the spinless case \cite{Ha},
we call the function $F_\beta$ the minimal form factor
of the $SU(2)$ spin CS model
(with integer coupling parameter).
The physical interpretation
of the minimal form factor has been discussed
in ref. \cite{KYA}.
\section{Conclusion}
\label{conclusion}
In this work,
we have taken the thermodynamic limit
of dynamical density and spin-density two-point
correlation functions of the spin CS model.
We have obtained the exact formulae
(\ref{dd-td}) and (\ref{ss-td})
of the density and spin-density two-point correlation
functions, respectively.
We have exactly shown that,
with appropriate numbers of quasiparticles and quasiholes,
the dynamical correlation functions of the spin CS model
can be described by
the unique function $F_\beta$ (\ref{form-factor})
which is called the minimal form factor.
These results are consistent
with our previous work \cite{KYA}.
\section*{Appendix A}
\label{appendixA}
In this appendix, we prove the following lemma:
for a colored partition
$\lambda\in\Lambda_n$ with even (resp. odd) weight,
$S_\lambda=0\quad(\mbox{resp. } \pm 1)
\Leftrightarrow
|W_\lambda|=|H_{2}(\lambda)|\quad
(\mbox{resp. } |H_{2}(\lambda)|+1)$.
In this appendix, we do not assume that
$n$ is even.
We introduce the notations.
The partition $\lambda=(\lambda_1,\lambda_2,\cdots\,)$
can be represented by the notation
$\lambda=(1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots\,)$ where
$m_i(\lambda)=\#\{j\,|\,\lambda_j=i\}$ \cite{Macd}.
Using this notation,
we define the following transformations
$\tau_i$ and $\tau'_i$ for $i\in\mbox{{\bf Z}}_{>0}$:
\begin{eqnarray}
&&
\tau_i:\,
\lambda=(1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots i^{m_i(\lambda)}\cdots)
\longrightarrow
\left\{
\begin{array}{ll}
(1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots i^{m_i(\lambda)-2}\cdots),
\quad & m_i(\lambda)\geq 2,
\\
\lambda,
\quad & m_i(\lambda)< 2,
\end{array}
\right.
\\
&&
\tau'_i:\,
\lambda
\longrightarrow
(\tau_i(\lambda'))'.
\end{eqnarray}
That is, $\tau_i$ ($\tau'_i$) is the following transformation:
if there exist two rows (columns)
which have same number of squares $i$
then $\tau_i$ ($\tau'_i$) removes these rows (columns),
if not then $\tau_i$ ($\tau'_i$) is the identity.
If $\lambda$ has even (odd) weight
then both $\tau_i(\lambda)$ and $\tau'_i(\lambda)$
have even (odd) weights.
We introduce the special partition $\delta(k)\in\Lambda_n$ by
\begin{equation}
\delta(k)
=
\left\{
\begin{array}{ll}
\O,
\quad & k=0,
\\
(k,k-1,\cdots,1)=(1^12^1\cdots k^1),
\quad & k=1,\cdots,n.
\end{array}
\right.
\end{equation}
The partitions $\delta(k)$ for $k=0,1,\cdots,n$
are the fixed points of the transformations
$\tau_i$ and $\tau'_i$.
We see that, applying $\tau_i$'s and $\tau'_j$'s
sufficiently many times,
any partition $\lambda\in\Lambda_n$ is mapped to
one of $\delta(k)$'s.
We denote the resultant mapping by
$\tau:\,\Lambda_n
\longrightarrow
\{\delta(k)\,|\,k=0,1,\cdots,n\}$.
{}From the definition,
the transformations $\tau_i$, $\tau'_i$ and $\tau$
can be defined on
the set of all colored partitions.
For instance,
$$
\tau
\left(
\,\,
\Yvcentermath1
\young(
\hfill\bullet\hfil\bullet,%
\bullet\hfil\bullet,%
\hfil\bullet\hfil,%
\bullet\hfil,%
\hfil\bullet,%
\bullet
)
\,\,
\right)
=
\Yvcentermath1
\young(
\hfill\bullet,%
\bullet
)
=\delta(2)
$$
We define the following numbers:
for a colored partition $\lambda\in \Lambda_n$,
$wb(\lambda)=|W_\lambda|-|B_\lambda|(=S_\lambda)$ and
$wh(\lambda)=|W_\lambda|-|H_2(\lambda)|$.
It is easy to see that
these numbers are invariant under the transformations
$\tau_i$ and $\tau'_i$,
{\it i.e.},
$wb(\tau_i(\lambda))=wb(\lambda)$,
$wb(\tau'_i(\lambda))=wb(\lambda)$ and same formulae
for $wh$.
Therefore
$wb(\tau(\lambda))=wb(\lambda)$ and $wh(\tau(\lambda))=wh(\lambda)$.
Moreover we have
\begin{eqnarray}
&&
\label{wb}
wb(\delta(k))
=
\left\{
\begin{array}{rl}
0,
\quad & k=0,
\\
l,
\quad & k=2l-1,\,\quad(l=1,2,\cdots),
\\
-l,
\quad & k=2l,\,\quad(l=1,2,\cdots),
\end{array}
\right.
\\
&&
\label{wh}
wh(\delta(k))
=
\left\{
\begin{array}{rl}
0,
\quad & k=0,
\\
l^2,
\quad & k=2l-1,\,\quad(l=1,2,\cdots),
\\
l^2,
\quad & k=2l,\,\quad(l=1,2,\cdots).
\end{array}
\right.
\end{eqnarray}
(Notice that $H_2(\delta(k))=\O$ for all $k$).
We define the subset $\Lambda_n(\delta(k))$ of
$\Lambda_n$ by
\begin{equation}
\Lambda_n(\delta(k))
=
\{
\lambda\in\Lambda_n\,|\,
\tau(\lambda)=\delta(k)
\}.
\end{equation}
It is important to note the following fact:
if $\lambda \in\Lambda_n(\delta(k))$ then
$wb(\lambda)=wb(\delta(k))$ and $wh(\lambda)=wh(\delta(k))$.
Therefore, from the formulae (\ref{wb}) and (\ref{wh}),
$\Lambda_n(\delta(k))\cap\Lambda_n(\delta(k'))=\O$
if $k\ne k'$.
This fact proves the lemma.
We have
the following decomposition
for the set of all {\it colored} partitions $\Lambda_n$
\begin{equation}
\Lambda_n
=
\sqcup_{k=0}^n\Lambda_n(\delta(k)).
\end{equation}
We see that
the set of all d-d (resp. s-s) admissible colored partitions is
the set $\Lambda_n(\delta(0))\cap\Lambda_n^{(\gamma)}$
(resp.
$(\Lambda_n(\delta(1))\sqcup\Lambda_n(\delta(2)))
\cap\Lambda_n^{(\gamma)}$).
\section*{Appendix B}
\label{appendixB}
In this appendix, we give examples of
the explicit formula for $Y_{{\cal A}_\lambda}$ and $Y_{{\cal B}_\lambda}$.
\bigskip
\noindent
a) Example for the d-d admissible colored partition
We consider the d-d admissible colored partition
$\lambda\in\Lambda_n^{(\gamma)}$ with parity
$\rho(\lambda)
=(\overbrace{1,1,\cdots,1}^{\gamma+1\ \mbox{{\footnotesize times}}})$.
In this case,
$I_1(\rho(\lambda))=\{j\in I \,|\, j : \mbox{odd}\}$
and
$I_2(\rho(\lambda))=\{j\in I \,|\, j : \mbox{even}\}$.
We have the explicit formula for $Y_{{\cal A}_\lambda}$ and $Y_{{\cal B}_\lambda}$
\begin{eqnarray}
&&
Y_{{\cal A}_\lambda}
=
\xi^{-(\gamma+1)}
\prod_{j\in I_1(\rho(\lambda))}
(p/\gamma+w_j(\lambda')-1/2-\xi(j-2))
(p/\gamma+w_j(\lambda')-\xi(j-1)),
\\
&&
Y_{{\cal B}_\lambda}
=
\xi^{\gamma+1}{\mit \Gamma}\big((1+1/\gamma)/2\big)^{-\gamma}
\prod_{j=1}^\gamma\xi^{-2w_j(\lambda')}
\nonumber\\
&&
\times
\prod_{j\in I_1(\rho(\lambda))}
{\mit \Gamma}\big(w_j(\lambda')-\xi(j-1)\big){\mit \Gamma}\big(w_j(\lambda')-\xi j+1/2\big)
\nonumber\\
&&
\times
\prod_{j\in I_2(\rho(\lambda))}
{\mit \Gamma}\big(w_j(\lambda')-\xi(j-1)+1/2\big){\mit \Gamma}\big(w_j(\lambda')-\xi j+1\big)
\nonumber\\
&&
\times
\prod_{{\scriptstyle j,k\in I_1(\rho(\lambda))}
\atop
{\scriptstyle j<k}}
\frac{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-\xi+1/2\big)
}
{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j,k\in I_2(\rho(\lambda))}
\atop
{\scriptstyle j<k}}
\frac{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-\xi+1/2\big)
}
{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_1(\rho(\lambda)),k\in I_2(\rho(\lambda))}
\atop
{\scriptstyle j<k}}
\frac{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-1/2\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-\xi\big)
}
{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+\xi-1\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-1/2\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_2(\rho(\lambda)),k\in I_1(\rho(\lambda))}
\atop
{\scriptstyle j<k}}
\frac{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+3/2\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)-\xi+2\big)
}
{
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+\xi+1\big)
{\mit \Gamma}\big(w_j(\lambda')-w_k(\lambda')+\xi(k-j)+3/2\big)
}.
\end{eqnarray}
\bigskip
\noindent
b) Example for the type I s-s admissible colored partition
We consider the type I s-s admissible colored partition
$\mu\in\Lambda_n^{(\gamma)}$ with parity
$\rho(\mu)
=(\overbrace{1,1,\cdots,1}^{\gamma\ \mbox{{\footnotesize times}}},0)$.
In this case,
$I_1(\rho(\mu))=\{j\in I \,|\, j : \mbox{even}\}$
and
$I_2(\rho(\mu))=\{j\in I \,|\, j : \mbox{odd}\}$.
The formula for
$Y_{{\cal A}_\mu}$ is given by
\begin{equation}
Y_{{\cal A}_\mu}
=
\xi^{-(\gamma-1)}
\prod_{j\in I_1(\rho(\mu))}
(p/\gamma+w_j(\mu')+1/2-\xi(j-1))
(p/\gamma+w_j(\mu')+1-\xi j).
\end{equation}
The explicit form of $Y_{{\cal B}_\lambda}$ is given by the same formula in a)
with replacement of
$I_1(\rho(\lambda))$ and $I_2(\rho(\lambda))$
by $I_2(\rho(\mu))$ and $I_1(\rho(\mu))$, respectively.
\bigskip
\noindent
c) Example for the type II s-s admissible colored partition
Finally, we consider the type II
s-s admissible colored partition
$\eta\in\Lambda_n^{(\gamma)}$ with parity
$$
\rho(\eta)
=
(\overbrace{0,1,0,1,\cdots,0,1,0}^{(\gamma+3)/2}
\overbrace{0,1,0,1,\cdots,0,1}^{(\gamma-3)/2},
0),
\quad
((\gamma+3)/2:\, \mbox{odd}).
$$
In this case,
$I_1(\rho(\eta))=\{1,\cdots,\beta+2\}$
and
$I_2(\rho(\eta))=\{\beta+3,\cdots,2\beta+1\}$.
We have the explicit formulae
for $Y_{{\cal A}_\eta}$ and $Y_{{\cal B}_\eta}$
\begin{eqnarray}
&&
Y_{{\cal A}_\eta}
=
\xi^{-(\gamma+3)}
\prod_{{\scriptstyle j\in I_1(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
(p/\gamma+w_j(\eta')-\xi(j-1))
(p/\gamma+w_j(\eta')+1/2-\xi j)
\nonumber\\
&&\ \ \ \ \ \ \
\times
\prod_{{\scriptstyle j\in I_1(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}}}}
(p/\gamma+w_j(\eta')+1/2-\xi(j-1))
(p/\gamma+w_j(\eta')+1-\xi j),
\\
&&
Y_{{\cal B}_\eta}
=
\xi^{\gamma+5}{\mit \Gamma}\big((1+1/\gamma)/2\big)^{-\gamma}
\prod_{j=1}^\gamma\xi^{-2w_j(\eta')}
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{\scriptstyle j\in I_s(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}}}}
{\mit \Gamma}\big(w_j(\eta')-\xi(j-1)\big){\mit \Gamma}\big(w_j(\eta')-\xi j+1/2\big)
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{\scriptstyle j\in I_s(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}}}}
{\mit \Gamma}\big(w_j(\eta')-\xi(j-1)+1/2\big){\mit \Gamma}\big(w_j(\eta')-\xi j+1\big)
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{{\scriptstyle j,k\in I_s(\rho(\eta))}
\atop
{\scriptstyle j<k}}
\atop
{\scriptstyle j,k:\, \mbox{{\scriptsize odd}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+1/2\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{{\scriptstyle j,k\in I_s(\rho(\eta))}
\atop
{\scriptstyle j<k}}
\atop
{\scriptstyle j,k:\, \mbox{{\scriptsize even}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+1/2\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{{\scriptstyle j,k\in I_s(\rho(\eta))}
\atop
{\scriptstyle j<k}}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}},\
k:\, \mbox{{\scriptsize even}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1/2\big)
}
\nonumber\\
&&
\times
\prod_{s=1,2}
\prod_{{{\scriptstyle j,k\in I_s(\rho(\eta))}
\atop
{\scriptstyle j<k}}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}},\
k:\, \mbox{{\scriptsize even}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+1\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+3/2\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_1(\rho(\eta)),
k\in I_2(\rho(\eta))}
\atop
{\scriptstyle j,k:\, \mbox{{\scriptsize odd}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+3/2\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_1(\rho(\eta)),
k\in I_2(\rho(\eta))}
\atop
{\scriptstyle j,k:\, \mbox{{\scriptsize even}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+3/2\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_1(\rho(\eta)),
k\in I_2(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize odd}},\,
k:\, \mbox{{\scriptsize even}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+1\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+1/2\big)
}
\nonumber\\
&&
\times
\prod_{{\scriptstyle j\in I_1(\rho(\eta)),
k\in I_2(\rho(\eta))}
\atop
{\scriptstyle j:\, \mbox{{\scriptsize even}},\,
k:\, \mbox{{\scriptsize odd}}}}
\frac{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+3/2\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)-\xi+2\big)
}
{
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+\xi+1\big)
{\mit \Gamma}\big(w_j(\eta')-w_k(\eta')+\xi(k-j)+3/2\big)
}.
\end{eqnarray}
\noindent {\bf Acknowledgement:}
The authors thank Y. Kato and Y. Kuramoto
for discussions.
TY is grateful to K. Takemura for useful comments.
TY was supported by
the Core Research for Evolutional Science and Technology (CREST)
program of the Science and Technology Agency of Japan.
\vspace{24pt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 169 |
The Tour-Bus Trainer
by Ryan Halvorson on Aug 27, 2008
Personal Trainer Profile
Gregg Miele's determination, charisma and passion for fitness have him seeing stars.
Subject: Gregg Miele
Company: New York Strength & Conditioning Inc.
Career Orientation. Like many fitness professionals, Gregg Miele—owner of New York Strength & Conditioning Inc. in Hillburn, New York—fell in love with the fitness industry after achieving his own fitness-related successes.
Miele had always been athletic, but he didn't fully realize his own potential until a bit of friendly rivalry sparked his interest. "One of my best friends was into strength training and became a power lifter," he recalls. "I remember going to visit him and being awestruck because I couldn't lift nearly as much as he could. I swore that the next time I visited I would be able to challenge him." As Miele trained hard to make vast physical and mental improvements, that initial spark met tinder and his passion for health and fitness caught fire. Miele comes from a family of teachers and has a knack for educating others, so a career combining teaching and fitness was the perfect match.
Setting Priorities. After becoming a certified personal trainer, Miele was drawn to working with high-profile clients. In order to provide those clients with superior service, he began working at an upscale fitness facility in New York City. "They offered what many gyms didn't," he says. "They had beach volleyball courts that I used for sand drills; indoor basketball courts; a rock climbing wall; and more. They also had in-house education, so I could continue to learn."
As Miele's knowledge and aptitude grew, he set his sights on his target clientele; however, instead of directly courting those clients, he decided to make a more strategic move. "I knew that the only way I could move forward was to offer my services to gym management," he says. By doing so, he believed management would feel comfortable referring clients to him. Miele's gamble paid off, and facility management began to funnel "high-profile" clients his way.
Taking It to the Top. Eventually word spread about Miele's effective services and positive demeanor, and his client base grew to include some of the top names in fashion, acting and music. "Early on I linked up with some musicians, and they asked me to go on tour with them," he recalls. While the experience was exciting, Miele notes that his eagerness to tour with his music clients taught him a hard lesson. "My first tour was both good and bad. I had to refer out all my current clients, and I didn't get paid very well for the tour. That was when I learned how to bid contracts." Despite rocky beginnings, Miele continued to impress and eventually landed a major gig working with Grammy Award winner Mary J. Blige during her 2006 world tour.
Training the Entourage. Miele believes that one of the most demanding aspects for trainers accompanying their clients on tour is creating an atmosphere of health and fitness. "You have one client, but you've also got to consider her entourage," he says. "I always ask, 'Who's on your support team?' I know that when I'm not with the client, the entourage is going to have an influence." Miele's strategy is to make small, manageable changes that affect all members of the tour. "I couldn't create a healthy environment right off the bat," he admits. "Instead of buying big sodas, I bought small ones. Even though the main focus is to keep the principal client healthy and fit, the trick is to get everyone around involved in the challenge so the client sees she's not alone."
The Celebrity Conundrum. Miele has experienced success with this very specific niche because his chief interest is in helping others make lasting lifestyle changes. Celebrities are naturally cautious, he says, as they're continuously bombarded by people who want something from them. "One of the reasons these people choose to work with me is because I'm not trying to be an actor or get a record deal," he says. "I'm a career personal trainer who simply works to help others achieve new heights in health and fitness. My clients recognize this and are then able to trust me."
calling all trainers
Do you own a business that breaks the mold? If so, e-mail rhalvorson@ideafit.com and let us know why you think your personal training business is unique.
People/ProfilesPFT Profile
Want more from Ryan Halvorson?
Fitness Journal, Volume 5, Issue 9
Find the Perfect Job
More jobs, more applicants and more visits than any other fitness industry job board.
- Select All Profession Type (All) - Personal Trainer Group Fitness Instructor Personal Trainer Manager Group Fitness Manager Fitness Director Yoga Instructor Pilates Instructor Athletic Trainer Life Coach Sales / Manager Other
© 2008 by IDEA Health & Fitness Inc. All rights reserved. Reproduction without permission is strictly prohibited.
Ryan Halvorson IDEA Author/Presenter
Ryan Halvorson is an award-winning writer and editor. He is a long-time author and presenter for IDEA Health & Fitness Association, fitness industry consultant and former director of group training for Bird Rock Fit. He is also a Master Trainer for TriggerPoint. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,902 |
4.56.2.2 4.18.1.2709
BBC iPlayer 4.18.1.2709 (Android 4.1+)
BBC BBC iPlayer 4.18.1.2709
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Full details of Media AT are available on the Companies House website at: http://data.companieshouse.gov.uk/doc/company/07100235.
Package: bbc.iplayer.android
Version: 4.18.1.2709 (41812709)
CPU Architecture: armeabi, armeabi-v7a, mips, x86
Uploaded: June 3, 2016 at 7:53 am UTC
MD5: 050e4ba05fd6d95cf4ae21856cf084d5
Recent and old versions of BBC iPlayer
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BBC iPlayer 4.20.1.5292 beta (Android 4.1+) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,876 |
\section{Introduction}
\label{Intro}
\setcounter{equation}{0}
Let $\Omega\subset {\mathbb{R}}^3$ denote an open, bounded, and connected set with smooth
boundary $\,\Gamma\,$ and outward normal derivative $\,\partial_{\bf n}$, let $T>0$ be a
final time, and let $H:=L^2(\Omega)$ denote the Hilbert space of square-integrable
real-valued functions defined on $\Omega$, endowed with the standard inner product
$(\cdot,\cdot)$ and norm $\,\|\cdot\|$, respectively. We denote
\,$Q_t:=\Omega\times (0,t)\,$ for $0<t<T$ and \,$Q:=\Omega\times (0,T)$.
We investigate in this paper the approximation and optimal control of an
abstract system of evolutionary variational (in)equalities. More precisely, the variational
state system has the following form: we look
for functions
$(\mu,y)$ such that
\begin{align}
\label{regy}
&y\in H^1(0,T;V_A^{-r})\cap L^\infty(0,T;V_B^\sigma) \quad\mbox{and} \quad \tau\partial_t y\in L^2(0,T;H)\\[0.5mm]
\label{regmu}
&\mu\in L^2(0,T;V_A^r),\\[0.5mm]
\label{L1}
&f_1(y)\in L^1(Q),
\end{align}
and \pier{satisfying}
\begin{align}
\label{weak1}
&\langle \partial_t y(t),v\rangle_{A,r}\,+\,(A^r\mu(t),A^r v)\,=\,0 \quad
\mbox{for every $\,v\in V_A^r\,$ and a.e. $t\in (0,T)$,} \\[1mm]
\noalign{\allowbreak}
\label{weak2}
&(\tau\partial_t y(t),y(t)-v)\,+\,(B^\sigma y(t),B^\sigma(y(t)-v))\,+\int_\Omega f_1(y(t))\nonumber\\
&\quad {}+\,(f_2'(y(t))-u(t), y(t)-v)\,\le\,(\mu(t),y(t)-v)\,+\int_\Omega f_1(v)\nonumber\\[1mm]
\noalign{\allowbreak}
&\mbox{for every \,$v\in V_B^\sigma\,$ and a.e. \,$t\in (0,T)$},\\[1mm]
\label{weak3}
&y(0)=y_0 \quad\mbox{in \,$\Omega$}.
\end{align}
Here, it is understood that
$\,\int_\Omega f_1(v)=+\infty\quad\mbox{whenever}\quad f_1(v)\not\in\Lx 1.$
The precise meaning of the involved quantities and spaces will be given below. Notice that
\eqref{weak1}--\eqref{weak3} is a generalized version of the evolutionary system
\begin{align}
& \partial_t y + A^{2r} \mu = 0\quad\mbox{in \,$Q$},
\label{Iprima}
\\
& \tau \partial_t y + B^{2\sigma} y + \pier{\partial f_1}(y)+f_2'(y) \pier{{}\ni{}} \mu + u\quad\mbox{in \,$Q$},
\label{Iseconda}
\\
& y(0) = y_0 \quad\mbox{in \,$\Omega$}.
\label{Icauchy}
\end{align}
Here, $\tau\ge 0$ is a constant, $f_2:{\mathbb{R}}\to{\mathbb{R}}$ is a smooth function, and $f_1:{\mathbb{R}}\to [0,+\infty]$ denotes a proper, convex,
and lower semicontinuous function with $f_1(0)=0$, whose effective domain $D(f_1)$ is a closed interval in ${\mathbb{R}}$ (possibly ${\mathbb{R}}$ itself)
and which is smooth in the interior of $D(f_1)$. \pier{In \eqref{Iseconda}, $\partial f_1$ denotes the subdifferential of $f_1$, which is a
multivalued operator, in general, so that the inclusion replaces the equality.}
The linear operators $A^{2r}$, and $B^{2\sigma}$,
with $r>0$ and $\sigma>0$,
denote fractional powers (in the spectral sense) of operators $A$ and $B$. We will
give a proper definition of such operators in the next section. Throughout this
paper, we generally assume:
\vspace{1mm}\noindent
{\bf (A1)} \,\,\,$A:D(A)\subset H\to H$\, and \,$B:D(B)\subset H\to H\,$ are
unbounded, monotone, and selfadjoint linear operators with compact resolvents.
\vspace{1mm}\noindent
This assumption implies that there are sequences
$\{\lambda_j\}$ and $\{\lambda'_j\}$ of eigenvalues
and orthonormal sequences $\{e_j\}$ and $\{e'_j\}$ of corresponding eigenvectors,
that~is,
\Begin{equation}
A e_j = \lambda_j e_j, \quad
B e'_j = \lambda'_j e'_j,
\quad\hbox{and}\quad
(e_i,e_j) = (e'_i,e'_j) = \delta_{ij},
\quad \hbox{for $i,j=1,2,\dots$,}
\label{eigen}
\End{equation}
\pier{with $\delta_{ij}$ denoting the Kronecker index,} such that
\begin{align}
& 0 \leq \lambda_1 \leq \lambda_2 \leq \dots,
\quad\hbox{and}\quad
0 \leq \lambda'_1 \leq \lambda'_2 \leq \dots,
\quad \hbox{with} \quad
\lim_{j\to\infty} \lambda_j
= \lim_{j\to\infty} \lambda'_j
= + \infty,
\label{eigenvalues}
\\[1mm]
& \hbox{$\{e_j\}$ and $\{e'_j\}$ are complete systems in $H$}.
\label{complete}
\end{align}
The state system \eqref{Iprima}--\eqref{Icauchy} (and thus also \eqref{weak1}--\eqref{weak3})
can be seen as a generalization
of the famous Cahn--Hilliard system which models a
phase separation process taking place in the container $\Omega$. In this case,
one typically has $A^{2r}=B^{2\sigma}=-\Delta$ with zero Neumann or Dirichlet boundary conditions, and the unknown
functions $\,y\,$ and $\,\mu\,$ stand for the \emph{order parameter} (usually a scaled
density of one of the involved phases) and the \emph{chemical potential}
associated with the phase transition, respectively. Moreover, $f:=f_1+f_2$ is a double-well potential. Typical cases
are the {\em classical regular potential}, the {\em logarithmic potential\/}, and the {\em double obstacle potential}, which (in this order)
are given by
\begin{align}
\label{regpot}
& f_{{\rm reg}}(v) := \frac 14 \, (v^2-1)^2 \,,
\quad v \in {\mathbb{R}},
\\[2mm]
\label{logpot}
& f_{{\rm log}}(v) := \left\{
\begin{array}{ll}
(1+v)\ln (1+v)+(1-v)\ln (1-v) - c_1 v^2
&\quad\mbox{for }\, v \in (-1,1)\\
2\ln(2)-c_1&\quad\mbox{for }\,v\in\{-1,1\}\\
+\infty&\quad\mbox{for }\,v\not\in [-1,1]
\end{array}
\right.
\\[2mm]
\label{obspot}
&f_{\rm obs}(v):= \left\{
\begin{array}{ll}
-c_1\,v^2 &\quad\mbox{if }\,|v|\le 1\\
+\infty &\quad\mbox{otherwise}
\end{array}
\right.
\end{align}
Here the constant $c_1>0$ is such that the above potentials are nonconvex.
Recently, in \cite[Thm.~2.6~and~2.8]{CGS18}, it was shown that the system \eqref{weak1}--\eqref{weak3}
admits a solution $(\mu,y)$ satisfying \eqref{regy}--\eqref{L1}, where the admissible nonlinearities include all of the three cases
\eqref{regpot}--\eqref{obspot}. In the analysis, it turned out that the first eigenvalue $\lambda_1$ of $\,A\,$ plays an important role.
Indeed, the main assumption for the operators $A,B$ besides {\bf (A1)} was the following:
\vspace{1mm}\noindent
{\bf (A2)} \,\,\,Either\\
\hspace*{20mm}(i) \,\,\,$\lambda_1>0$\\
\hspace*{12.5mm} or\\
\hspace*{20mm}(ii) \,\,$0=\lambda_1<\lambda_2$, and $\,e_1\,$ is a constant and belongs to the domain of $\,B^\sigma$.
\vspace{1mm}\noindent
The existence proof in \cite{CGS18} was based on Moreau--Yosida approximation, which is generally applicable to all of
the three cases \eqref{regpot}--\eqref{obspot}. It turned out that the second component \,$y$\, of the solutions $(\mu,y)$
is always uniquely determined, while this is not necessarily so for the chemical potential $\mu$ (for cases in which also
$\,\mu\,$ is unique, see \cite[Rem.~4.1]{CGS18} and \cite[Rem.~3.4]{CGS19}).
In this paper, we focus on the case when $\,f=f_{{\rm obs}}$, that is, when $\,f_1=I_{[-1,1]}\,$
is the indicator function of the interval $[-1,1]$, given by $I_{[-1,1]}(v)=0$ if $v\in [-1,1]$ and
$I_{[-1,1]}(v)=+\infty$ otherwise. In this case, any solution $(\mu,y)$ of \eqref{weak1}--\eqref{weak3} must satisfy \,$\int_Q I_{[-1,1]}(y) <+\infty$,
which entails that $y\in [-1,1]$ almost everywhere in $Q$ and thus \,$\int_\Omega f_1(y(t))=0\,$ for almost every $t\in (0,T)$ in \eqref{weak2}.
While the question of well-posedness was settled in \cite[Thm.~2.6~and~2.8]{CGS18} for $\,f_1=I_{[-1,1]}$, the matter of optimal control is still open.
Indeed, the optimal control theory recently developed in \cite{CGS19} applies to certain classes of differentiable potentials only. In this paper,
we aim at extending this theory to the case $\,f=f_{{\rm obs}}$. More precisely, we investigate the following
optimal control problem:
\vspace{1mm}
\noindent (${\mathcal{CP}}_0$) \quad Minimize the tracking-type cost functional
\begin{align}
\label{cost}
{\cal J}(y,u):=& \frac{\beta_1}2\,\|y(T)-y_\Omega\|^2\,+\,\frac {\beta_2}2\int_0^T\!\|y(t)-y_Q(t)\|^2\,dt
\,+\,\frac{\beta_3}2\int_0^T\!
\|u(t)\|^2\,dt
\end{align}
over the admissible set
\begin{equation}
\label{uad}
{\mathcal{U}}_{\rm ad}:=\left\{u\in H^1(0,T;\Lx 2):\,|u|\le\rho_1 \mbox{ \,a.\,e. in }\,Q, \,\,\,\,\|u\|_
{H^1(0,T;\Lx 2)}\,\le\,\rho_2\right\},
\end{equation}
subject to \eqref{weak1}--\eqref{weak3} with $f_1=I_{[-1,1]}$.
Here, $\rho_1>0$ and $\rho_2>0$ are such that ${\mathcal{U}}_{\rm ad}\not= \emptyset$, $\beta_i$, $i=1,2,3$, are
nonnegative but not all zero, and the given target functions satisfy $y_\Omega\in \pier{L^2(\Omega)} $ and $y_Q\in L^2(Q)$. Note
that (${\mathcal{CP}}_0$) is well defined, since the component \,$y$\, of the solutions to the state system is uniquely determined.
The main difficulty inherent in (${\mathcal{CP}}_0$) is the nondifferentiability of the nonlinearity $I_{[-1,1]}$, which
entails that standard constraint qualifications from optimal control theory are violated, so that
suitable Lagrange multipliers cannot easily be constructed. In such situations, the
so-called ``deep quench'' approximation
has proved to be a useful tool in a number of cases in the framework
of Cahn--Hilliard systems (see, e.g.,\cite{CFGS1, CFGS3, CGSEECT, CGSConvex, CS}). In all of these works, the starting point was that the optimal control problem (we will later
denote this problem by (${\mathcal{CP}}_\alpha$)) had been successfully treated (by proving the Fr\'echet differentiability of the control-to-state operator and establishing
first-order necessary optimality conditions in terms of a variational inequality and the adjoint
state system)
for the case when in the state system \eqref{weak1}--\eqref{weak3} the nonlinearity
$f_1=I_{[-1,1]}$ is \juerg{for $\alpha>0$} replaced by $f_1=h^\alpha:=\varphi(\alpha)h$, with the functions
\begin{eqnarray}\label{defphi}
\hspace*{-10.5mm}\mbox{\bf (A3)} \,&&\phi\in C^1[0,+\infty) \,\mbox{ is strictly increasing and satisfies \,}
\lim_{\alpha\searrow0}\phi(\alpha)=0\pier{;} \hspace*{16mm}\\
\label{defh}
&&h(v)=\left\{
\begin{array}{ll}
(1+v)\ln(1+v)+(1-\juerg{v})\ln(1-v), \,\,\,&\pier{v\in(-1,1)}\\
2\ln(2),\,\,\,&v\in\{-1,1\}\\
+\infty, \,\,\,&v\not\in [-1,1]
\end{array}
\right. \pier{.}
\end{eqnarray}
\juerg{We obviously have that}
\begin{align}
\label{monoal}
&0\,\le\,h^{\alpha_1}(v)\,\le\,h^{\alpha_2}(v)\quad\forall\,v\in{\mathbb{R}}, \quad\mbox{if }\,0<\alpha_1<\alpha_2, \\[1mm]
\label{limh}
&\lim_{\alpha\searrow0} h^\alpha(v)\,=\,I_{[-1,1]}(v)\quad\forall\,v\in {\mathbb{R}}.
\end{align}
In addition, $\,h'(v)=\ln\left(\frac{1+v}{1-v}\right)$ \,and\, $h''(v)=\frac 2{1-v^2}>0$\, for
$v\in (-1,1)$, and thus, in particular,
\begin{align}
\label{limh1}
&\lim_{\alpha\searrow0}\,\phi(\alpha)h'(v)=0 \quad\mbox{for }\,-1<v<1,\\
\label{limh2}
&\lim_{\alpha\searrow0} \Bigl(\phi(\alpha)\,\lim_{v\searrow -1}h'(v)\Bigr)=-\infty, \quad
\lim_{\alpha\searrow0} \Bigl(\phi(\alpha)\,\lim_{v\nearrow +1}h'(v)\Bigr)=+\infty.
\end{align}
We may therefore regard the graphs of the single-valued functions
\begin{equation}
(h^\alpha)'(v)\,=\, \phi(\alpha)h'(v), \quad \mbox{for}\quad v\in (-1,1)\quad\mbox{and}\quad \alpha>0,
\end{equation}
as approximations to the graph of the multi-valued subdifferential $\dI$.
Now the
well-posedness results of \cite{CGS18,CGS19} apply,
yielding a solution pair $(\mu^{\alpha},y^{\alpha})$ for every $\alpha>0$, where the component $y^{\alpha}$ is uniquely
determined. It is a natural
question whether we have $y^\alpha\to y$ as $\alpha\searrow0$ in a suitable topology.
Below (cf.~Theorem 3.5), we will show that this is actually true; in Corollary 3.6, we will show that in a
very special case with some global constant $K_2>0$ a quantitative error estimate of the form
\begin{equation}
\|y^{\alpha}-y \|_{C^0([0,T];\Lx 2)\cap L^2(0,T;\Hx 1)}\,\le\,K_2\,|\alpha|^{1/2}
\end{equation}
is valid. Also, owing to the construction, the \pier{approximate functions} $\,y^\alpha\,$ automatically attain values in
the domain of $I_{[-1,1]}$; that is, we have $\|y^\alpha \|_{L^\infty(Q)}\,\le\,1$ \,for
all $\alpha>0$.
As far as the optimal control problem is concerned, the general strategy is then to derive
uniform (with respect to $\alpha\in (0,1]$) a priori estimates for the state and adjoint state variables of
\juerg{an ``adapted'' version of} (${\mathcal{CP}}_\alpha$)\ that are sufficiently strong as to permit a passage to the limit as $\alpha\searrow0$ in order to derive meaningful first-order necessary optimality conditions also for (${\mathcal{CP}}_0$).
We can follow this strategy in this paper, since in \cite{CGS19} a
corresponding theory for (${\mathcal{CP}}_\alpha$)\ with the logarithmic potential \eqref{logpot} has been developed.
However, while the approximation results for the solutions of the state system hold true under essentially the same
assumptions as those imposed in \cite{CGS18} for the well-posedness results, it seems impossible
to prove that the control-to-state operator is Fr\'echet differentiable
between suitable Banach spaces without having at disposal suitable uniform $L^\infty(Q)$ bounds for both the
state component $y$ and the functions $\,f^{(i)}(y)$,
for $i=1,2,3$. In the case of the logarithmic potentials \,$h^\alpha$, which we intend to use for the
deep quench approximation, this means that we need to separate $\,y^{\alpha}\,$ away
from the critical arguments $\pm 1$. Unfortunately, this postulate has the unpleasant consequence that
Fr\'echet differentiability (and thus satisfactory first-order necessary optimality conditions) can only
be established under rather restrictive conditions on the operators $A$ and $B$. A particular
case in which our analysis will work is given if $A=B=-\Delta$ with zero Neumann boundary condition,
$\sigma=1/2$, and $r\ge 3/8$.
Let us add a few remarks on the existing literature. \pier{One can find} numerous contributions on viscous/nonviscous,
local/nonlocal, convective/nonconvective Cahn--Hilliard systems for the classical (non-fractional) case
$A=B=-\Delta$, $2r=2\sigma=1$, where various types of boundary
conditions (e.g., Dirichlet, Neumann, dynamic) and different assumptions on the nonlinearity were
considered. We refer the
interested reader to the recent paper \cite{CGSAnnali} for a selection of associated references. Some papers
also address the coupled Cahn--Hilliard/Navier--Stokes system (see, e.g, \cite{FGG}, \cite{FGGS}, and the
references given therein).
The literature on optimal control problems for non-fractional Cahn--Hilliard systems is still rather scarce.
The case of Dirichlet and/or Neumann boundary conditions for various types of such systems were the
subject of, e.g., the works\cite{CGRS, CGSAIMS, CGSAMO, CGSEECT, Duan, WN, Z, ZW}, while the case of
dynamic boundary conditions was studied in\cite{CFGS1, CFGS2, CFGS3, CGSAdvan, CGSAMO, CGSSIAM18, CGSConvex,
CS, Fukao}. The optimal control of convective Cahn--Hilliard systems was addressed in\cite{CGSSIAM18,
CGSConvex,RS, ZL1, ZL2},
while the papers\cite{BDM1, BDM2, FGS, FRS, HHCK, HKW, HW1, HW2, HW3, Medjo} were concerned with
coupled Cahn--Hilliard/Navier--Stokes systems.
There are only a few contributions to the theory of Cahn--Hilliard systems involving fractional operators.
In the connection of well-posedness and regularity results, we refer to \cite{AM,AkSeSchi} for the case of
the fractional negative Laplacian with zero Dirichlet boundary conditions; general operators other than the
negative Laplacian
have apparently only been
studied in\gianni{\cite{CGS18,GalDCDS,GalEJAM,GalAIHP}}.
As of now, aspects of optimal control have been scarcely
dealt with even for simpler linear evolutionary systems involving fractional operators; for such systems,
some identification problems were addressed in the recent contributions \cite{GV, SV}, while
for optimal control problems for such cases we refer to~\cite{HAS} (for the stationary (elliptic) case, see
also \juerg{\cite{AKW,HO1, HO2,APR,AW1,AW2})}.
However, to the authors' best
knowledge, the present paper appears to be the first contribution that addresses optimal control problems
for Cahn--Hilliard systems with general fractional order operators and potentials of double obstacle
type.
The paper is organized as follows: the subsequent Section~2 brings some auxiliary functional analytic
material on fractional order operators, while in Section~3 we establish some general
convergence results for the deep quench
approximation of the state system \juerg{\eqref{weak1}--\eqref{weak3}}. In particular, an error
estimate is proved.
In Section~4, we investigate the relations between the solutions to the optimal control problems
(${\mathcal{P}}_0$) and the solutions to the corresponding optimal control problems for the deep
quench approximations. In the final Section~5, we then employ the results from \cite{CGS19}
to establish the first-order necessary optimality conditions
for (${\mathcal P}_0$).
Throughout the paper, we denote for a general Banach space $\,X\,$ other than $H=\Lx 2$ by
$\,\|\cdot\|_X\,$ and $\,X^*\,$ its norm
and dual space, respectively; the dual pairing between elements of $X^*$ and $X$ is denoted by $\langle\cdot,\cdot
\rangle_X$.
\section{Fractional powers and auxiliary results}
\label{FPAM}
\setcounter{equation}{0}
In this section, we collect some auxiliary material concerning functional analytic notions.
To this end, we generally assume that the conditions {\bf (A1)} and {\bf (A2)} are satisfied.
At this point, some remarks on the assumption {\bf (A2)} are in order.
\Begin{remark}\rm
The condition $\lambda_1>0$ is satisfied
for many standard elliptic operators of second or higher order with zero Dirichlet boundary conditions
(however, also zero mixed boundary conditions could be considered,
with proper definitions of the domains of the operators); typical cases are
the (negative) Laplacian \,$A=-\Delta$ with the domain $D(-\Delta)=\Hx 2\cap{H^1_0(\Omega)}$ or
the bi-harmonic operator $A=\Delta^2$ with the domain\,
$D(\Delta^2)=\Hx4\cap H^2_0(\Omega)$. On the other hand, we have $0=\lambda_1<\lambda_2$ and
$e_1\equiv {\rm const.}$ \,for important problems with zero Neumann boundary conditions;
typical examples are \,$A=-\Delta$\, with the domain $D(-\Delta)=\{v\in\Hx 2:\ \partial_{\bf n} v=0 \mbox{ \,on\, }\Gamma\}$\, and \,$\,A=\Delta^2$ with the domain
$D(\Delta^2)=\{v\in\Hx4:\ \partial_{\bf n} v=\partial_{\bf n}\Delta v=0\mbox{ \,on\, }\Gamma\}$. We also point out that
$A$ and $B$ can be completely unrelated if $\lambda_1>0$, while in the other case
the constant functions have to belong to $D(B^\sigma)$. The latter holds true if $B=-\Delta$
with the domain $D(-\Delta)=\{v\in\Hx 2:\ \partial_{\bf n} v=0 \mbox{ \,on\, }\Gamma\}$, while in the Dirichlet case
$D(-\Delta)=\Hx 2\cap H_0^1(\Omega)$ no nontrivial constant functions are contained in
$D(B)$; however, \gianni{if $0<\sigma<1/4$, \juerg{then} $D(B^\sigma)$~coincides with the usual Sobolev--Slobodeckij space $\Hx{2\sigma}$
and thus contains all constant functions}.
\End{remark}
Using the facts summarized in \eqref{eigen}--\eqref{complete},
we can define the powers of $A$ and $B$
for an arbitrary positive real exponent.
For the first operator, we have
\Begin{eqnarray}
&& \VA r := D(A^r)
= \Bigl\{ v\in H:\ \somma j1\infty |\lambda_j^r (v,e_j)|^2 < +\infty \Bigr\}
\quad\hbox{and}\quad
\label{defdomAr}
\\
&& A^r v = \somma j1\infty \lambda_j^r (v,e_j) e_j
\quad \hbox{for $v\in\VA r$},
\label{defAr}
\End{eqnarray}
the series being convergent in the strong topology of~$H$,
due to the properties \eqref{defdomAr} of the coefficients.
In principle, we can endow $\VA r$ with the (graph) norm and inner product
\Begin{equation}
\norma v_{gr,A,r}^2 := (v,v)_{gr,A,r}
\quad\hbox{and}\quad
(v,w)_{gr,A,r} := (v,w) + (A^r v , A^r w)
\quad \hbox{for $v,w\in\VA r$}.
\label{defnormagrAr}
\End{equation}
This makes $\VA r$ a Hilbert space.
However, we can choose any equivalent Hilbert norm.
Indeed, in view of assumption {\bf (A2)}, it is more convenient to work with the Hilbert norm
\Begin{equation}
\norma v_{A,r}^2 := \left\{
\begin{aligned}
& \norma{A^r v}^2
= \somma j1\infty |\lambda_j^r (v,e_j)|^2
\qquad \hbox{if $\lambda_1>0$,}
\\
& |(v,e_1)|^2 + \norma{A^r v}^2
= |(v,e_1)|^2 + \somma j2\infty |\lambda_j^r (v,e_j)|^2
\qquad \hbox{if $\lambda_1=0$}.
\end{aligned}
\right.
\label{defnormaAr}
\End{equation}
In \cite[Prop.~3.1]{CGS18} it has been shown that this norm is equivalent
to the graph norm defined in~\eqref{defnormagrAr},
and we always will work with the norm \eqref{defnormaAr} instead of with \eqref{defnormagrAr}.
We also use the corresponding inner product in $\VA r$
given~by
\Begin{eqnarray}
&& (v,w)_{A,r}
= (A^r v,A^r w)
\quad \hbox{or} \quad
(v,w)_{A,r}
= (v,e_1)(w,e_1) + (A^r v,A^r w),
\qquad
\non
\\
\label{inpro}
&& \hbox{depending on whether $\lambda_1>0$ or $\lambda_1=0$,\quad
for $v,w\in\VA r$.}
\End{eqnarray}
\Begin{remark}\rm
Observe that in the case $\lambda_1=0$
the constant value of $e_1$
equals one of the numbers $\pm|\Omega|^{-1/2}$,
where $|\Omega|$ is the volume of~$\Omega$.
It follows for every $v\in H$ that the first term $(v,e_1)e_1$ of the Fourier series of $v$
is the constant function whose value~is the mean value of~$v$, which is defined by
\Begin{equation}
\mean (v) := \frac 1 {|\Omega|} \int_\Omega v\,.
\label{defmean}
\End{equation}
\End{remark}
In the same way as for $A$, starting from \accorpa{eigen}{complete} for~$B$,
we can define the power $B^\sigma$ of $B$ for every $\sigma>0$, where for
$V_B^\sigma$ we choose the graph norm.
We therefore set
\Begin{eqnarray}
&& \VB\sigma := D(B^\sigma),
\quad \hbox{with the norm $\norma\,\cdot\,_{B,\sigma}$ associated to the inner product}
\label{defBs}
\non
\\
&& (v,w)_{B,\sigma} := (v,w) + (B^\sigma v,B^\sigma w)
\quad \hbox{for $v,w\in \VB\sigma$}.
\label{defprodBs}
\End{eqnarray}
To resume our preparations, we observe that if $r_i$ and $\sigma_i$ are arbitrary positive exponents,
then it is easily seen that we have the ``Green type'' formulas
\Begin{eqnarray}
&& (A^{r_1+r_2} v,w)
= (A^{r_1} v, A^{r_2} w)
\quad \hbox{for every $v\in\VA{r_1+r_2}$ and $w\in\VA{r_2}$},
\label{propA}
\\[1mm]
&& (B^{\sigma_1+\sigma_2} v,w)
= (B^{\sigma_1} v, B^{\sigma_2} w)
\quad \hbox{for every $v\in\VB{\sigma_1+\sigma_2}$ and $w\in\VB{\sigma_2}$}.
\label{propB}
\End{eqnarray}
The next step is the introduction of some spaces with negative exponents.
We set
\Begin{equation}
\VA{-r} := (\VA r)^*
\quad \hbox{for $r>0$},
\label{defVAneg}
\End{equation}
and endow $\VA{-r}$ with the dual norm $\,\|\cdot\|_{A,-r}\,$ of $\,\|\cdot\|_{A,r}$.
We use the symbol $\langle\,\cdot\,,\,\cdot\,\rangle_{A,r}$ for the duality pairing
between $\VA{-r}$ and~$\VA r$ and identify $H$ with a subspace of $\VA{-r}$
in the usual way, i.e., such that
$\,\langle v,w \rangle_{A,r} = (v,w)\,$ for every $v\in H$ and $w\in\VA r$. Likewise,
we~set
\Begin{equation}
V_B^{-\sigma}:=(V_B^\sigma)^* \quad\mbox{for }\,\sigma>0.
\End{equation}
As $\,V_B^\sigma\,$ is dense in $\,H$, we have the analogous embedding
\Begin{equation}
H\subset V_B^{-\sigma}.
\End{equation}
Observe that the following embedding results are valid:
\begin{align}
& \hbox{The embeddings $\VA{r_2} \subset \VA{r_1} \subset H$ are dense and compact for $0<r_1<r_2$}.
\label{compembA}
\\
& \hbox{The embeddings $H \subset \VA{-r_1} \subset \VA{-r_2}$ are dense and compact for $0<r_1<r_2$}.
\qquad
\label{compembAneg}
\\
& \hbox{The embeddings $\VB{\sigma_2} \subset \VB{\sigma_1} \subset H$ are dense and compact for $0<\sigma_1<\sigma_2$}.
\label{compembB}
\end{align}
At this point, we introduce the Riesz isomorphism $\calR_r:\VA r\to\VA{-r}$
associated with the inner product~\eqref{inpro}, which is given by
\Begin{equation}
\< \calR_r v , w >_{A,r}
= (v,w)_{A,r}
\quad \hbox{for every $v,w\in\VA r$}.
\label{riesz}
\End{equation}
Moreover, we~set
\begin{align}
& \Vz r := \VA r
\quad\hbox{and}\quad
\Vz{-r} := \VA{-r}
\quad \hbox{if $\lambda_1>0$},
\non
\\[1mm]
& \Vz r := \{v\in \VA r :\ \mean (v)=0\}
\quad\hbox{and}\quad
\Vz{-r} := \{v \in \VA{-r} :\ \<v,1>_{A,r}=0 \}
\quad \hbox{if $\lambda_1=0$} \,.
\label{defVrpos}
\end{align}
According to \cite[Prop.~3.2]{CGS18}, $\calR_r$ maps $\Vz r$ onto $\Vz{-r}$
and extends to $\Vz r$ the restriction of $A^{2r}$ to $\Vz{2r}$.
In view of this result, it is reasonable to use a proper notation
for the restrictions of $\calR_r$ and $\calR_r^{-1}$ to the subspaces
$\Vz r$ and $\Vz{-r}$, respectively.
We~set
\Begin{equation}
\Az{2r} := (\calR_r)_{|\Vz r}
\quad\hbox{and}\quad
\Az{-2r} := (\calR_r^{-1})_{|\Vz{-r}}\,,
\label{defAz}
\End{equation}
where the index $0$ has no meaning if $\lambda_1>0$ (since then $\Vz{\pm r}=\VA{\pm r}$),
while it reflects the zero mean value condition in the case $\lambda_1=0$.
We thus have
\Begin{eqnarray}
&& \Az{2r} \in \calL(\Vz r,\Vz{-r}) , \quad
\Az{-2r} \in \calL(\Vz{-r},\Vz r)
\quad\hbox{and}\quad
\Az{-2r} = (\Az{2r})^{-1}\,,
\label{contlinAz}
\\[1mm]
&& \< \Az{2r} v,w >_{A,r} = (v,w)_{A,r} = (A^r v,A^r w)
\quad \hbox{for every $v\in\Vz r$ and $w\in\VA r$}\,,
\qquad
\label{identityAz}
\\[1mm]
&& \< f , \Az{-2r} f >_{A,r}
= \norma{\Az{-2r} f}_{A,r}^2
= \norma f_{A,-r}^2
\quad \hbox{for every $f\in\Vz{-r}$}.
\label{normaAz}
\End{eqnarray}
\section{Deep quench approximation of the state system}
\setcounter{equation}{0}
In this section, we state our general assumptions and discuss the deep quench approximation
of the state system \juerg{\eqref{weak1}--\eqref{weak3}}.
Besides \gianni{{\bf (A1)}--{\bf (A3)}}, we generally assume for \gianni{the structure and} the data of the state system:
\vspace{2mm}\noindent
{\bf (A4)} \,\,$r>0$, $\sigma>0$, and $\tau\ge 0$ are fixed real numbers.
\vspace{2mm}\noindent
{\bf (A5)} \,\,$f_2\in C^3({\mathbb{R}})$, and $\,f_2'\,$ is Lipschitz continuous on ${\mathbb{R}}$ with Lipschitz constant
$L>0$.
\vspace{2mm}\noindent
{\bf (A6)} \,\,\pier{$y_0\in V^{\sigma}_B$}, \, and \, $-1<\mathop{\rm inf\,ess}_{x\in\Omega}y_0(x),
\quad \mathop{\rm sup\,ess}_{x\in\Omega}y_0(x)<+1$.
\vspace{2mm}\noindent
{\bf (A7)} \,\,$u\in {\cal X}\,:= \,H^1(0,T;\Lx 2)\cap L^\infty(Q)$.
\vspace{2mm}\noindent
We draw a few consequences from {\bf (A6)}. Namely, the mean value of $y_0$ belongs to the interior
of both $D(\dI)$ and $D((h^\alpha)')$, for all $\alpha>0$. Moreover, we have $I_{[-1,1]}(y_0)\in\Lx 1$ and
$h(y_0)\in\Lx 1$, and
$h'(y_0)$ belongs to $\Lx 2$. Thus, the conditions \cite[(2.27), (2.28)]{CGS18}
on $y_0$ for the application of \cite[Thm.~2.6]{CGS18}
are satisfied, where we note that
\begin{equation}
\label{estini}
\|h^\alpha(y_0)\|_{\Lx 1}\,+\,\|(h^\alpha)'(y_0)\|_{\Lx 2}\,\le\,\hat c \quad\forall\,\alpha\in (0,1],
\end{equation}
with some constant $\hat c>0$ which is independent of $\alpha\in (0,1]$.
\vspace{2mm
We now consider the state system \eqref{weak1}--\eqref{weak3} for the cases $f_1=I_{[-1,1]}$ and $f_1=h^\alpha$ ($\alpha
\in (0,1]$), respectively. By virtue of \cite[Thm.~2.6]{CGS18}, there exist solution pairs \,$(\mu,y)$\, and
$\,(\mu^{\alpha},y^{\alpha})$, respectively, which enjoy the properties \eqref{regy}--\eqref{L1}, and the
(uniquely determined) second components satisfy
\begin{equation}
\label{interv}
-1\le y\le 1\, \mbox{ a.e. in $\,Q$,}\quad -1\le y^{\alpha}\le 1\,\mbox{ a.e. in \,$Q$.}
\end{equation}
We are now going to investigate the behavior of the family
$\{(\mu^{\alpha},y^{\alpha})\}_{\alpha>0}$ of deep quench approximations for $\alpha\searrow0$. We begin our analysis with the derivation of
general a priori estimates.
\Begin{theorem}
Suppose that the general assumptions {\bf (A1)}--{\bf \gianni{(A7)}} are
fulfilled, and assume that $(\mu^{\alpha},y^{\alpha})$ are solution pairs to the problem \eqref{weak1}--\eqref{weak3} with
$f_1=h^\alpha$ for $\alpha\in (0,1]$ as established in {\rm \cite[Thm.~2.6]{CGS18}}.
Then there exists a constant $K_1>0$, which only depends on the data of the system
\eqref{weak1}--\eqref{weak3}, such that
\begin{align}
\label{albound1}
&\|\mu^{\alpha}\|_{L^2(0,T;V_A^{r})}\,+\,\|y^{\alpha}\|_{H^1(0,T;V_A^{-r})\cap L^\infty(0,T;V_B^\sigma)\cap
L^\infty(Q)}
\,+\,\|\phi(\alpha)h(y^{\alpha})\|_{L^\infty(0,T;\Lx 1)}\nonumber\\
&+\,\|\tau^{1/2}\partial_ty^{\alpha}\|_{L^2(0,T;H)}\,\le\,K_1\quad\forall\,\alpha\in (0,1].
\end{align}
If, in addition,
\begin{equation}
\label{adco}
\tau>0 \quad\mbox{and}\quad y_0\in V_B^{2\sigma},
\end{equation}
then we have the additional bounds
\begin{align}
\label{albound2}
&\|y^{\alpha}\|_{W^{1,\infty}(0,T;H)\cap H^1(0,T;V_B^\sigma)}\,+\,\|\mu^{\alpha}\|_{L^\infty(0,T;V_A^{2r})}
\nonumber\\
&+\int_0^T\!\!\int_\Omega\phi(\alpha)h''(y^{\alpha})\left|\partial_ty^{\alpha}\right|^2\,\le\,K_1
\quad\forall\,\alpha\in (0,1].
\end{align}
\End{theorem}
\Begin{proof}
To establish the validity of \eqref{albound1}, we have to follow the lines of the proof of \cite[Thm.~2.6]{CGS18}.
The method of proof of \cite[Thm.~2.6]{CGS18}, specified to our situation where the convex
part of the nonlinearity is given by $h^\alpha$, was the following:
\noindent {\sc Step 1:} Replace in \eqref{weak2} the function $\,f_1=h^\alpha$\, by its Moreau--Yosida
approximation~$\,h^\alpha_\lambda$, where $\lambda>0$.\\[0.5mm]
{\sc Step 2:} Approximate the resulting system of variational inequalities (which on the level of
Moreau--Yosida approximations become variational equalities) via time discretization.\\[0.5mm]
{\sc Step 3:} Show unique solvability for the discrete system and derive a priori estimates for
the discrete approximations.\\[0.5mm]
{\sc Step 4:} Take the time step-size to zero in the time-discrete system to establish
unique solvability of the system governing the Moreau--Yosida approximations.\\[0.5mm]
{\sc Step 5:} Take the limit as $\lambda\searrow0$ to obtain the solvability of the system \eqref{weak1}--\eqref{weak3}
for $f_1=h^\alpha$.
\\[0.5mm]
{\sc Step 6:} Show the uniqueness of the second solution component $y^{\alpha}$.
Now, a closer inspection reveals that in our case all of the bounds established in the a priori estimates
performed in {\sc Step 3} are uniform with respect to $\alpha\in (0,1]$, and due to the semicontinuity properties
of norms, they persist under the limit processes as the step-size of the time discretization and
the Moreau--Yosida parameter $\lambda$ approach zero. The validity of the estimate \eqref{albound1} is thus a consequence
of the estimate \cite[Eq.~(6.1)]{CGS18}.
To \pier{offer to} the reader a little flavor of the argument, we give a formal derivation of
a part of \eqref{albound1} (which
becomes rigorous on the level of the time-discrete approximation). To this end, let us assume that $\partial_ty^{\alpha}\in
L^2(0,T;V_B^\sigma)$\, (which is satisfied under the assumption \eqref{adco}) and that the variational inequality
\eqref{weak2} with $f_1=h^\alpha$ is equivalent to the variational equation
\begin{align}
\label{al2new}
&(\tau\partial_t y^{\alpha}(t),v)\,+\,(B^\sigma y^{\alpha}(t),B^\sigma v)\,+\,((h^\alpha)'(y^{\alpha}(t))+f_2'(y^{\alpha}(t)),v)\,=\,
(\mu^{\alpha}(t)+u(t),v)\nonumber\\[1mm]
&\mbox{for every \,$v\in V_B^\sigma\,$ and a.e. \,$t\in (0,T)$}.
\end{align}
The latter is certainly satisfied on the level of the Moreau--Yosida approximations to the deep
quench approximations $(\mu^{\alpha},y^{\alpha})$. We then insert $v=\mu^{\alpha}(t)$ in \eqref{weak1} (written for
$(\mu,y)=(\mu^{\alpha},y^{\alpha})$) and $v=\partial_ty^{\alpha}(t)$ in \eqref{al2new},
add the resulting equations, and integrate with respect to time over $[0,t]$, where $t\in (0,T]$ is arbitrary.
It then follows after an obvious cancellation of terms that
\begin{align*}
&\tau\int_0^t\!\!\int_\Omega|\partial_ty^{\alpha}|^2\,+\,\frac 12\,\|B^\sigmay^{\alpha}(t)\|^2\,+\int_0^t\!\|A^r\mu^{\alpha}(s)\|^2\,ds
\,+\int_\Omega h^\alpha(y^{\alpha}(t))\\
&=\,\frac 12\,\|B^\sigma y_0\|^2\,+\int_\Omega(h^\alpha(y_0)-f_2(y^{\alpha}(t))
+f_2(y_0))\,+\int_0^t\!\!\int_\Omega u\,y^{\alpha}.
\end{align*}
Now we recall \eqref{estini} and the fact that $y_0\in V_B^\sigma$ (cf.~{\bf \gianni{(A6)}}).
We thus can infer from \eqref{interv}, {\bf \gianni{(A5)}}, and {\bf \gianni{(A7)}},
that all of the terms on the right-hand side\ are bounded independently of $\alpha\in (0,1]$
by a constant that depends in a continuous and monotone way on $\|u\|_{L^1(Q)}$. But this means
that
\begin{align*}
&\|\pier{A^r\mu^{\alpha}}\|_{L^2(0,T;H)}+\|y^{\alpha}\|_{L^\infty(0,T;V_B^\sigma)\cap L^\infty(Q)}
+\|h^\alpha(y^{\alpha})\|_{\juerg{L^\infty(0,T;}\Lx 1)}\\
&+\|\tau^{1/2}\partial_ty^{\alpha}\|_{L^2(0,T;H)}\,\le\,
C,
\end{align*}
where $C>0$ is independent of $\alpha\in (0,1]$. This is already a part of the asserted bound
\eqref{albound1}. Now, if $\lambda_1>0$, then \eqref{defnormaAr} and the above estimate
immediately entail that $\{\mu^{\alpha}\}_{\alpha\in(0,1]}$ is bounded in $L^2(0,T;V_A^r)$, and
comparison in \eqref{Iprima} yields a uniform bound for $\{\partial_ty^{\alpha}\}_{\alpha\in (0,1]}$
in $L^2(0,T;V_A^{-r})$, which then shows that \eqref{albound1} is valid. In the case
when $\lambda_1=0$, the boundednes of $\{\mu^{\alpha}\}_{\alpha\in(0,1]}$ in $L^2(0,T;V_A^r)$
is shown by proving that the mean values of $\{\mu^{\alpha}(t)\}_{\alpha\in(0,1]}$ are
uniformly bounded in $L^2(0,T)$.
For this argument, we refer the reader to the proof of \cite[Thm.~2.6]{CGS18}.
Assume now that also the condition \eqref{adco} is fulfilled.
In order to prove the bounds \eqref{albound2}, we follow the proof
of \cite[Thm.~2.8]{CGS18}, which again uses the time-discrete approximation scheme for the
system governing the Moreau--Yosida approximations mentioned above in describing {\sc Step 3} in the proof
of \cite[Thm.~2.6]{CGS18}. At this point, we recall the estimate \eqref{estini}. With this estimate in mind,
it turns out that all of the estimates performed in the proof
of \cite[Thm.~2.8]{CGS18} on the discrete approximations yield bounds that do not depend on $\alpha\in (0,1]$
and persist under the limit processes of taking the time step-size and the Moreau--Yosida parameter
$\lambda$ to zero. Since \eqref{albound2} exactly reflects the bounds established there,
the assertion is proved.
For the reader's convenience, we again provide a formal sketch of the argument. To this end, we formally
differentiate \eqref{al2new} with respect to
$t$, obtaining the identity
\begin{eqnarray}
\label{diffi1}
&&(\tau\partial^2_{tt}y^{\alpha},v)+(B^\sigma\partial_ty^{\alpha},B^\sigma v)+ (\phi(\alpha)h''(y^{\alpha})\partial_ty^{\alpha}+f_2''(y^{\alpha})\partial_ty^{\alpha},v)
=(\partial_t\mu^{\alpha}+\partial_t u,v)\non\\
&&\mbox{for every $\,v\in V_B^\sigma\,$ and a.e. in $\,(0,T)$}.
\end{eqnarray}
Then we formally test \eqref{weak1} by $v=\partial_t\mu^{\alpha}$ and \eqref{diffi1} by $v=\partial_ty^{\alpha}$, and add the
resulting identities. After an obvious cancellation \juerg{of terms}, we arrive at
\begin{align}
\label{diffi2}
&\frac \tau 2\,\|\partial_ty^{\alpha}(t)\|^2\,+\,\frac 12\,\|A^r\mu^{\alpha}(t)\|^2\,+\int_0^t\!\!\int_\Omega |B^\sigma\partial_ty^{\alpha}|^2
\,+\int_0^t\!\!\phi(\alpha)h''(y^{\alpha})|\partial_ty^{\alpha}|^2 \nonumber\\
&=\,\frac \tau 2\,\|\partial_ty^{\alpha}(0)\|^2\,+\,\frac 12\,\|A^r\mu^{\alpha}(0)\|^2
\,-\int_0^t\!\!f_2''(y^{\alpha})|\partial_ty^{\alpha}|^2\,+\int_0^t\!\!\int_\Omega \partial_t u\,\partial_ty^{\alpha}\,,
\end{align}
where the last summand on the left-hand side is nonnegative and the last two terms on the right-hand
side can be estimated by an expression of the form
$$C_1\int_0^t\!\!\int_\Omega\left(|\partial_t u|^2\,+\,|\partial_ty^{\alpha}|^2\right)\,,$$
where $C_1>0$ is independent of $\alpha\in (0,1]$. We thus are left to estimate the initial value terms.
To this end, we formally write \eqref{weak1} and \eqref{al2new} for $t=0$, obtaining the
identities
\begin{align}
\label{diffi3}
&\langle\partial_ty^{\alpha}(0),v\rangle_{A,r}+(A^r\mu^{\alpha}(0),A^r v)=0 \quad\forall\,v\in V_A^r,\\[0.5mm]
\label{diffi4}
&(\tau\partial_ty^{\alpha}(0),v)+(B^{2\sigma} y_0+(h^\alpha)'(y_0)+f_2'(y_0),v)\,=\,(\mu^{\alpha}(0)+u(0),v)
\quad\forall\,v\in V_B^\sigma.
\end{align}
Now observe that, by virtue of \eqref{adco}, \eqref{estini}, and {\bf \gianni{(A5)}},
the sum $\,B^{2\sigma} y_0+ (h^\alpha)'(y_0)+f_2'(y_0)\,$ is bounded in $\Lx 2$,
uniformly with respect to $\alpha\in (0,1]$.
Hence, if we (formally) test \eqref{diffi3} by $\mu^{\alpha}(0)$ and \eqref{diffi4} by $\partial_ty^{\alpha}(0)$, add
the resulting identities, and apply Young's inequality (note that we have $\tau>0$ by
assumption \eqref{adco}), then we arrive at an estimate of the form
$$
\|A^r\mu^{\alpha}(0)\|^2\,+\,\frac\tau 2\,\|\partial_ty^{\alpha}(0)\|^2\,\le\, C_2\,\tau^{-1}(1+\|u(0)\|^2),
$$
where $C_2>0$ is independent of $\alpha\in (0,1]$. We may then combine this estimate with
\eqref{diffi2} to conclude from Gronwall's lemma that
\begin{align}
\|\partial_ty^{\alpha}\|_{L^\infty(0,T;H)\cap L^2(0,T;V_B^\sigma)}\,+\,\|A^r\mu^{\alpha}\|_{L^\infty(0,T;H)}
\,+\int_0^T\!\!\int_\Omega\phi(\alpha)h''(y^{\alpha})\left|\partial_ty^{\alpha}\right|^2\,\le\,C_3,
\end{align}
where $C_3>0$ is independent of $\alpha\in (0,1]$. With this, the first and third summands on
the left of \eqref{albound2} are uniformly bounded, which then, by comparison in \eqref{Iprima},
also holds true for $\,\|A^{2r}\mu^{\alpha}\|_{L^\infty(0,T;H)}$. Hence, \eqref{albound2}
is proved if $\lambda_1>0$. In the case $\lambda_1=0$, it is necessary to derive
a uniform $L^\infty(0,T)$ bound for the mean values of $\,\{\mu^{\alpha}(t)\}_{\alpha\in (0,1]}$.
\pier{About} this, we again refer the reader to the proof of \cite[Thm.~2.8]{CGS18}.
\End{proof}
\Begin{remark}\rm
A closer inspection of the a priori estimates for the time-discretized
systems mentioned above \juerg{reveals} that the constant $K_1$ depends in a monotone and continuous way
on the norm $\|u\|_{\cal X}$. Hence, for any bounded subset
${\cal U}$ of ${\cal X}$ (in particular, for ${\cal U}={\mathcal{U}}_{\rm ad}$) it follows that there is a constant, which is
again denoted by $K_1$, such that the estimates \eqref{albound1} and \eqref{albound2}, respectively, hold true whenever $u$ is an arbitrary element of ${\cal U}$.
\End{remark}
\vspace{2mm} Next, we show the convergence of the deep quench approximations. Before formulating the result, we notice
that the following control-to-state operators are well defined on the space ${\cal X}$:
\begin{align}
\label{defcs0}
\mathcal{S}_0&:\,{\cal X}\ni u\mapsto \mathcal{S}_0(u):= y,\\
\label{defcsal}
\mathcal{S}_{\alpha}&:\,{\cal X}\ni u\mapsto \mathcal{S}_{\alpha}(u):=y^{\alpha},
\end{align}
where $(\mu,y)$ and $(\mu^{\alpha},y^{\alpha})$ denote solutions to the systems \eqref{weak1}--\eqref{weak3} for $f_1=I_{[-1,1]}$ and
$f_1=h^\alpha$, $\alpha\in (0,1]$, respectively, as established in \cite[Thm.~2.6]{CGS18}. We have the following result.
\Begin{theorem}
Suppose that the assumptions {\bf (A1)}--{\bf \gianni{(A7)}} are fulfilled, and let sequences $\{\alpha_n\}\subset (0,1]$ and
$\{u^{\alpha_n}\}\subset{\cal X}$ be given such that $\alpha_n\searrow0$ and $u^{\alpha_n}\to u$ weakly-star in ${\cal X}$ as $n\to\infty$
for some $u\in{\cal X}$. Moreover, let $(\mu^{\alpha_n},y^{\alpha_n})$
be solutions to \eqref{weak1}--\eqref{weak3} for $f_1=h^{\alpha_n}$ and $u=u_n$, $n\in{\mathbb{N}}$, as established in
{\rm \cite[Thm.~2.6]{CGS18}}. Then there are a solution $(\mu,y)$ with $y=\mathcal{S}_0(u)$ to the problem
\eqref{weak1}--\eqref{weak3} with $f_1=I_{[-1,1]}$ and a subsequence $\{\alpha_{n_k}\}_{k\in{\mathbb{N}}}$ of
$\{\alpha_n\}$ such that,
as $k\to\infty$,
\begin{align}
\label{conmu}
\mu^{\alpha_{n_k}}\to\mu&\quad\mbox{weakly in }\,L^2(0,T;V_A^r),\\
\label{cony}
y^{\alpha_{n_k}}\to y&\quad\mbox{weakly-star in }\,H^1(0,T;V_A^{-r})\cap L^\infty(0,T;V_B^\sigma)\nonumber\\
&\quad\mbox{and strongly in }\,C^0([0,T];H), \\
\label{conyt}
\partial_t y^{\alpha_{n_k}} \to \partial_t y&\quad\mbox{weakly in }\,L^2(0,T;H) \quad\mbox{if }\,\tau>0.
\end{align}
Moreover, if \eqref{adco} is fulfilled, then the above solution $(\mu,y)$ also satisfies
\begin{align}
\label{conmup}
\mu^{\alpha_{n_k}}\to\mu&\quad\mbox{weakly-star in }\,L^\infty(0,T;V_A^{2r}),\\
\label{conyp}
y^{\alpha_{n_k}}\to y&\quad\mbox{weakly-star in }\,W^{1,\infty}(0,T;H)\cap H^1(0,T;V_B^\sigma).
\end{align}
\End{theorem}
\Begin{proof}
The sequence $\{u^{\alpha_n}\}$ converges weakly-star in ${\cal X}$ and thus forms a bounded subset of ${\cal X}$.
According to Remark~3.2, the bounds \eqref{albound1} and \eqref{albound2} (the latter if \eqref{adco}
is satisfied) apply, where the constant $K_1$ is independent of $n$. Therefore, there are limits $(\bar \mu,\bar y)$ and a subsequence of $\{(\mu^{\alpha_n},y^{\alpha_n})\}$,
which is for convenience again indexed by $n$, such that, as~$n\to\infty$,
\begin{align}
\label{p1}
\mu^{\alpha_n}\to\bar \mu&\quad\mbox{weakly in }\,L^2(0,T;V_A^r),\\
\label{p2}
y^{\alpha_n}\to\bar y&\quad\mbox{weakly-star in }\,H^1(0,T;V_A^{-r})\cap L^\infty(0,T;V_B^\sigma),\\
\label{p2bis}
\pier{y^{\alpha_n}\to\bar y}
&\quad\mbox{strongly in }\,C^0([0,T];H) \, \mbox{ and pointwise a.e. in }\,Q,\\
\label{p3}
\partial_t y^{\alpha_n} \to \partial_t \bar y&\quad\mbox{weakly in }\,L^2(0,T;H) \quad\mbox{if }\,\tau>0,
\end{align}
and, if \eqref{adco} is satisfied,
\begin{align}
\label{p4}
\mu^{\alpha_n}\to\bar \mu&\quad\mbox{weakly-star in }\,L^\infty(0,T;V_A^{2r}),\\
\label{p5}
y^{\alpha_n}\to \bar y&\quad\mbox{weakly-star in }\,W^{1,\infty}(0,T;H)\cap H^1(0,T;V_B^\sigma).
\end{align}
Notice that the strong convergence result in \eqref{p2bis} follows from \cite[Sect.~8,~Cor.~4]{Simon}, since,
by \eqref{compembB}, $V_B^\sigma$ is compactly emdedded in $H$; we thus may without loss of generality
assume that $y^{\alpha_n}\to\bar y$ pointwise a.e. in $Q$. Since, by virtue of \eqref{interv}, $-1\ley^{\alpha_n}\le+1$ a.e. in $Q$,
we infer that $-1\le\bar y\le 1$ a.e. in $Q$, and thus $I_{[-1,1]}(\bar y)\in L^1(Q)$ with
$$\int_0^T\!\!\int_\OmegaI_{[-1,1]}(\bar y)=0.$$
It remains to show that $(\bar \mu,\bar y)$ is a solution to \eqref{weak1}--\eqref{weak3} in the sense of
\cite[Thm.~2.6]{CGS18} for $f_1=I_{[-1,1]}$ and control $\,u$. To this end, we pass to the limit as $n\to\infty$ in the system \eqref{weak1}--\eqref{weak3},
written for $f_1=h^{\alpha_n}$ and $u=u_n$, for $n\in{\mathbb{N}}$. We immediately see that
$\bar y(0)=y_0$ and that \eqref{weak1} holds true for $(\bar \mu,\bar y)$. Also, the Lipschitz continuity of $f_2'$
and \eqref{p2bis} imply that $\,f_2'(y^{\alpha_n})\to f_2'(\bar y)\,$ strongly in \,$C^0([0,T];H)$.
Now, recall that $B^\sigma y^{\alpha_n}\to B^\sigma\bar y$ weakly in $L^2(0,T;H)$, by virtue of \eqref{p2}. We thus have, by
\pier{lower} semicontinuity,
\begin{align}
&\int_0^T\left(B^\sigma\bar y(t),B^\sigma(\bar y(t)-v(t))\right)\, dt
\non
\\
& \leq \liminf_{n\to\infty} \int_0^T \bigl( B^\sigma y^{\alpha_n}(t) , B^\sigma y^{\alpha_n}(t) \bigr) \, dt
- \lim_{n\to\infty} \int_0^T \bigl( B^\sigma y^{\alpha_n}(t) , B^\sigma v(t) \bigr) \, dt
\non
\\
& = \liminf_{n\to\infty} \int_0^T \bigl( B^\sigma y^{\alpha_n}(t) , B^\sigma (y^{\alpha_n}(t) - v(t)) \bigr) \, dt
\non
\end{align}
for every $v\in\L2{\VB\sigma}$.
In conclusion, \gianni{owing to~\eqref{limh} as well}, we have that
\begin{align}
& \int_Q I_{[-1,1]}(\bar y)
+ \int_0^T \bigl( B^\sigma \bar y(t) , B^\sigma (\bar y(t) - v(t)) \bigr) \, dt
= \int_0^T \bigl( B^\sigma \bar y(t) , B^\sigma (\bar y(t) - v(t)) \bigr) \, dt
\non
\\
& \leq \,\liminf_{n\to\infty} \int_Q h^{\alpha_n}(y^{\alpha_n})
+ \liminf_{n\to\infty} \int_0^T \bigl( B^\sigma y^{\alpha_n}(t) , B^\sigma (y^{\alpha_n}(t) - v(t)) \bigr) \, dt
\non
\\
& \leq \,\liminf_{n\to\infty} \Bigl(
\int_Q h^{\alpha_n}(y^{\alpha_n})
+ \int_0^T \bigl( B^\sigma y^{\alpha_n}(t) , B^\sigma (y^{\alpha_n}(t) - v(t)) \bigr) \, dt
\Bigr)
\non
\\
& \leq \,\lim_{n\to\infty} \,\Bigl( \int_0^T \bigl(
- \tau \partial_t y^{\alpha_n}(t) - f_2'(y^{\alpha_n}(t)) + u(t) + \mu^{\alpha_n}(t) , y^{\alpha_n}(t) - v(t)
\bigr) \, dt
+ \int_Q h^{\alpha_n}(v)\Bigr)
\non
\\
& = \,\int_0^T \bigl(-\tau \partial_t \bar y(t)
-f_2'(\bar y(t)) + u(t) {\,+\,} \bar \mu(t) , \bar y(t)-v(t) \bigr) \, dt
+ \int_Q I_{[-1,1]}(v),
\non
\end{align}
for all $v\in L^2(0,T;V_B^\sigma)$. Thus the time-integrated version of \eqref{weak2}, with time-dependent
test functions, holds true. Since this version is equivalent to \eqref{weak2},
we see that $(\bar \mu,\bar y)$ is indeed a solution in the sense of \cite[Thm.~2.6]{CGS18} to \eqref{weak1}--\eqref{weak3} for $f_1=I_{[-1,1]}$. The assertion is thus proved.
\End{proof}
\Begin{remark}\rm
According to \cite[Thm.~2.6]{CGS18}, the second solution component $y$ and the expression $A^r\mu$ are uniquely determined.
This entails that $\bar y=\mathcal{S}_0(u)$ and that the convergence properties \pier{\eqref{p2}, \eqref{p3}} and \eqref{p5} are valid for the
entire sequence $\{\alpha_n\}$ and not only for a subsequence. In addition, we can infer from \eqref{p1} that
$A^r\mu^{\alpha_n} \to A^r\bar \mu$ weakly in $L^2(0,T;H)$ as $n\to\infty$. If \,$\lambda_1>0$\pier{, then even $\mu^{\alpha_n}$ converges to} $\bar \mu$ weakly in $L^2(0,T;V_A^r)$.
\End{remark}
In the following theorem, we prove a quantitative estimate that yields information on the order of convergence as
$\alpha\searrow0$ in a very special (but important) situation. To this end, we need further assumptions that will
also be needed in the derivation of first-order necessary optimality conditions in Section~5.
\Begin{theorem}
Suppose that in addition to \pier{{\bf (A1)}--{\bf (A6)}} the following assumptions are fulfilled:
\noindent
{\bf \gianni{(A8)}} \,\,\,The condition \eqref{adco} is satisfied.
\noindent
{\bf \gianni{(A9)}} \,\,\,$B=-\Delta$ with the domain $D(B)=\{v\in\Hx 2: \partial_{\bf n} v=0 \,\mbox{ on }\,\Gamma\}$, \,$\sigma=\frac 12$,
and \hspace*{12mm} $\,V_A^{2r}\subset L^\infty(\Omega)$.
\noindent
Moreover, assume that \,$u^{\alpha_1}, u^{\alpha_2}\in{\cal X}$ are given,
where \,$0<\alpha_1<\alpha_2<1$, and that $(\mu^{\alpha_i},y^{\alpha_i})$ are solutions to \eqref{weak1}--\eqref{weak3}
for $f_1=h^{\alpha_i}$ and $u=u_i$ in the sense of {\rm \cite[Thm.~2.6]{CGS18}}, for
$i=1,2$.
Then there is a constant $K_2>0$, which
depends only on the data of the problem, such that it holds, for all $t\in (0,T]$,
\begin{align}
\label{esti1}
&\|y^{\alpha_1}-y^{\alpha_2}\|_{C^0([0,t];\Lx 2)\cap L^2(0,t;\Hx 1)}\,+\,\Bigl\|\int_0^{\bullet} A^r(\mu^{\alpha_1}-\mu^{\alpha_2})(s)ds\Bigr
\|_{C^0([0,t];\Lx 2)}\non\\
&\le\,K_2\,\Bigl(|\alpha_1-\alpha_2|^{1/2}\,+\,\|u^{\alpha_1}-u^{\alpha_2}\|_{L^2(0,t;H)}\Bigr)\,.
\end{align}
\End{theorem}
\Begin{proof}
We first observe that in \cite[Example~1]{CGS19} it has been shown that a uniform separation property is valid
for the solutions to \eqref{weak1}--\eqref{weak3} with $f_1=h^\alpha$ under the assumptions {\bf (A1)--\gianni{(A9)}};
that is, there are constants $r_*,r^*\in (-1,1)$ \gianni{(depending on~$\alpha$)} such that
\begin{equation}
\label{separa}
\gianni{r_* \le y^\alpha \le r^* \quad\mbox{a.e. in \,$Q$}}.
\end{equation}
Moreover, we have $V_{-\Delta}^{1/2}=\Hx 1$, and thus we can infer from \cite[\pier{Remarks}~3.4,~3.5~and~3.6]{CGS19} that
for any $\alpha>0$ the solution $(\mu^{\alpha},y^{\alpha})$ to \eqref{weak1}--\eqref{weak3} in the sense of \cite[Thm.~2.6]{CGS18} for
$f_1=h^\alpha$ is in fact uniquely determined and satisfies the variational equality (which in this special case
turns out to be equivalent to \eqref{weak2})
\begin{align}
\label{weak2new}
&(\tau\partial_t y^\alpha(t),v) + (\nabla y^{\alpha}(t),\nabla v)+( (h^\alpha)'(y^{\alpha}(t)),v)+(f_2'(y^{\alpha}(t)),v)
=(\mu^{\alpha}(t)+u(t),v)\nonumber\\
&\mbox{for a.e. $t\in (0,T)$ and every \,$v\in \Hx 1$.}
\end{align}
Now\pier{,} let $u:=u^{\alpha_1}-u^{\alpha_2}$, $\mu:=\mu^{\alpha_1}-\mu^{\alpha_2}$, and $y:=y^{\alpha_1}-y^{\alpha_2}$.
Then, taking the difference
in \eqref{weak1} for the two different cases $\alpha=\alpha_1$, $\alpha=\alpha_2$, and integrating the resulting
equality over $[0,t]$ with respect to time, where $t\in (0,T]$, we obtain the identity
\begin{equation*}
\langle y(t),v\rangle_{A,r}\,+\,\Bigl(A^r\mbox{$\int_0^t\mu(s)ds$},A^r v\Bigr)\,=\,0
\quad\mbox{for all $t\in (0,T]$ and $v\in V_A^r$}.
\end{equation*}
Testing this identity by $v=\mu(t)$, and noting that $\langle y(t),\mu(t)\rangle_{A,r}=(y(t),\mu(t))$ for almost
every $t\in(0,T)$, we thus obtain that
\begin{align}
\label{diff1}
&\int_0^t\!\!\int_\Omega y\mu\,=\,- \int_0^t\Bigl(A^r\mu(s), \int_0^s A^r\mu(\rho)d\rho\Bigr)ds
\,=\,-\frac 12\,\Bigl\|\int_0^t A^r\mu(s)ds\Bigr\|^2\,.
\end{align}
Next, we insert $v=\gianni{-y}$ in the variational equality \eqref{weak2new} for
$\alpha=\alpha_2$, and $v=\gianni y$ in \eqref{weak2new} for $\alpha=\alpha_1$. Summation of the resulting
\pier{identities} then yields the equality
\begin{align}
\label{diff2}
&\frac\tau 2\,\|y(t)\|^2\,+\,\int_0^t\|\nabla y(s)\|^2\,ds\,+\int_0^t\!\!\int_\Omega \varphi(\alpha_1)(h'(y^{\alpha_1})-h'(y^{\alpha_2}))
(y^{\alpha_1}-y^{\alpha_2})\nonumber\\
&\gianni =\,-\int_0^t\!\!\int_\Omega(\varphi(\alpha_1)-\varphi(\alpha_2))h'(y^{\alpha_2})(y^{\alpha_1}-y^{\alpha_2})
\,-\int_0^t\!\!\int_\Omega(f_2'(y^{\alpha_1})-f_2'(y^{\alpha_2}))\,y\nonumber\\
&\,+\,\int_0^t\!\!\int_\Omega y\mu\,+\int_0^t\!\!\int_\Omega u\,y.
\end{align}
Owing to the monotonicity of $\,h'$, the third summand on the left-hand side of \eqref{diff2} is nonnegative.
Moreover,
$h'(y^{\alpha_2})(y^{\alpha_1}-y^{\alpha_2})\,\le\,h(y^{\alpha_1})-h(y^{\alpha_2})$ almost
everywhere in $Q$, since $\,h\in C^1(-1,1)\,$ is convex, and \,$\phi(\alpha_1)<\phi(\alpha_2)$. So
the first summand on the right-hand side\ of \eqref{diff2}, which we denote by $I$, satisfies
\begin{align}
I\,&\le\,(\phi(\alpha_2)-\phi(\alpha_1))\int_0^t\!\!\int_\Omega(|h(y^{\alpha_1})|\,+\,|h(y^{\alpha_2})|)\,\le\,C_1\,
(\alpha_2-\alpha_1)\,,
\end{align}
with $\,C_1:=4\ln(2)\,|\Omega|\,T\,\|\phi'\|_{C^0([0,1])}\,$, where $|\Omega|$ denotes the volume of $\Omega$.
Therefore, adding \eqref{diff1}
and \eqref{diff2}, and using the Lipschitz continuity of $f_2'$, we obtain from Young's inequality an
estimate of the form
\begin{align}
&\frac\tau 2\,\|y(t)\|^2\,+\,\int_0^t\|\nabla y(s)\|^2\,ds\,+\,
\frac 12\,\Bigl\|\int_0^t A^r\mu(s)ds\Bigr\|^2\non\\
&\le\,C_1\,|\alpha_1-\alpha_2|\,+\,(L+1)\int_0^t\!\!\int_\Omega|y|^2\,+\,\frac 14\int_0^t\!\!\int_\Omega|u|^2\,,
\end{align}
and \eqref{esti1} follows from Gronwall's lemma.
\End{proof}
\Begin{corollary}
Suppose that {\bf (A1)}--{\bf \gianni{(A9)}} are fulfilled and that $\alpha\in (0,1]$.
Moreover, let $y=\mathcal{S}_0(u)$ and $y^{\alpha}=\mathcal{S}_{\alpha}(u)$. Then
\begin{align}
\label{esti2}
&\|y^{\alpha}-y\|_{C^0([0,t];\Lx 2)\cap L^2(0,t;\Hx 1)}\,\le\,K_2\,|\alpha|^{1/2}\,.
\end{align}
\End{corollary}
\Begin{proof}
We apply \eqref{esti1} with $\alpha_1=\alpha$, $\alpha_2=\alpha_n$, where $\alpha_n\searrow0$, and
$u^{\alpha_1}=u^{\alpha_2}=u$, which with $y^{\alpha_n}=\mathcal{S}_{\alpha_n}(u)$ yields the estimate
\begin{align*}
&\|y^{\alpha}-y^{\alpha_n}\|_{C^0([0,t];\Lx 2)\cap L^2(0,t;\Hx 1)}\,+\,\Bigl\|\int_0^{\bullet} A^r(\mu^{\alpha}-\mu^{\alpha_n}
)(s)ds\Bigr
\|_{C^0([0,t];\Lx 2)}\non\\
&\le\,K_2\,|\alpha-\alpha_n|^{1/2}\,.
\end{align*}
The assertion now follows from \eqref{cony} in Theorem 3.2 by taking the
limit as $n\to\infty$, invoking Remark 3.4
and the semicontinuity of norms.
\End{proof}
\section{Existence and approximation of optimal controls}
\setcounter{equation}{0}
Beginning with this section, we investigate the optimal control problem (${\mathcal{CP}}_0$) of minimizing the cost
functional \eqref{cost} over the admissible set ${\mathcal{U}}_{\rm ad}$ subject to state system \eqref{weak1}--\eqref{weak3}
where $f_1=I_{[-1,1]}$. In comparison with (${\mathcal{CP}}_0$),
we consider for $\alpha>0$ the following control problem:
\vspace{1mm}\noindent
(${\mathcal{CP}}_\alpha$) \,\,\,Minimize $\,{\cal J}(y,u)\,$
for $\,u\in{\mathcal{U}}_{\rm ad}$, subject to the condition that $y=\mathcal{S}_\alpha(u)$ for some solution $(\mu,y)$ to the state system \eqref{weak1}--\eqref{weak3} with $f_1=h^\alpha$\pier{,} in the sense of \cite[Thm.~2.6]{CGS18}.
\vspace{1mm}\noindent
We expect that the minimizers of (${\mathcal{CP}}_\alpha$)\ are for $\alpha\searrow0$ related to minimizers of (${\mathcal{CP}}_0$).
Prior to giving an affirmative answer to this conjecture, we first show an existence result for (${\mathcal{CP}}_\alpha$).
\Begin{proposition}
Suppose that \pier{{\bf (A1)}--{\bf (A6)}} are satisfied. Then \mbox{{\rm(}${\cal CP}_{\alpha}${\rm)}} has for
every $\alpha>0$ a solution.
\End{proposition}
\Begin{proof}
Let $\alpha>0$ be fixed, and assume that a minimizing sequence $\,\{(y_n,u_n)\}$ for (${\mathcal{CP}}_\alpha$)\ is given,
where $y_n=\mathcal{S}_\alpha(u_n)$ for some solution
pair $(\mu_n,y_n)$ to the state system with $u=u_n\in{\mathcal{U}}_{\rm ad}$ and $f_1=h^\alpha$, for $n\in{\mathbb{N}}$. Then it holds for every $n\in{\mathbb{N}}$ that
\begin{align}
\label{exi1}
&\langle \partial_t y_n(t),v\rangle_{A,r}+(A^r\mu_n(t),A^r v)=0\quad \mbox{for a.e. \,$t\in (0,T)$\, and every $\,v\in V_A^r$,}
\\[0.5mm]
\label{exi2}
&(\tau\partial_t y_n(t),y_n(t)-v)+(B^\sigma y_n(t),B^\sigma(y_n(t)-v))+h^\alpha(y_n(t))\non\\
&\quad \le\,(\mu_n(t)+u_n(t)-f_2'(y_n(t)),y_n(t)-v) +h^\alpha(v) \non\\
& \mbox{for a.e. \,$t\in (0,T)$\, and every $\,v\in V_B^\sigma$,}
\\[0.5mm]
\label{exi3}
&y_n(0)=y_0.
\end{align}
Taking \gianni{estimate \eqref{albound1}} into account, we may without loss of generality assume that there
are $\,\bar u\in{\mathcal{U}}_{\rm ad}\,$ and $(\bar \mu,\bar y)$ such that
\begin{align}
\label{exiu}
u_n\to\bar u&\quad\mbox{weakly-star in }\,{\cal X},\\
\label{eximu}
\mu_n\to\bar \mu&\quad\mbox{weakly in }\,L^2(0,T;V_A^r),\\
\label{exiy}
y_n\to\bar y&\quad\mbox{weakly-star in }\,H^1(0,T;V_A^{-r})\cap L^\infty(0,T;V_B^\sigma),\nonumber\\
&\quad\mbox{strongly in }\,C^0([0,T];H), \,\,\mbox{ and pointwise a.e. in }\,\,Q.
\end{align}
Then also $\,f_2'(y_n)\to f_2'(\bar y)\,$ strongly in $C^0([0,T];H)$. Moreover, it holds
\begin{equation}\label{elvis}
\int_0^T \!\!\int_\Omega h^\alpha(y_n)\,\le\,K_1\quad\mbox{for every $n\in{\mathbb{N}}$}.
\end{equation}
Therefore, we have $\,y_n\in [-1,1]\,$ almost everywhere in $Q$, and since $h^\alpha$ is continuous
in $[-1,1]$, it follows that $\,h^\alpha(y_n)\to h^\alpha(\bar y)$\, pointwise almost everywhere in $Q$.
Lebesgue's dominated convergence theorem then yields that
$$\int_0^T\!\!\int_\Omega h^\alpha(y_n)\to \int_0^T\!\!\int_\Omega h^\alpha(\bar y)\,.$$
In addition, by lower semicontinuity, \pier{we have that}
$$\int_0^T (B^\sigma \bar y(t)),B^\sigma \bar y(t))\,dt\,\le\,\liminf_{n\to\infty}\int_0^T
(B^\sigma y_n(t),B^\sigma y_n(t))\,dt\,.$$
Combining the convergence results shown above, we obtain by passage to the limit as $n\to\infty$ that
\begin{align}
\label{exi4}
&\int_0^T\langle \partial_t \bar y(t),v(t)\rangle_{A,r}\,dt\,+\int_0^T(A^r\bar \mu(t),A^r v(t))\,dt\,=\,0\quad\forall\,
v\in L^2(0,T;V_A^r),
\\[0.5mm]
\label{exi5}
&\int_0^T(\tau\partial_t \bar y(t),\bar y(t)-v(t))\,dt\,+\int_0^T(B^\sigma \bar y(t),B^\sigma(\bar y(t)-v(t)))\,dt
\,+\int_0^T\!\!\int_\Omega h^\alpha(\bar y)\non\\
&\le\int_0^T(\bar \mu(t)+\bar u(t)-f_2'(\bar y(t)),\bar y(t)-v(t))\,dt\,+\int_0^T\!\!\int_\Omega h^\alpha(v)\non\\[0.5mm]
& \mbox{for every $\,v\in L^2(0,T;V_B^\sigma)$,}
\\[0.5mm]
\label{exi6}
&\bar y(0)=y_0.
\end{align}
Apparently, \eqref{exi4}--\eqref{exi5} is just the time-integrated version of
\eqref{weak1}--\eqref{weak2} for $u=\bar u$ and $f_1=h^\alpha$,
written with time-dependent test functions, which is equivalent to
\eqref{weak1}--\eqref{weak2}.
Hence, $(\bar \mu,\bar y)$ solves \eqref{weak1}--\eqref{weak2} for $u=\bar u$ and $f_1=h^\alpha$ in the sense of
\cite[Thm.~2.6]{CGS18}. In particular, we have
$\bar y=\mathcal{S}_\alpha(\bar u)$. But this means that $\,(\bar y,\bar u)\,$ is admissible
for (${\mathcal{CP}}_\alpha$). From the semicontinuity properties
of the cost functional \eqref{cost} it then follows that $\,(\bar y,\bar u)\,$
is an optimal pair, which concludes the proof of the assertion.
\End{proof}
\Begin{proposition}
Let \pier{{\bf (A1)}--{\bf (A6)}} be fulfilled, and suppose that sequences $\,\{\alpha_n\}\subset (0,1]\,$ and
$\,\{u_n\}\subset{\mathcal{U}}_{\rm ad}\,$ are given such that $\,\alpha_n\searrow0\,$ and $\,u_n\to u\,$ weakly-star in ${\cal X}$
for some $\,u\in{\mathcal{U}}_{\rm ad}$. Then, with the solution operators defined in \eqref{defcs0} and \eqref{defcsal},
\begin{align}\label{cesareuno}
&{\mathcal{J}}(\mathcal{S}_0(u),u)\,\le\,\liminf_{n\to\infty}\,{\mathcal{J}}(\mathcal{S}_{\alpha_n}(u^{\alpha_n}),u^{\alpha_n}),\\[0.5mm]
\label{cesaredue}
&{\mathcal{J}}(\mathcal{S}_0(v),v)\,=\,\lim_{n\to\infty}\,{\mathcal{J}}(\mathcal{S}_{\alpha_n}(v),v) \quad\forall\,v\in{\mathcal{U}}_{\rm ad}.
\end{align}
\End{proposition}
\Begin{proof}
Under the given assumptions, we may apply \eqref{cony} in Theorem 3.3 and Remark 3.4 to infer
that $\gianni{\mathcal{S}_{\alpha_n}(u^{\alpha_n})\to\mathcal{S}_0(u)}$ strongly in $C^0([0,T];H)$.
The validity of \eqref{cesareuno} is then a direct consequence of the weak and weak-star
sequential semicontinuity properties of the cost
functional~\gianni{\eqref{cost}}.
Now suppose that $v\in{\mathcal{U}}_{\rm ad}$ is arbitrarily chosen, and put $y^{\alpha_n}:=\mathcal{S}_{\alpha_n}(v)$ for all $n\in{\mathbb{N}}$.
Then, again by Theorem~3.3 and Remark~3.4, $\,y^{\alpha_n}\to \mathcal{S}_0(v)\,$ strongly in $C^0([0,T];H)$. \pier{Next,}
observe that the first two summands of the cost functional are
obviously continuous with respect to the strong topology of $C^0([0,T];H)$, which then shows the validity of
\eqref{cesaredue}.
\End{proof}
We are now in a position to prove the existence of minimizers for the problem (${\mathcal{CP}}_0$). We have the following result.
\Begin{corollary}
Under the assumptions of Proposition~4.2, the optimal control problem {\rm ($\mathcal{CP}_0$)} has at least
one solution.
\End{corollary}
\Begin{proof}
Pick an arbitrary sequence $\{\alpha_n\}\subset (0,1]$ such that $\alpha_n\searrow0$ as $n\to\infty$.
By virtue of Proposition 4.1, the optimal control problem (${\mathcal{CP}}_{\alpha_n}$) has for every $n\in{\mathbb{N}}$ a solution $(y^{\alpha_n},u^{\alpha_n})$, where $y^{\alpha_n}=\mathcal{S}_{\alpha_n}(u^{\alpha_n})$ for a solution $(\mu^{\alpha_n},y^{\alpha_n})$ to the corresponding state system.
Since ${\mathcal{U}}_{\rm ad}$ is bounded in ${\cal X}$, we may without loss of generality assume that $u^{\alpha_n}\to u$ weakly-star in ${\cal X}$
for some $u\in{\mathcal{U}}_{\rm ad}$. At this point, we apply Theorem 3.3 to the present situation. We then infer that the convergence
results \eqref{conmu} and \eqref{cony} hold true for some subsequence $\,\{\alpha_{n_k}\}\,$
with a pair $(\mu,y)$ satisfying $y=\mathcal{S}_0(u)$. Invoking the optimality of
$\,(y^{\alpha_n},u^{\alpha_n})\,$ for (${\mathcal{CP}}_{\alpha_n}$), we then find for every $\,v\in{\mathcal{U}}_{\rm ad}\,$ the chain of (in)equalities
\begin{align}
&{\cal J}(y,u)\,=\,{\cal J}(\mathcal{S}_0(u),u)\,\le\,\liminf_{k\to\infty}\,{\cal J}(\mathcal{S}_{\alpha_{n_k}}(u^{\alpha_{n_k}}),
u^{\alpha_{n_k}})\non\\
&\le\,\liminf_{k\to\infty}\,{\cal J}(\mathcal{S}_{\alpha_{n_k}}(v),v)\,\le\,{\cal J}(\mathcal{S}_0(v),v),
\end{align}
which yields that $\,(y,u)\,$ is an optimal pair for (${\mathcal{CP}}_0$). The assertion is thus proved.
\End{proof}
Theorem 3.3 and the proof of Corollary 4.3 indicate that optimal controls of (${\mathcal{CP}}_\alpha$)\ are ``close'' to optimal
controls of (${\mathcal{CP}}_0$). However, they do not yield any information on whether every optimal control
of (${\mathcal{CP}}_0$)\ can be approximated in this way. In fact, such a global result cannot be expected to hold true.
However, a local answer can be given. For this purpose, we employ a trick introduced in \cite{Barbu}. To this end,
let $\bar u\in{\mathcal{U}}_{\rm ad}$ be an optimal control for (${\mathcal{CP}}_0$)\ with the associated state $(\bar \mu,\bar y)$ where $\bar y=\mathcal{S}_0(\bar u)$. We associate
with this optimal control the {\em adapted cost functional}
\begin{equation}
\label{adcost}
\widetilde{\cal J}(y,u):={\cal J}(y,u)\,+\,\frac 12\,\|u-\bar u\|^2_{L^2(Q)}
\end{equation}
and a corresponding \emph{adapted optimal control problem} for $\alpha>0$, namely:
\vspace{2mm}\noindent
($\widetilde{\mathcal{CP}}_{\alpha}$)\quad Minimize $\,\, \widetilde {\cal J}(y,u)\,\,$
for $\,u\in{\mathcal{U}}_{\rm ad}$, subject to the condition that $y=\mathcal{S}_\alpha(u)$ for some solution $(\mu,y)$ to the state system \eqref{weak1}--\eqref{weak3} with $f_1=h^\alpha$ in the sense of \cite[Thm.~2.6]{CGS18}.
\vspace{2mm}
With essentially the same proof as that of Proposition 4.1 (which needs no repetition here), we can show the following
result.
\Begin{lemma}
Suppose that the assumptions \pier{{\bf (A1)}--{\bf {(A6)}}} are fulfilled. Then the optimal control problem
$(\widetilde{\cal CP}_\alpha)$ has for every $\alpha>0$ at least one solution.
\End{lemma}
\vspace{1mm}
We are now in the position to give a partial answer to the question raised above. We have the following result.
\Begin{theorem}
Let the assumptions \pier{{\bf (A1)}--{\bf {(A6)}}} be fulfilled, suppose that
$\bar u\in {\mathcal{U}}_{\rm ad}$ is an arbitrary optimal control of {\rm $({\mathcal{CP}}_{0})$} with associated state
$(\bar\mu,\bar y)$ where $\bar y=\mathcal{S}_0(\bar u)$, and let $\,\{\alpha_n\}\subset (0,1]\,$ be
any sequence such that $\,\alpha_n\searrow 0\,$ as $\,n\to\infty$. Then there exist a
subsequence $\{\alpha_{n_k}\}$ of $\{\alpha_n\}$, and, for every $k\in{\mathbb{N}}$, an optimal control
$\,u^{\alpha_{n_k}}\in {\mathcal{U}}_{\rm ad}\,$ of the adapted problem {\rm $(\widetilde{\mathcal{CP}}_{\alpha_{n_k}})$}
with associated state $(\mu^{\alpha_{n_k}},y^{\alpha_{n_k}})$, where $\,y^{\alpha_{n_k}}=\mathcal{S}_{\alpha_{n_k}}(u^{\alpha_{n_k}})$,
such that, as $k\to\infty$,
\begin{align}
\label{tr3.4}
&u^{\alpha_{n_k}}\to \bar u\quad\mbox{strongly in }\,L^2(Q),
\end{align}
and such that the property \eqref{cony} is satisfied with $\,y\,$ replaced
by $\,\bar y\,$. Moreover, we have
\begin{align}
\label{tr3.5}
&\lim_{k\to\infty}\,\widetilde{{\cal J}}(y^{\alpha_{n_k}},u^{\alpha_{n_k}})\,=\, {\cal J}(\bar y,\bar u)\,.
\end{align}
\End{theorem}
\Begin{proof}
Let $\alpha_n \searrow 0$ as $n\to\infty$. For any $ n\in{\mathbb{N}}$, we pick an optimal control
$u^{\alpha_n} \in {\mathcal{U}}_{\rm ad}\,$ for the adapted problem $(\widetilde{\cal CP}_{\alpha_n})$ and denote by
$(\mu^{\alpha_n},y^{\alpha_n})$, where $y^{\alpha_n}=\mathcal{S}_{\alpha_n}(u^{\alpha_n})$, an associated solution to
(\ref{weak1})--(\ref{weak3}) with $f_1=h^{\alpha_n}$ and $u=u^{\alpha_n}$.
By the boundedness of ${\mathcal{U}}_{\rm ad}$ in $\calX$, there is some subsequence $\{\alpha_{n_k}\}$ of $\{\alpha_n\}$ such that
\begin{equation}
\label{ugam}
u^{\alpha_{n_k}}\to u\quad\mbox{weakly-star in }\,{\cal X}
\quad\mbox{as }\,k\to\infty,
\end{equation}
with some $u\in{\mathcal{U}}_{\rm ad}$.
Thanks to Theorem~3.3, the convergence \gianni{properties \accorpa{conmu}{cony} hold} true
with some pair $(\mu,y)$ satisfying $y=\mathcal{S}_0(u)$. In particular, the pair $(y,u)$
is admissible for (${\cal CP}_0$).
We now aim to prove that $u=\bar u$. Once this is shown, it follows from the uniqueness of the second
solution component to the state system
\eqref{weak1}--\eqref{weak3} that also $y=\bar y$,
which implies that \gianni{\eqref{cony}} holds true with $\,y\,$ replaced by $\,\bar y$.
Now observe that, owing to the weak sequential lower semicontinuity of
$\widetilde {\cal J}$,
and in view of the optimality property of $(\bar y,\bar u)$ for problem $({\cal CP}_0)$,
\begin{align}
\label{tr3.6}
&\liminf_{k\to\infty}\, \widetilde{\cal J}(y^{\alpha_{n_k}},
u^{\alpha_{n_k}})
\ge \,{\cal J}(y,u)\,+\,\frac{1}{2}\,
\|u-\bar{u}\|^2_{L^2(Q)}\nonumber\\[1mm]
&\geq \, {\cal J}(\bar y,\bar u)\,+\,\frac{1}{2}\,\|u-\bar{u}\|^2_{L^2(Q)}\,.
\end{align}
On the other hand, the optimality property of $\,(y^{\alpha_{n_k}},u^{\alpha_{n_k}})
\,$ for problem $(\widetilde {\cal CP}_{\alpha_{n_k}})$ yields that
for any $k\in{\mathbb{N}}$ we have
\begin{equation}
\label{tr3.7}
\widetilde {\cal J}(y^{\alpha_{n_k}},u^{\alpha_{n_k}})\, =\,
\widetilde {\cal J}({\cal S}_{\alpha_{n_k}}(u^{\alpha_{n_k}}),
u^{\alpha_{n_k}})\,\le\,\widetilde {\cal J}({\cal S}_{\alpha_{n_k}}
(\bar u),\bar u)\,,
\end{equation}
whence, taking the limit superior as $k\to\infty$ on both sides and invoking (\ref{cesaredue}) in
Proposition 4.2,
\begin{align}
\label{tr3.8}
&\limsup_{k\to\infty}\,\widetilde {\cal J}(y^{\alpha_{n_k}},
u^{\alpha_{n_k}})\,\le\,\widetilde {\cal J}({\calS}_0(\bar u),\bar u)
\,=\,\widetilde {\cal J}(\bar y,\bar u)
\,=\,{\cal J}(\bar y,\bar u)\,.
\end{align}
Combining (\ref{tr3.6}) with (\ref{tr3.8}), we have thus shown that
$\,\frac{1}{2}\,\|u-\bar{u}\|^2_{L^2(Q)}=0$\,,
so that $\,u=\bar u\,$ and thus also $\,y=\bar y$.
Moreover, (\ref{tr3.6}) and (\ref{tr3.8}) also imply that
\begin{align}
\label{tr3.9}
&{\cal J}(\bar y,\bar u) \, =\,\widetilde{\cal J}(\bar y,\bar u)
\,=\,\liminf_{k\to\infty}\, \widetilde{\cal J}(y^{\alpha_{n_k}},
u^{\alpha_{n_k}})\nonumber\\[1mm]
&\,=\,\limsup_{k\to\infty}\,\widetilde{\cal J}(y^{\alpha_{n_k}},
u^{\alpha_{n_k}}) \,
=\,\lim_{k\to\infty}\, \widetilde{\cal J}(y^{\alpha_{n_k}},
u^{\alpha_{n_k}})\,,
\end{align}
which proves {(\ref{tr3.5})} and, at the same time, also (\ref{tr3.4}). This concludes the proof
of the assertion.
\End{proof}
\section{Adjoint system and first-order optimality conditions}
\setcounter{equation}{0}
In this section, we aim at deriving first-order necessary optimality conditions for the optimal
control problem (${\mathcal{CP}}_0$) using the deep quench approximation. Throughout the section, we assume
that $\bar u\in{\mathcal{U}}_{\rm ad}$ is an optimal control of (${\mathcal{CP}}_0$) with associated state
$(\bar \mu,\bar y)$\pier{, with $\bar y=\mathcal{S}_0(\bar u)$.} The derivation will be achieved by a passage to the
limit as $\alpha\searrow0$ in the first-order optimality conditions for the adapted optimal control
problems ($\widetilde{\mathcal{CP}}_\alpha$) that can be derived as in \cite{CGS19} with only minor
and obvious changes. This approach will not be possible in full generality. In fact, we have to assume
that, besides {\bf (A1)--\gianni{(A7)}}, the assumptions {\bf \gianni{(A8)}--\gianni{(A9)}} from Theorem 3.5 are fulfilled.
\Begin{remark}\rm
Observe that {\bf \gianni{(A8)}} yields the validity of the stronger regularity properties \eqref{albound2} from Theorem~3.1.
Also, {\bf \gianni{(A9)}} implies that the constant functions belong to $V_{-\Delta}^{1/2}=\Hx 1$, so that {\bf (A2)}
is automatically fulfilled.
In addition, since $\Hx 1\cap L^\infty(\Omega)$ is dense in $\Hx 1$ and
$\Hx 1$ is continuously embedded in $L^4(\Omega)$, the conditions \cite[{\bf (A8)}~and~{\bf (A9)}]{CGS19} are satisfied.
\End{remark}
\Begin{remark}\rm
The condition that $\,V_A^{2r}\subset L^\infty(\Omega)\,$ is, for instance,
satisfied if $A=-\Delta$ with zero Dirichlet or Neumann
boundary condition and $r> \frac 3 8$. Indeed, we have in this case that $V_A^{2r}
\subset H^{4r}(\Omega)\subset L^\infty(\Omega)$, since $\,\,4r>\frac 32$. Likewise, if $A=\Delta^2$ with
domain $D(A)\subset H^4(\Omega)$, then $V_A^{2r}\subset L^\infty(\Omega)$ provided that $r>\frac 3{16}$.
In this sense, while the
improvement obtained in the following results over previously known results for the
classical case $A=B=-\Delta$, $r=\sigma=
\frac 12$, is not too large, the results are entirely new for other operators \pier{$A$}; in fact, to our best knowledge,
they constitute the first ever first-order necessary
optimality conditions for Cahn--Hilliard type systems with fractional operators and nondifferentiable nonlinearities of
double obstacle type.
\End{remark}
As already mentioned in the proof of Theorem 3.5, it follows under the assumptions {\bf (A1)--\gianni{(A9)}} that also the
solution component $\,\mu^{\alpha}\,$ of the solutions $(\mu^{\alpha},y^{\alpha})$ to \eqref{weak1}--\eqref{weak3} in the sense of \cite[Thm.~2.6]{CGS18} for $f_1=h^\alpha$ is uniquely determined, so that a corresponding solution operator
$$\widetilde\mathcal{S}_\alpha=(\widetilde \mathcal{S}_\alpha^1,\widetilde\mathcal{S}_\alpha^2):u\ni{\mathcal{U}}_{\rm ad}\mapsto\widetilde\mathcal{S}_\alpha(u)=
(\widetilde \mathcal{S}_\alpha^1(u),\widetilde\mathcal{S}_\alpha^2(u)):=(\mu^{\alpha},y^{\alpha})$$
is well defined. Clearly, we have $\widetilde\mathcal{S}_\alpha^2=\mathcal{S}_\alpha$. Moreover, $(\mu^{\alpha},y^{\alpha})$ satisfies the variational equality \eqref{weak2new}, which in this situation is equivalent to \eqref{weak2}. In addition, a
uniform separation property is satisfied; indeed, \pier{thanks to {\bf (A6)}}, for every $\alpha>0$ and every bounded set ${\cal U}\subset {\cal X}$,
there exist constants $r_*(\alpha),r^*(\alpha)\in (-1,1)$, which depend only on ${\cal U}$,
such that the following holds true: whenever $(\mu^{\alpha},y^{\alpha})=\widetilde\mathcal{S}_\alpha(u)$
for some $u\in {\cal U}$, then
\begin{equation}
\label{howgh}
r_*(\alpha) \le y^{\alpha} \le r^*(\alpha) \quad\mbox{a.e. in \,$Q$}, \quad r_*(\alpha)\le y_0\le r^*(\alpha)\quad
\mbox{a.e. in $\,\Omega$.}
\end{equation}
In particular, the condition \cite[{\bf (GB)}]{CGS19}, which was crucial for the analysis carried out in
\cite{CGS19}, is fulfilled for the potentials $f_1=h^\alpha$, $\alpha>0$, and we may take advantage of the results derived there.
\Begin{remark}\rm Owing to the separation property \eqref{howgh}, there is, for every $\alpha>0$ and every
bounded ${\cal U}\subset {\cal X}$, some constant $K_\alpha>0$, which depends only on ${\cal U}$, such
that
\begin{equation}
\label{albound3}
\max_{0\le i\le 3}\,\|(h^\alpha)^{(i)}(y^{\alpha})\|_{L^\infty(Q)}\,\le\,K_\alpha \quad\mbox{whenever $\,y^{\alpha}=\mathcal{S}_\alpha(u)\,$
for some $\,u\in {\cal U}$}.
\end{equation}
Now we have $V_A^{2r}
\subset L^\infty(\Omega)$ and thus, by \eqref{albound2}, $\mu^{\alpha}\in L^\infty(Q)$. Since also
$\partial_ty^{\alpha}\in L^\infty(0,T;H)$, comparison in \eqref{weak2new} shows that then $y^{\alpha}\in L^\infty(0,T;\Hx 2)$, which means
that the state equations \eqref{weak1}, \eqref{weak2} for $f_1=h^\alpha$ are even satisfied in the strong sense,
that is, we have
\begin{align}
\label{pw1}
&\partial_ty^{\alpha}+A^{2r}\mu^{\alpha}=0 \quad\mbox{a.e. in }\,Q,\\
\label{pw2}
&\tau\partial_t y^{\alpha}-\Deltay^{\alpha}+(h^\alpha)'(y^{\alpha})+f_2'(y^{\alpha})=\mu^{\alpha}+u \quad\mbox{a.e. in }\,Q.
\end{align}
\End{remark}
At this point, we observe that that the state systems associated with (${\mathcal{CP}}_\alpha$) and
($\widetilde{\mathcal{CP}}_\alpha$) are exactly the same. Hence, if $\bar u^{\alpha}\in{\mathcal{U}}_{\rm ad}$ is an optimal control
of ($\widetilde{\mathcal{CP}}_\alpha$) with associated state $(\bar \mu^{\alpha},\bar y^{\alpha})=\widetilde\mathcal{S}_\alpha(\bar u^{\alpha})$
for some $\alpha>0$, then
$(\bar \mu^{\alpha},\bar y^{\alpha})$ satisfies the global bounds \eqref{albound1}, \eqref{albound2}, \eqref{albound3}, as well as the
separation property \eqref{howgh}, and the state equations hold true in the form \eqref{pw1}, \eqref{pw2}. Moreover,
introducing for $\alpha>0$ the abbreviating notation
\begin{equation}
\label{defgal}
g_1^\alpha:=\beta_1(\bar y^{\alpha}(T)-y_\Omega),\quad g_2^\alpha:=\beta_2(\bar y^{\alpha}-y_Q),\quad \psi_1^\alpha:=f_2''(\bar y^{\alpha}),
\quad\psi_2^\alpha:=\phi(\alpha)
h''(\bar y^{\alpha}),
\end{equation}
we observe that \eqref{albound1}, \eqref{albound2}, \eqref{interv}, and {\bf \gianni{(A5)}} imply the global bound
\begin{equation}
\label{galbound}
\|g_1^\alpha\|_{\Lx 2}\,+\,\|g_2^\alpha\|_{L^2(Q)}\,+\,\|\psi_1^\alpha\|_{L^\infty(Q)}\,\le\,C_1\quad\forall\,\alpha \in (0,1],
\end{equation}
where, here and in the following, $C_i$, $i\in{\mathbb{N}}$, denote positive constants that may depend on the data of the state
system but not on $\alpha\in (0,1]$.
Observe that a corresponding bound for $\psi^\alpha_2$ cannot be expected: indeed, it may well happen that
the separation constants $r_*(\alpha)$ and/or $r^*(\alpha)$ introduced in \eqref{howgh} approach $\pm 1$ as
$\alpha\searrow0$, so that $\,\psi_2^\alpha=\frac {2\phi(\alpha)}{1-(\bar y^{\alpha})^2}\,$ may become unbounded as
$\alpha\searrow0$.
\vspace{3mm}
Next, we consider the adjoint system associated with the
adapted optimal control problem ($\widetilde{\mathcal{CP}}_\alpha$). According to \cite[Sect.~5]{CGS19}, it has the
following form:
\begin{align}
\label{adj1}
&(A^rp^\alpha(t),A^r v)-(q^\alpha(t),v)\,=\,0\quad\mbox{for a.e. $\,t\in (0,T)\,$ and every }\,v\in V_A^r,\\[0.5mm]
\label{adj2}
&\langle -\partial_t (p^\alpha+\tauq^\alpha)(t),v\rangle +(\nabla q^\alpha(t),\nabla v)+((\psi_1^\alpha(t)+\psi_2^\alpha(t))\,q^\alpha(t),v)\non\\
&=\,(g_2^\alpha(t),v) \quad\mbox{for a.e. $\,t\in (0,T)\,$ and every }\,v\in
\Hx 1,\\[0.5mm]
\label{adj3}
&(p^\alpha+\tauq^\alpha)(T)\,=\,g_1^\alpha \quad\mbox{in }\,\Omega.
\end{align}
Here, for the sake of simplicity, we have denoted by $\langle\cdot,\cdot\rangle$ the dual pairing between
${\Hx 1}^*$ and $\Hx 1$.
The system \eqref{adj1}--\eqref{adj3} is a special case of the type of systems that has been analyzed in
\cite[Sect.~5]{CGS19}. We briefly summarize some of the results established there
(cf., \cite[Prop.~5.2,~Lem.~5.3,~Lem.~5.4, Rem.~5.7,~Thm.~5.8]{CGS19}),
where we have to distinguish the following cases:
\vspace{2mm}\noindent
\underline{{\sc Case 1:} $\lambda_1>0$.}\\[1mm]
In this case, the system \eqref{adj1}--\eqref{adj3}
admits a unique solution $(p^\alpha,q^\alpha)$ satisfying
\begin{align}
\label{regp}
&p^\alpha\in L^2(0,T;V_A^{2r}),\\
\label{req}
&q^\alpha\in L^2(0,T;\Hx 1),\\
\label{regpq}
&p^\alpha+\tau q^\alpha\in H^1(0,T;{\Hx 1}^*).
\end{align}
Notice that \eqref{regpq} implies that $p+\tau q\in C^0([0,T]; {\Hx 1}^*)$, so that the endpoint condition
\eqref{adj3} is meaningful.
Now observe that the operator $A^{2r}\in {\cal L}(V_A^{2r},H)$ is for $\lambda_1>0$ a topological isomorphism,
and with $\,A^{-2r}:=(A^{2r})^{-1}:
H\to V_A^{2r}$ the variational equation \eqref{adj1} takes the simple form $\,p^\alpha=A^{-2r}q^\alpha$. Inserting this in
\eqref{adj2} and \eqref{adj3}, we obtain that
\begin{align}
\label{lg1}
&\langle -\partial_t \bigl((A^{-2r}+\tau I)q^\alpha\bigr)(t),v\rangle +\int_\Omega\nabla q^\alpha(t)\cdot\nabla v\,+(\psi_1^\alpha(t)\,q^\alpha(t),v)\non
\\[0.5mm]
&\quad+(\psi_2^\alpha(t)q^\alpha(t),v)\,=\,(g_2^\alpha(t),v) \quad\mbox{for a.e. $\,t\in (0,T)\,$ and all }\,v\in
\Hx 1,\\[1mm]
\label{lg2}
&(A^{-2r}+\tau I)q^\alpha(T)\,\,=\,g_1^\alpha\qquad\mbox{a.e. in }\,\Omega,
\end{align}
where $\,I\,$ denotes the identity operator in $\,H$.
Moreover, since also the linear operator $\,A^{-2r}+\tau I \in {\cal L}(H,H)$ is obviously
a topological isomorphism, \eqref{lg2} can be equivalently written as
\begin{equation}
\label{lg2neu}
q^\alpha(T)\,=\,(A^{-2r}+\tau I)^{-1} g_1^\alpha,
\end{equation}
which gives $q^\alpha(T)$ a proper meaning as well.
We now derive an estimate for the adjoint variables that is uniform in $\alpha>0$.
Testing \eqref{lg1} by $q^\alpha(t)$ and integrating with respect to time over $[t,T]$, where $t\in[0,T)$, we then
conclude the equation
\begin{align}
\label{lg3}
&\int_t^T\!\langle -\partial_t \bigl((A^{-2r}+\tau I)q^\alpha\bigr)(\rho),q^\alpha(\rho)\rangle\,d\rho\,
+\int_t^T\!\|\nabla q^\alpha(\rho)\|^2\,d\rho
\, +\int_t^T\!\!\int_\Omega\!\psi_2^\alpha \,|q^\alpha|^2\non
\\[0.5mm]
&=\,\int_t^T\!\!\int_\Omega\bigl(-\psi_1^\alpha q^\alpha\,+\,g_2^\alpha\bigr)\,q^\alpha\,,
\end{align}
where the last term on the left-hand side\ is nonnegative and, owing to \eqref{galbound}, the right-hand side\ is bounded by an expression of the form
\begin{equation}
\label{Paul}
C_2\,+\,C_3\int_t^T\!\!\int_\Omega|q^\alpha|^2\,.
\end{equation}
Now observe that, by definition \eqref{defAr}, and since $\lambda_1>0$, it holds for every $v\in H$ that
\begin{equation}
\label{lg4}
(A^{-2r}+\tau I)^{1/2}v\,=\,\sum_{j=1}^\infty \left(\lambda_j^{-2r}+\tau\right)^{1/2}(v,e_j)e_j,
\end{equation}
and we have the estimates
\begin{align}
\label{lg5}
&\left\|(A^{-2r}+\tau I)^{1/2}v\right\|^2\,=\,\sum_{j=1}^\infty\left(\lambda_j^{-2r}+\tau\right)|(v,e_j)|^2
\,\ge\,\tau\,\|v\|^2,
\\
\label{lg6}
&\left\|(A^{-2r}+\tau I)^{1/2}v\right\|^2\,\gianni{{=\,\sum_{j=1}^\infty\left(\lambda_j^{-2r}+\tau\right)|(v,e_j)|^2}}
\,\le\,\gianni{(\lambda_1^{-2r}+\tau)}\,\|v\|^2,
\\
\label{lg7}
&\left\|(A^{-2r}+\tau I)^{-1}v\right\|^2\,=\,\sum_{j=1}^\infty(\lambda_j^{-2r}+\tau)^{-2}|(v,e_j)|^2
\,\le\,\tau^{-2}\,\|v\|^2.
\end{align}
Moreover, it is easily verified that
\begin{align}
\label{lg8}
-\,\langle \partial_t(A^{-2r}+\tau I)q^\alpha(t),q^\alpha(t)\rangle\,=\,-\,\frac 12\,\frac d{dt}\left\|(A^{-2r}+\tau I)^{1/2}
q^\alpha(t)\right\|^2.
\end{align}
Therefore, by virtue of \eqref{lg2neu}, the first term on the left-hand side\ of \eqref{lg3} is equal to
\begin{equation}
\label{lg9}
\frac 12\left\|(A^{-2r}+\tau I)^{1/2}q^\alpha(t)\right\|^2\,-\,\frac 12\left\|(A^{-2r}+\tau I)^{1/2}
(A^{-2r}+\tau I)^{-1} g_1^\alpha\right\|^2,
\end{equation}
which, by \eqref{galbound} and \eqref{lg5}--\eqref{lg7}, is bounded from below by $\,\frac \tau 2 \,\|q^\alpha(t)\|^2-C_4$,
with some global constant $C_4>0$. At this point, we invoke Gronwall's lemma, taken backward in time,
as well as the fact that $p=A^{-2r}q$, to conclude that
\begin{equation}
\label{lgq}
\|p^\alpha\|_{L^\infty(0,T;V_A^{2r})}\,+\,\|q^\alpha\|_{L^\infty(0,T;H)\cap L^2(0,T;\Hx 1)}\,\le\,C_5\quad\forall\,
\juerg{\alpha\in (0,1]}.
\end{equation}
\vspace{2mm}\noindent
\underline{{\sc Case 2:} $\lambda_1=0.$}\\[1mm]
This case is considerably more difficult to handle. To motivate this, we
denote by $\,\mathbf{1}\,$ both the functions that are identically equal to \,$1$\,
in either $\,\Omega\,$ or $\,Q$.
Then, by {\bf (A2)}(ii), $\,A^r{\mathbf{1}}=0$, and insertion
of $v={\mathbf{1}}$ in \eqref{adj1} yields that
\Begin{equation}
\label{meanqal}
\mbox{mean}\,(q^\alpha(t))=0 \quad\mbox{for a.e. $\,t\in (0,T)$.}
\End{equation}
At this point, and also for later use, we recall an integration-by-parts formula that was proved in
\cite[Lem.~4.5]{CGSSIAM18}: if $({\cal V},{\cal H},{\cal V}^*)$ is a Hilbert triple and
\begin{equation}
\label{leibniz1}
w\in H^1(0,T;{\cal H})\cap L^2(0,T;{\cal V})\quad\mbox{and}\quad z\in H^1(0,T;{\cal V}^*)\cap L^2(0,T;{\cal H}),
\end{equation}
then the function $\,t\mapsto (w(t),z(t))_{\cal H}\,$ is absolutely continuous, and for every $t_1,t_2\in [0,T]$ it holds
the formula
\begin{equation}
\label{leibniz2}
\int_{t_1}^{t_2}\bigl[(\partial_t w(t),z(t))_{\cal H}\,+\,\langle \partial_t z(t),w(t)\rangle_{{\cal V}}\bigr]\,dt
\,=\,(w(t_2),z(t_2))_{{\cal H}}-(w(t_1),z(t_1))_{{\cal H}},
\end{equation}
where $\,(\cdot,\cdot)_{\cal H}\,$ denotes the inner product in ${\cal H}$.
We now insert $v=\mathbf{1}$ in \eqref{adj2} and integrate the resulting identity with respect to time
over $[t,T]$. Using \eqref{leibniz2} formally (this will later be justified by the regularity properties of the involved
functions), we then obtain for every $t\in [0,T]$ the
representation formula
\gianni{%
\begin{align}
\label{meanpq}
\mbox{mean}\,(p^\alpha(t)+\tauq^\alpha(t))\,&= \,\mbox{mean}\,(g_1^\alpha)\,+\,\frac 1{|\Omega|}
\int_t^T\!\!\int_\Omega(g_2^\alpha-\psi_1^\alpha\,q^\alpha-\psi_2^\alpha\,q^\alpha),
\end{align}
}%
where the left-hand side\ equals $\,\mbox{mean}\,(p^\alpha(t))\,$ for almost every $\,t\in (0,T)$ \gianni{by~\eqref{meanqal}}.
In view of this identity,
we cannot expect the bound \eqref{lgq} to hold also in this case: indeed, due to the presence of the
term $\,-\int_t^T\!\!\int_\Omega\psi_2^\alpha\,q^\alpha$ on the right-hand side\ of \eqref{meanpq}, we cannot hope
to be able to control the mean value of $\,p^\alpha$ independently of $\alpha>0$.
Nevertheless, a proper solution to \eqref{adj1}--\eqref{adj3} exists also in this case.
To this end, we eliminate $\,\mbox{mean}\,(p^\alpha)\,$ from the problem, following a strategy
introduced in \cite{CFGS1} and~\cite{CGSAMO}. We put
\begin{equation}\label{defH0}
H_0:=\{v\in H: \mbox{\,mean}(v)=|\Omega|^{-1}(v,{\mathbf{1}})=0\}.
\end{equation}
Then $\,H=H_0\oplus \mbox{span}\{{\mathbf{1}}\}$, and we have (cf.~\eqref{defVrpos}) that
$\,V_0^r=V_A^r\cap H_0\,$ for $\lambda_1=0$. Moreover, the
linear operator $\,A_0^{2r}=A^{2r}_{|V_0^{2r}}\,$ is a topological isomorphism from
$V_0^{2r}$ onto~$H_0$, where we have the representation formulas
\begin{align}
\label{arnull}
&A_0^{2r}v\,=\,A^{2r}v\,=\,\sum_{j=2}^\infty \lambda_j^{2r}(v,e_j)e_j \quad\forall \,v\in V_0^{2r},\\
\label{a-rnull}
&A_0^{-2r}v\,:=\,(A_0^{2r})^{-1}v\,=\, \sum_{j=2}^\infty \lambda_j^{-2r}(v,e_j)e_j \quad\forall \,v\in H_0.
\end{align}
Moreover, with
\Begin{equation}\label{defh10}
H^{1,0}(\Omega):=\Hx 1 \cap H_0,
\End{equation}
we have (cf.~\cite[Sect.~5]{CGS19}) that $\,(H^{1,0}(\Omega),H_0, (H^{1,0}(\Omega))^*)\,$ is a
Hilbert triple with dense, continuous, and compact embeddings.
Now observe that\, $A^r(\mbox{mean}\,(p^\alpha(t))\mathbf{1})=\mbox{mean}\,(p^\alpha(t))\, A^r{\mathbf{1}}=0$,
and thus \gianni{\eqref{adj1} becomes}
\begin{align}\label{ll5}
&(A^r(p^\alpha(t)-\mbox{mean}(p^\alpha(t))\mathbf{1}),A^r v)\,=\,(q^\alpha(t),v) \non\\
&\quad\pier{\mbox{for a.e. \,$t\in (0,T)$\, and every $v\in V_A^r$}.}
\end{align}
Since $\,p^\alpha(t)-\mbox{mean}(p^\alpha(t))\mathbf{1}\in H_0\,$ for almost every $t\in (0,T)$, this is equivalent to
\begin{equation}
\label{ll6}
A_0^{2r}(p^\alpha-\mbox{mean}(p^\alpha)\mathbf{1})=q^\alpha \quad\mbox{and}\quad p^\alpha-\mbox{mean}(p^\alpha)\mathbf{1}\,=\,A_0^{-2r}q^\alpha .
\end{equation}
At this point, we are able to state the existence result for the system \eqref{adj1}--\eqref{adj2}
in the case $\,\lambda_1=0$ by adapting the results established in \cite[Sect.~5]{CGS19} to the
present situation. We then can infer that there
exists a unique solution $(p^\alpha,q^\alpha)$ such that
\begin{align}
\label{hugo1}
&A_0^{-2r}q^\alpha\in L^\infty(0,T;V_0^{2r}),\\[1mm]
\label{hugo2}
&q^\alpha\in L^\infty(0,T;H_0)\cap L^2(0,T;H^{1,0}(\Omega)),\\[1mm]
\label{hugo3}
&(A_0^{-2r}+\tau I)q^\alpha\in H^1(0,T;(H^{1,0}(\Omega))^*),
\end{align}
as well as
\begin{align}
&\mbox{mean}(p^\alpha+\tauq^\alpha) \,\mbox{ satisfies }\,\eqref{meanpq}\,\mbox{ for every }\,t\in [0,T],\\[2mm]
\label{hugo5}
&p^\alpha-\mbox{mean}(p^\alpha)\mathbf{1}=A_0^{-2r}q^\alpha,\\[1mm]
\label{hugo6}
&\left\langle -\partial_t\bigl(A_0^{-2r}+\tau I\bigr)q^\alpha(t),v\right\rangle_{H^{1,0}(\Omega)}
\,+\,\int_\Omega\nablaq^\alpha(t)\cdot\nabla v \,+\,\bigl((\psi_1^\alpha(t)+\gianni{\psi_2^\alpha}(t))\,q^\alpha(t),v\bigr)
\non\\[1mm]
&\quad =\,(g_2^\alpha(t),v)\quad\mbox{for a.e. \,$t\in (0,T)$\, and every }\,v\in H^{1,0}(\Omega),
\\[2mm]
\label{hugo7}
&\left\langle (A_0^{-2r}+\tau I)q^\alpha(T),v \right\rangle_{H^{1,0}(\Omega)}
\,=\,\bigl(g_1^\alpha-\mbox{mean}(g_1^\alpha)\mathbf{1},v)\quad\mbox{for all }\,v\in H^{1,0}
(\Omega).
\end{align}
Notice that, by \eqref{hugo3}, we have $\,(A_0^{-2r}+\tau I)q^\alpha \in C^0([0,T];(H^{1,0}(\Omega))^*)$,
which gives the endpoint condition \eqref{hugo7} a proper meaning: indeed, \eqref{hugo7}
means that $\,(A_0^{-2r}+\tau I)q^\alpha(T)=g_1^\alpha-\mbox{mean}(g_1^\alpha)\mathbf{1}\,$ in
$\,(H^{1,0}(\Omega))^*$, where the right-hand side\ belongs to~$H_0$. Now observe that the operator
\Begin{equation}
\label{hugo8}
(A_0^{-2r}+\tau I)v\,=\,\sum_{j=2}^\infty (\lambda_j^{-2r}+\tau)(v,e_j)e_j \quad\forall \,v\in H_0
\End{equation}
is a topological isomorphism from $H_0$ into itself with the inverse
\Begin{equation}
\label{hugo9}
(A_0^{-2r}+\tau I)^{-1}v\,=\,\sum_{j=2}^\infty (\lambda_j^{-2r}+\tau)^{-1}(v,e_j)e_j
\quad\forall \,v\in H_0.
\End{equation}
Hence, also $\,q^\alpha(T)=(A_0^{-2r}+\tau I)^{-1}(g_1^\alpha-\mbox{mean}(g_1^\alpha)\mathbf{1})\,$
has a proper meaning as an element of~$H_0$.
Next, we consider the mapping
\begin{align}
\label{hugo10}
&(A_0^{-2r}+\tau I)^{1/2}v\,=\,\sum_{j=2}^\infty (\lambda_j^{-2r}+\tau)^{1/2}(v,e_j)e_j\quad\forall\,v\in H_0.
\end{align}
It is readily seen that the estimates \eqref{lg5}--\eqref{lg7} have the analogues
\begin{align}
\label{hugo11}
&\|(A_0^{-2r}+\tau I)^{1/2}v\|^2\,\ge\,\tau\,\|v\|^2\quad\forall\,v\in H_0,\\[1mm]
\label{hugo12}
&\|(A_0^{-2r}+\tau I)^{1/2}v\|^2\,\le\,\gianni{(\lambda_2^{-2r}+\tau)}\,\|v\|^2\quad\forall\,v\in H_0,\\[1mm]
\label{hugo13}
&\|(A_0^{-2r}+\tau I)^{-1}v\|^2\,\le\,\tau^{-2}\,\|v\|^2\quad\forall\,v\in H_0.
\end{align}
Now observe that for a.e. $t\in(0,T)$ it holds that
\begin{equation}
\label{hugo14}
-\langle (A_0^{-2r}+\tau I)\,q^\alpha(t),q^\alpha(t)\rangle_{H^{1,0}(\Omega)}\,=
\,-\,\frac 12\,\frac d{dt}\left\|(A^{-2r}_0+\tau I)^{1/2}
q^\alpha(t)\right\|^2 .
\end{equation}
At this point, we insert $\,v=q^\alpha(t)\in H^{1,0}(\Omega)$ in \eqref{hugo6} and integrate over $[t,T]$, where $t\in [0,T)$, to recover the
identity \eqref{lg3}, only that in the first term the expression $A^{-2r}$ and
the dual pairing between $\Hx 1^*$ and $\Hx 1$ are replaced by $A_0^{-2r}$ and the
dual pairing between $(H^{1,0}(\Omega))^*$ and $H^{1,0}(\Omega)$, respectively.
Again, the third summand on the left-hand side\ is nonnegative, and the right-hand side\ is bounded by the expression
\eqref{Paul}. Moreover, the first summand on the left-hand side,
which we denote by $I_1^\alpha(t)$, can by \eqref{hugo12} and \eqref{hugo13} be estimated as follows:
\begin{align}
\label{hugo15}
I_1^\alpha(t)&\,=\,\frac 12 \left\|(A_0^{-2r}+\tau I)^{1/2}q^\alpha(t)\right\|^2 \,-\,
\frac 12 \left\|(A_0^{-2r}+\tau I)^{1/2}q^\alpha(T)\right\|^2 \,\ge\,
\frac \tau 2\,\|q^\alpha(t)\|^2-\frac 12\,C_6.
\end{align}
At this point, we can again employ Gronwall's lemma to conclude the
estimate
\begin{equation}
\label{llq}
\|p^\alpha-\mbox{mean}(p^\alpha)\mathbf{1}\|_{L^\infty(0,T;V_0^{2r})}\,+\,
\|q^\alpha\|_{L^\infty(0,T;H_0)\cap L^2(0,T;H^{1,0}(\Omega))}\,
\le\,C_7 \quad\forall\,\juerg{\alpha\in (0,1]},
\end{equation}
which is the sought analogue of \eqref{lgq}.
\vspace{3mm}
In the following, we complement \eqref{lgq} and \eqref{llq} by further estimates.
We treat the two cases $\lambda_1>0$ and $\lambda_1=0$ simultaneously, where it is understood that
the spaces $V_0^r$ and the operators $A_0^r$ are defined as in \eqref{defVrpos} and \eqref{defAz}, respectively.
We now introduce the space
\begin{equation}
\label{defZ}
{\cal Z}:=\left\{
\begin{array}{ll}
\{v\in H^1(0,T;H)\cap L^2(0,T;\Hx 1):v(0)=0\}&\mbox{if $\,\lambda_1>0$}\\[1mm]
\{v\in H^1(0,T;H_0)\cap L^2(0,T;H^1(\Omega)):v(0)=0\}&\mbox{if $\,\lambda_1=0$}
\end{array}
\right. ,
\end{equation}
which is a Hilbert space when endowed with its natural inner product and norm.
Moreover, setting
\begin{equation}
\label{defG}
G=H \quad\mbox{for }\,\lambda_1>0\quad\mbox{and}\quad G=H_0\quad\mbox{for }\, \lambda_1=0,
\end{equation}
we see that the embedding $\,{\cal Z}\subset C^0([0,T];G)\,$ is continuous.
\gianni{\pier{Furthermore,} we also have the dense and continuous embeddings ${\cal Z}\subset L^2(0,T;G)\subset{\cal Z}^*$},
where it is understood that
\begin{equation}
\label{dualZ}
\langle v,z\rangle_{\cal Z}=\int_0^T(v(t),z(t))\,dt \quad\mbox{for all $\,z\in {\cal Z}\,$ and \,$v\in
L^2(0,T;G)$.}
\end{equation}
In order to avoid to have to distinguish between the two cases, we employ in the following
the same notation $\,\langle\cdot,\cdot\rangle\,$ for the dual pairings
$\,\langle\cdot,\cdot\rangle_{\Hx 1}\,$ and $\,\langle\cdot,\cdot\rangle_{H^{1,0}(\Omega)}$,
where the former corresponds to the case $\lambda_1>0$ and the latter to the case $\lambda_1=0$.
At this point, \gianni{by recalling \accorpa{regp}{regpq} for $\lambda_1>0$ and \accorpa{hugo1}{hugo3} for $\lambda_1=0$},
we may employ the integration-by-parts formula \eqref{leibniz2} \gianni{with $z=A_0^{-2r}q^\alpha+\tauq^\alpha$}
to conclude that for every $v\in {\cal Z}$ it holds that
\begin{align}
\label{absch1}
\langle -\partial_t(p^\alpha+\tauq^\alpha\pier{)},v\rangle_{{\cal Z}}\,&=\,-\int_0^T\langle \partial_t(A_0^{-2r}q^\alpha(t)+\tauq^\alpha(t),v(t)\rangle\,dt
\non\\
&=\,\int_0^T(\partial_t v(t),(A_0^{-2r}+\tau I)q^\alpha(t))\,dt\,-\,(g_1^\alpha,v(T))\non\\[1mm]
&\le\,\|\partial_t v\|_{L^2(0,T;H)}\,\|(A_0^{-2r}+\tau I)q^\alpha\|_{L^2(0,T;H)}\,+\,\|g_1^\alpha\|_H\,\|v(T)\|_H\non\\[1mm]
&\le\,C_7\,\|v\|_{\cal Z},
\end{align}
which implies that
\begin{equation}
\label{absch2}
\|\partial_t(p^\alpha+\tauq^\alpha)\|_{{\cal Z}^*}\,\le\,C_7 \quad\forall\,\juerg{\alpha\in(0,1]}.
\end{equation}
Now observe that for any $v\in {\cal Z}$ it holds that
\begin{align*}
&\int_0^T(\nablaq^\alpha(t),\nabla v(t))\,dt\,+\int_0^T(\psi_1^\alpha(t)q^\alpha(t),v(t))\,dt\,
-\int_0^T (g_2^\alpha(t),v(t))\,dt
\\[1.5mm]
&\le\,\|q^\alpha\|_{L^2(0,T;\Hx 1)}\,\|v\|_{\cal Z}\,+\,C_8\,\|q^\alpha\|_{L^2(0,T;H)}\,\|v\|_{L^2(0,T;H)}\,
+\,C_9\,\|v\|_{L^2(0,T;H)}\\[1.5mm]
&\le\,C_{10}\,\|v\|_{\cal Z},
\end{align*}
and it follows from comparison in \eqref{adj2} that, with $\,\Lambda^\alpha:=\psi_2^\alpha q^\alpha=
\phi(\alpha)h''(\bar y^{\alpha})q^\alpha$,
\begin{equation}
\label{lamal}
\|\Lambda^\alpha\|_{{\cal Z}^*}\,\le\,C_{11} \quad\forall\,\juerg{\alpha\in(0,1]}.
\end{equation}
At this point, we choose any sequence $\{\alpha_n\}$ such that $\alpha_n\searrow0$. We infer from Theorem 3.3
and Theorem 4.5 that, at least for a subsequence which is again indexed by~$\,n$,
\begin{align}
\label{ustark}
&{\bar u}^{\alpha_n}\to\bar u\quad\mbox{strongly in }\,L^2(Q),\\[0.5mm]
\label{yweak}
&{\bar y}^{\alpha_n}\to\bar y \quad\mbox{weakly-star in }\,W^{1,\infty}(0,T;H)\cap H^1(0,T;\Hx 1).
\end{align}
By virtue of \cite[Sect.~8,~Cor.~4]{Simon}, we may also assume that
\begin{align}
\label{stark1}
&{\bar y}^{\alpha_n}\to\bar y \quad\mbox{strongly in }\,C^0([0,T];L^p(\Omega)) \quad\forall\,p\in [1,6),
\end{align}
which entails, in particular, that
\begin{align}
\label{stark2}
&f_2''({\bar y}^{\alpha_n})\to f_2''(\bar y) \quad\mbox{strongly in }\,C^0([0,T];L^p(\Omega)) \quad\forall
\,p\in [1,6),\\[0.5mm]
\label{stark3}
&g_1^{\alpha_n}\to \beta_1(\bar y(T)-y_\Omega)\quad\mbox{strongly in }\,H,\\[0.5mm]
\label{stark4}
&g_2^{\alpha_n}\to \beta_2(\bar y-y_Q)\quad\mbox{strongly in }\,L^2(Q).
\end{align}
Moreover, by virtue of the estimates \eqref{lgq}, \eqref{llq}, \eqref{absch2}, and \eqref{lamal},
there are limits $\,\zeta,\bar q,\Lambda$ such that, at least for another subsequence which is again
indexed by $n$,
\begin{align}
\label{limzeta}
\partial_t(A_0^{-2r}+\tau I)q^{\alpha_n}&\to\zeta\quad\mbox{weakly in }\,{\cal Z}^*,\\[0.5mm]
\label{limq}
q^{\alpha_n}&\to \bar q\quad\mbox{weakly-star in }\,L^\infty(0,T;G)\cap L^2(0,T;\Hx 1),\\[0.5mm]
\label{limA0q}
A_0^{-2r}q^{\alpha_n}&\to A_0^{-2r}\bar q\quad\mbox{weakly-star in }\,L^\infty(0,T;V_0^{2r}),\\[0.5mm]
\label{limlam}
\Lambda^{\alpha_n}&\to\Lambda\quad\mbox{weakly in }\,{\cal Z}^*.
\end{align}
The limit $\zeta\in {\cal Z}^*$ is readily identified. Indeed, by formula \eqref{leibniz2} we have,
for every $v\in{\cal Z}$,
\begin{align}
\label{findzeta}
&\lim_{n\to\infty}\int_0^T\langle\partial_t(A_0^{-2r}+\tau I)q^{\alpha_n}(t),v(t)\rangle\,dt\non\\
&=\,\lim_{n\to\infty}\Bigl[-\int_0^T\bigl(\partial_t v(t),(A_0^{-2r}+\tau I)q^{\alpha_n}(t)\bigr)\,dt
+\,(g_1^{\alpha_n},v(T))\Bigr]\non\\
&=\,-\int_0^T\bigl(\partial_t v(t),(A_0^{-2r}+\tau I)\bar q(t)\bigr)\,dt\,+\,(\beta_1(\bar y(T)-y_\Omega)
,v(T))\,=:\,\langle\zeta,v\rangle_{{\cal Z}}.
\end{align}
Moreover, by combining the strong convergence \eqref{stark2} with \eqref{limq}, it is easily checked that
\begin{equation}
\label{yeah}
f_2''({\bar y}^{\alpha_n})\,q^{\alpha_n}\to f_2''(\bar y)\,\bar q\quad\mbox{weakly in }\,L^2(Q).
\end{equation}
At this point, we recall that
$$
\langle -\partial_t(p^\alpha+\tauq^\alpha)(t),v(t)\rangle\,=\,\langle -\partial_t(A_0^{-2r}+\tau I)q^\alpha(t),v(t)\rangle
\quad\mbox{for a.e.$\,t\in (0,T)\,$ and \,$v\in{\cal Z}$.}
$$
We now write the adjoint system \eqref{adj1}--\eqref{adj2} for $\alpha=\alpha_n$, insert $v=v(t)$
for an arbitrary $v\in {\cal Z}$, integrate the resulting identity with respect to time over $[0,T]$, and pass to
the limit as $n\to\infty$. It then results the following equation:
\begin{align}
\label{limeq}
\langle \Lambda,v\rangle_{\cal Z}\,&=\,-\int_0^T\bigl(\partial_t v(t),(A_0^{-2r}+\tau I)\bar q(t)\bigr)\,dt
\,+\beta_1(\bar y(T)-y_\Omega,v(T))\,\non\\
&\quad\,\,\pier{{}-\int_0^T\!\!\int_\Omega\nabla\bar q\cdot\nabla v}\,+\int_0^T\!\!\int_\Omega\bigl(\beta_2(\bar y-y_Q)-f_2''(\bar y)\bar q\bigr)\,v \quad\forall\,v\in{\cal Z}.
\end{align}
Finally, we need to identify the variational inequality relating the optimal control to the adjoint variables.
In this regard, we can infer, with the same argument as in the proof of \cite[Thm.~5.9]{CGS19}, that
the optimal control $\bar u^{\alpha_n}$ satisfies the variational inequality
\begin{align}\label{vugtil}
\int_0^T\!\!\int_\Omega(q^{\alpha_n}+\beta_3\bar u^{\alpha_n} + \bar u^{\alpha_n}-\bar u)
(v-\bar u^{\alpha_n})\,\ge\,0 \quad\forall\,v\in{\mathcal{U}}_{\rm ad}.
\end{align}
Taking the limit as $n\to\infty$ in \eqref{vugtil}, and using \eqref{ustark} and \gianni{\eqref{limq}}, we
arrive at the necessary optimality condition
\begin{align}
\label{vugo}
\int_0^T\!\!\int_\Omega (\bar q\,+\,\beta_3\,\bar u)(v-\bar u)\,\ge\,0\quad\forall\,v\in{\mathcal{U}}_{\rm ad}.
\end{align}
From the above considerations, we can conclude the following first-order necessary
\gianni{optimality} conditions for the optimal control problem~(${\mathcal{CP}}_0$):
\Begin{theorem}
Suppose that the conditions \pier{{\bf (A1)}--{\bf {(A6)}}, {\bf (A8)}, {\bf {(A9)}}} are satisfied, and let $\,\bar u\in{\mathcal{U}}_{\rm ad}\,$ be an optimal control
for {\rm (${\mathcal{CP}}_0$)} with associated state $\,(\bar \mu,\bar y)\,$ where $\,\bar y=\mathcal{S}_0(\bar u)$.
Then there exist $(\bar q,\Lambda)$ such that the
following statements hold true:
\noindent
{\rm (i)} We have the regularity properties
\begin{align}
\bar q\in L^\infty(0,T;G)\cap L^2(0,T;\Hx 1), \quad \Lambda\in {\cal Z}^*.
\end{align}
\noindent
{\rm (ii)} \,\,The adjoint equation \eqref{limeq} is fulfilled.
\noindent
{\rm (iii)} \,The necessary optimality condition \eqref{vugo} is satisfied.
\End{theorem}
\Begin{remark}\rm
From \eqref{vugo} we infer that\pier{, in the case $\beta_3>0$, $\,\bar u\,$ is} nothing but the
$L^2(Q)$-orthogonal projection of \,$-\beta_3^{-1}\,q\,$ onto ${\mathcal{U}}_{\rm ad}$.
\End{remark}
\Begin{remark}\rm
Unfortunately, we are unable to derive any complementarity slackness
conditions for the Lagrange multiplier $\Lambda$. Indeed, while it is easily seen that
$$
\liminf_{n\to\infty}\int_0^T\!\!\int_\Omega\Lambda^{\alpha_n}\,q^{\alpha_n}\,=\,\liminf_{n\to\infty}\int_0^T\!\!\int_\Omega\frac
{2 \phi(\alpha_n)}{1-(\bar y^{\alpha_n})^2}\,|q^{\alpha_n}|^2\,\ge\,0
\quad \forall\,n\in{\mathbb{N}},
$$
the convergence properties \eqref{yweak} and \eqref{limq} do not suffice to conclude that
$\,\langle \Lambda,\bar q\rangle_{\cal Z}\,\ge\,0$.
\End{remark}
\section*{Acknowledgments}
\pier{This research was supported by the Italian Ministry of Education,
University and Research~(MIUR): Dipartimenti di Eccellenza Program (2018--2022)
-- Dept.~of Mathematics ``F.~Casorati'', University of Pavia.
In addition, PC and CG gratefully acknowledge some other
financial support from the MIUR-PRIN Grant 2015PA5MP7 ``Calculus of Variations'',}
the GNAMPA (Gruppo Nazionale per l'Analisi Matematica,
la Probabilit\`a e le loro Applicazioni) of INdAM (Isti\-tuto
Nazionale di Alta Matematica) and the IMATI -- C.N.R. Pavia.
\vspace{3truemm}
{\small
\Begin{thebibliography}{10}
\bibitem{AM}
M. Ainsworth, Z. Mao, Analysis and approximation of a fractional
Cahn--Hilliard equation. SIAM J. Numer. Anal. {\bf 55} (2017),
1689-1718.
\bibitem{AkSeSchi}
G. Akagi, G. Schimperna, A. Segatti,
Fractional Cahn--Hilliard, Allen--Cahn and porous medium equations.
J. Differ. Equations {\bf 261} (2016), 2935-2985.
\juerg{
\bibitem{AKW}
H. Antil, R. Khatri, M. Warma, External optimal control of nonlocal PDEs.
Preprint arXIv:1811.04515 [math.OC] (2018), pp. 1-30.
}
\bibitem{HO1}
H. Antil, E. Ot\'arola,
A FEM for an optimal control problem of fractional
powers of elliptic operators. SIAM J. Control Optim. {\bf 53} (2015), 3432-3456.
\bibitem{HO2}
H. Antil, E. Ot\'arola, An a posteriori error analysis
for an optimal control problem involving the fractional Laplacian.
IMA J. Numer. Anal. {\bf 38} (2018), 198-226.
\bibitem{HAS}
H. Antil, E. Ot\'arola, A.J. Salgado, A space-time fractional optimal control problem:
analysis and discretization.
SIAM J. Control Optim. {\bf 54} (2016), 1295-1328.
\juerg{
\bibitem{APR}
H. Antil, J. Pfefferer, S. Rogers, Fractional operators with inhomogeneous boundary conditions:
analysis, control and discretization. Preprint arXiv:1703.05256 [math.NA] (2017), pp. 1-38.%
}
\juerg{%
\bibitem{AW1}
H. Antil, M. Warma, Optimal control of fractional semilinear PDEs. Preprint arXiv:1712.04336
[math.OC] (2017), pp. 1-27.%
}
\juerg{
\bibitem{AW2}
H. Antil, M. Warma, Optimal control of the coefficient for the regional fractional $p$--Laplace equation: approximation and convergence. AIMS Math. Control Related Fields (2018),
https://doi.org/10.3934/mcrf.2019001.
}
\bibitem{Barbu}
V. Barbu,
``Nonlinear Differential Equations of Monotone Type in Banach Spaces''.
Springer, London, New York, 2010.
\bibitem{BDM1}
T. Biswas, S. Dharmatti, M.T. Mohan, Pontryagin's maximum principle for optimal
control of the nonlocal Cahn--Hilliard--Navier--Stokes systems in two dimensions.
Preprint arXiv:1802.08413 [math.OC] (2018), pp.~1-36.
\bibitem{BDM2}
T. Biswas, S. Dharmatti, M.T. Mohan, Maximum principle and data assimilation problem
for the optimal control problems governed by 2D nonlocal Cahn--Hilliard--Navier--Stokes equations.
Preprint arXiv:1803.11337 [math.AP] (2018), pp.~1-26.
\bibitem{CFGS1}
P. Colli, M.H. Farshbaf-Shaker, G. Gilardi, J. Sprekels,
Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition
and double obstacle potentials. SIAM J.
Control Optim. {\bf 53} (2015), 2696-2721.
\bibitem{CFGS2}
P. Colli, M.H. Farshbaf-Shaker, G. Gilardi, J. Sprekels,
Second-order analysis of a boundary
control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions.
Ann. Acad. Rom. Sci. Ser. Math. Appl. {\bf 7}
(2015), 41-66.
\bibitem{CFGS3}
P. Colli, M.H. Farshbaf-Shaker, J. Sprekels,
A deep quench approach to the optimal control of
an Allen--Cahn equation with dynamic boundary conditions and double obstacles.
Appl. Math. Optim. {\bf 71}
(2015), 1-24.
\bibitem{CGRS}
P. Colli, G. Gilardi, E. Rocca, J. Sprekels,
Optimal distributed control of a diffuse interface model
of tumor growth.
Nonlinearity {\bf 30}
(2017), 2518-2546.
\bibitem{CGSAdvan}
P. Colli, G. Gilardi, J. Sprekels,
A boundary control problem for the pure Cahn--Hilliard equation
with dynamic boundary conditions.
Adv. Nonlinear Anal. {\bf 4}
(2015), 311-325.
\bibitem{CGSAIMS}
P. Colli, G. Gilardi, J. Sprekels,
Distributed optimal control of a nonstandard nonlocal phase field
system. AIMS Mathematics {\bf 1}
(2016), 246-281.
\bibitem{CGSAMO}
P. Colli, G. Gilardi, J. Sprekels,
A boundary control problem for the viscous Cahn--Hilliard equation
with dynamic boundary conditions.
Appl. Math. Optim. {\bf 72}
(2016), 195-225.
\bibitem{CGSEECT}
P. Colli, G. Gilardi, J. Sprekels,
Distributed optimal control of a nonstandard nonlocal phase field
system with double obstacle potential.
Evol. Equ. Control Theory {\bf 6}
(2017), 35-58.
\bibitem{CGSAnnali}
P. Colli, G. Gilardi, J. Sprekels, On a Cahn--Hilliard system with convection
and dynamic boundary conditions. Ann. Mat. Pura Appl.~(4) {\bf 197}
(2018), 1445-1475.
\bibitem{CGSSIAM18}
P. Colli, G. Gilardi, J. Sprekels,
Optimal velocity control of a viscous Cahn--Hilliard system with convection and dynamic boundary conditions.
SIAM J. Control Optim. {\bf 56} (2018), 1665-1691.
\bibitem{CGSConvex}
P. Colli, G. Gilardi, J. Sprekels,
Optimal velocity control of a convective Cahn--Hilliard system with double
obstacles and dynamic boundary conditions: a `deep quench' approach.
\pier{To appear in J. Convex Anal. {\bf 26} (2019) (see also Preprint arXiv: 1709.03892
[math.AP] (2017), pp.~1-30).}
\bibitem{CGS18}
P. Colli, G. Gilardi, J. Sprekels,
Well-posedness and regularity for a generalized fractional Cahn--Hilliard system.
Preprint arXiv:1804.11290 [math.AP] (2018), \pier{pp.~1-36}.
\bibitem{CGS19}
P. Colli, G. Gilardi, J. Sprekels,
Optimal distributed control of a generalized fractional Cahn--Hilliard system.
\juerg{Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7.}
\bibitem{CS}
P. Colli, J. Sprekels, Optimal boundary control of a nonstandard Cahn--Hilliard system
with dynamic boundary condition and double obstacle inclusions. In:
``Solvability, Regularity, \juerg{and}
Optimal Control of Boundary Value Problems for PDEs'' (P. Colli, A. Favini, E. Rocca,
G. Schimperna, J. Sprekels, eds.), Springer INdAM Series Vol. {\bf 22}, pp. 151-182,
Springer 2017.
\bibitem{Duan}
N. Duan, X. Zhao,
Optimal control for the multi-dimensional viscous Cahn--Hilliard equation.
Electron. J. Differ. Equations 2015, Paper No. 165, 13 pp.
\bibitem{FGG}
S. Frigeri, C.G. Gal, M. Grasselli,
On nonlocal Cahn--Hilliard--Navier--Stokes systems in two dimensions.
J. Nonlinear Sci. {\bf 26}
(2016), 847-893.
\bibitem{FGGS}
S. Frigeri, C.G. Gal, M. Grasselli, J. Sprekels,
\juerg{Two-dimensional nonlocal Cahn--Hilliard--Navier--Stokes
systems with variable viscosity, degenerate mobility and singular potential. To appear in
Nonlinearity (see also WIAS Preprint Series No. 2309, Berlin 2016, \pier{pp.~1-56}).}
\bibitem{FGS}
S. Frigeri, M. Grasselli, J. Sprekels, Optimal distributed control of two-dimensional
nonlocal Cahn--Hilliard--Navier--Stokes systems with degenerate mobility and singular potential.
\juerg{Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9524-7.}
\bibitem{FRS}
S. Frigeri, E. Rocca, J. Sprekels,
Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes
system in two dimensions. SIAM J. Control Optim.
{\bf 54} (2016), 221-250.
\bibitem{Fukao}
T. Fukao, N. Yamazaki, A boundary control problem for the equation and dynamic
boundary condition of Cahn--Hilliard type. In: ``Solvability, Regularity, \juerg{and}
Optimal Control of Boundary Value Problems for PDEs'' (P. Colli, A. Favini, E. Rocca,
G. Schimperna, J. Sprekels, eds.), Springer INdAM Series Vol. {\bf 22}, pp. 255-280, Springer 2017.
\bibitem{GalDCDS}
C.G. Gal,
On the strong-to-strong interaction case
for doubly nonlocal Cahn--Hilliard equations.
Discrete Contin. Dyn. Syst. {\bf 37} (2017), 131-167.
\bibitem{GalEJAM}
C.G. Gal,
Non-local Cahn--Hilliard equations with fractional dynamic boundary conditions. European J. Appl. Math. {\bf 28} (2017), 736-788.
\bibitem{GalAIHP}
C.G. Gal,
Doubly nonlocal Cahn--Hilliard equations.
Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire {\bf 35} (2018), 357-392.
\bibitem{GV}
C. Geldhauser, E. Valdinoci,
Optimizing the fractional power in a model with stochastic PDE constraints.
Preprint arXiv:1703.09329v1 [math.AP] (2017), pp. 1-18.
\bibitem{HHCK}
M. Hinterm\"uller, M. Hinze, C. Kahle, T. Keil,
A goal-oriented dual-weighted adaptive finite element approach for the optimal control
of a nonsmooth Cahn--Hilliard--Navier--Stokes system.
Optimization and Engineering (2018), https://doi.org/10.1007/s11081-018-9393-6.
\bibitem{HKW}
M. Hinterm\"uller, T. Keil, D. Wegner,
Optimal control of a semidiscrete Cahn--Hilliard--Navier--Stokes
system with non-matched fluid densities. SIAM J. Control Optim. {\bf 55} (2018),
1954-1989.
\bibitem{HW1}
M. Hinterm\"uller, D. Wegner,
Distributed optimal control of the Cahn--Hilliard system including the case
of a double obstacle homogeneous free energy density. SIAM J. Control Optim.
{\bf 50} (2012), 388-418.
\bibitem{HW2}
M. Hinterm\"uller, D. Wegner,
Optimal control of a semidiscrete Cahn--Hilliard--Navier--Stokes system.
SIAM J. Control Optim. {\bf 52} (2014), 747-772.
\bibitem{HW3}
M. Hinterm\"uller, D. Wegner,
Distributed and boundary control problems for the semidiscrete Cahn--Hilliard/Navier--Stokes
system with nonsmooth Ginzburg--Landau energies. In: ``Topological Optimization
and Optimal Transport'', Radon Series on Computational and Applied Mathematics vol.
{\bf 17} (2017), pp. 40-63.
\bibitem{Medjo}
T. Tachim Medjo,
Optimal control of a Cahn--Hilliard--Navier--Stokes model
with state constraints.
J. Convex Anal.
{\bf 22} (2015), 1135-1172.
\bibitem{RS}
E. Rocca, J. Sprekels,
Optimal distributed control of a nonlocal convective Cahn--Hilliard equation by
the velocity in three dimensions. SIAM J. Control Optim. {\bf 53} (2015), 1654-1680.
\bibitem{Simon}
J. Simon,
Compact sets in the space $L^p(0,T; B)$.
Ann. Mat. Pura Appl.~(4)
{\bf 146}, (1987), 65-96.
\bibitem{SV}
J. Sprekels, E. Valdinoci, A new class of identification problems: optimizing the
fractional order in a nonlocal evolution equation. SIAM J. Control Optim. {\bf 55}
(2017), 70-93.
\bibitem{WN}
Q. F. Wang, S.-i. Nakagiri,
Weak solutions of Cahn--Hilliard equations having forcing terms and
optimal control problems. Mathematical models in functional equations
(Kyoto, 1999), Surikaisekikenkyusho Kokyuroku No. 1128 (2000), 172-180.
\bibitem{ZL1}
X.P. Zhao, C.C. Liu,
Optimal control of the convective Cahn--Hilliard equation. Appl. Anal.
{\bf 92} (2013), 1028-1045.
\bibitem{ZL2}
X.P. Zhao, C.C. Liu,
Optimal control of the convective Cahn--Hilliard equation in 2D case. Appl.
Math. Optim. {\bf 70}
(2014), 61-82.
\bibitem{Z}
J. Zheng, Time optimal controls of the Cahn--Hilliard equation with
internal control. Optimal Control Appl. Methods {\bf 36}
(2015), 566-582.
\bibitem{ZW}
J. Zheng, Y. Wang, Optimal control problem for Cahn--Hilliard equations with
state constraint. J. Dyn. Control Syst. {\bf 21} (2015), 257-272.
\End{thebibliography}}
\End{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,348 |
CMA Stakeholders' Consultative Paper
August 29, 2018 - 5 Minutes Read - By Mutindi Muema
The Capital Markets Authority (CMA) Stakeholders' Consultative Paper on Policy Framework for Implementation of a Regulatory Sandbox to Support Financial Technology ("Fintech") Innovation in the capital markets in Kenya ("Fintech Regulatory Sandbox Policy Paper") seeks to introduce the concept of a Fintech regulatory sandbox in Kenya.
It provides the background and rationale for a regulatory sandbox for fintech innovations for the Capital Markets ("FintechCM") and a global outlook on the adoption of the Regulatory Sandbox for Fintech. Notably, while the regulated services may differ from one jurisdiction to the other, the Regulatory Sandbox has been adopted in the United Kingdom (the "UK"), Australia, Malaysia, Singapore, Abu Dhabi, Indonesia and Hong Kong, among others.
The proposed scope of the regulatory sandbox for Kenya is for it to be a tailored framework that allows firms deploying innovative technology in the financial services sector to conduct their activities in a controlled and cost-effective environment.
The benefits of such a "relaxed" framework would include: reduced time to market at potentially lower cost; better access to finance; more innovative products reaching the market; contained consequences of failure; and reaching the best solution for the customer.
In preparing this Fintech Regulatory Sandbox Policy Paper, the CMA notes that it has: jointly with other stakeholders, prepared a policy advisory paper on approach to crowdfunding oversight for the East African region; engaged with ICT business incubators; and engaged with other regulatory authorities and international standard setting bodies such as the Financial Stability Board.
The CMA's initiative is worthy to be applauded. It is hoped that it will ignite more collaboration among all the regulators of financial services in respect of fintech such that the framework is expanded to cover all innovative fintech solutions and help Kenya augment its position as Africa's most nurturing environments for financial innovation.
Notably, and a fact which is appreciated by the CMA, the impact of fintech has been more pronounced in other financial services sectors such as money remittance, insurtech, payments and e-money issuance, than in the capital markets.
Nonetheless, the CMA has highlighted the following FintechCM that could benefit from the proposed regulatory sandbox: regtech, big-data, artificial intelligence and investment tech, distributed ledger technology (blockchain technology), cryptocurrency and peer to peer finance (also referred to as crowdfunding).
Given that most financial services are not exclusively regulated by the CMA, the Fintech Regulatory Sandbox Policy Framework proposed by the CMA would be limited to fintech companies developing solutions for CMA's existing licensees or authorized institutions or solutions by CMA's existing licensees or authorized institutions.
Proposed Regulatory Sandbox Models
From the Fintech Regulatory Sandbox Policy Paper, the CMA has considered 4 regulatory sandbox models and highlighted their features, benefits and shortcomings.
The relevant models are as follows:
Industry-led Models
Virtual Sandbox – this would be an environment to enable firms to test their solutions virtually without entering the real market. An example of a virtual sandbox could be a cloud-based solution set up and equipped in collaboration between the industry, which businesses could then customize for their products or services, run tests with public data sets or data provided by other firms through the virtual sandbox, and then invite firms or even customers to try their new solution. In this environment, there would be no risk of consumer detriment, risk to market integrity or financial stability while testing. A virtual sandbox could be used by all innovators regardless of the size or whether they are authorized or not. CMA is of the view that this model would work best if introduced by industry (noting that a large number of firms already have similar testing technologies although they mostly operate separately from each other and with data only from the owners of these sandboxes), but the CMA would facilitate collaboration between interested parties and provide support when the virtual sandbox is being developed.
Sandbox Umbrella – this could be a not-for-profit company set up by industry to act as a sandbox umbrella that allows unauthorized innovators to offer their services under its shelter as appointed representatives. The umbrella company would need to be authorized with appropriate permissions and then supervised by the CMA as other authorized firms. The umbrella company would monitor its appointed representatives. Innovators would not have to apply for authorization and meet authorization requirements in their own right. The umbrella company would assess whether the firms applying to become appointed representatives are ready to test their solutions.
CMA-led Models
The CMA is considering which of the two models below it should adopt for its regulatory sandbox:
Developing a regulatory sandbox as a new regulated activity – this approach would see CMA create a sandbox regime (with new authorization requirements and rules) that would be flexible and in line with in-principle approaches to regulation. The CMA would do this through the development of a Policy Guidance Note to facilitate introduction of a regulatory sandbox into the market. This approach could allow for a streamlined authorization process and potentially less regulatory requirements to comply with when testing. The drawbacks to this option are that firms would still need to become authorized before being able to test, and it would not apply to activities that are not regulated under the Capital Markets Act (e.g. payment services, e-money and insurance services). Thus, this change could have a limited effect.
Amending the waiver test contained in specific sections of the Capital Markets Act –Under this approach, the CMA could consider changing the waiver conditions in the Capital Markets Act (if need be) to make it easier to waive rules for a firm within the sandbox. This could be achieved by introducing a new test for sandbox firms. Under this approach, waivers would enable firms to test concepts before ensuring compliance with all relevant rules (firms can start testing quicker). On the downside, the CMA's power to issue waivers is limited to the provisions of the Capital Markets Act. Legislative changes would be necessary to broaden the scope for waivers; legislative changes take significant time and resource to introduce. This approach would similarly require the CMA to develop a Policy Guidance Note to facilitate introduction of a regulatory sandbox.
High Level Comments on the Proposed Regulatory Framework
The CMA's proposed regulatory framework has adopted the essential features of the regulatory frameworks being used in the UK, Australia, Malaysia, Singapore and other jurisdictions. The proposed regulatory framework sets out the following steps: the application process, acceptance and progression, operationalization, exit strategy, expiry of approval and revocation of approval. The proposed regulatory framework is generally in line with the current best practices. However, the table below contains comments on sections that could be improved by the CMA in drafting the Policy Guidance Note: | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,673 |
\section{Introduction}
Autonomous driving has gained significant attention within the automotive research community in recent years \cite{eskandarian2019research, montanaro2018towards, kuutti2019deep}. The potential benefits in improved fuel efficiency, passenger safety, traffic flow, and ride sharing mean self-driving cars could have a significant impact on issues such as climate change, road safety, and passenger productivity \cite{eskandarian2012handbook, thrun2010toward, DepartmentforTransport2017}. Deep learning techniques have been demonstrated to be powerful tools for autonomous vehicle control, due to their capability to learn complex behaviours from data and generalise these learned rules to completely new scenarios \cite{kuutti2020survey}. These techniques can be divided in two categories based on the modularity of the system. On one hand, modular systems divide the driving task into multiple sub-tasks such as perception, planning, and control and deploy a number of sub-systems and algorithms to solve each of these tasks. On the other hand, end-to-end systems aim to learn the driving task directly from sensory measurements (e.g., camera, radar) by predicting low-level control actions.
End-to-end approaches have recently risen in popularity, due to the ease of implementation and better leverage of the function approximation of Deep Neural Networks (DNNs). However, the high level of opaqueness in DNNs is one of the main limiting factors in the use of neural network-based control techniques in safety-critical applications, such as autonomous vehicles \cite{borg2018safely, burton2017making, varshney2017safety, rajabli2020software}. As the DNNs used to control the autonomous vehicles become deeper and more complex, their learned control policies and any potential safety issues within them become increasingly difficult to evaluate. This is made further challenging by the complex environment in which autonomous vehicles have to operate in, as it is impossible to evaluate the safety of these systems in all possible scenarios they may encounter once deployed \cite{kalra2016driving, wachenfeld2017new, koopman2016challenges}. One class of solutions to introduce safety in machine learning enabled autonomous vehicles is to utilise a modular approach to autonomous driving, where the machine learning systems are used mainly in the decision making layer, whilst low-level control is handled by more interpretable rule-based systems \cite{xu2018zero, wang2019lane, arbabi2020lane}. Alternatively, safety can be guaranteed through redundancy for end-to-end approaches, where machine learning can be used for vehicle motion control with an additional rule-based virtual safety cage acting as a supervisory controller \cite{heckemann2011safe, vom2020fail}. The purpose of the rule-based virtual safety cage is to check the safety of the control actions of the machine learning system, and to intervene in the control of the vehicle if the safety rules imposed by the safety cages are breached. Therefore, during normal operation, the more intelligent machine learning-based controller is in control of the vehicle. However, if the safety of the vehicle is compromised the safety cages can step in and attempt to bring the vehicle back to a safe state through a more conservative rule-based control policy.
This work extends our previously developed Reinforcement Learning (RL) based vehicle following model \cite{kuutti2019end} and virtual safety cages \cite{kuutti2019safe}. We make important extensions to our previous works, by integrating our safety cages into the RL algorithm. The safety cages not only act as a safety function enhancing the vehicle's safety, but are also used to provide weak supervision during training, by limiting the amount of unnecessary exploration and providing penalties to the agent when breached. In this way, the vehicle can be safe during training by avoiding collisions when the RL agent takes unsafe actions. More importantly, the efficiency of the training process is improved, as the agent converges to an optimal control policy with less samples. We also compare our proposed framework on less safe agents with smaller neural networks, and show significant improvement in the final learned policies when used to train these shallow models. Our contributions can be summarised as follows:
\begin{itemize}
\item We combine the safety cages with reinforcement learning by intervening on unsafe control actions, as well as providing an additional learning signal for the agent to enable safe and efficient exploration.
\item We compare the effect of the safety cages during training for both models with optimised hyperparameters, as well as less optimised models which may require additional safety considerations.
\item We test all trained agents without safety cages enabled, in both naturalistic and adversarial driving scenarios, showing that even if the safety cages are only used during training, the models exhibit safer driving behaviour.
\item We demonstrate that by using the weak supervision from the safety cages during training, the shallow model which otherwise could not learn to drive can be enabled to learn to drive without collisions.
\end{itemize}
The remainder of this paper is structured as follows. Section \ref{sec2} discusses related work and explains the novelty of our approach. Section \ref{sec3} provides the necessary theoretical background for the reader and describes the methodology used for the safety cages and reinforcement learning technique. The results from the simulated experiments are presented and discussed in Section \ref{sec4}. Finally, the concluding remarks are presented in Section \ref{sec5}.
\section{Related Work}\label{sec2}
\subsection{Autonomous Driving}
A brief overview of relevant works in this field is given in this Section. For a more in-depth view of deep learning based autonomous driving techniques, we refer the interested readers to the review in \cite{kuutti2020survey}. One of the earliest works in neural control for autonomous driving was Pomerleau's Autonomous Land Vehicle In a Neural Network (ALVINN) \cite{pomerleau1989alvinn}, which learned to steer an autonomous vehicle by observing images from a front facing camera, using the recorded steering commands of a human driver as training data. Among the first to adapt techniques such as ALVINN to use deep neural networks, was NVIDIA's PilotNet \cite{bojarski2016end}. PilotNet was trained for lane keeping using supervised learning with a total of 72 h of recorded human driving as training data.
Since then, these works have inspired a number of deep learning techniques, with imitation learning often being the preferred learning technique. For instance, \mbox{Zhang et al. \cite{zhang2016query}} and Pan et al. \cite{pan2018agile} extended the popular Dataset Aggregation (DAgger) \cite{ross2011reduction} imitation learning algorithm to the autonomous driving domain, demonstrating that autonomous vehicle control can be learned from vision. While imitation learning based approaches have shown important progress in autonomous driving \cite{codevilla2018end, bansal2018chauffeurnet, wang2018deep, hecker2018end}, they present limitations when deployed in environments beyond the training distribution \cite{codevilla2019exploring}. These driving models relying on supervised techniques are often evaluated on performance metrics on pre-collected validation datasets \cite{xu2017end}, however low prediction error on offline testing is not necessarily correlated with driving quality \cite{codevilla2018offline}. Even when demonstrating desirable performance during closed-loop testing in naturalistic driving scenarios, imitation learning models often degrade in performance due to distributional shift \cite{ross2011reduction}, unpredictable road users \cite{kuutti2020training}, or causal confusion \cite{de2019causal} when exposed to a variety of driving scenarios.
{However, RL-based techniques have shown promising results for autonomous vehicle applications \cite{kiran2020deep, wu2020battery, wu2020batteryb}. These RL approaches are advantageous for autonomous vehicle motion control, as they can learn general driving rules, which can also adapt to new environments.} Indeed, many recent works have utilised RL for longitudinal control in autonomous vehicles with great success \cite{puccetti2019actor, chae2017autonomous, zhao2017model, li2019ecological, huang2017parameterized}. This is largely due to the fact that longitudinal control can be learned from low-dimensional observations (e.g., relative distances, velocities), which partially overcomes the sample-efficiency problem inherent in RL. Moreover, the reward function for RL is easier to define in the longitudinal control case (e.g., based on safety distances to vehicles in front). For these reasons, we focus on longitudinal control and extend on our previous work on RL-based longitudinal control in a highway driving environment \cite{kuutti2019end}.
\subsection{Safety Cages}
Virtual safety cages have been used in several cyber-physical systems to provide safety guarantees when the controller is not interpretable. The most straightforward application of such safety cages is to limit the possible actions of the controller to ensure the system is bounded to a safe operational envelope. If the controller issues commands that breach the safety cages, the safety cages step in and attempt to recover the system back to a safe state. This type of approach has been used to guarantee the safety of complex controllers in different domains such as robotics \cite{kurien1998model, crestani2015enhancing, haddadin2012making, kuffner2016virtual}, aerospace \cite{polycarpou2004neural}, and automotive \mbox{applications \cite{adler2016safety, jackson2019certified, pek2020using}.} Heckemann et al. \cite{heckemann2011safe} suggested that these safety cages could be used to ensure the safety of black box systems in autonomous vehicles by utilising the vehicle's sensors to monitor the state of the environment, and then limiting the actions of the vehicle in safety-critical scenarios. Demonstrating the effectiveness of this approach, \mbox{Adler et al. \cite{adler2016safety}} proposed five safety cages based on the Automotive Safety Integrity Levels (ASIL) defined by \mbox{ISO26262 \cite{iso26262}} to improve the safety of an autonomous vehicle with machine learning based controllers. Focusing on path planning in urban environments, Yurtsever et al. \cite{yurtsever2020integrating} combined RL with rule-based path planning to provide safety guarantees in autonomous driving. Similar approaches have also been used for highway driving, by combining rule-based systems with machine learning based controllers for enhanced driving safety \cite{likmeta2020combining, baheri2019deep}.
In our previous work \cite{kuutti2019safe}, we developed two safety cages for highway driving, and demonstrated these safety cages can be used to prevent collisions when the neural network controllers make unpredictable decisions. Furthermore, we demonstrated that the interventions by the safety cages can be used to re-train the neural networks in a supervised learning approach, enabling the system to learn from its own mistakes and further making the controller more robust. However, the main limitation of the safety cage approach was that the re-training happened in an offline manner, where the learning was broken down into three stages: (i) supervised training, (ii) closed-loop evaluation with safety cages, and (iii) re-training using the safety cage interventions as labels for supervised learning.
Here, we extend on this approach by utilising the safety cages to improve the safety of a RL based vehicle motion controller, and using the interventions of the safety cages as weak supervision which enables the system to learn to drive more safely in an online manner. Weak supervision has been shown to improve the efficiency of exploration in \mbox{RL \cite{lee2020weakly}} by guiding the agent towards useful directions during exploration. Here, the weak supervision enhances the exploration process in two ways; the safety cages stop the vehicle from taking unsafe actions thereby eliminating the unsafe part of the action space from the exploration, while also maintaining the vehicle in a safe state and thereby reducing the amount of states that need to be explored. {Reinforcement learning algorithms often struggle to learn efficiently at the beginning of training, since initially the agent is taking largely random actions, and it can take a significant amount of training before the agent starts to take the correct actions which are needed to learn its task. Therefore, by utilising weak supervision to guide the agent to the correct actions and states, the efficiency of the early training stage can be improved.} We show that eliminating the unsafe parts of the exploration space improves convergence during training, which can be a significant advantage considering the low sample efficiency of RL. Furthermore, we show that the safety cages eliminate the collisions that would normally happen during training, which could be a further advantage of our technique, should the training occur in a real-world system where collisions are undesirable.
\section{Materials and Methods}\label{sec3}
\subsection{Reinforcement Learning}
\end{paracol}
\begin{figure}[H]
\widefigure
\includegraphics[width=16 cm]{Figures/rl}
\caption{Reinforcement learning process.\label{fig:rl}}
\end{figure}
\begin{paracol}{2}
\switchcolumn
Reinforcement learning can be formally described as a Markov Decision Process (MDP). The MDP is denoted by a tuple \{$\mathcal{S}, \mathcal{A}, \mathcal{P}, \mathcal{R}$\}, where $\mathcal{S}$ represents the state space, $\mathcal{A}$ represents the the action space, $\mathcal{P}$ denotes the state transition probability model, and $\mathcal{R}$ is the reward function \cite{sutton1998introduction}. As shown in Figure~\ref{fig:rl}, at each time step $t$, the agent takes an action $a_t$ from the possible set of actions $\mathcal{A}$, according to its policy $\pi$ which is a mapping from states $s_t$ to actions $a_t$. Based on the action taken in the current state, the environment then transitions to the next state $s_{t+1}$ according to the transition dynamics $p(s_{t+1}|s_t, a_t)$ as given by the transition probability model $\mathcal{P}$. The agent then observes the new state $s_{t+1}$ and receives a scalar reward $r_t$ according to the reward function $\mathcal{R}$. The aim of the agent in the RL setting is to maximise the total accumulated returns $R_t$:
\begin{equation}
R_t = \sum_{i=t}^{\infty}\gamma^{(i-t)}r(s_i,a_i)
\end{equation}
where $\gamma \in [0,1]$ is the discount factor used to prioritise immediate rewards over \mbox{future rewards.}
\subsection{Deep Deterministic Policy Gradient}
Deep Deterministic Policy Gradient (DDPG) \cite{lillicrap2015continuous} extends the Deterministic Policy Gradient algorithm by Silver et al. \cite{silver2014deterministic} by utilising DNNs for function approximation. It is an actor-critic based off-policy RL algorithm, which can scale to high-dimensional and continuous state and action spaces. The DDPG uses the state-action value function, or Q-function, $Q(s, a)$, which estimates the expected returns after taking an action $a_t$ in \mbox{state $s_t$} under policy $\pi$. Therefore, given a state visitation distribution $\rho^\pi$ under policy $\pi$ in environment $E$ the Q-function is denoted by:
\begin{equation}
Q^\pi(s_t, a_t) = \mathbb{E}_{r_{i \geq t}, s_{i>t}\sim E,a_{i>t} \sim \pi}[R_t|s_t, a_t]
\end{equation}
The Q-function can be estimated by the Bellman equation for deterministic policies as:
\begin{equation}
Q^\pi(s_t, a_t) = \mathbb{E}_{r_t,s_{t+1} \sim E}[r(s_t, a_t) + \gamma Q^\pi (s_{t+1}, \pi(s_{t+1}))]
\end{equation}
As the expectations depend only on the environment, the critic network can be trained off-policy, using transitions from a different stochastic policy with the state visitation distribution $\rho^\beta$. The parameters of the critic network $\theta^Q$ can then be updated by minimising the critic loss $\mathcal{L}_Q$:
\begin{equation}
\mathcal{L}_Q = \mathbb{E}_{s_t\sim\rho^\beta,r_t\sim E}[(Q(s_t, a_t| \theta^Q) - y_t)^2]
\end{equation}
where
\begin{equation}
y_t = r(s_t, a_t) + \gamma Q(s_{t+1}, \pi(s_{t+1})|\theta^Q)
\end{equation}
The actor network parameters $\theta^\pi$ are then updated using the policy gradient \cite{silver2014deterministic} from the expected returns from a start distribution $J$ with respect to the actor parameters $\theta^\pi$:
\begin{multline}
\triangledown_{\theta^\pi}J \approx \mathbb{E}_{s_t\sim\rho^\beta}[\triangledown_{\theta^\pi}Q(s, a|\theta^Q)|_{s=s_t,a=\pi(s_t|\theta^\pi)}] \\
= \mathbb{E}_{s_t\sim\rho^\beta}[\triangledown_{a}Q(s,a|\theta^Q)|_{s=s_t, a=\pi(s_t)} \triangledown_{\theta^\pi}\pi(s|\theta^\pi)|_{s=s_t}]
\end{multline}
For updating the networks, mini-batches are drawn from a replay memory $\mathcal{D}$, which is a finite sized buffer storing state transitions $e = [s_t, a_t, r_t, s_{t+1}]$. To avoid divergence and improve stability of training, DDPG utilises target networks \cite{mnih2013playing}, which copy the parameters of the actor and critic networks. These target networks, target actor $\pi ' (s|\theta^{\pi '})$ and target critic network $Q'(s,a|\theta^{Q'})$, are updated slowly based on the learned network parameters to improve stability:
\begin{equation}
\theta' \leftarrow \tau \theta + (1 - \tau)\theta'
\end{equation}
where $\tau \ll 1$ is the mixing factor, a hyperparameter controlling the speed of target network updates.
To encourage the agent to explore the possible actions for continuous action spaces, noise is added to the actions of the deterministic policy $\pi(s_t|\theta^\pi)$. This exploration policy $\pi^e(s_t)$, samples noise from a noise process $\mathcal{N}$ which is added to the actor policy:
\begin{equation}
\pi^e(s_t) = \pi(s_t|\theta_t^\pi) + \mathcal{N}
\end{equation}
Here, the chosen noise process $\mathcal{N}$ is the Ornstein-Uhlenbeck process \cite{uhlenbeck1930theory}, which generates temporally correlated noise for efficient exploration in physical control problems.
\subsection{Safety Cages}
Virtual safety cages provide interpretable rule-based safety for complex cyber-physical systems. The purpose of these safety cages is to limit the actions of the system to a safe operational envelope. The simple way to achieve this, would be to limit the upper or lower limits of the system's action space. However, by using run-time monitoring to observe the state of the environment, the safety cages can dynamically select the control limits based on the current states. Therefore, the system can be limited in its possible courses of action when faced with a safety-critical scenario, such as a near-accident situation on a highway. We utilise our previously presented safety cages \cite{kuutti2019safe}, which limit the longitudinal control actions of a vehicle based on the Time Headway (TH) and Time-To-Collision (TTC) relative to the vehicle in front. The TH and TTC metrics represent the risk of potential forward collision with the vehicle in front, and are calculated as:
\begin{equation}
TH = \frac{x_{rel}}{v}
\end{equation}
\begin{equation}
TTC = \frac{x_{rel}}{v_{rel}}
\end{equation}
where $x_{rel}$ is the distance between the two vehicles in m, $v$ is the velocity of the host vehicle in m/s, and $v_{rel}$ is the relative velocity between the two vehicles in m/s.
The TTC and TH metrics were chosen as the states monitored by the safety cages as they represent the risk of potential collision with the vehicle in front, thereby providing effective safety measurements for our vehicle following use-case. We utilise two metrics as the TTC and TH provide complimentary information; the TTC measures time to a forward collision assuming both vehicles continue at their current speeds, whilst TH measures distance to the vehicle in front in time and makes no assumptions about the lead vehicle's actions. For example, when the host vehicle is driving significantly faster than the vehicle in front, as the distance between the vehicles gets closer the TTC approaches zero and correctly captures the risk of a forward collision. However, in a scenario where both vehicles are driving close to each other but at the same speed, the TTC will not signal a high risk of collision even though in this scenario if the lead vehicle begins to break, the two vehicles would be in a likely collision. In such a scenario, the two vehicles will have a low headway, therefore monitoring the TH will correctly inform the safety monitors of a collision risk.
The risk levels for both safety cages are as defined in \cite{kuutti2019safe}, where the aim was to identify potential collisions in time to prevent them, whilst minimising unnecessary interventions on the control of the vehicle. The different risk levels and associated minimum braking values are illustrated in Figure~\ref{fig:cages}. For each safety cage, there are three risk levels for which the safety cages will enforce a minimum braking value on the vehicle, with higher risk levels using increased rate of braking relative to the associated safety metric. When the vehicle is in the low risk region, no minimum braking is necessary and the RL agent is in full control of the vehicle. The minimum braking values enforced by the safety cages can be formally defined as shown in (\ref{eqn:sc_th})--(\ref{eqn:sc_ttc}). The braking value is normalised to the range [0, 1] where 0 is no braking and 1 is maximum braking value. In this framework, both safety cages provide a recommended braking value, which is then compared to the current braking action from the RL agent. The final braking value used for the vehicle motion control, $b$, is then chosen as the largest braking value between the two safety cages and the RL agent as given by (\ref{eqn:min_b}).
\begin{equation}\label{eqn:sc_th}
b_{TH} =
\begin{cases}
0.0, & \text{for TH $>$ 1.6} \\
-0.5TH + 1.0, & \text{for 1.0 $<$ TH $\leq$ 1.6} \\
-1.0TH + 1.5, & \text{for 0.5 $<$ TH $\leq$ 1.0} \\
1.0, & \text{for TH $\leq$ 0.5} \\
\end{cases}
\end{equation}
\begin{equation}\label{eqn:sc_ttc}
b_{TTC} =
\begin{cases}
0.0, & \text{for TTC $>$ 2.5} \\
-0.5TTC + 1.25, & \text{for 1.5 $<$ TTC $\leq$ 2.5} \\
-1.0TTC + 2.0, & \text{for 1.0 $<$ TTC $\leq$ 1.5} \\
1.0, & \text{for TTC $\leq$ 1.0} \\
\end{cases}
\end{equation}
\begin{equation}\label{eqn:min_b}
b = max(b_{TH}, b_{TTC}, b_{RL})
\end{equation}
where $b_{TH}$ is the minimum braking value from the TH safety cage, $b_{TTC}$ is the minimum braking value from the TTC safety cage, and $b_{RL}$ is the current braking value from the \mbox{RL agent.}
\end{paracol}
\nointerlineskip
\begin{figure}[H]
\widefigure
\includegraphics[width=18 cm]{Figures/safety_cages_2}
\caption{Risk-levels and minimum braking values for each safety cage.}
\label{fig:cages}
\end{figure}
\begin{paracol}{2}
\switchcolumn
\subsection{Highway Vehicle Following Use-Case}
The vehicle following use-case was framed as a scenario on a straight highway, with two vehicles travelling in a single lane. The host vehicle controlled by RL is the follower vehicle, and its aim is to maintain a 2 s headway from the lead vehicle. The lead vehicle velocities were limited to $v_{lead} \in$ [17, 40] m/s, and coefficient of friction values were chosen from the range [0.4, 1.0] for each episode. During training, the lead vehicle's acceleration was limited to $\dot{v}_{lead} \in$ [$-$2, 2] m/s\textsuperscript{2}, except for emergency braking manoeuvrers, which occurred on average once an hour, which used an acceleration in the range \mbox{$\dot{v}_{lead} \in$ [$-$6, $-$3] m/s\textsuperscript{2}}. The output from the RL agent is the gas and brake pedal values, which are continuous action values used to control the vehicle. As in \cite{kuutti2019end}, a neural network is used to estimate the longitudinal vehicle dynamics, by inferring the vehicle response to the pedal actions from the RL agent. This neural network acts as a type of World \mbox{Model \cite{ha2018world}}, providing an estimation of the simulator environment. This has the advantage that the neural network can be deployed on the same GPU as the RL network during training, thereby speeding up training time significantly. The World Model was trained with 2,364,041 time-steps from the IPG CarMaker simulator under different driving policies combining a total of 45 h of simulated driving. This approach was shown in \cite{kuutti2019end} to speed up training by up to a factor of 20, compared to training with the IPG CarMaker simulator. However, to ensure the accuracy of all results, we also evaluate all trained policies in IPG CarMaker (Section~\ref{sec4.1}).
\subsection{Training}
The DDPG model is trained in the vehicle following environment for 5000 episodes, where each episode lasts up to 5 min or until a collision occurs. The simulation is sampled at 25 Hz, therefore each time-step has a duration of 40 ms. The training parameters of the DDPG were tuned heuristically, and the final values can be found in Table \ref{tbl:netarch}. The critic uses a single hidden layer, followed by the output layer estimating the $Q$ value. The actor network utilises 3 hidden feedforward layers, followed by a Long Short-Term Memory (LSTM) \cite{hochreiter1997long} and then the action layer. {The actor network outputs the vehicle control action, for which the action space is represented by a single continuous value $a_t \in$ [$-$1, 1], where positive values represent the use of the gas pedal and negative values represent the use of the brake pedal. The observations of the agent are composed of 4 continuous state-values, which are the host vehicle velocity $v$, host vehicle \mbox{acceleration $\dot{v}$}, relative velocity $v_{rel}$, and time headway $TH$, such that $s_t = [v, \dot{v}, v_{rel}, TH]^T$.} To enable the LSTM to learn temporal correlations, the mini-batches for training were sampled as consecutive time-steps, with the LSTM cell state reset between each training update. To encourage the agent to learn a safe vehicle following policy, a reward function based on its current headway and headway derivative was defined in \cite{kuutti2019end} based on the reward function by Desjardins \& \mbox{Chaib-Draa \cite{desjardins2011cooperative}}, as shown in Figure~\ref{fig:reward}. The agent gains the maximum reward when it is close to the target headway of 2 s, whilst straying further from the target headway results in smaller rewards. The headway derivative is used in the reward function to encourage the vehicle to move towards the target headway, by giving small positive rewards as it moves closer to the target and penalising the agent when it is moving further away from the target region. For further comparison, we compare training the model with an additional penalty for breaching the safety cages, such that the final reward is given \mbox{as follows:}
\begin{equation}
r_t = r_{th} + r_{sc}
\end{equation}
where $r_t$ is the reward for time-step $t$, $r_{th}$ is the headway based reward function as shown in Figure~\ref{fig:reward}, and $r_{sc}$ is the safety cages penalty equal to -0.1 if the safety cage is breached and 0 otherwise.
\begin{specialtable}[H]
\caption{Network hyperparameters.}
\label{tbl:netarch}
\setlength{\cellWidtha}{\columnwidth/2-2\tabcolsep+0.8in}
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\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Mini-batch size & 64 \\
Hidden neurons in feedforward layers & 50 \\
LSTM units & 16 \\
Discount factor $\gamma$ & 0.99 \\
Actor learning rate $\eta_{\pi}$ & 10\textsuperscript{$-$4} \\
Critic learning rate $\eta_{q}$ & 10\textsuperscript{$-$2} \\
Replay memory size & 10\textsuperscript{6} \\
Mixing factor $\tau$ & 10\textsuperscript{$-$3} \\
Initial exploration noise scale & 1.0 \\
Max gradient norm for clipping & 0.5 \\
Exploration noise decay & 0.997 \\
Exploration mean $\mu$ & 0.0 \\
Exploration scale factor $\theta$ & 0.15 \\
Exploration variance $\sigma$ & 0.2 \\
\bottomrule
\end{tabularx}}
\end{specialtable}
\begin{figure}[H]
\includegraphics[width=0.5\textwidth]{Figures/rewardfunc}
\caption{Time headway based reward function for vehicle following.}
\label{fig:reward}
\end{figure}
The episode rewards during training can be seen in Figure~\ref{fig:training}, where three models are compared. The three models are DDPG only, DDPG+SC which is DDPG with safety cages, and DDPG+SC (no penalty) which is the DDPG with safety cages but without the $r_{sc}$ penalty. As can be seen, the DDPG+SC model has lower rewards at the beginning of training as it receives additional penalties compared to the other two models. However, after the initial exploration the DDPG+SC is the first model to reach the optimal rewards per episode ($\sim$7500 rewards), demonstrating improved convergence. Comparing the DDPG+SC models with and without penalties from the safety cages shows the model with the penalties converges to the optimal solution sooner, suggesting the penalty improves convergence during training. An additional benefit of the safety cages here is the safety of exploration, as the DDPG model collided 30 times during training, whilst the DDPG+SC model had no collisions during training. However, it can be seen that all three models converge to the same level of performance, therefore no significant difference in the trained policies can be concluded from the training rewards alone.
\begin{figure}[H]
\includegraphics[width=0.5\textwidth]{Figures/train_deep}
\caption{Episode rewards during DDPG training. Darker lines represent the moving average reward plot, whilst the actual reward values are seen in the transparent region.}
\label{fig:training}
\end{figure}
As an additional investigation of the effect of the safety cages on less safe control policies, we train two further models utilising smaller neural networks with constrained parameters. These models use the same parameters as in Table \ref{tbl:netarch}, except they only have 1 single hidden layer with 50 neurons and no LSTM layer. We refer to these models as Shallow DDPG and Shallow DDPG+SC. It should be noted that the parameters of these models were not tuned for better performance, and indeed sub-optimal parameters were chosen on purpose to enable better insight into the effect of the safety cages in unsafe systems. The episode rewards for the two shallow models during training are shown \mbox{in Figure~\ref{fig:training2}}. As can be seen, these two models have a more significant difference in training performance. The Shallow DDPG struggles to learn a feasible training policy, whilst the Shallow DDPG+SC learns to drive without collisions, although at a lower level of overall performance compared to the deeper models.
\begin{figure}[H]
\includegraphics[width=0.5\textwidth]{Figures/train_shallow}
\caption{Episode rewards during DDPG training of shallow models. Darker lines represent the moving average reward plot, whilst the actual reward values are seen in the transparent region.}
\label{fig:training2}
\end{figure}
\section{Results}\label{sec4}
To investigate the performance of the learned control policies, we evaluate the vehicle follower models in various highway driving scenarios. We utilise two types of testing for this evaluation. Naturalistic testing tests the control policies in typical driving scenarios, giving an idea of how the control policies perform in everyday driving. Adversarial testing utilises an adversarial agent to create safety-critical scenarios, showing how the vehicle performs in dangerous edge cases where collisions are likely to occur. The controller performance in both types of scenario is important, since most driving scenarios on the road fall into naturalistic driving the controller must be able to drive efficiently and safely in these scenarios, however the controller must also be able remain safe in dangerous edge cases in order to avoid collisions. To enable better analysis of the performance of the RL-based control policies, no safety cages are used during testing so the vehicle follower models must depend on their own learned knowledge to keep the vehicle safe. This also enables better understanding on the effect of using the safety cages during training on the final learned control policy.
\subsection{Naturalistic Testing}\label{sec4.1}
For the naturalistic driving, similar lead vehicle behaviours were used to those during training, with velocities in the range [17, 40] m/s and acceleration [$-$2, 2] m/s\textsuperscript{2}. The exception to this was the harsh braking manoeuvres which occurred, on average, once an hour with deceleration [$-$6, $-$3] m/s\textsuperscript{2}. At the start of the episode, the coefficient of friction is randomly chosen in the range $[0.4, 1.0]$ and each episode lasts until 5 min has passed or a collision occurs. For each driving model, a total of 120 test scenarios were completed, totalling up to 10 h of testing. All driving for these tests occurred in the IPG CarMaker simulation environment to ensure accuracy of the results. Two types of baselines are provided for comparison; the IPG Driver is the default driver in the CarMaker Simulator and A2C is the Advantage Actor Critic \cite{mnih2016asynchronous} based vehicle follower model in \cite{kuutti2019end}.
The results from the naturalistic driving scenarios are summarised in Table \ref{tbl:simresults}. The table shows the RL based models outperform the default IPG Driver, with the exception of the shallow models. The results demonstrate that both DDPG based models outperform the previous A2C-based vehicle follower model. However, comparing the DDPG and DDPG+SC models shows the benefit of using the safety cages during RL training. While in most scenarios the two models have similar performance (the mean values seen are approximately equal), the minimum headway by the DDPG+SC during testing is higher, showing it can maintain a safer distance from the lead vehicle. However, as both models can maintain a safe distance without collisions this difference is not significant by itself. Therefore, investigating the difference between the Shallow DDPG and Shallow DDPG+SC models provides further insight into the role the safety cages play in supervision during RL training. Similar to the training rewards, the shallow models show a more extreme difference between the two models. The results show the Shallow DDPG model without safety cages fails to learn to drive safely, whilst the Shallow DDPG+SC model avoids collisions safely, although it comes relatively close to collisions with a minimum time headway at 0.79 s. This shows the benefit of the safety cages in guiding the model towards a safe control policy during training.
\end{paracol}
\nointerlineskip
\begin{specialtable}[H]
\widetable
\caption{10-hour driving test under naturalistic driving conditions in IPG CarMaker.}
\label{tbl:simresults}
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\toprule
\textbf{Parameter} & \textbf{IPG Driver} & \textbf{A2C} \cite{kuutti2019end} & \textbf{DDPG} & \textbf{DDPG+SC} & \textbf{Shallow DDPG} & \textbf{Shallow DDPG+SC} \\
\midrule
min. x\textsubscript{rel} [m] & 10.737 & 7.780 & 15.252 & 13.403 & 0.000 & 5.840 \\
mean x\textsubscript{rel} [m] & 75.16 & 58.01 & 58.19 & 58.24 & 41.45 & 59.34 \\
max. v\textsubscript{rel} [m/s] & 13.90 & 7.89 & 10.74 & 9.33 & 13.43 & 6.97 \\
mean v\textsubscript{rel} [m/s] & 0.187 & 0.0289 & 0.0281 & 0.0271 & 4.59 & 0.0328 \\
min. TH [s] & 1.046 & 1.114 & 1.530 & 1.693 & 0.000 & 0.787 \\
mean TH [s] & 2.546 & 2.007 & 2.015 & 2.015 & 1.313 & 2.034 \\
collisions & 0 & 0 & 0 & 0 & 120 & 0 \\
\bottomrule
\end{tabularx}}
\end{specialtable}
\begin{paracol}{2}
\switchcolumn
\subsection{Adversarial Testing}
Utilising machine learning to expose weaknesses in safety-critical cyber-physical systems has been shown to be an effective method for finding failure cases \mbox{effectively \cite{corso2020survey, riedmaier2020survey}}. We utilise the Adversarial Testing Framework (ATF) presented in \cite{kuutti2020training}, which utilised an adversarial agent trained through RL to expose over 11,000 collision cases in machine learning based autonomous vehicle control systems. The adversarial agent is trained through A2C \cite{mnih2016asynchronous} with a reward function $r_A$ based on the inverse headway:
\begin{equation}
r_A = min\left(\frac{1}{TH}, 100\right)
\end{equation}
This reward function encourages the adversarial agent to minimise the headway and make collisions happen, while capping the reward at 100 ensures that the reward does not tend to infinity as the headway reaches zero.
As this lead vehicle used in the adversarial testing can behave very differently to those seen during training, this testing focuses on investigating the models' generalisation capability as well as their response to hazardous scenarios. Each DDPG model is tested under two different velocity ranges; the first limits the lead vehicle's velocity to the same as the training scenarios with $v_{lead} \in [17, 40]$ m/s, and the second uses a lower velocity range which enables the ATF to expose collisions more easily at a velocity range of \mbox{$v_{lead} \in [12, 30]$ m/s.} For each model, 3 different adversarial agents were trained, such that results can be averaged between these 3 training runs. The minimum episode TH during training can be seen for both deep models over the 2500 training episodes in Figures~\ref{fig:at_deep} and \ref{fig:at_deep_low}. These tests show that both deep models can maintain a safe distance from the lead vehicle even when the lead vehicle is attempting to cause collisions intentionally. Although a slight difference in the two models can be seen, as the DDPG+SC model has a slightly higher headway on average as well as significantly less variance. However as both deep models remain at a safe distance from the adversarial agent, these models can be considered safe even in safety-critical edge cases. Comparing the two shallow models in Figures~\ref{fig:at_shallow} and \ref{fig:at_shallow_lw}, a more significant difference can be seen. While both models are worse in performance than the deep models, the Shallow DDPG is significantly easier to exploit than the Shallow DDPG+SC model. The Shallow DDPG model continues to cause collisions during the adversarial testing, whilst the Shallow DDPG+SC model remains at a safer distance. In the training conditions, the Shallow DDPG+SC remains relatively safe, with no decrease in the minimum headway during the training of the adversarial agent, although it can be seen that the variance increases as the training progresses. In the lower velocity case, the Shallow DDPG+SC still avoids collisions, but the adversarial agent is able to reduce the minimum headway significantly better. This shows that the safety cages have helped the model learn a significantly more robust control policy, even when the model uses sub-optimal parameters. Without the additional weak supervision from the safety cages, it can be seen that these shallow models would not have been able to learn a reasonable driving policy. Therefore, the weak-supervision by the safety cages can be used to train models with sub-optimal parameters. In addition, for models with optimal parameters they provide improved convergence during training and slightly improved safety in the final trained policy.
\begin{figure}[H]
\includegraphics[width=0.45\textwidth]{Figures/at2_deep2}
\caption{Comparison of the deep vehicle following agents' minimum TH per episode over adversarial training runs with lead vehicle velocity limits $v_{lead} \in [17, 40]$ m/s. Averaged over 3 runs,
with standard deviation shown in shaded colour.}
\label{fig:at_deep}
\end{figure}
\begin{figure}[H]
\includegraphics[width=0.45\textwidth]{Figures/at2_deep}
\caption{Comparison of the deep vehicle following agents' minimum TH per episode over adversarial training runs with lead vehicle velocity limits $v_{lead} \in [12, 30]$ m/s. Averaged over 3 runs,
with standard deviation shown in shaded colour.}
\label{fig:at_deep_low}
\end{figure}
\begin{figure}[H]
\includegraphics[width=0.45\textwidth]{Figures/at2_shallow2}
\caption{Comparison of the shallow vehicle following agents' minimum TH per episode over adversarial training runs with lead vehicle velocity limits $v_{lead} \in [17, 40]$ m/s. Averaged over 3 runs,
with standard deviation shown in shaded colour.}
\label{fig:at_shallow}
\end{figure}
\begin{figure}[H]
\includegraphics[width=0.45\textwidth]{Figures/at2_shallow}
\caption{Comparison of the shallow vehicle following agents' minimum TH per episode over adversarial training runs with lead vehicle velocity limits $v_{lead} \in [12, 30]$ m/s. Averaged over 3 runs,
with standard deviation shown in shaded colour.}
\label{fig:at_shallow_lw}
\end{figure}
\section{Conclusions}\label{sec5}
In this paper, a reinforcement learning technique combining rule-based safety cages was presented. The safety cages provide a safety mechanism for the autonomous vehicle in case the neural network-based controller makes unsafe decisions, thereby enhancing the safety of the vehicle and providing interpretability in the vehicle motion control system. In addition, the safety cages are used as weak supervision during training, by guiding the agent towards useful actions and avoiding dangerous states.
We compared the model with safety cages to a model without them, and show improvements in safety of exploration, speed of convergence, and the safety of the final control policy. {In addition to improved training efficiency, simulated testing scenarios demonstrated} that even with the safety cages disabled, the model which used them during training has learned a safer control policy {by maintaining a minimum headway of 1.69 s in a safety-critical scenario, compared to 1.53 s without safety cage training}. We additionally tested the proposed approach on shallow models with constrained parameters, and {showed that the shallow model with safety cage training was able to drive without collisions, whilst the shallow model without safety cage training collided in every test scenario. These results }demonstrate that the safety cages enabled the shallow models to learn a safe control policy while otherwise the shallow models were not able to learn a feasible driving policy. This showed that the safety cages add beneficial supervision during training, enabling the model to learn from the environment more effectively.
Therefore, this work provides an effective way to combine reinforcement learning based control with rule-based safety mechanisms not only to improve the safety of the vehicle, but also incorporating weak supervision in the training process for improved convergence and performance.
{This work opens up multiple potential avenues for future work. The use-case in this study was a simplified vehicle following scenario. However, extending the safety cages to consider both longitudinal and lateral control actions, as well as potential objects on other lanes, would allow the technique to be applied to more complex use-cases such as urban driving. Moreover, comparing the use of the weak supervision for different use-cases or learning algorithms (e.g., on-policy vs. off-policy RL) would help with understanding the most efficient use of weak supervision in reinforcement learning. Furthermore, extending the reinforcement learning agent to use more high dimensional inputs, such as images, would allow investigation into how the increased speed of convergence helps in cases where the sample inefficient reinforcement learning algorithms struggle. Finally, using the safety cages presented here in real-world training could better demonstrate the benefit in both safety and efficiency of exploration, compared to the simulated scenario presented in this work.}
\vspace{6pt}
\authorcontributions{Conceptualization, S.K., R.B. and S.F.; methodology, S.K.; software, S.K.; validation, S.K.; formal analysis, S.K.; investigation, S.K.; resources, S.K.; data curation, S.K.; writing---original draft preparation, S.K.; writing---review and editing, R.B and S.F.; visualization, S.K.; supervision, R.B. and S.F.; project administration, R.B. and S.F.; funding acquisition, R.B. and S.F. All authors have read and agreed to the published version of the manuscript.}
\funding{This research was supported by the U.K.-Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/R512217.}
\institutionalreview{Not applicable.}
\informedconsent{Not applicable.}
\dataavailability{Data sharing not applicable.}
\conflictsofinterest{The authors declare no conflict of interest.}
\newpage
\end{paracol}
\reftitle{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,611 |
{"url":"http:\/\/function-of-time.blogspot.ca\/","text":"## Monday, July 14, 2014\n\n### Essential Questions for Algebra 2\n\nPssst....Anna....here's what we have so far.\n\nSequences and Series\n\u2022 What kinds of patterns commonly arise in our world?\n\u2022 Why is it sometimes desirable to describe a pattern mathematically?\n\u2022 When we notice a real-world or mathematical pattern, what are some different ways in which we can describe it?\n\u2022 How is it possible to keep getting closer and closer to something, but never actually touch it?\nProbability and Statistics\n\u2022 How can I use probability and statistics to make predictions and decisions that will benefit me in life?\n\u2022 How should I interpret statistical information about myself and that I see in the news?\n\u2022 What is the bell curve, why does it appear in many aspects of society, why is understanding it so important to our society?\n\u2022 What are are some more sophisticated ways of counting, and when are they useful?\nIntro to Functions\n\u2022 How are functions used to represent\/simulate the world we live in, and why are they so important?\n\u2022 How do functions help us to make the best decision?\n\u2022 What are some different kinds of functions, and what sorts of real-world situations can they model?\n\u2022 Why is the idea of \"inverse\" so important in mathematics?\n\u2022 How are quadratic functions used to understand\/represent the Universe we live in?\n\u2022 How can writing a mathematical statement in different but equivalent ways highlight its various features?\n\u2022 Often, solving problems involves making choices. How can we make smart choices for any problem?\nPolynomials\n\u2022 How are polynomial functions used to understand\/represent the Universe we live in?\n\u2022 How are all the different representations of a polynomial function related?\nRationals\n\u2022 How are rational functions and different types of variation used to understand\/represent the Universe we live in?\n\u2022 How is it possible to keep getting closer and closer to something, but never actually touch it?\n\u2022 How are radical functions used to understand\/represent the Universe we live in?\n\u2022 How can something that \"doesn't exist\" affect our world?\n\u2022 How can we make sense of exponents that are not integers?\nExponentials and Logs\n\u2022 How are exponential and logarithmic functions used to understand\/represent the Universe we live in?\n\u2022 Why does the graph of an exponential function have its shape? How is it possible to get closer and closer to something and never touch it?\n\u2022 Why is the idea of \"inverse\" important in mathematics?\nModeling with Data\n\u2022 How do you decide if a mathematical model is \"good\"?\n\u2022 How can we use existing measurements to make predictions?\n\u2022 What are some possible pitfalls of using mathematical models to make predictions?\n\n## Saturday, June 21, 2014\n\n### There's This Book. You Probably Want It.\n\nIf you don't know Sue VanHattum, you're missing out. She's a community college math teacher and math circle leader, and just one of the warmest, most thoughtful people I know. (If you're reading this, you probably know her blog, Math Mama Writes.)\n\nFor the past few years, Sue has been assembling and editing an anthology called Playing with Math, featuring writing by people who like to play with math. (Stories from Math Circles, Homeschoolers, and Passionate Teachers is the subtitle.) I appear in it, as do many of my heroes. It's stories, but it's also games and puzzles and fun things to play around with.\n\nThe book is finally ready to go! If you like f(t) and others of the genre, this book is probably right up your alley. The initial print run will need some funding. Donate as little as you like, but $25 gets you a print copy. (Oh, and don't put it off! The campaign only runs for a month.) Better yet, get one for you and a friend! Maybe that friend who is a parent and wants their kids to like math, but is afraid of it. You know the one. This is how we change people's minds about learning mathematics. One positive interaction at a time. ## Thursday, May 22, 2014 ### The AMS Published a Kids' Book, and It's Really Good The American Mathematical Society does some pretty great things. Now, a kids' book! It's called Really Big Numbers, by Richard Evan Schwartz. This trailer is a great intro: Here are particular things I really dig about it: \u2022 The subject matter is prime kid-bait. This book will give them many ways to think about how big a million, trillion, etc really is. It starts small with grokkable quantities (ladder rungs, cutting a cube into 1000 little cubes) and the thread of making really big numbers concrete runs through the entire book. It does this with distances (a million people joining hands would stretch from Providence to Chicago), volumes (100 billion basketballs would fill New York City roughly to the height of a person), and arrangements (between 5 and 6 trillion 9-letter \"words\"), among other constructs. \u2022 The bright, humorous, borderline-psychedelic illustrations. \u2022 The conversational and non-threatening invitations to think about mathematics that go past the words on the page. \"There are about 20,000 ways to color a tic-tac-toe board with three colors.\" (following page) \"You know, when I said that there are about 20,000 patterns like this, I was hoping that you would try to figure out exactly how many patterns there are.\" There are many such pages that set up tantalizing problems, that could launch some great explorations and conversations, particularly in combinatorics, geometric sequences, and graph theory. \u2022 It introduces concepts and notation only as needed. Exponents come up organically, as do special names for them. (As a kid, I would have been particularly tickled to learn names for powers of ten past a decillion. I was a weird little nerd, but still.) \u2022 It invites kids to read as far as makes sense to them. Schwartz compares reading the book to a game of bucking bronco: hold on as long as you can, and when you get thrown off, come back any time. You might pick the book up for your third grader, and the concept of exponents (about 1\/3 of the way through) might stretch her mind. But the ideas and problems beyond could entertain and challenge her if she picks it up again in middle school and then high school. I can't wait until my nieces come at me with \"Infinity!\" for the first time. I'll be ready for them. ## Monday, April 14, 2014 ### NCTM 2014 Presentation --- One of Us, Every Teacher a Blogging Teacher Mine was one of several on Friday billed in the \"Leveraging Technology\" strand. All the presenters appeared on a panel discussion at the end of the day. I give the NCTM program committee huge props for putting the strand together. In previous years, there were complaints that NCTM had not tapped into the voices, experience, and community going on in MTBoS every day. My interpretation is, this criticism was heard, and this strand was their response. My overall thesis was that in many places, the professional development opportunities offered to teachers are not good enough to result in consistent improvement in their practice over time. And that starting a blog about what is going on in their classes can be a really effective way for teachers to take responsibility for their own professional growth. I owe a big public thank you to Ashli Black, Sadie Estrella, Chris Lusto, and Meg Lane for sitting through a rehearsal Thursday night and offering thoughtful criticism and feedback. And an extra thank you to Ashli for suggesting some very positive changes in the weeks leading up. I didn't want to just post slides, even though that would have been way easier than writing this all out, because they wouldn't make any sense by themselves. So you're welcome. I got many kind compliments afterwards, but I think if we are all being honest, this talk was a 6, maybe a 7. I was not as nervous as I feared I would be, and I was well-prepared with what I had prepared, but I think I could have made my points better if I had more experience planning this sort of thing. Here are the conclusions I've come to about the very surreal scenario of speaking to a big room with a microphone and a slideshow: 1. I have the capacity to do this well. 2. Learning to do it really well would require focused, sustained, intentional preparation and practice. 3. I am undecided whether this skill is important enough to me to take the time to develop it. But that's where I am on that. This is a snapshot of why people came to see me. My Polleverywhere account only goes up to 50 responses, but here is the sample made up of the first 50 to respond: I was glad that the second and third options were the most frequent, because that was what I was prepared to talk about. Hooray! And here is a snapshot of some of the rock stars who came to see me and sat in the front row. I am a lucky one, indeed. Photo Credit: Avery Pickford Part 1: Why I Started Reading Blogs I started with a little story about how terrible I was at teachin' school in my first few years. The intention was mostly to be funny and get the audience to like me. But every word was true. In order to do something about that problem, I started poking around the internet. I found that there were people sharing their lessons -- like, what was actually going on in their classes -- with narratives and media and supporting documents included. And they were good. I said an especial thank you to Dan for being a pioneer in this area, except I didn't address it directly at him, even though he was in the audience, because I spazzed. Also mentioned: Jackie and Sam. Part 2: Why I Started Writing a Blog I started writing because I had to. I was coming up with some good stuff, or had ideas for some good stuff, and I needed to share them. I'm not sure if this itch can be learned, but I tried to get the audience to experience it. First, I said, think of something that went really well this year in your class (or a class you're involved in). I put up a 30-second timer to give people a chance to think of something good. Then, turn to a neighbor for two minutes and tell them about it. People were game. It got satisfyingly loud and animated. After four minutes (two for each partner), we regrouped, and I asked a volunteer to share what she talked about with the room. Then, I asked her how that felt. The point I was trying to make was, it's satisfying when you try something that goes well in your classroom, but it's deeply satisfying when you know that your insight and planning was put to good use by hundreds of people all over the world. Part 3: What Blogging is Good For I gave a few examples of some of my favorite posts: We know this sort of reflective summary of our work is a valuable activity for learning. We know it because we assign portfolio assessments to students. We know it as a profession, because National Board Certification for teachers requires four portfolios. Four! Also important: most teachers do not spend their whole career in the same place. Having a record of your professional growth and a collection of your best work is very helpful when you apply for new jobs. Then I cited one example of posting a lesson that I wasn't happy with, soliciting feedback, and then posting the update. I had initially planned on using the example of the introduction to right triangle trig (original) (revised). However, I was afraid it would take too long to explain the gist of the lesson, so instead I went with a problem set used to help Algebra 2 students review what they know about equations of lines and extend it to point slope form (original) (revised). In hindsight, I should have stuck with the trig stuff; it was meatier content and had better visuals. The upshot: once you have a critical mass of readers who are active in this community, you can tap into a hive mind of experienced, generous, knowledgable folks who will help you work out the kinks in a lesson. Part 4: Typical PD Offerings are Not That Effective, but Blogging Is A poll showed that, for this audience, professional development offerings skew unhelpful: Then I asked them to choose one or two of these statements to best characterize the PD they've participated in this year, and discuss them with a neighbor: Then I revealed that these statements were characteristics of effective and ineffective PD, because research: And this was the thing I wanted to communicate most loudly and most clearly. Dear Teacher: if the professional development offered by your school or district is not helping you improve your practice in clear, consistent, measurable ways, then it's up to you to take responsibility for your professional growth. Blogging isn't the only mechanism for that, but it is fun, and it does exhibit the three characteristics of effective PD outlined by Linda Darling-Hammond and Nikole Richardson. It's sustained over time because you're doing it at least a few times a month, it's linked to curricula and applied to practice because you are reflecting on your planning and what went down in your classroom, and it is done in collaboration with commenters and other bloggers. Part 5: Tips for Blogging Teachers I used Bree's blog as an example: \u2022 Use your real name, and bonus points for including a picture. This is your professional work and should have your name on it. People resist trusting and interacting with a pseudonym. You have no reason to hide. \u2022 Put a creative commons badge on your work. It won't stop nefarious ne'er-do-wells from publishing your stuff under their own name, but it does give you some recourse should that ever happen. \u2022 Put pictures in your posts. For whatever reason, people are more likely to read and interact when there are pictures. I suppose readers are averse to an unbroken wall of text. \u2022 The arrow at the top points to links to Bree's short stories. I'm not suggesting you should publish short works of fiction, but this message is more, you do you. Be your authentic self. Bree is a good writer and storyteller, and that talent comes through in all of her writing. I am kind of dry and practical and occasionally sarcastic. Hedge does this stream-of-consciousness thing that goes with her personality. Sam enthuses. Fawn swears. People respond to writers, in this genre, who are being themselves. \u2022 This is a comment left by Tina Cardone on Bree's blog. Tina's name is a link. If I clicked on it, it would take me to Tina's blog. If you would like more people to read your stuff, it helps to read their stuff, and let them know you did in this way. If your comments are quality, people will be curious about you and go check you out. You just include your blog's URL in a special spot when you leave a comment. \u2022 Also shown is Bree's response to Tina. This is a tip to engage commenters, answer questions, and have a dialog with them. \u2022 Final thing, for which there is no graphic, is that it's okay to reach out. If you have been writing for several months and feel like you're talking to yourself, shoot an email to a few people you read and admire. Ask them to consider sharing a link to your blog, if they like what you're doing. This is a gracious and generous community. Part 6: Tips for Admins and Others Supporting Teachers in Blogging This is a screenshot that a teacher took for me at her school. AT HER SCHOOL. WORDPRESS IS BLOCKED AT SCHOOL. Admins, you have to find a way to unblock everything you can for teachers' accounts. Same goes for Youtube, and Google Hangout, and just as much as you can. If your mission statement includes the phrase \"21st century learning,\" and you are denying teachers access to the tools of 21st century learning, you're not really doing it. I know there are privacy concerns and bandwidth concerns, but this is a problem you need to solve. This is a picture from the blog of Jonathan, a high school teacher in Texas. He posts photos from his classroom, but edits students' faces. This practice is, presumably, within the acceptable use guidelines of his district. Some teachers are reluctant to start blogging because they don't know what it's okay to share from their classrooms. Having a clear and well-publicized policy can help. Final tip: find a way to make it count. If maintaining a blog is an additional thing a teacher has to find time for, on top of everything else she has to do, it's hard to keep up with. A decent blog post a few times a month is a time-consuming effort. If you can make that activity count in your district's existing inservice credit program, or find a way for blogging to replace another PD commitment in which the teacher already participates, it would be a huge encouragement. For practical reasons, but also to communicate that you recognize blogging as a worthwhile pursuit. I ended on a link to the Welcome to MTBoS page, for people looking for blogs to start reading, and that was that. I had about fifteen minutes to spare, so I took some questions. I'd tweak it a bit next time. First, it needed a better ending. The ending just kind of appeared abruptly with a thud. I need to end with an inspirational quote or a call to arms or somesuch. Second, Sadie suggested offering some examples of easier onramps to blogging, like a 180 blog. If this is all too intimidating, just commit to seeking out, taking, and posting one photo a day from school. That is an excellent idea. Because, boom, you're blogging -- the task is easy and you can figure out all the technical stuff in the process. And more important, it gets you in the habit of paying attention to worthwhile things to share. I wish I had thought of that. If you want to talk about any of this, you know the way. ## Monday, April 7, 2014 ### Dear Reader From the mailbag, a kind of question I hear a whole lot. I don't know that I'm all that qualified to respond, but, I want to state clearly my belief that the vast, vast majority of our teaching force is made up of smart, well-intentioned, hard-working people who want to do a good job, and are willing to entertain the adoption of CCSS-M as an opportunity to do better. I know that there are valid complaints about the way tests have been implemented and used for ridiculous purposes, and I don't want to have that argument. I do want to spend my energy on productive changes in classrooms. Hi Kate! When you have time, can you give me an idea of how your class ran when you taught? I guess I should start with some explanation\u2026 I am a fairly traditional teacher. My students come into class. I have some sort of warm up on the SMART Board and I check their homework. We go over the homework from the previous day \u2013 I give them the answers and ask for questions. Then I go into the lesson for the day and give them time to work on the assignment when possible. Some days we do stations or other stuff in my class to practice, but by in large, many of my classroom days are spent \u201cteaching\u201d or \u201clecturing.\u201d Now, I know there is a time and place for that, but I also know and am really coming to realize that I need and want to make changes in my classroom next year. I want my students to not be so dependent on me. I know that this will be a tough thing \u2013 it will move my students out of their comfort zone and it will move me out of mine. However, I have seen the Field Test (we are a PARCC state) and I can also see that there is a larger emphasis on being able to read the problem and apply the concepts rather than problems like \u201csolve this system of equations.\u201d I am trying to figure out how to get there. What I mean is, I am trying to figure out what I need to do differently in my classroom to not only \u201ccover\u201d the material but to prepare my students to better think on their own. Prepare them so they can read a problem and say \u201cwell, I haven\u2019t seen this before exactly but I do know I can do or try x, y, or z\u201d and then they\u2019ll (hopefully) do that. I am fairly certain that if I were to give my students an assessment that did not look like their review sheets and their practice problems, they would pretty much freak out. I know and realize that it is not a change I can make abruptly, but what I would really like is that at this point in the year (well, maybe a lot earlier), that I would feel like my students are capable of looking at an unfamiliar problem and knowing how to begin. That they can think and apply on their own without totally freaking out in the process. I was hoping that you would be so kind to share with me how your class was structured and if you had any great words of wisdom to get me pointed in the right direction. I know it will be a huge shift for me and my students, but in the end, I also know that it is the best thing to prepare them for life beyond school. Hi Reader! Here we go: 1. Your class doesn't sound that different from the way mine ran. The details that may have been different were: 1a. When you \"go into the lesson for the day\" what that looks like. If we could watch a time-lapse of my classes over the years, we'd see too much of me talking to silent kids at the beginning, and much more of kids talking to each other at the end. It was a slow process for me, but whenever I discovered or thought up a way to replace me-telling with them-experiencing, I replaced a lesson with a better lesson. For example, for log laws, traditionally some of the most bewildering of the Algebra 2 content, I wrote up a series of questions that started kids with things they already knew and prompted them to look for patterns. I didn't do it \"with them,\" I said, okay, read carefully, work together, I'd like you to try to get through question 8 before we stop and discuss, ten minutes from now. An old lesson plan would look like the notes I would write and the things I would say. A new lesson plan would look like either a series of questions, or 2-3 tasks that kids were meant to work though independently or in small groups and my anticipation of their responses, and then how I planned to tie it all together. In practice, I'd only let them work in 5-10 minute chunks before I interrupted them to share what they found and re-cap. (See also: the 5 Practices book. If I were teaching right now, I'd also be paying attention to Assessment for Learning.) The most important change is the shift to giving them questions and problems unlike ones they'd seen before. Whether they're working through sort-of familiar or unfamiliar tasks to develop a whole new concept, or you're asking them to apply something previously-learned in a different way, I think the principle is unavoidable if you want them to be able to think for themselves. The only way to learn to do it is to do it. It's less scary and more do-able if you build off something they already know, or start with something concrete that they can count or measure. Count stuff, make a table, look for a pattern, generalize, solve new problems is as good a lesson flow as any. Alot of this work was just reading blogs and journal articles and saving useful things in a digital filing cabinet. Then, when I went to plan a unit, I would go look at everything I had saved for that unit, to see if I could replace any of the mathematical\/pedagogical flow of the unit with something better. Unfortunately I don't have my filing cabinet organized in a sharable way, but lots of other people do. Start slow -- see if you can replace one lesson every unit -- if you try to change everything all at once it will be overwhelming and you'll give up on it. 1b. You also said you give homework every day. I gave homework, but I gave less at the end than at the beginning, and I became more intentional about its purpose. A person can really learn the content of a high school math course in 45 minutes a day. If your class time is firing on all cylinders, you can give less homework. You need to stop thinking of it as an insurance policy. And when you do, there should be a reason. It's okay if the reason is fluency with some specific thing -- \"I want kids to be able to identify the correct triangle congruence theorem by looking at a diagram with congruent things marked\" for example. But there can be other good reasons, too, like \"Kids are going to estimate the coordinates of a bunch of points on the unit circle to two decimal places -- I don't want to suck up 15 minutes of class time with this exercise that they can complete independently, but it's a necessary part of tomorrow's unit circle introduction.\" I found more kids completing homework when it had a clearly articulated purpose. Corollary to 1b: Don't flip. They won't watch the videos. 2. Try to have this conversation with your whole department. The best possible thing would be for your school to adopt, and get your colleagues committed to, a research-based, field-tested, high-quality curriculum with lots of instructional supports like CME or Core-Plus. You still may have to supplement; for example, I think both of these are light on authentic real-world applications. But developing a curriculum from scratch is not the work of teaching -- both roles are too complicated to be done well by one person. But even if that kind of systemic change is a non-starter, still see if you can get others in your department to start making some changes, because it's hard to make big changes isolated to your classroom. Let's say you try more problem-based investigation in small groups, and expect kids to articulate their reasoning in writing, and make it count on assessments. Kids will perceive your class as harder than other teachers'. They will complain, which is annoying, but also they will resist trying what you are asking of them, because it's soooo not fair! Or let's say you implement some flavor of standards-based grading. No matter what you do, kids will interpret that as a million retakes and second chances -we don't have to study, woo! Your class will be perceived as easier, and guidance counselors will start shunting more of the struggling learners into your classes, because they are more likely to be able to pass. This isn't fair to you. I hope that helps! You can do it. A little at a time. ## Sunday, February 23, 2014 ### CCSS Geometry and Proof through Transformations Congruence and similarity proofs through transformations are new to most teachers with CCSS-M. I have noticed instances of them making teachers freak out. But they are actually delightful, once you understand what is expected. I find transformations a much clearer way to show why figures must be congruent and similar than with the axiomatic approach most of us used to use. In the two triangles pictured below m(A)=m(D) and m(B)=m(E)Using a sequence of translations, rotations, reflections, and\/or dilations, show that ABC is similar to DEF. This is a task called Similar Triangles from Illustrative Mathematics. If you can do it, you've proven AA, so, hey, pretty darn useful, too. If you give it to teachers, they will mostly freak out. You'll hear things like \"This would be so much easier on a coordinate grid.\" \"I don't know what I can assume.\" \"What definition of similarity are we using?\" So, clarify that the only givens are the congruent angles. And that similarity is defined as one figure being the product of a sequence of transformations of the other figure. Still, they are freaking out because they don't know what to put. Here is a magical, yet still general, procedure that will unlock their willingness to engage with this task. It was shared with me by Dr. Kristin Umland from University of New Mexico, who is a badass. \u2022 state the transformation precisely \u2022 state the object you are applying the transformation to \u2022 draw the intermediate step Here is what it will look like. Colored pencils are very helpful. 1. I'm going to do a translation by vector AD. I'm going to apply it to triangle ABC. Here is the intermediate step: 2. Then, I will do a rotation, about point D, clockwise, by the number of degrees in angle B'DE. The object I'll rotate is triangle A'B'C'. Here is the intermediate step: As a result of the rotation, A''B'' will lie right on top of DE, and A''C'' will lie right on top of DF. I know this because it was given that angles A and D are congruent, and angle measures are preserved in translations and rotations. 3. Finally, I will do a dilation, with center D, with scale factor that is equal to DE\/AB. The object I'll dilate is triangle A''B''C''. Here is the final step: I know this final transformation to triangle A'''B'''C''' will land precisely on triangle DEF. The dilated segments will lie right on top of DE and DF, because the center of dilation was D, so dilated segments that include D will grow in their original direction. B''' will land right on top of E, because of the scale factor I chose, and because distance was preserved in the translation and rotation. Since it was given that angles B and E are congruent, and angle measures were preserved in all the transformations, B'''C''' will be right on top of EF. D''' will land right on F, because the two dilated segments have no choice but to intersect at F. Okay, I am going to anticipate some of your objections and pre-emptively respond to them: Objection: \"Aren't you just teaching another procedure? Isn't that what we are trying to get away from?\" Response: State the transformation, state the object, draw the intermediate step doesn't feel like an evil procedure in the evil sense of the word. It's not actually dictating what you'll write. It's more a framework. The shallowest of footholds - something that can be used to gain purchase, and free you up to talk about the details of the proof. I wouldn't feel evil for teaching it. Objection: \"This is so stupid! Why are we making kids learn this?! At no time in my adult life have I had to prove that two shapes are similar!\" Response: I believe you, and I don't think anyone is suggesting that writing Euclidean proofs, itself, is a necessary life skill. The thing we really want kids to learn is how to reason logically and communicate their reasoning clearly. Geometry is a context for that, and has been for over 2000 years, because it is a pure abstraction, and you don't have to account for the limitations of measuring devices or friction or any other real-world complicating factors, so we can focus on the argument itself without any distractions. If you don't think that learning to reason logically and communicate your reasoning is a desirable skill, then you and I aren't going to have a very productive conversation. Objection: \"The students I work with might be able to follow those three steps, but no way can they complete the argument like you did. The bit about preserving distances and the argument for why C''' and F are coincident.\" Response: Yeah, maybe. But the three steps are a good chunk of the proof, and if all of my students could do that, I'd be about 75% happy. Let's think about what students need to be able to do to complete just the three steps: visualize and draw the outcome of a transformation, communicate the details of a transformation clearly, and understand which attributes are preserved under transformation (whether they actually state them or not -- but with practice, I think most could make progress stating the argument with less formal language). For test-taking purposes, I'd be willing to bet that communicating the three steps clearly will count for at least half the available credit. The rest is difficult, and comes with practice and intellectual maturity, I'll totally concede that. But, if we're being honest, all practicing teachers make our peace with the reality that not 100% of the kids learn 100% of what they are supposed to learn, so I don't see why this should be any different. Objection: There is too much going on here! The kids have to remember all the details for describing transformations, and then they have to draw them accurately, on top of constructing this argument. Too hard! Response: Yeah, it's a lot. You'll need to probably spend an entire unit before this where they just learn how to describe transformations with precision. However, technology can alleviate some of this burden. For example, Khan Academy is creating some interactive modules where the transformations can be performed through button clicks. I'd cite this as possibly the first example I've seen of KA doing something right. I'd definitely use these modules, selectively, while teaching this, especially for any students with motor-control difficulties, but also for all students to have opportunities to focus on the reasoning and not worry about drawing. I've heard rumors that Desmos is cooking up something similar, so stay tuned for that. An exercise for you: if you'd like to practice a transformational proof with a task that's suitable for, say, advanced high school students and up, try this: Show that when a rectangle is dilated by a factor of k, its perimeter changes by a factor of k, and its area changes by a factor of k2 ## Monday, January 13, 2014 ### Our Favorites... Tuesday, Tuesday, Tuesday! Sometimes you read a MTBoS post and you're like, \"Dag. I want to sit down and buy this person a burrito and get them to tell me all their secrets.\" That happens to me pretty frequently, and I normally try to share my enthusiasm on Twitter, or whatever, because I don't live within burrito-sharing distance of most of these people, but I don't do this in any organized kind of way. At the same time I'm thinking about this, I've been involved in some conversations about how Global Math Department is going through minor growing pains, as most successful endeavors do. Don't worry, we have some really smart and dedicated people who are ON IT. These two things have converged! And resulted in yours very truly hosting an edition of Global Math tomorrow night, where I got to invite three of my favorite bloggers to come talk about three of my favorite posts from the past year. This one is geared toward high school content, but I have no doubt there will be good takeaways for middle school and post-secondary folks, as well. I hope you can join us! (Unfortunately you will have to provide your own burrito.) If you'd like to do some reading ahead, check out the posts we'll be discussing: Shireen D, Math Teacher Mambo, Related Rates and Crowdsourcing in which frustrations with related rates problems are shared (including the dreaded cheese factor), and a plan to address them. Shireen's going to tell us how it went. Mimi Yang, I Hope This Old Train Breaks Down, Graphically Analyzing Inequalities and Equations Flexibly Here is a quote that will get your blood pumping: \"On the quiz for inequalities, my kids had 100% accuracy on all the equations and inequality questions because they were asked to show their work two ways, one by hand and one by the graph, for each equation or inequality.\" Breedeen Murray, The Space Between the Numbers, More Projects, Please I got sincerely excited by the combination of accessibility and mathematical depth going on in Bree's projects. And she's going to share some student work! Woo! ## Saturday, December 21, 2013 ### My Geogebra Fancy Pants The #1 thing you should learn in Geogebra if you want to make cool-as-in-cool things is the Sequence command. If you're already familiar with the sequence command on TI's it works the same: Sequence[expression, variable, start, end, interval]. As a simple example, here is a tool I made to subdivide a segment with evenly-spaced points. But you don't have to just make sequences of points, you can make sequences of anything! Segments, so you can make a grid with a variable number of gridlines. Or an ice cube chopped up into smaller ice cubes: Sectors, so you can make a customizable circle graph. And this thing for Pandemic that is just too fun: ## Sunday, December 1, 2013 ### Catching Fire\/Hell Warning: cranky old lady rant coming. Avert your eyes if you don't like this sort of thing. I saw Catching Fire last night, on its second weekend, at an 8 PM show on a Saturday. Normally, I go to matinees, because I am cheap. But, for complicated reasons, I was there. This is all to say, I haven't been to a crowded showing where the audience skews young in quite a while. During the previews, I could see lots of phone screens. Maybe a dozen. They were the brightest things in the theater. Far brighter than the projected image on the screen. I thought, surely, everyone would put away his phone when the movie started. Wrong! The young man (age hard to tell...I put him at 16-20) sitting right next to me, in fact, was looking at his phone more than he was watching the movie. He was reading his Facebook feed, composing status updates, and tagging lots of people. I know because it kept distracting me from the film, so I read over his shoulder. He was RUINING. The MOVIE. Which cost ELEVEN DOLLARS. I noticed, though, that he kept logging out of Facebook, so when he went back to check it again, he had to log back in. I reasoned that he was trying to deter himself from checking his phone. Each time he finished, he thought, \"I know, I'll log out. That way, it will be a pain to get back on, which will make me less likely to check it again.\" (I often use the same logic when polishing off a pint of ice cream.) Before the film had started, he had to leave and come back twice, awkwardly stepping over my companion and me. Each time, he apologized for inconveniencing us and said thank you, and we were, of course, very polite and accommodating. This was not a rude kid. So about twenty minutes into the film, when I couldn't take it anymore, I leaned to him and loud-whispered, \"YOU KNOW. THAT SCREEN IS REALLY BRIGHT.\" He apologized and put his phone away. I thanked him. I was afraid that might not last very long, but no. He didn't turn it back on for the rest of the movie. There were a couple others in rows further down, and they were annoying, but they were too far away for me to yell at. And the one right next to me was the one really ruining the movie. Which cost eleven dollars. The kids in the theater didn't make me mad. They're kids, and they need to be told. What made me mad, after the fact (I was not ruminating on this during the movie, mind you. It was great. You should go.) was remembering every dummy on the Internet, who inevitably is not someone who spends much time in a classroom, who suggests that if teachers' lessons were interesting enough, kids wouldn't be tempted to distract themselves with their phones. Therefore we shouldn't have to require kids to put away their phones sometimes during math class. Um, Francis Lawrence can't keep kids from being distracted by their phones. With a crazy-good story about reluctant teen revolutionaries. And a$78 million budget.\n\nLearning takes focus. Focus takes practice. Kids might never know how jaw-droppingly cool are the things we are trying to teach them, if their focus is interrupted by Snapchat every two minutes. There are some things they need to be told.\n\n## Saturday, November 9, 2013\n\n### Tell Me Why You Blog\n\nSo, as much to my surprise as anyone's, I'm not only talking at NCTM in April but they made me a featured speaker? Only freaking out a little. I applied in response to a few people who shall not be named (unless they want to out themselves) proposing \"a possible blogging strand with maybe a panel or something.\" So as you can imagine, the idea for what I'm talking about is super well thought-out and fully baked right now. (That was sarcasm, if that wasn't clear.)\n\nThe benefits to written, public reflection are, to me, by this point, so internalized that I find them hard to articulate. And, \"reflective practice\" as The Thing to Do seems to have gone out of fashion. Now it's all about data. Data is the new Reflection.\n\nAhem. If you would, comment on this post and share with me some things:\n1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?\n2. What keeps you coming back? What's the biggest thing you get out of reading and\/or commenting?\n3. If you write, why do you write? What's the biggest thing you get out of it?\n4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and\/or tourism? How-to's? Stories?\n\nAnd, please, link your reply back to your blog, if you have one. I'll make every effort to cite appropriately. Feeling a little weird about crowdsourcing this but I should get over that already. This community has already helped me crowdsource lessons, units, math research, and recommendation letters. Lots of us like to say we got involved and stay involved in this so we can suck a little less. I only get one chance to not suck in New Orleans, and I'd love your help. 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Price's Dakar title hopes 'starting to get out of reach' despite stage win
> News > Offroad > Dakar
Thursday 13th January, 2022 - 8:53am
Toby Price admits he is a longshot to win this year's Dakar Rally, despite taking honours on Stage 10.
The Red Bull KTM rider jumped from ninth to sixth overall with his victory on the Wadi Ad Dawasir-Bisha special, the highest he has been at any point in the event.
However, Price is still 27:43s away from the overall lead, now held by Yamaha's Adrien Van Beveren, and more than 20 minutes away from the podium placings with just two days to go.
He is essentially still trying to make up for the loss of over 40 minutes when a controversial roadbook note caught out several on Stage 1B but, as he has maintained for days now, he has not given up just yet.
"Still two days to come and anything can happen," said the two-time Dakar champion.
"It's not over until the finish, but it's definitely slowly starting to get out of reach.
"But we won't give up, we'll keep trying till that last day and stay healthy, and that's the main thing. We're looking forward to the finish."
The stage win is Price's first of Dakar 2022, after he lost one on Stage 5 due to a speeding penalty.
"It's been a really good day," he said.
"Only a couple of little small mistakes, but we tried to stay on the roadbook as much as we could and pray and hope that the guys at the front would make a mistake, but it didn't quite happen."
There have now been three different rally leaders at the end of the last three days, with Price's KTM team-mate Matthias Walkner dropping from first to fourth overall on Stage 10, although he is 8:24s behind Van Beveren.
One rider who is no longer a contender is fellow KTM rider Kevin Benavides, who suffered a mechanical problem midway through the stage.
"Honestly, anything can happen these last two days," added Price.
"As you could see, it was Kevin who unfortunately today stopped, so it's not over until it's done.
"We've just got to keep charging and hopefully tomorrow we'll have a really good day, leading the pack out, and we'll go from there."
Stage 11 is a 501km loop around Bisha which includes a 346km special, starting this afternoon (AEDT). | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,660 |
1. FESTIVALS: Dhanu Sankranti is the sankranti (transmigration of sun from one rashi – constellation to another) related to Dhanu rashi and occurs on 16th December.
It is celebrated as Dhanu Yatra - a colourful festival related to Bhagwan Krishna's visit to Mathura, at Bargarh, Odisha. Different acts of puranic descriptions are performed at specific locations and the spectators move from place to place with the action to follow the performance. During this festival the Bhagwan Krishna is offered sweetened rice flakes which are specially prepared in a Conical shape.
3. BHARAT LAUNCHES THIRD NAVIGATION SATELLITE: Bharat's Polar Satellite Launch Vehicle (PSLV-C26) lifted off with aplomb from the Satish Dhawan Space Centre (SDSC) in Sriharikota at 1.32 a.m. on 16th October and precisely put the Bharatiya navigation satellite, IRNSS-1C into its perfect, pre-designated orbit. This was the 28th successful launch of the Indian Space Research Organisation.
The IRNSS-1C, the third of the seven navigation satellites in the Indian Navigation Satellite System (IRNSS), has wide-ranging applications in terrestrial, aerial and marine navigation. From vehicle tracking to fleet management and from disaster management to mapping, the satellite extends services to its clients.
The IRNSS-C1 carried two types of payloads, one for transmitting navigation service signals to the users and another consisting of a C-band transponder to facilitate Cube Retro Reflectors for laser ranging.
4. NANAJI BROUGHT INTEGRAL HUMANISM IN ACTION: "Nanaji's thinking was out of box, he always tried to do something extra, beyond his given responsibility. Whatever he did he did with some value addition. He always insisted on 'self-sufficiency' approach. Optimum utilisation of local resources was the basis of his vision. He perfectly blended universal science with local technologies," said Prime Minister Narendra Modi while releasing a book on Nanaji Deshmukh in New Delhi on October 11, the 98th birth anniversary of Nanaji.
RSS Sahsarkaryavah Shri Dattatreya Hosabale, veteran Sangh Pracharak Shri Madan Das, DRI president Shri Virendrajeet Singh and general secretary of DRI Shri Bharat Pathak also shared the dais. Many distinguished personalities including RSS Sarkaryavah Shri Bhaiyaji Joshi, former deputy Prime Minister Shri LK Advani, Governor of Guajrat Prof. Omprakash Kohli, etc were present at the jam-packed Vigyan Bhavan.
5. BHARATIYAS CELEBRATE DESI FESTIVALS IN DENMARK: Vijayadashmi was celebrated by Bharatiyas in Copenhagen on September 28 in a traditional way, thanks to the efforts put in by members of Hindu Swayamsevak Sangh (HSS)-Denmark. The most interesting part of these celebrations was the play: Ramleela. Done with very few props, the characters were able to convey Lord Rama's story to the audience.
"The programme was designed for all age groups and involved a drawing contest based on Ramayana characters and a discussion on the main characters of Ramayana. There were 15 people in the play and we spent more than four weekends preparing for it," said Hemant Dubey, one of the active members of HSS-Denmark.
Dattatreya Hosabale, Rashtriya Swayamsevak Sangh's joint general secretary, had travelled to Copenhagen from Bharat to attend the Vijayadashmi celebrations. "He spoke on the significance of Vijayadashmi as well as key learnings from Ramayana for an hour, before ending it with a moral for the children.
The 1000th year of coronation of renowned King Rajendra I of the famous Chola Dynasty of Bharat is a matter of great pride and inspiration to all of us. Crowned in 10I4 CE, King Rajendra I had his rule extending not only from the banks of river Ganga to the whole of Southern Bharat but also up-to Sri Lanka, Lakshadweep, Maldives, Myanmar, Indonesia, Malaysia, Laos, Cambodia and Vietnam. By virtue of his well managed administration and a well organized military, trade, commerce, art, culture, architecture and sculpture flourished under his reign in this entire region.
Literature and pursuit of knowledge also flourished in his times and several books and volumes were written, both in Sanskrit and Tamil. During his rule, several grand temples and Stupas were constructed in Bharat, Sri Lanka and South East Asia which stand testimony to our living cultural heritage. It is testified from his Charter comprising 21 Copper Inscriptions with his royal insignia written in both Sanskrit and Tamil and paying obeisance at the very beginning to Lord Vishnu which are kept by Leiden University of Netherlands.
At the time of Mahmud Ghazni's attack on our north-western frontier and the turmoil of Euro-Arab conflict, King Rajendra I provided stable rule to ensure peace, prosperity and unhindered trade in the entire South East Asian Region to Bharatiyas, especially Tamil traders and their trade associations. He also established diplomatic Mission in China to promote trade.
He also set up a University for study of Vedas and other disciplines at Ennayirum. Besides ensuring political stability, in order to promote cultural unity and emotional integrity he sent his General Aryan Rajrajan to bring holy water of Ganga, ceremonially received it, mixed it with Kaveri waters and built a grand lake due to which he came to be known as Gangai Konda Cholan [i.e. Chola who brought Ganga].
Remembering such glorious period of our history will be inspirational to the people in the work for national resurgence. All the people of Bharat including Swyamasevaks are called upon to remind Bharat and the world about the achievement of the Chola king Rajendra for providing benign rule with all round progress in such a vast area and to ensure their support as well as participation in all the events related to this incidence.
7. DUBAI-BASED BHARATIYA-ORIGIN EYE SURGEON HONOURED IN LONDON: Dr. Vinod Gauba, Dubai-based Bharatiya-origin doctor has been conferred the prestigious Mahatma Gandhi Pravasi Samman for his contributions to healthcare. Dr. Gauba, 36, who has worked with the less fortunate and visually impaired, was presented with the award earlier this month by Baroness Verma at the House of Lords in London. He was awarded for his pioneering role in the field of ophthalmology.
'Mahatma Gandhi Pravasi Samman' is presented to 20 recipients selected from over 30 million Non-Resident Indians or people of Indian origin for exceptional achievements in various fields.
8. STATE DEPARTMENT CELEBRATES DIWALI AS GLOBAL AFFAIR: Bharatiya tapestries, silver lamps, and traditional delicacies adorned the Benjamin Franklin State Dining Room as Secretary John Kerry inaugurated the U.S. Department of State's annual Diwali celebration. He lit a diya while a Hindu priest from the Sri Shiva Vishnu Temple (SSVT) of Maryland sang Vedic hymns amidst hundreds of well-wishers, including senior administration officials, Ambassadors, and community leaders. "As we celebrate Diwali this evening, we also hail the accomplishments of the many hundreds of thousands of Hindu, Sikh, Buddhist, and Jain Americans who live now all across our country in every community," said Secretary Kerry. "And we honor their faith and their traditions, and the indispensable contributions that they make every single day to our prosperity, to our freedom, and to our culture - to this new chapter of American history that they are helping to write." Kerry was joined on stage by Subrahmanyam Jaishankar, Ambassador of Bharat, who highlighted the joyous festivities taking place around the world.
9. BHARAT TO BUILD 1800-KM HIGHWAY ALONG CHINA BORDER IN ARUNACHAL: To counter China, Bharat is all set to embark on an ambitious road project along the McMahon Line in Arunachal Pradesh. The proposed Indo-China frontier highway will run parallel along the China border and will be 1,800 km long.
10. BHARAT IGNORES CHINA'S FROWN, OFFERS DEFENCE BOOST TO VIETNAM: Bharat took a decisive step towards countering China's assertive poweron 28th October by committing to help Vietnam's defence modernization, a move that will resonate unpleasantly in Beijing.
After his meeting with visiting Vietnamese Prime Minister Nguyen Tan Dung, Bharatiya PM Narendra Modi said , "Our defence cooperation with Vietnam is among our most important ones. India remains committed to the modernization of Vietnam's defence and security forces. This will include expansion of our training programme, which is already very substantial, joint-exercises and cooperation in defence equipment. We will quickly operationalise the $100 million line of credit that will enable Vietnam acquire new naval vessels from India."
11. MASSIVE RELIEF AND RESCUE OPERATION BY SWAYAMSEVAKS IN CYCLONE HIT AP: The cyclone Hudhud severely hit three districts of Andhra Pradesh, Visakhapatnam, Vizianagaram and Srikakulam. Although, the north-east monsoon affects every year the coastal areas of Bay of Bengal during October-November, this year the devastation is huge. Visakhapatnam is the worst hit city.
RSS swayamsevaks started relief and rescue operation on October 12 itself by cutting the trees fallen across the streets and clearing the garbage to facilitate the movement of people and the vehicles. RSS with its several associated organisations like Jana Samkshema Samithi, Sewa Bharati, ABVP, Vanvasi Kalyan Ashram, Vishwa Hindu Parishad, Bharatiya Vidya Kendram Educational institutions, etc, plunged into service activities from the moment they could came out of their houses.
12. RSS VOLUNTEERS CLEAN MOSQUE IN CYCLONE AFFECTED VISAKHAPATNAM: Volunteers of the Rashtriya Swayamsevak Sangh (RSS) cleaned mosques in Visakhapatnam, after cyclone Hudhud left behind a trail of destruction when it hit the region last week.
Coordinator of Visakhapatnam region of RSS, Ram Bahadur, said that the political leaders in the country had created a wrong perception that RSS is against Muslims and Christians.
13. WIDOWS PERFORM LAKSHMI PUJA AT KUDROLI TEMPLE: A large number of widows were allowed on 22nd October to perform Lakshmi Pooja (workship) on the occasion of Deepavali in the Kudroli Sri Gokarnanatheshwara Temple – established in the city by social reformer Narayana Guru and renovated by Senior Congress Leader B. Janardhana Poojary. Allowing widows to do the pooja is among a string of measures taken by the temple aimed at social reformation. The temple has, earlier appointed widows, a couple of them from Dalit communities, as priests.
Around 2,000 widows from Mangalore and other parts of the state participated in the programme. They were brought into the temple in a procession accompanied by beating of drums and other musical instruments.
Indira, Lakshmi and Chandravati, the three priests at the temple, performed Lakshmi Pooja that was witnessed by the large number of widows gathered in the temple. After the pooja, widows were allowed to perform 'Aarati'. The women priests sat along with idols of presiding deity on the silver chariot that was taken around sanctum sactorum by these women three times.
15. MALAYSIAN HINDUS CALL TO SAVE CENTURY-OLD VIVEKANANDA ASHRAM: An organization of Hindus in Malaysia Hindraf recently urged the National Heritage Department of Malaysian Government to take immediate steps to protect the Swami Vivekananda Ashram at Jalan Tun Sambanthan, in Brickfields.
The century-old ashram has been earmarked for major redevelopment that will see a 23-storey residential tower with 264 units and an eight-storey car park built at the site. "Hindraf strongly urge the National Heritage Department steps in to object on the proposed development to protect and preserve this cultural heritage as part of the rich Malaysian heritage," said its chief P. Waytha Moorthy.
16. BHARATIYA ARMY'S EASTERN COMMAND TURNS 94: Army's biggest operational command - the Eastern Command - responsible for military operations along the international borders with Bangladesh, Bhutan, China, Myanmar and Nepal, turned 94 on 31st Oct.
The Eastern Command will continue to strive for "unparalleled professional competence to ensure the highest standards of operational readiness", Lt. Gen. M.M.S. Rai, general officer commanding-in-chief, said on the occasion at its headquarters at Fort William Kolkata.
The Eastern Command was formed Nov 1, 1920, with its summer headquarters in Nainital and winter headquarters in Lucknow. The command was designated as Eastern Army in April 1942, and its headquarters moved to Barrackpore. The Eastern Command theatre consists of three distinct geographical regions - the mountainous sectors of Sikkim and Arunachal Pradesh in the north, the jungle-clad hill tracts of Nagaland, Manipur, Mizoram, Tripura and Meghalaya in the east and the south, and the plains of Assam and Bengal.
17. NATIONWIDE BLOOD DONATION CAMP OF VHP: 1 LAKH UNITS COLLECTED: In an overwhelming response to nationwide blood donation camp of VHP on 2nd November , thousands of youth donated blood across Bharat, crossing a collection of a total of 100,000 units of blood across the nation. The blood donation drive was organised by VHP and its youth wing Bajarangadal to commemorate the Golden Jubilee year of Vishwa Hindu Parishad.
18. BHARATIYA COP WINS UN'S FEMALE PEACEKEEPER AWARD: A Bharatiya police inspector has been named recipient of a prestigious international female peacekeeper award by the UN's police division for her "exceptional achievements" in her duty with the UN mission in Afghanistan.
Inspector Shakti Devi of the Jammu & Kashmir Police, currently deployed in the UN Assistance Mission in Afghanistan (UNAMA), has been awarded the International Female Police Peacekeeper Award 2014.
Devi has been honoured for her "exceptional achievements" in leading the establishment of Women Police Councils in several parts of Afghanistan. She has also contributed to the improvement of the status of female police and has effectively helped the police of Afghanistan move towards achieving their goals of fully adopting democratic principles of policing.
19. BHARAT - BORN SCIENTIST AWARDED WORLD FOOD PRIZE: Bharat -born Mexican scientist Sanjaya Rajaram has been presented with the prestigious World Food Prize 2014 for his agricultural research that led to a remarkable increase in world wheat production building on the successes of the Green Revolution. "It is a collective achievement, rather than that of a single person," Rajaram told while accepting the award at the Iowa State University in USA. The award "honours the innovative spirit of farmers", he said adding that "without their contributions, my research wouldn't have been possible".
I just cannot remember the day. But I can definitely recall that on that day Eknathji (Shri Eknath Ranade) was coming to Kolkata from Chennai by a morning flight. I went to the airport to receive him. We were heading towards Vivekananda Kendra Karyalaya. All of a sudden Eknathji asked me, "What do you think? What should I do now?"
So sudden was the question, I was taken aback. If a giant personality like Eknathji asks such a question to a most junior worker like me what should I say? I had no answer for a moment. Then I thought when Eknathji was assigned the work of Vivekananda Memorial, he was Sarkaryavah of RSS and as a Sangh swayamsevak we have been missing him. So much so, the time before the country (1971 or 1972) was also bad so I said we hardly can afford to miss you any more from the Sangh work.
It was my honest and spontaneous reply. As a seasoned 'karyakarta', Eknathji immediately read my mind and started explaining the situation. By that time the first phase of the work of Vivekananda Memorial work was over and the then President of India VV Giri had inaugurated the grand memorial of Swami Vivekananda. In a way, the job assigned by Shri Guruji to Eknathi was successfully done. Now, puting forward a question to one and all of his acquaintances was obvious. What is the purpose of setting up of a temple? Should it not become a centre of activities to fulfill the ideal of 'Serve people, serve God'?
"Some people ask, 'Could this stone-structure alone be a fitting memorial to that great patriot saint?' To them I may humbly say that I myself have never been much of a believer in merely putting stone upon stone. The real urge in the Committee workers, including me, to work out the present memorial-plan springs forth from their faith and confidence that this granite structure would not only be an abiding source of great inspiration to posterity, but should also provide a nucleus a round which, over a period, great activities which the Swamiji preached and envisaged would spring up.
"Kanyakumari is a place, which may be considered as great symbol of purity and unity. It is a meeting point of three oceans. If the Sangam at Prayag is sacred because of the mingling of the holy waters of the three sacred rivers, this holy spot is a Mahasangam where the waters of the Ganga and Sindhu, Narmada and Goda, Krishna and Kaveri ultimately meet in the form of those three oceans the Gangasagar, the Sindhusagar and Hindu Mahasagar.
The place is, in a way, a meeting point of the East and the West also Standing at the topering end of our country, at Kanyakumari we can notice the disc of the sun rising from the sea in the East and taking a dip in the sea in the West."
"It is again the meeting point of the North and the South. The Goddess Kanya stands there at the southern most tip of our land, with a garland in Her hand to meet Her Lord, Shiva, whose abode is in Kailash, in the regions of the northernmost Himalayas.
"What a unique symbol of unity and purity! I have faith and confidence that the great memorial that is taking shape at that holy spot will, in times to come, generate a powerful current of Swamiji's thought and will envelop the entire country."
This lecture of Eknathji led me to understand him from altogether a different angle. His vision was so clear at the same time so down to earth as people of Bharat were eager to accept without any hesitation and time has proved its perfectibility. I was associated with the construction of the Vivekananda Rock Memorial at the very initial stage i.e. from 1963 to 1970. I visited Kanyakumari in the year 1963 as Eknathji wanted me to go there to get first hand experience.
Eknathji used to tell one thing that the idea of setting up of a memorial for Vivekananda would be successful only when mass participation in the construction work was assured. He said, through one-rupee coupons money should be collected so that even a poor man could participate in this Mahayajna. Mrs. Kamal Basu, wife of late Jyoti Basu, former Marxist chief minister of West Bengal, collected more that eleven thousand rupees by selling One-Rupee Coupons and deposited them to Vivekananda Rock Memorial Fund.
Eknathji was a perfectionist. When the question of constructing the bronze statue came he had engaged almost all the famous and renowned sculptors of the country. While engaging them he did not give any undertaking as to whose sculpture would be set up at the memorial temple on the rock. He only said the committee which was assigned to do this work selects the statue to be set up there at the Shilamandir. In Calcutta, he engaged Shri Debi Prasad Roychoudhury who was world famous sculptor and whose sculptures of Gandhiji or Freedom Movement of India were set up there in Calcutta, Delhi, Patna and many other places. Eknathji asked me to regularly visit the house of Shri Roychoudhury to watch the progress of the work and report it to Eknathji. Whenever I went to his house, Shri Rouchoudhury used tell me, "It was a mystery that I had accepted the proposal put forward to me by Eknathji. You know this proposal itself was derogatory for me as it had put me on the same rank of other general sculptors, which I was not. But Shri Ranade must have some kind of spiritual power by which he influenced me." The fact remained that the statue made by him was not put up at the Shilamandir at Kanyakumari. But it was placed somewhere else and in that case Eknathji abided by the condition of contract rules.
Eknathji had faced two almost insurmountable hurdles. Those hurdles were put up by (1) Prof. Humayun Kabir, Central Cultural Affairs minister and (2) Shri Bhaktavatsalam, the then Chief Minister of Tamil Nadu. Both were very tough personalities. How Eknathji managed those two guys is an interesting history. Recently, some 4/5 months back at a meeting held at Chhatubabur Rsjbati, retired Justice Shyamal Sen of Calcutta High Court, while mentioning the role of Chpalakanta Bhattacharya, former editor of 'Ananda Bazar Patrika' and Congress MP said Chapalababu was of great help to Shri Eknath Ranade in collecting signatures of Members of Parliament in support of constructing the grand temple of Swami Vivekananda at the rock at Kanyakumari. Eknathji was in Calcutta at that time to organise opposition against the comment of Humayun Kabir as "temple on the Rock would spoil the scenic beauty".
Swami Sambuddhananda was the General Secretary of Ramakrishna Mission at that time. He asked Debu Maharaj who was looking after the press section of the Mission to accompany with Eknathji and help him out in this campaign. The Press of Calcutta had given all out support to the cause of Vivekananda Temple at Kanyakumari.
In Calcutta the sentiment about Swami Vivekananda was tremendous. But the work started by the Vivekananda Rock Memorial at Kanyakumari under the leadership of Shri Eknathji was unknown to them. At the same time Eknathji himself was also not so much acquainted with the prominent personalities of Bengal. Hence he chalked out a unique programme of writing personal letters to all those important personalities regularly apprising them of progress of the work at Kanyakumari.
Apart from this, every time he came to Calcutta he used to meet the chief minister, Mayor of Calcutta, sometimes the Governor of Bengal and the leaders of different political parties and social organizations. Personalities like the great historian Dr. Ramesh Chandra Majumdar, Speaker of state assembly Keshav Chandra Basu, Principal Dr. Amiya Kumar Majumdar, Bharat Maharaj of Belur Math and several other Swamijis of Ramakrishna Mission etc. Before his visit to Calcutta, every time he used to write letters to all those expressing his desire to meet them.
I know these things because as Calcutta Representative Eknathji used to keep me informed by sending copies of those letters to me. I am fortunate enough to get to work with such a visionary as a representative in Kolkata.
अब युद्ध अटल हो जाता है। और युद्ध में विजय प्राप्त करना अनिवार्य रहता है। कुरुक्षेत्र के रण में 18 दिनों तक पाण्डवों और कौरवों के बीच घनघोर युद्ध होता है। इतना घनघोर की 18 अक्षौहिणी की विशाल सेना में से केवल दस लोग जीवित बचते हैं। पाण्डवों में सात और कौरवों में तीन. बस्स! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,421 |
\section{Introduction}
H~{\scshape i} emission lines are one of the defining characteristics of the classification of pre-main-sequence Sun-like sources known at T Tauri stars (TTS). Still in the midst of formation, the less evolved TTS, known as classical TTS (cTTS), are surrounded by optically thick disks of gas and dust. In most cases, these young pre-main-sequence stars are still interacting with and accreting matter from the innermost regions of their disks via stellar magnetic fields. In this magnetospheric accretion paradigm, the stellar magnetosphere guides disk material from the inner disk onto the stellar surface through magnetic channels. The gas travels along these channels or so-called accretion columns near free-fall velocities, terminating in an accretion shock at the stellar surface. It is generally accepted that the gas is heated and ionized prior to and after reaching the stellar surface and that the characteristic H~{\scshape i} emission lines result, in part, from recombining and accreting hydrogen gas confined to these magnetic channels \citep{lynd1974, uchi1984, bert1988}.
Balmer H$\alpha$ emission is the dominant H~{\scshape i} feature present in the optical spectra of cTTS, and the emission line strength (or line width) is often invoked as a measure of accretion rates for these sources \citep{muze1998a}. Spatially resolved observations of H$\alpha$ emission lines in these young stellar objects (YSOs) also show it to be a strong component of the optical line emission from outflows, indicating that the underlying stimulation mechanisms for the H~{\scshape i} lines is likely to be a combination of phenomena. Magnetospheric accretion models can successfully reproduce many aspects of the H~{\scshape i} emission features detected in the spectra of young stars. However, they notably fail to account for the highest velocity gas in the H~{\scshape i} line wings \citep{muze1998a}. Spectro-astrometric observations of the Pa$\beta$ emission feature in the cTTS DG~Tau show that the high velocity blue-shifted gas ($v$~$>$~-200~km~s$^{-1}$) is extended in the same direction as [Fe~{\scshape ii}] at 1.644~$\mu$m, a forbidden emission line feature that traces the known outflow in the DG~Tau system. This illustrates why a model producing H~{\scshape i} emission from accretion funnels does not account for high velocity gas forming the H~{\scshape i} line wings \citep{whel2004}. While the high velocity gas is spatially shifted by as much as 0.$"$5 around DG Tau (or $\sim$~70~AU at the distance of Taurus-Auriga), the majority of the Pa$\beta$ emission remains coincident with the source. Although these results demonstrate that there are multiple processes for stimulating H~{\scshape i} emission lines, the bulk of the emitting gas does seem to arise from radii within 14~AU of the central source constraining the emission to the magnetospheric accretion columns, inner gaseous disk, and the base of disk winds and outflows.
In the infrared, Br$\gamma$ (2.16~$\mu$m) emission serves as a surrogate for H$\alpha$ as a signpost for circumstellar disk accretion in TTS \citep{naji1996}. Br$\gamma$ line luminosities appear to correlate with mass accretion luminosity in brown dwarfs, cTTS, and Herbig Ae/Be stars \citep[HAEBEs;][]{muze1998a, natt2004, moha2003, moha2005}. Moreover, Br$\gamma$ and other lines in the infrared are less affected by optical depth effects that have proven problematic for using Balmer series lines to infer temperatures, densities, and geometries of the emitting gas. Historically, the problem of predicting the spectra emerging from a recombining hydrogen gas has been divided in to two distinct cases, A and B \citep{bake1938}. Case A theory applies to very low density gases that are optically thin to all transitions of the hydrogen atom, including the ultraviolet photons associated with the Lyman series transitions. Case B theory, which applies to higher density gases which are optically thick to UV photons but optically thin to all n $\ge$ 2 transitions, is often applied to the environments of T Tauri stars. In a Case B model of a recombining atomic hydrogen gas, Br$\gamma$ is $\sim$0.8-1\% of the flux of H$\alpha$ for a wide range of densities and temperatures (10$^2$ $<$ n$_e$ $<$ 10$^6$~cm$^{-3}$, 5000K $<$ $T$ $<$ 20000K). H$\alpha$ is known to be a strong component in optical line emission from YSO outflows \citep{naji1996}, it seems natural that a corresponding component of the Br$\gamma$ emission would also arise from the outflows. Yet, to date, there has been little evidence in the literature for spatially resolved Br$\gamma$ emission in the vicinity of YSOs, and hence nearly all Br$\gamma$ emission is assumed to arise from magnetospheric processes within the inner accretion zone. In fact, infrared interferometric observations of HAeBe stars reveal that the Br$\gamma$ often arises from very compact locations within the dust sublimation radius of the circumstellar disk \citep{eisn2009, krau2008}. Though, a small extended Br$\gamma$ emission component can not be ruled out based on these observations. Further interferometric and spectro-astrometric programs that seek to reveal the inner $\sim$1 AU environments show that the compact Br$\gamma$ is not always well modeled by disk emission alone \citep{eisn2010, malb2010}. The analysis implies that a non-negligible component from outflowing gas needs to be incorporated into the models to explain the Br$\gamma$ emission structure.
As an extension of the spectro-astrometric and interferometric studies mentioned above, imaging spectroscopy of these TTS can provide us with considerable insight into the geometric distribution of H~{\scshape i} emitting gas in accreting systems. Three-Dimensional imaging spectroscopy techniques can help to disentangle the respective contributions to the H~{\scshape i} emission features. With just one pointing of a telescope, imaging spectroscopy with integral field units (IFUs) can provide three-dimensional x, y, $\lambda$ datacubes at high spatial resolution with simultaneous coverage of many emission lines of interest. There has recently been an increase in the capabilities for adaptive optics (AO) fed near IR integral field spectroscopy at 8-10 meter class observatories \citep{eise2000, mcgr2003, lark2006}. IFUs optimized for AO spectroscopy have the power to spatially resolve emission line structures with less than 0.$''$1 extents over the full wavelength ranges sampled by typical IR spectrographs. As such, the new generation of IFUs provides the means to study the accretion and outflow environments in cTTSs.
In this paper, we present detections of spatially resolved Br$\gamma$ emission in YSO environments from data acquired using the Near IR Integral Field Spectrograph at the Gemini North Observatory. We report on Br$\gamma$ arising from eight classical TTS systems, and particularly highlight the spatially extended emission detected in four of these: DG Tau, Haro 6-10 (also known as GV Tau), HL Tau and HV Tau C.
\section{Observations}
Observations of the eight CTTSs listed in Table~1 were obtained using the Near IR Integral Field Spectrograph (NIFS) at the Frederick C. Gillette Gemini North Telescope on Mauna Kea, Hawaii. NIFS is an image slicing IFU fed by Gemini's Near IR adaptive optics system, Altair, that is used to obtain integral field spectroscopy at spatial resolutions of $\le$0.$''$1 with a spectral resolving power of R$\sim$5300 at 2.2~$\mu$m \citep[as measured from arc and sky lines;][]{mcgr2003}. The NIFS field has a spatial extent of 3$''\times3''$, and the individual IFU pixels are 0.$''$1$\times$0.$''$04 on the sky. Data were obtained at the standard K-band wavelength setting for a spectral range of 2.003-2.447~$\mu$m. All observations were acquired in natural seeing of better than 0.$''$7 for excellent AO correction.
The data sets for this study were acquired for commissioning and system verification of NIFS in October 2005 and February 2006, GTO time in December 2006, and in queue mode in February 2007 (see Table~1). For each observation, a standard set of calibrations were acquired using the Gemini facility calibration unit, GCAL. The raw IFU frames were reduced into datacubes using the NIFS tasks in the Gemini IRAF package\footnote{Information on the Gemini IRAF package is available at http://www.gemini.edu/sciops/data/dataIRAFIndex.html}.
\cite{beck2008} discuss these NIFS data on DG~Tau, HL~Tau, HV~Tau~C, RW~Aur, T~Tau and XZ~Tau in the context of spatially resolved molecular hydrogen emission lines. Hence the observational details, calibration, and data reduction information is described in great detail in that paper and excluded here. To study the Br$\gamma$ emission in the young stars, the absorption features from the A0 stellar type telluric calibration stars were removed by fitting and dividing Voigt absorption profiles in the 2.16~$\mu$m spectral region and cleaning the calibration spectra for any small residuals. Observations of Haro~6-10 were acquired with the laser-fed AO system using the R$\sim$16.5 mag southern component in this 1$.''2$ binary as the laser tip-tilt reference star. These data were obtained in excellent laser-quality weather, photometric with better than 0$.''5$ seeing. The CW~Tau and Haro~6-10 data were processed in a similar manner to all other data, as described above and in \cite{beck2008}. All or part of the data for each source in this project was observed during photometric conditions and were flux calibrated using K-band magnitudes estimated by comparison to the brightness of the A0 standard star used for telluric correction. With the exception of Haro 6-10, the derived fluxes of the systems (combined in the case of multiples) were within 10-15\% of published or 2MASS magnitudes. For Haro~6-10, the NIFS IFU spectra were compared to infrared K-band images acquired nearby in time for a complimentary project, the difference in flux was less than 0.2 magnitudes between the two observations. We estimate that our overall data flux calibration is good to $\pm$10-15\%. The data cubes were interpolated onto a square pixel grid with 0.$''$05 spatial sampling, and the velocity channel steps through the IFU cubes are $\sim$29~km~s$^{-1}$ pixels in extent at 2.20~$\mu$m ($\sim$56~km~s$^{-1}$ 2 pixel resolution). The final reduced, combined, telluric corrected, and flux-calibrated datacubes of the Br$\gamma$ line emission for each target are discussed in detail in the following sections.
\section{Spatially Extended Br$\gamma$ Emission from YSO Environments}
The data acquired for this project, described in detail in the preceding section, were obtained with the goal of studying the K-band features of molecular hydrogen emission. The IFU spectral data were discussed by \citet{beck2008} in this context. We never expected to detect spatially extended Br$\gamma$ emission in any CTTS. This was a serendipitous discovery, revealed as we stepped through the raw velocity cube at Br$\gamma$ wavelengths in the Haro~6-10~S data. The clear detection of spatially extended emission from the Haro~6-10~S jet prompted us to take a closer look at the Br$\gamma$ emission in all TTS for which K-band IFU spectra had been obtained. As presented and discussed in the following, we have found significant spatially extended Br$\gamma$ emission in four of the eight stars presented here: DG~Tau, Haro~6-10, HL~Tau and HV~Tau~C. We do not find appreciable spatially extended Br$\gamma$ in CW~Tau, T~Tau, XZ~Tau, or RW~Aur.
Figures~1 through 4 show the images of the spatially extended Br$\gamma$ emission from DG~Tau (Figure~1), Haro~6-10 (Figure~2), HL~Tau (Figure~3) and HV~Tau~C (Figure~4). The panels in these figures show: a) - the continuum emission with contours of [Fe~{\scshape ii}] emission overplotted to demonstrate the outflow position and geometry, b) - continuum subtracted Br$\gamma$ line emission maps with contours of the continuum overplotted, c) - ``point-source subtracted'' spatially extended maps of Br$\gamma$ emission with the contours of the continuum-subtracted Br$\gamma$ overplotted, and d) - images of the integrated blue-shifted Br$\gamma$ emission only, with contours of the point-source subtracted Br$\gamma$ emission overplotted. At the right side of each image is the key correlating the image intensity to flux. The continuum images in panels (a) were constructed by fitting a straight line to the continuum around the Br$\gamma$ emission feature and integrating the linear fit through the velocity channels that correspond to the Br$\gamma$ emission. The images of the total Br$\gamma$ emission presented in the b) panels were constructed by subtracting the linear fit to the continuum from the datacubes, then integrating in velocity over the Br$\gamma$ emission feature. The "point-source subtracted" Br$\gamma$ images shown in the c) panels were derived by normalizing the continuum (PSF) image to the peak flux in each velocity channel through the Br$\gamma$ emission datacube, subtracting this scaled continuum image off of the Br$\gamma$ cube, and integrating over the velocity extent to form an image of only the extended emission. The image of spatially extended blue-shifted Br$\gamma$ emission shown in panel d) was made by integrating the 'point-source subtracted' image in 2-3 velocity channels of blue-shifted emission only. The d) panels show that the spatially extended Br $\gamma$ emission is stronger in the blue-shifted velocity channels. Because of Poisson statistics associated with the subtraction process, detection of spatially extended Br$\gamma$ emission is less robust at distances of $<$0.$"$1 from the central point-source. The Br$\gamma$ emission in the vicinity of HV~Tau~C is quite weak, and all of the detected line emission is spatially extended. As a result, Figure~4 presents only the (a) and (b) panels for HV~Tau~C.
For DG~Tau and Haro~6-10 S, the majority of the Br$\gamma$ emission that we detect, integrated over wavelength, is consistent with the stellar point-source image. For HL Tau and HV Tau C, the majority of the Br$\gamma$ is not coincident with the central TTS. The integrated Br$\gamma$ emission from DG Tau follows the point-source continuum contours with little deviation (Figure~1b). The spatially extended Br$\gamma$ emission from DG~Tau is detected in panel 1(c), and seen clearly in the blue-shifted emission shown in panel 1(d). The Blue-shifted emission (Figure~1d) extends to the south-west of DG~Tau, at an orientation and velocity consistent with the known, collimated blue-shifted jet. Haro~6-10 also shows the bulk of the Br$\gamma$ emission arising from the two stellar point-sources. However, strong emission comes from the same location as the [Fe~{\scshape ii}] outflow emission from Haro~6-10~S (Figure~2c). In fact, for Haro~6-10~S, the Br$\gamma$ arising from the outflow is $\sim$20\% of the total spatially integrated Br$\gamma$ emission in some blue-shifted velocity channels (Figure~2d). No spatially extended Br$\gamma$ is detected toward the Haro~6-10~N component.
Data on HL~Tau were acquired with the 0.$"$2 occulting disk blocking the central stellar point-source, however an appreciable amount of the detected Br$\gamma$ emission is seen to deviate from the central stellar position. HL~Tau shows strong spatially extended Br$\gamma$ that follows the scattered light nebulosity revealed in the continuum emission \citep[Figure~3b, 3c;][]{taka2007, clos1997}. The integrated blue-shifted Br$\gamma$ from HL~Tau (Figure~2d) shows very weak emission detected at a 4$\sigma$ level of significance that corresponds precisely with the spatial location of the [Fe~{\scshape ii}] emission. HV~Tau~C is a system with a known circumstellar disk viewed nearly edge-on \citep{stap2003} and the stellar continuum flux is much fainter than the other sources. Curiously, we find no significant Br$\gamma$ emission associated with the locations of the continuum flux. The Br$\gamma$ emission from HV~Tau~C is quite weak. It is spatially extended from the continuum and it appears to only follow the location of the outflow seen in [Fe~{\scshape ii}] (Figure~4).
The spatially extended Br$\gamma$ emission revealed in Figures~1, 2, and 4 for DG~Tau, Haro~6-10~S, and HV~Tau~C, lies precisely along the outflow axis of the known Herbig-Haro energy flows associated with these young stars. For HL~Tau, the detected (low signal-to-noise) Br$\gamma$ emission in the blue-shifted component of the emission (Figure~3d) arises from the same spatial location as the known blue-shifted outflow \citep{taka2007}. However in HL~Tau the majority of the spatially extended Br$\gamma$ emission (Figure~3c) is not appreciably shifted in velocity from the nominal stellar radial velocity; it is detected at much higher S/N, and appears to arise from Br$\gamma$ emission from the central point-source that has been scattered off of the wall of the outflow cavity \citep{clos1997, taka2007}.
Figure~5 shows the velocity profiles of the Br$\gamma$ flux associated with the central point source for DG~Tau, Haro~6-10, HL~Tau and HV~Tau~C at the location of the peak continuum emission (upper panel) and the profiles of Br$\gamma$ emission extracted in 0.$"$2 diameter apertures at ``Position B" and ``Position C" as designated in Figures~1 through 4 for each star. The spatially extended Br$\gamma$ from DG~Tau and Haro~6-10~S is blue-shifted in velocity by $>$~100~km~s$^{-1}$ with respect to the central point-source flux. HL~Tau shows strong emission at the velocity of the point-source flux in ``Position B," and very slight $\sim$3-4$\sigma$ detection of flux from blue-shifted ($\sim$-200~km~s$^{-1}$) emission from the outflow.
In DG~Tau, Haro~6-10~S and HL~Tau, the spatially extended Br$\gamma$ emission that we detect from the outflows corresponds to the blue-shifted regions of the jets. This is consistent with the fact that blue-shifted jet components are flowing into our line of sight, and are hence less obscured by intervening circumstellar disk material. HV~Tau~C is the only source where both blue and red-shifted Br$\gamma$ is detected from opposite sides of the outflow with respect to the (estimated) position of the central star. The extended Br$\gamma$ from HV Tau C is quite weak in some regions, it is detected at a low signal-to-noise but follows the location of the extended outflow. Curiously, we find that the blue-shifted outflow lies to the northeast, and the red-shifted emission is to the southwest, which \citep[as predicted by][]{stap2003} is at odds with the relative brightnesses of the lobes of the scattered light edge-on disk reflection nebula associated with this source. The southwestern lobe of the HV Tau C scattered light nebula is brighter, and was thus thought to be associated with the blue-shifted (closer) lobe of the outflow. Our data reveal that this is not the case, and the north-eastern, fainter lobe seems to be associated with more blue-shifted outflow emission.
Figure~6 presents the standard deviation of continuum subtracted Br$\gamma$ flux with increasing distance from the central star (plotted as a solid line) for HL~Tau (a), DG~Tau (b), Haro~6-10 (c) and HV~Tau C (d). These curves were derived for each source by calculating the mean flux within 0.$"$08 wide pixel annuli from the central stellar position and computing the standard deviation from the mean. This standard deviation of continuum subtracted line emission provides a measure of the uncertainty in the subtraction process, and thus can be used to gauge the S/N of the spatially extended line emission. Though, the continuum subtracted plots do have the extended Br$\gamma$ flux included within the field. As a result, the measured standard deviations in the spatial bins with appreciable Br$\gamma$ flux are correspondingly higher, and the accuracy of the subtraction process is thus under-estimated. Overplotted in each of the four panels is a dashed line that shows the magnitude of the point-source subtracted Br$\gamma$ flux with increasing distance from the central stellar position. Comparison of the dashed curve with the solid curve provides an estimate of the measured S/N of the extended Br$\gamma$ emission. The peak S/N for extended Br$\gamma$ is $\sim$7 for HL~Tau, $\sim$5 for DG~Tau, $\sim$25 for Haro~6-10 S and $\sim$4 for HV~Tau~C.
Figure~7 plots the same standard deviation of continuum subtracted Br$\gamma$ flux with increasing distance from CW~Tau, T~Tau, XZ~Tau, and RW~Aur, and a dashed line is also over-plotted that presents the peak magnitude of the point-source subtracted Br$\gamma$ emission from the central star (in the case of the multiple systems, the central star is assumed to be the brightest stellar component). The stellar companion position locations are apparent in the plots for XZ~Tau and T~Tau, while RW~Aur's companion has a slightly greater separation than presented in Figure~7d. For all four sources presented in Figure~7, no excess was seen in spatially extended Br$\gamma$ emission at the location of the known Herbig-Haro outflows. For the cases of CW~Tau, XZ~Tau and RW~Aur, no strong evidence of extended Br$\gamma$ emission beyond a S/N of $\sim$3 is found. Curiously, XZ~Tau~B showed weak Br$\gamma$ emission associated with the position of the star, but XZ~Tau~A had no detectable Br$\gamma$. This result is seemingly at odds with the proposition that XZ~Tau~A is the more actively accreting star and the main driving source of the Herbig-Haro flow associated with this system \citep{kris2008} Although RW~Aur exhibits very strong, centralized Br$\gamma$ emission flux, RW~Aur~B showed no measurable Br$\gamma$ emission associated with stellar mass accretion from its circumstellar disk.
The central region of Br$\gamma$ emission for T~Tau~N was saturated in the data, making a proper measurement of the continuum subtracted Br$\gamma$ flux and point-source subtracted flux difficult. The surrounding spatial and spectral regions nearby are not saturated, and were used to estimate the 2.16~$\mu$m flux level based on the shape of the PSF. A cursory fitting analysis done at velocities on and off of the emission line using the A 0 spectral type telluric calibrator as a PSF reference showed that the Br$\gamma$ emission associated with T~Tau~N is not appreciably extended compared to the point-source continuum emission. The line emission is asymmetric around the T~Tau~S PSF, and is brighter to the northwest at the location of T~Tau~Sb. Hence, the Br$\gamma$ line emission associated with the nearby 0$."$1 T~Tau~S a+b binary seems to arise preferentially from the $\it{b}$ component, and it is stronger than the continuum flux (i.e., the T Tau Sb / Sa flux ratio is greater in Br$\gamma$). This causes the apparent enhancement in Br$\gamma$ emission at the position of the companion in Figure~7b. Extraction of the individual spectra of the blended components was not done because of the saturation of T~Tau~N, which would need to serve as a PSF calibrator.
\section{Br$\gamma$ Estimates of YSO Mass Accretion and Mass Outflow Rates}
\citet{muze1998b} showed that the Br$\gamma$ line luminosity from young stars correlates with the mass accretion rate, as determined from UV excess emission. Hence, we now use our detected Br$\gamma$ line fluxes to derive mass accretion rate for the observed targets. Table~2 presents the total Br$\gamma$ emission line fluxes (i.e., integrated over all velocity channels) for each of the young stars in this study. The relation from \citet{muze1998b} assumed all detected Br$\gamma$ emission was associated with the stellar point-sources, so the line fluxes that we have included in Table~2 are the total integrated flux values, including emission detected in the outflowing gas. Also included in Table~2 are the adopted stellar parameters used to derive the mass accretion rates: the stellar mass, temperature, luminosity and visual extinction \citep{keny1995, whit2001, hart2003, dopp2008}. The stellar parameters seem consistent with our K-band spectra, so we do not rederive them from our data.
XZ~Tau~B, T~Tau~S and Haro~6-10~N are ``infrared luminous companions'' (IRCs) to their respective primaries, and the stellar parameters for these sources are much less certain \citep{kore1997,whit2001}. The accretion activity and line of sight visual extinction toward the IRCs could also be variable \citep{ghez1991, beck2001, lein2001, beck2004}. Moreover, T~Tau~S is itself a binary, and the bulk of the Br$\gamma$ emission that we detect arises not from the IRC but from the $\sim$M-type T~Tau~Sb companion \citep{duch2005}. Haro~6-10~N was found by \citet{dopp2008} to have weak evidence of photospheric Na absorption features at 2.20~$\mu$m, with a late spectral type and an infrared veiling value estimated to be in the range of 12-15. Similarly, XZ~Tau~B also has strong infrared veiling and a poorly determined spectral type. As a result, we do not try to estimate mass accretion rates for T~Tau~S, Haro~6-10~N or XZ~Tau~B. Column~7 of Table~2 presents the mass accretion rates ($\dot{M}_{acc}$) derived for all of the other stars, using the relation from \citet{muze1998b} for the Br$\gamma$ line luminosity to accretion luminosity and the Virial Theorem treatment put forth by \citet{gull1998}.
The mass accretion rates derived for the eight stars in this study that have well determined stellar parameters lie in the range from less than 4$\times$10$^{-10}$~M$_{\odot}$ for RW~Aur~B, to 1.5$\times$10$^{-7}$~M$_{\odot}$ for T~Tau~N, with most sources in the range of 10$^{-8}$ to 10$^{-7}$~M$_{\odot}$. These values compare lie within the overall range of mass accretion rates derived for cTTSs \citep{muze1998b, gull1998}. For the most part, the mass accretion rates that we derive here are similar to $\dot{M}_{acc}$ values for these stars that have been derived from previous studies, within the associated uncertainties (e.g., Hartigan et al.\ 1995; Muzerolle et al.\ 1998; White et al.\ 2004). However, direct comparisons between mass accretion values for DG~Tau, particularly, show a large discrepancy in our study. DG~Tau has in the past exhibited a mass accretion value on the high side of the range for TTSs; around or just under 10$^{-6}$~M$_{\odot}$~yr$^{-1}$. Our derived mass accretion rate is an order of magnitude less than many previously published estimates \citep{hart1995,muze1998b}. DG~Tau is known to vary in flux and spectral characteristics on short time scales \citep{bisc1997, hess1997, bary2008}. Bary et al.\ (2008) found a nightly average Br$\gamma$ equivalent width of 6.4 \AA\ for seven nights of observations with a standard deviation of 20\%. Therefore, the discrepancy we see in mass accretion rate toward DG Tau might be a result of both intrinsic stellar and accretion variability.
Also presented in column~8 of Table~2 is the Br$\gamma$ flux from the spatially extended outflows from DG~Tau and Haro~6-10~S. For our analysis of Br$\gamma$ emission seen in the outflows, we extracted a region of Br$\gamma$ flux from the spatially extended jets from DG~Tau and Haro~6-10 (Figures~1d and 2d). Multi-epoch observations have shown that the average transverse proper motion of an outflow is $\sim$0$."$2 to 0$."$3 per year, or 28-42~AU at the $\sim$140pc distance of Taurus \citep{torr2009}. Both the DG~Tau and Haro~6-10~S outflows are viewed at 60-70$^{\circ}$ inclination angles \citep{movs1999, pyo2003}, and the 28-42 AU extent in the outflow is also roughly consistent with the corresponding annual motion of an average flow velocity of $\sim$200~km~s$^{-1}$; e.g., material in the flow would move 28-42AU/yr at the estimated geometry and velocity. For this reason, to capture the emitting flux from approximately one year of jet motion, we have extracted a segment, 0$."$3 in extent, of the spatially resolved Br$\gamma$ emission along the jet axes for the DG~Tau and Haro~6-10 outflows. The width of the extraction box corresponds to a 0$."$2 radius encircling the jet. The Br$\gamma$ emitting volume, V$_{\rm {HI}}$, roughly corresponds to this area surrounding the jet, and is approximated by a cylinder of 28~AU (0$."$2) radius and 42~AU (0$."$3) height. The ``one year" integrated Br$\gamma$ outflow emission fluxes for DG~Tau and Haro~6-10~S extracted from these apertures are 6.8$\times$10$^{-16}$ and 8.5$\times$10$^{-16}$~erg~s$^{-1}$~cm$^{-2}$, respectively.
Under the assumption that the spatially extended Br$\gamma$ emission seen from DG~Tau and Haro~6-10~S arises from thermally excited emission from optically-thin, post-shock regions of the outflows and behave as a Case~B recombining gas \citep{bake1938, broc1971, humm1987, stor1995}, we use a simple analysis to measure the mass outflow rate from these stars. Using the H~{\scshape i} emission coefficients from \cite{oste1989}, the expression for the detected Br$\gamma$ flux can be described as:
\begin{equation}
\rm F_{\rm Br\gamma}=1.2\times10^{-28}(\rm N_e^2V_{\rm {HI}}/D^2)
\end{equation}
\noindent
where D is the distance to the emitting region in centimeters, N$_e$ is the electron density, and V$_{\rm HI}$ is the volume of the emitting region (MKS units). The electron temperature is assumed to be $\sim$10$^4$~K, which is a reasonable estimate for the inner regions of YSO outflows. The mass of the emitting hydrogen in the spatially resolved regions of the outflow can be estimated as M$_{\rm HI}$~=~$m_p$N$_e$V$_{\rm HI}$. Merging this equation for the mass with the above equation for the flux gives:
\begin{equation}
M_{\rm {HI}} = 1.5\times10^{-13} (\rm F_{\rm Br\gamma}V_{\rm HI})^{1/2}D
\end{equation}
\noindent
We can thus solve for the mass of emitting hydrogen gas using the parameters of our measured Br$\gamma$ flux, the selected emission volume, and the assumed 140~pc distance to the stars, which is based on distances derived toward TTS within the Taurus star forming complex \citep{torr2009}. For DG~Tau, an H~{\scshape i} mass of 1.2$\times$10$^{-8}$~M$_{\odot}$ is derived, and for Haro~6-10~S this value is 1.4$\times$10$^{-8}$~M$_{\odot}$. The flux extraction volumes were chosen to be the same, so the difference in the atomic hydrogen outflow masses between the two stars is determined by the difference in their detected Br$\gamma$ flux values. Having chosen the flux extraction volumes to correspond to the average $\it{annual}$ proper motions of these jets, these estimates approximate the mass outflow rates for DG~Tau and Haro~6-10~S in solar masses per year.
Based on this analysis, we derive average electron densities on the order of a few $\times$10$^4$~cm$^{-3}$ for the DG~Tau and Haro~6-10~S outflows. These densities are consistent with the large values of 10$^4$ to 10$^6$~cm$^{-3}$ for N$_e$ that are often found within the inner one hundred AU regions of young star outflows derived by inspecting [Fe II] and other forbidden emission species \citep{hart1995, bacc1999, hart2007, hart2009}. These electron densities are also very consistent with past values found in the inner DG~Tau high velocity blue-shifted jet \citep{bacc1999, coff2008}.
The hydrogen outflow rates of 1.2$\times$10$^{-8}$~M$_{\odot}$~yr$^{-1}$ for DG~Tau and 1.4$\times$10$^{-8}$~M$_{\odot}$~yr$^{-1}$ for Haro~6-10~S represent lower limits for the true mass outflow levels derived in this manner. These mass outflow rates are underestimates of the true mass flow because only the fraction of the gas that has recently been heated by the shock radiates in the emission species that is studied. We also only detect the gas in the high velocity component of the outflow, not in lower velocity flow surrounding the jet axis \citep{bacc2000, pyo2003}. Moreover, the derived Br$\gamma$ flux in the outflowing volume was not corrected for any line of sight extinction effects which might further serve to increase the derived mass outflow rate, particularly in the obscured Class~I star, Haro~6-10~S. Thus, we predict that the true $\dot{M}_{out}$ values are greater than 1.2$\times$10$^{-8}$M$_{\odot}$~yr$^{-1}$ for DG~Tau and more than 1.4$\times$10$^{-8}$M$_{\odot}$~yr$^{-1}$ for Haro~6-10~S.
\section{Discussion}
This study shows that not all H~{\scshape i} Br$\gamma$ emission from classical TTS arises from magnetospheric accretion processes within a few radii from the central star. We detect spatially resolved Br$\gamma$ from DG~Tau, Haro~6-10~S, HL~Tau and HV~Tau~C, which represents 50\% of the TTS systems in our sample. We do not spatially resolve Br$\gamma$ line emission in the environments of XZ~Tau, RW~Aur, T~Tau and CW~Tau. Two stars within these latter systems, RW~Aur~B and XZ~Tau~A, exhibit no Br$\gamma$ emission at all.
In two of the blue-shifted velocity channels, the spatially extended Br$\gamma$ from Haro~6-10~S makes up $\sim$20\% of the total spatially integrated line flux. Integrated over the whole velocity width, about 10$\pm$2\% of the Br$\gamma$ emission from Haro~6-10~S is spatially resolved at distances of greater than 0.$"$1 (14 AU) in the extended outflow (entirely from the blue-shifted velocity component). The spatially extended Br$\gamma$ emission makes up $\sim$2\% of the total line flux for DG~Tau. All of the detected Br$\gamma$ flux seems to arise from the outflow for HV~Tau~C, but the star is not seen directly because the continuum flux is measured only from the scattered light nebulosity. The inner magnetospheric accretion region of HV~Tau~C may be shielded by the inner rim of the central circumstellar dust disk in our edge-on viewing orientation, hence in this case we might not see the strong central Br$\gamma$ component in the scattered light nebulosity. HL~Tau has considerable Br$\gamma$ emission scattered off of its surrounding outflow cavity walls, and the fraction of spatially extended Br$\gamma$ emission is estimated to be 18$\pm$7\% of the line flux from the central position. Observations of HL~Tau were acquired with an occulting disk in the beam, so the total point source flux is estimated from the PSF shape in the target acquisition setup images and the resulting value is significantly less certain.
The majority of the integrated Br$\gamma$ flux that we measure is spatially unresolved from the position of the central stellar sources in our data, with the noted exception of the edge-on disk system HV~Tau~C. NIFS data can spatially resolve the bright Br$\gamma$ emission beyond $\sim$0.$"$1 from the star, or about $\sim$14~AU. Additional Br$\gamma$ emission likely arises from the outflows in regions closer than our resolution limit, where it cannot be detected. Overall, the extended Br$\gamma$ line emission that we detect beyond $\sim$14~AU distances from the parent stars comprises anywhere from a few percent (DG~Tau) to all of the detected line flux from these systems (HV~Tau~C).
All of the sources that we have studied here are known to drive Herbig-Haro outflows. It is not clear why some stars exhibit extended Br$\gamma$ emission, while other sources with strong and collimated outflows do not. However, from information presented in Table~2, we see that the stars that have appreciable spatially extended Br$\gamma$ emission also have stronger estimated levels of optical visual extinction, A$_{\rm v}$, toward the stellar photosphere. The stars where we do not find extended Br$\gamma$ emission all have lower estimated optical obscurations. In the systems where extended Br$\gamma$ emission is seen, the stronger levels of stellar continuum flux attenuation from the high visual extinction may make the weak spatially extended emission easier to detect. The stars that do not exhibit extended Br $\gamma$ emission have brighter stellar continuum flux, which may be a result of less obscuration by natal material because of a slightly older evolutionary state, or a perhaps a more inclined viewing geometry that directly reveals more of the central photosphere of the star. Thus, spatially extended Br$\gamma$ emission may exist toward the other stars, but the bright continuum flux might prevent us from detecting it.
Several recent spectro-astrometric and interferometric investigations of inner YSO disks have sought to spatially resolve the atomic hydrogen associated with the magnetospheric recombination regions within 1~AU from the target stars \citep{whel2004, eisn2009, krau2008}. The spectro-astrometric study of \citet{whel2004} revealed that an appreciable amount of Pa$\beta$ emission from the blue and red-shifted velocity profiles arises from spatially extended distances from a TTS. Three of their four targets exhibited spatially resolved Pa$\beta$ emission, two revealed evidence for spatially extended bi-polar red and blue-shifted components. This is an important finding, since the broadened emission line wings seen in the H~{\scshape i} profiles from cTTS have never been well explained by magnetospheric accretion models. Evidence from this Pa$\beta$ study and now our Br$\gamma$ project suggest that the H~{\scshape i} emission from TTS does have non-negligible components from the outflow on spatial scales that were unresolvable prior to the current generation of sensitive instrumentation on 8-10 meter class telescopes.
Thus far, spectro-interferometric studies of Br$\gamma$ that can spatially resolve the inner magnetospheric accretion region have largely concentrated on the brighter Herbig Ae/Be stars (HAEBEs; \cite{eisn2009, krau2008, malb2007}). Interestingly, most interferometric data reveals that the Br$\gamma$ emission surrounding the HAEBEs arises from a centralized location that is spatially more compact than the infrared continuum emission, with perhaps some contribution from an outflowing wind \citep{malb2007}. The compact H~{\scshape i} likely arises predominantly from the accretion-driven processes within the star-disk boundary; either from the central stellar mass accretion engine or from a stellar or disk wind \citep{krau2008, malb2007, eisn2010}. Further work is warranted to better reveal the central emission location of H~{\scshape i} from TTSs, and the fraction of H~{\scshape i} emission that is truly spatially extended from the parent star in comparison to the higher mass HAEBEs.
Although we do detect spatially extended Br$\gamma$ emission from 50\% of our sample targets, the bulk of the line emission from most stars does arise within $\sim$14~AU from the central unresolved point sources. While the spatially extended H~{\scshape i} seems to affect the wings of the velocity line profile shapes \citep{whel2004}, it is likely that the majority of the low velocity H~{\scshape i} is emitted via magnetospheric processes in the central disk-star accretion region. \citet{bary2008} presented a large-scale multi-epoch near-infrared spectroscopic survey of 15 actively accreting TTS in Taurus-Auriga. They compared the H~{\scshape i} line ratios of 16 Paschen and Brackett series emission features and found little variations in these ratios from epoch to epoch and from source to source. For the first time, a statistically significant correlation was found between the observed H~{\scshape i} line ratios in TTS systems and those predicted by Case~B line recombination theory \citep{bake1938,humm1987,stor1995} for a tightly constrained gas temperature and electron density. While the range of electron densities Bary et al.\ find for the emitting H~{\scshape i} gas agrees with that predicted by magnetospheric accretion models, the temperatures are substantially lower than expected for accreting gas\footnote{\citet{bary2008} found the most likely temperature and density ranges of $T$~$<$~2000~K and 10$^9$~cm$^{-3}$~$<$~$n_e$~$\le$~10$^{10}$~cm$^{-3}$.}. These results suggest that the emission is from a recombining gas not in local thermodynamic equilibrium and is instead being stimulated by a non-thermal process. In addition, these results may also suggest a combination of emission sources for the emitting H~{\scshape i} gas in the central regions and that the observed H~{\scshape i} line fluxes are a superposition of emitting gas with different physical conditions. As such, one might not necessarily expect a linear correlation between detected Br$\gamma$ flux from the central star and the liberated accretion luminosity measured from the UV excess emission, as found by \citet{muze1998b}. However, the correlation does exist and the mass accretion rates we derived from our measured Br$\gamma$ line fluxes are consistent with those found in past studies of TTS. Taken in combination, these results further confirm the connection between accretion and outflow activity.
Deriving accurate mass outflow values is notoriously difficult. This is in part because the emission line regions in outflows measure only material in the post-shock area of a jet and do not sample outflow material ahead of the shock. Shocks from outflows and their associated optical and infrared emission lines are obvious observational manifestations of mass outflow activity, but they are the result of fast material plowing into slower clumps of gas in the outflow path. Hence, material can (and does) exist in the outflow without these detectable emission features if there is no slower material in the flow path to cause the shock. Methods described in the literature for measuring mass outflow rates from classical TTSs rely largely on measuring the mass flux from forbidden emission line species and converting this to an outflow rate based on estimates of the gas ionization fraction \citep{hart1995, cabr2002, doug2002, bacc2002, coff2008, agra2009}. The different observational techniques derive mass outflow to mass accretion ratios in the range of 1 to 10\%. In some cases, multi-wavelength or emission line studies of the same system find $\dot{M}_{out}$/$\dot{M}_{acc}$ values that differ by factors of $\sim$10, even for the same system (e.g., DG~Tau, see below). Measurement of mass outflow rates using atomic hydrogen emission species has been difficult in the past because of the strong optical depth effects in optical Balmer emission species. The infrared Brackett features are less optically thick, and may serve as a more robust tracer of the atomic hydrogen gas mass in the young star outflows.
In the previous section, we derived the first masses of ionized hydrogen using Br$\gamma$ emission for the DG~Tau and Haro~6-10~S outflows. Comparison of the mass accretion rates with our derived atomic hydrogen mass outflow rates for DG~Tau and Haro~6-10~S (see Table~2) reveals that the outflowing H~{\scshape i} mass is estimated at $\sim$10-15\% of the mass accreting onto the central star. Moreover, our derived mass outflow rates are also lower limits, implying that the actual ratio of $\dot{M}_{out}$/$\dot{M}_{acc}$ may be larger than 10-15\% for these two sources. These ratios are consistent with $\dot{M}_{out}$/$\dot{M}_{acc}$ values predicted by the theory of mass accretion efficiency and magnetocentrifugal launching mechanisms (i.e., disk winds and accretion-driven stellar winds) for young star outflows \citep{koen1991, matt2005, cran2008}, and are on the high side in comparison with ratios found from other observational investigations \citep{edwa1987, hart1995, cabr2002, doug2002, bacc1999, coff2008, agra2009}.
It is interesting that we find $\dot{M}_{out}$/$\dot{M}_{acc}$ values for DG~Tau and Haro~6-10~S that seem in the high range compared to ratios derived from other observations. Additionally, the $\dot{M}_{out}$ rates we find are lower limits, as discussed in $\S$5. However, the uncertainties in the derivation of both the $\dot{M}_{out}$ and $\dot{M}_{acc}$ values can be very large. Our lower limit for the value of M$_{out}$ in DG~Tau and Haro~6-10~S is derived from straightforward assumptions on the physics of LTE atomic hydrogen in recombination regions. As noted above, the hydrogen recombination emission area very close to the central star is possibly excited by non-thermal processes in the inner magnetospheric region \citep{bary2008}. Hence, it seems feasible that these non-LTE conditions may extend into the inner regions of the outflow, making our H~{\scshape ii}-like analysis of hydrogen emitting mass correspondingly uncertain. This is likely one of the largest sources of uncertainty in our analysis, but it is also difficult to characterize. We also estimate an additional source of error in our selection of the emission volume that was chosen to represent one year worth of jet motion. Moreover, the correlation of integrated Br$\gamma$ line luminosity with stellar mass accretion luminosity has an intrinsic scatter \citep{muze1998b}, and deviation in $\dot{M}_{out}$/$\dot{M}_{acc}$ by a factor of several for a given target can result from using this method to derive the stellar mass accretion rate. Multiple observations of accretion indicators from large sample of TTS demonstrate the highly variable nature of accretion activity in these young stars \citep[e.g.,][]{bary2008,nguy2009}. However, the effects from intrinsic time variation in the accretion and outflow properties are not an issue in our study because the mass outflow and mass accretion rates are derived simultaneously from the same data set.
The blue-shifted jet emerging from DG~Tau is arguably one of the best studied outflows associated with a young star. It was among the first cTTSs for which a collimated jet-like outflow was discovered \citep{mund1983}. Since its discovery, the HH~158 outflow from DG~Tau has been investigated with high spatial resolution imaging and spectroscopy with $\it{HST}$, ground-based adaptive optics systems, and spectro-imaging techniques \citep{lava1997, lava2000, bacc2000, doug2002, bacc2003, pyo2003, taka2004, coff2008}. This outflow has observationally revealed the structure of collimated YSO jets in great spatial detail; they are typically comprised of an on-axis high velocity component (HVC) at radial velocities greater than $\sim$50~km~s$^{-1}$ that can extend to spatial distances of hundreds of AU from the star, and an encompassing shell of lower velocity gas that extends only about $\sim$100~AU away from the central star \citep{bacc2000, pyo2003, taka2004}. Although He~I 10830 emission is commonly attributed only to stellar winds in the inner $\sim$10 AU environments of CTTS, \citet{taka2002} detect He~I 10830 emission from DG Tau at the jet velocity, spatially extended over ~ 0.5 arcsec ($\sim$70AU) of the jet, though their spectroastrometry did not find extension in the low-velocity He I 10830 emission. \citet{bacc2000} and \citet{pyo2003} described the blue-shifted HH~158 jet as having an ``onion-skin" structure, with high velocity low density gas on-axis, surrounded by successive layers of lower velocity, higher density gas. Integrated over the width of the jet axis, the $\dot{M}_{out}$/$\dot{M}_{acc}$ values derived for the DG~Tau jet from past studies lie in the range of 0.05-0.1 \citep{lava1997, bacc2000, coff2008}, though most of these estimates use large mass accretion rate of $\sim$10$^6$ M$_{\odot}$~yr$^{-1}$ \citep{hart1995}. \citet{bacc2003} analyzed the kinematics of DG~Tau's jet and found evidence for velocity asymmetries across the jet axis, and they interpret this result as evidence for rotation within the inner jet channel. This controversial finding challenges the theories of magnetospheric interaction at the inner star + disk + outflow boundary region.
The outflow from DG~Tau has served as a laboratory to test the physics of YSO jets, so it is fitting that our first discovery of spatially extended Br$\gamma$ from cTTSs includes this system. In DG~Tau, the spatially extended Br$\gamma$ emission that we detect arises from the high velocity, on-axis gas in the blue-shifted outflow. Interestingly, this H~{\scshape i} coincides very closely in projected location with the recently discovered X-ray emission in the jet \citep{gude2005, gude2008}. \citet{gude2008} found that the blue-shifted outflow from DG~Tau has emission from soft X-rays out to spatially extended distances of several hundred AU from the central star. The temperatures of the soft X-ray emission regions associated with the extended X-ray jet are on the order of a few times 10$^6$K. These temperatures pose a problem for understanding jet heating mechanisms. The standard jet shock models for YSO outflows cannot explain temperatures on the order of 10$^6$K, and this argues strongly in favor of some form of non-thermal excitation of the gas; for example by Ohmic heating from magnetic dissipative currents \citep{gude2008}. The detected Br$\gamma$ emission from the DG~Tau blue-shifted outflow seems to correspond to gas nearby to these super-heated regions, and thus if strong magnetic fields significantly effect the physical conditions in the inner flows then our simple method of deriving $\dot{M}_{out}$ assuming thermal Case~B recombination of atomic hydrogen at temperatures of $\sim$10$^4$K would be correspondingly inaccurate. Further tests and comparison of the physical conditions in Br$\gamma$ and X-ray emitting jets could help clarify the physics in the inner regions of the outflows. Unfortunately, as previously mentioned, HL~Tau, HV~Tau~C and Haro~6-10~S are Class~I protostars or cTTSs with circumstellar disks viewed in a nearly edge-on configuration, and all three of these systems have very strong optical obscuration (see Table~2). It is likely not feasible to detect the soft X-ray emission from these jets to further test the physical conditions of the Br$\gamma$ emission environments.
\section{Summary}
We have presented the results of our K-band integral field spectroscopy study of the Br$\gamma$ line emission from eight T Tauri star systems. The key points of our study are:
1) Using adaptive optics fed integral field spectroscopy, we have spatially resolved Br$\gamma$ line emission in the circumstellar environments around four of our eight survey target stars: DG~Tau, Haro~6-10~S, HL~Tau and HV~Tau~C.
2) The spatially extended Br$\gamma$ emission arises predominantly from the hydrogen recombination regions associated with the inner Herbig-Haro outflows from these young stars. Only HL~Tau shows a significant contribution of stellar Br$\gamma$ emission scattered into our line of sight off of the outflow cavity walls. This emission from HL~Tau has a similar morphology to the scattered light continuum flux, and it likely originated from the inner magnetospheric accretion region around the star.
3) At some blue-shifted velocities, the spatially extended Br$\gamma$ emission comprises $\sim$20\% of the detected Br$\gamma$ (e.g., Haro~6-10). Although we spatially resolve Br$\gamma$ emission from outflows in our high-contrast measurements, the majority of the integrated Br$\gamma$ from most systems is spatially unresolved and may arise from the magnetospheric accretion processes at the location of the central stellar source (with the exception of HV~Tau~C).
4) All of the Br$\gamma$ emission that we detect above the continuum flux from HV~Tau~C is from the spatially extended emission, consistent with the location of the known Herbig-Haro outflow. HV~Tau~C is seen in continuum light as a scattered light edge-on disk nebulosity. The inner magnetospheric component of Br$\gamma$ flux may be shielded from our line of sight by material in the inner edge of the circumstellar disk.
5) Derivation of the stellar mass accretion rates from the relationship between Br$\gamma$ line luminosity and mass accretion reveal $\dot{M}_{acc}$ values that are typical of cTTSs.
6) Detection of the spatially extended Br$\gamma$ emission from the outflows in the DG~Tau and Haro~6-10~S systems have allowed us to derive a value for the emitting hydrogen mass outflow rate using simple arguments applicable to hydrogen recombination regions. The corresponding values for $\dot{M}_{out}$/$\dot{M}_{acc}$ that we derive are on the order of $\sim$10-15\%, consistent with many prediction from accretion-driven stellar winds and disk winds, while on the high side in comparison to observationally derived mass outflow to accretion rate ratios from past studies.
7) We find that in some young protostars, Br$\gamma$ emission extended on spatial scales of greater than 0.$"$1 (14 AU) can contribute $\sim$ 10\% of the flux to the detected integrated line emission (or more, as in the case of HL Tau and HV~Tau~C).
\acknowledgments
Several NIFS datasets presented in this study were acquired during the early stages of instrument integration at Gemini North Observatory, and we are extremely grateful for the support of the NIFS teams at the Australian National University, Auspace, and Gemini Observatory for their tireless efforts during the instrument commissioning and system verification. This study is based on data from the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., on behalf of the international Gemini partnership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,420 |
<?php if ( ! defined('BASEPATH')) exit('No direct script access allowed');
/**
* Ionize, creative CMS
*
* Core controller
* Basic Ionize functions
*
* @package Ionize
* @author Ionize Dev Team
* @license http://doc.ionizecms.com/en/basic-infos/license-agreement
* @link http://ionizecms.com
* @since Version 0.9.0
*/
class Core extends MY_Admin {
/**
* Constructor
*
*/
public function __construct()
{
parent::__construct();
}
function index()
{
// By default, the controller will send the user to the login screen
$this->login();
}
/**
* Gets the website current structure and ouput it as a tree
*
*/
function get_structure()
{
// Text Helper
$this->load->helper('text_helper');
// Models
$this->load->model(
array(
'menu_model',
'structure_model',
'article_model'
), '', TRUE);
// Structure librairy
$this->load->library('structure');
// Get all articles from DB
$articles = $this->structure_model->get_articles();
// Menus : All menus
$menus = $this->menu_model->get_list(array('order_by'=>'ordering ASC'));
foreach($menus as &$menu)
{
$menu['items'] = array();
$menu_items = $this->structure_model->get($menu['id_menu']);
$this->structure->get_nested_structure($menu_items, $menu['items'], 0, 0, -1, $articles);
}
$this->template['menus'] = $menus;
$this->output('structure');
}
/**
* Get main informations about settings
* Used during developement
*
*
*/
function get_info()
{
// $this->output('info');
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,760 |
Refinery::AdminController.class_eval do
def require_refinery_users!
if current_spree_user && !current_spree_user.has_role?("Refinery")
flash[:error] = t(:authorization_failure)
redirect_to spree.unauthorized_path
elsif current_spree_user
true
else
redirect_to spree.login_path
end
end
def just_installed?
Spree::Role[:refinery].users.empty?
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,071 |
'use strict';
Object.defineProperty(exports, "__esModule", {
value: true
});
var _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; };
var _createClass = function () { function defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if ("value" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } } return function (Constructor, protoProps, staticProps) { if (protoProps) defineProperties(Constructor.prototype, protoProps); if (staticProps) defineProperties(Constructor, staticProps); return Constructor; }; }();
var _react = require('react');
var _react2 = _interopRequireDefault(_react);
var _understyle = require('understyle');
function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; }
function _objectWithoutProperties(obj, keys) { var target = {}; for (var i in obj) { if (keys.indexOf(i) >= 0) continue; if (!Object.prototype.hasOwnProperty.call(obj, i)) continue; target[i] = obj[i]; } return target; }
function _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError("Cannot call a class as a function"); } }
function _possibleConstructorReturn(self, call) { if (!self) { throw new ReferenceError("this hasn't been initialised - super() hasn't been called"); } return call && (typeof call === "object" || typeof call === "function") ? call : self; }
function _inherits(subClass, superClass) { if (typeof superClass !== "function" && superClass !== null) { throw new TypeError("Super expression must either be null or a function, not " + typeof superClass); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, enumerable: false, writable: true, configurable: true } }); if (superClass) Object.setPrototypeOf ? Object.setPrototypeOf(subClass, superClass) : subClass.__proto__ = superClass; }
var Robox = function Robox(Comp) {
var WrappedComponent = function (_React$Component) {
_inherits(WrappedComponent, _React$Component);
function WrappedComponent() {
_classCallCheck(this, WrappedComponent);
return _possibleConstructorReturn(this, Object.getPrototypeOf(WrappedComponent).apply(this, arguments));
}
_createClass(WrappedComponent, [{
key: 'render',
value: function render() {
var config = this.context.robox;
var _props = this.props;
var p = _props.p;
var pt = _props.pt;
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var pb = _props.pb;
var pl = _props.pl;
var px = _props.px;
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var mx = _props.mx;
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var gutter = _props.gutter;
var col = _props.col;
var block = _props.block;
var inlineBlock = _props.inlineBlock;
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var table = _props.table;
var tableRow = _props.tableRow;
var tableCell = _props.tableCell;
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var justify = _props.justify;
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var _style = (0, _understyle.createUnderstyle)(config);
var styleProps = {
p: p, pt: pt, pr: pr, pb: pb, pl: pl, px: px, py: py,
m: m, mt: mt, mr: mr, mb: mb, ml: ml, mx: mx, my: my,
gutter: gutter,
col: col,
block: block,
inlineBlock: inlineBlock,
inline: inline,
table: table,
tableRow: tableRow,
tableCell: tableCell,
flex: flex,
inlineFlex: inlineFlex,
wrap: wrap,
flexColumn: flexColumn,
align: align,
justify: justify,
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};
var sx = _extends({
boxSizing: 'border-box'
}, _style(styleProps), style);
return _react2.default.createElement(Comp, _extends({}, props, { style: sx }));
}
}]);
return WrappedComponent;
}(_react2.default.Component);
WrappedComponent.contextTypes = {
robox: _react2.default.PropTypes.shape({
scale: _react2.default.PropTypes.arrayOf(_react2.default.PropTypes.number),
columns: _react2.default.PropTypes.number
})
};
var spaceScale = [0, 1, 2, 3, 4, 5, 6];
WrappedComponent.propTypes = {
m: _react2.default.PropTypes.oneOf(spaceScale),
mt: _react2.default.PropTypes.oneOf(spaceScale),
mr: _react2.default.PropTypes.oneOf(spaceScale),
mb: _react2.default.PropTypes.oneOf(spaceScale),
ml: _react2.default.PropTypes.oneOf(spaceScale),
mx: _react2.default.PropTypes.oneOf(spaceScale),
my: _react2.default.PropTypes.oneOf(spaceScale),
gutter: _react2.default.PropTypes.oneOf(spaceScale),
p: _react2.default.PropTypes.oneOf(spaceScale),
pt: _react2.default.PropTypes.oneOf(spaceScale),
pr: _react2.default.PropTypes.oneOf(spaceScale),
pb: _react2.default.PropTypes.oneOf(spaceScale),
pl: _react2.default.PropTypes.oneOf(spaceScale),
px: _react2.default.PropTypes.oneOf(spaceScale),
py: _react2.default.PropTypes.oneOf(spaceScale),
col: _react2.default.PropTypes.number,
block: _react2.default.PropTypes.bool,
inlineBlock: _react2.default.PropTypes.bool,
inline: _react2.default.PropTypes.bool,
table: _react2.default.PropTypes.bool,
tableRow: _react2.default.PropTypes.bool,
tableCell: _react2.default.PropTypes.bool,
flex: _react2.default.PropTypes.bool,
inlineFlex: _react2.default.PropTypes.bool,
wrap: _react2.default.PropTypes.bool,
flexColumn: _react2.default.PropTypes.bool,
align: _react2.default.PropTypes.oneOf(['flex-start', 'flex-end', 'baseline', 'center', 'stretch']),
justify: _react2.default.PropTypes.oneOf(['flex-start', 'flex-end', 'space-between', 'space-around', 'center']),
flexAuto: _react2.default.PropTypes.bool,
flexNone: _react2.default.PropTypes.bool,
order: _react2.default.PropTypes.number,
// Warn against legacy prop name
column: function column(props, propName, componentName) {
if (props[propName]) {
return new Error('Warning! Invalid prop `' + propName + '` supplied to' + ' `' + componentName + '`. Use the `column` prop instead.');
}
}
};
return WrappedComponent;
};
exports.default = Robox; | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,050 |
This is a placeholder page for Sherrmar Kennedy, which means this person is not currently on this site. We do suggest using the tools below to find Sherrmar Kennedy.
You are visiting the placeholder page for Sherrmar Kennedy. This page is here because someone used our placeholder utility to look for Sherrmar Kennedy. We created this page automatically in hopes Sherrmar Kennedy would find it. If you are not Sherrmar Kennedy, but are an alumni of Tucker High School, register on this site for free now. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,795 |
\section{Introduction}\label{sec:intro}
The $\mathcal{N}=4$ super-Yang-Mills (SYM) theory is a toy model for QCD, with highly enhanced symmetry as can be seen by the high number of supersymmetry generators. Due to its vanishing beta function, $\mathcal{N}=4$ super-Yang-Mills theory is conformal even at the quantum level. In addition to the manifest superconformal symmetry, amplitudes in the planar limit possess a hidden \emph{dual} superconformal symmetry \cite{Drummond:2007au,Drummond:2008vq}. This hidden symmetry is related to the amplitude/Wilson loop duality (see ref.~\cite{Alday:2008yw} for a review). The dual conformal symmetry of the amplitude is nothing but the ordinary conformal symmetry of the dual Wilson loop.
Recently, a new duality was discovered between the six-particle MHV amplitude in parity-even kinematics and a three-particle form factor \cite{Dixon:2021tdw}, based on empirical evidence up to seven loops. (The amplitude and the form factor had previously been bootstrapped to seven and eight loops respectively \cite{Caron-Huot:2019vjl,Dixon:2022rse}.) The duality involves a mathematically nontrivial \emph{antipode} map, which at the symbol level of polylogarithmic functions reverses the order of symbol letters, and roughly speaking, exchanges the roles of discontinuities and derivatives. The physical origin of such a duality is still not clear, thanks no less to the novelty of the antipode map in physically important examples.
In this paper, we show that the antipode map appears in another context, that it is part of a new hidden symmetry of two-loop MHV amplitudes in planar $\mathcal{N}=4$ SYM when evaluated on parity-even kinematics. We explicitly find the symmetry map for amplitudes up to eight external particles at the symbol level, and conjecture its existence for all multiplicity. This is the first evidence for a role of the antipode map in full amplitudes beyond six particles. On the other hand, its fate beyond two loops is still not clear.
The paper is organized as follows. In Section~\ref{sec:review} we review the MHV amplitudes in planar $\mathcal{N}=4$ SYM, and the class of functions they belong to, the multiple polylogarithms, or MPL. We briefly review the \emph{antipode} operation associated with the Hopf algebra structure of MPLs, especially its action on their \emph{symbols}. In Section~\ref{sec:main} we give a statement of the conjectured antipodal symmetry for all two-loop MHV amplitudes in parity-even kinematics, where the symmetry operator is given by the antipode map plus a letter map, which is just a change of basis of the symbol letters; we also comment on the uniqueness of the letter map. Then, in Section~\ref{sec:fan} we discuss the action of the letter map on the tropical geometry of the symbol letters. Interestingly, it acts as an isomorphism on their tropical fans. Next, in Section~\ref{sec:number} we present further evidence for the existence of the antipodal symmetry at all multiplicities. In particular, we count the number of allowed first and final entry letters in the symbol, and show that they are the same in parity-even kinematics. We conclude in Section~\ref{sec:discuss}, and discuss future directions.
We also provide three ancillary files, \texttt{letter\_map\_R62.txt}, \texttt{letter\_map\_R72.txt} and \texttt{letter\_map\_R82.txt}, which give the letter maps associated with the antipodal symmetry at six-, seven- and eight-points respectively.
\section{Review} \label{sec:review}
\subsection*{MHV Amplitudes in Planar $\mathcal{N}=4$ SYM}
Amplitudes in the planar limit are color-ordered partial amplitudes, which are coefficients of $\text{Tr}(T^{a_1}T^{a_2}\cdots T^{a_n})$ in the color decomposition. The maximally helicity violating (MHV) amplitudes are those related by supersymmetric Ward identities to all-gluon amplitudes among which exactly two have negative helicities. At loop level, the amplitudes have infrared divergences, but the infrared divergent part is captured by the exponentiation of the one-loop amplitude \cite{Bern:2005iz}. In short, the $n$-point MHV amplitude takes the following factorized form,
\begin{equation}
\mathcal{A}_n^{\mathrm{MHV}} = \mathcal{A}_n^{\mathrm{BDS}} \cdot e^{R_n} \,,
\end{equation}
where $\mathcal{A}_n^{\mathrm{BDS}}$, the BDS ansatz \cite{Bern:2005iz}, captures all infrared divergences; and $R_n$, the \emph{remainder function}, is divergence-free. Since the BDS ansatz is the exponentiation of the one-loop amplitude, $R_n$ starts at two loops; that is,
\begin{equation}
R_n = g^4 R_n^{(2)} + O(g^6) \,
\end{equation}
where $g^2 \equiv {g_{\text{YM}}^{2}N_c}/{(16 \pi^{2})}$ is the planar coupling constant.
Due to the duality to Wilson loops, the remainder function is dual conformal invariant, and has no kinematic dependence for $n<6$. For $n\ge 6$, the kinematic degrees of freedom is the same as the number of independent conformal cross ratios, which is $3(n-5)$. This is reduced to $2(n-5)$ on the slice that preserves parity, see Appendix \ref{app:ope}. It is on this parity-even surface that the conjectured symmetry of this paper holds.
\subsection*{Multiple Polylogarithms and the Antipode Map}
The remainder function $R_n$ is believed to belong to a class of special functions called \emph{multiple polylogarithms} (MPL) \cite{Goncharov:2001iea}, based on analyses of the loop integrands \cite{Arkani-Hamed:2012zlh}. An MPL $F$ can be defined recursively by specifying its total derivative,
\begin{equation}
\label{eq:def_mpl}
dF = \sum_{\phi} F^{\phi} d\ln\phi \,,
\end{equation}
plus a base point of integration; the $F^{\phi}$ above are also MPLs. The \emph{symbol} of an MPL is also defined recursively,
\begin{equation}
\mathcal{S}(F) = \sum_{\phi} \mathcal{S}(F^{\phi}) \otimes \phi \,,
\end{equation}
where $F^{\phi}$ and $\phi$ are the same as in (\ref{eq:def_mpl}) above. By definition, the symbol of a constant is zero. Put another way, the symbol keeps all the information of an MPL in its definition (\ref{eq:def_mpl}), except for the base point at every step of integration. The $\phi$ that shows up in the entries of a symbol are called \emph{symbol letters}; collectively, all the symbol letters in the symbol are the \emph{symbol alphabet}. Since logarithms of products are additive, from (\ref{eq:def_mpl}) we have identities such as
\begin{equation} \nonumber
\ldots\otimes(\phi_1\phi_2)\otimes\ldots =
\left(\ldots\otimes\phi_1\otimes\ldots\right) +
\left(\ldots\otimes\phi_2\otimes\ldots\right)
\end{equation}
and
\begin{equation} \nonumber
\ldots\otimes\frac{1}{\phi}\otimes\ldots =
- \left(\ldots\otimes\phi\otimes\ldots\right) \,.
\end{equation}
MPLs are endowed with a Hopf algebra structure, which is a bialgebra together with an \emph{antipode} operation that satisfies certain axioms. In particular, the antipode map acts on each term of a symbol as,
\begin{equation} \label{eq:antipode}
S(\phi_1\otimes\phi_2\otimes \ldots \otimes\phi_{k-1}\otimes\phi_k) =
(-1)^k \,
\phi_k\otimes\phi_{k-1}\otimes \ldots \otimes\phi_2\otimes\phi_1 \,,
\end{equation}
and extends to the whole symbol by linearity.
\section{The Antipodal Symmetry} \label{sec:main}
The symbol of all two loop remainder functions $R_n^{(2)}$ are obtained in ref.~\cite{Caron-Huot:2011zgw},
\begin{equation}
\mathcal{S}(R_n^{(2)}) = r_n^{ijkl} \phi_i\otimes\phi_j\otimes\phi_k\otimes\phi_l \,,
\end{equation}
where summation over repeated indices is assumed on the right-hand side, and the summation is over a symbol alphabet that depends on $n$. We will look at the remainder function in parity-even kinematics. At symbol level, this is equivalent to projecting out all parity-odd letters,
\begin{equation} \label{eq:Rn2e}
\mathcal{S}(R_{n,e}^{(2)}) = r_{n,e}^{ijkl} \phi_i\otimes\phi_j\otimes\phi_k\otimes\phi_l \,,
\end{equation}
where the summation is now over a smaller alphabet consisting only of letters that are parity invariant.
Now we are ready to state the main conjecture of this paper: for any $n$, there exists a matrix $A_n^{ij}$ such that
\vspace{15pt}
\begin{equation} \label{eq:main}
\mathcal{S}(R_{n,e}^{(2)}) = \frac{1}{4} \, S\left(\mathcal{S}(R_{n,e}^{(2)})\Big|_{\ln\phi_i \mapsto A_n^{ij} \ln\phi_j}\right) \,,
\vspace{15pt}
\end{equation}
where $S$ is the antipode map (\ref{eq:antipode}). In terms of the tensor $r_{n,e}^{ijkl}$ in (\ref{eq:Rn2e}), this is,
\begin{equation} \label{eq:main_tensor}
r_{n,e}^{ijkl} = \frac{1}{4} \, r_{n,e}^{l'k'j'i'} A_n^{i'i} A_n^{j'j} A_n^{k'k} A_n^{l'l} \,,
\end{equation}
where again summation over repeated indices is assumed.
In words, this says that the two-loop remainder function in parity-even kinematics is invariant, up to an overall factor $1/4$, under a change of symbol letter basis plus the antipode map. The overall factor $1/4$ turns out to be hard to get rid of, as discussed below.
For the maps we have found through $n=8$, $A_n^{ij}$ satisfies,
\begin{equation} \label{eq:mapSq}
A_n^{ij} A_n^{jk} = 2\cdot\delta^{ik} \,.
\end{equation}
So if we apply the transformation on the right-hand side of (\ref{eq:main}) twice, we get,
\begin{equation}
\mathcal{S}(R_{n,e}^{(2)}) \mapsto \left(\frac{1}{4}\right)^2 \cdot 2^4 \cdot \mathcal{S}(R_{n,e}^{(2)}) = \mathcal{S}(R_{n,e}^{(2)}) \,,
\end{equation}
where $2^4$ comes from (\ref{eq:mapSq}) and the fact that it acts on each of the four entries in the symbol (\ref{eq:Rn2e}). The antipodal symmetry is therefore a $\mathbb{Z}_2$ symmetry.
Let us note here that the map $\ln\phi_i \mapsto A_n^{ij} \ln\phi_j$ should not be thought of as a map of the underlying kinematics. For example, for $n=6$, the parity-even kinematics is parametrized by two variables $(T,S)$ (see Appendix~\ref{app:ope}). It is easy to see that there exists no change of variables $(T,S) \mapsto (f_1(T,S), f_2(T,S))$ that results in the letter map (\ref{eq:R62map}). The letter map therefore is nothing but a linear algebraic manipulation on the letter basis; this is seen more obviously in the tensorial form (\ref{eq:main_tensor}).
We have explicitly found the letter map $A_n^{ij}$ through $n=8$. The maps for $n=6$ and $n=7$ are given in Appendix~\ref{app:map}, and we also give all the maps through $n=8$ in ancillary files to this paper.
\subsection*{Uniqueness of the Letter Map}
The letter map $A_n$ that satisfies (\ref{eq:main}) is by no means unique. Since $R_n$ is invariant under dihedral transformations, $OA_nO'$ would also satisfy (\ref{eq:main}) for any $O, O'$ in the dihedral group $D_n$\footnote{To be technical, $O,O'$ are in the representation of the dihedral group on the space of symbol letters.}. However, for $n=6$, it is uniquely fixed up to a scaling, either by requiring that the letter map respects dihedral symmetry, or by requiring that it acts as an isomorphism on the normal fans of symbol letters, as described in Section~\ref{sec:fan}. If we further require the overall scaling to be a rational number, then there exists no scaling which sets the 1/4 in (\ref{eq:main}) to unity. It would be interesting to understand more about this overall factor.
For $n=7,8$, the letter map is uniquely fixed by requiring the normal fan isomorphism and OPE factorization. That is, the map should reduce to a lower-point case for letters that are missing some $\{T_i,S_i\}$ for a subset of $i$.
\section{Letter Map and Tropical Fans} \label{sec:fan}
First let us recall the definition of Newton polytopes. The Newton polytope of a polynomial
\begin{equation}
f(x_1,x_2,\ldots,x_d) = \sum_i c_i x_1^{b_{i,1}} x_2^{b_{i,2}} \cdots x_d^{b_{i,d}}
\end{equation}
is the convex hull of $\{\mathbf{b}_i\}$, where each $\mathbf{b}_i = (b_{i,1},b_{i,2},\ldots,b_{i,d})$ is a point in $d$-dimensional space. The \emph{normal fan} of a polytope consists of a set of \emph{rays}, where each ray is a $d$-dimensional vector (modulo positive scaling), and a set of \emph{cones}, where each cone is a subset of the rays. The cones are in bijection with the faces of the polytope. For example, each facet (i.e. a $(d-1)$-dimensional face of the polytope) corresponds to a cone consisting of a single ray that is the inner normal vector of this facet.\footnote{Convex hulls and normal fans can be conveniently computed using computer programs such as \texttt{polymake} \cite{polymake:2000}.}
Let us look at some examples. $1+ST+T^2$ is a polynomial that shows up as a six-point symbol letter in (\ref{eq:R62letters}). The three monomials $1$, $ST$, $T^2$ correspond to three points $(0,0)$, $(1,1)$, $(2,0)$ respectively in the $T$-$S$ plane, as can be seen as the black dots in the left panel of Figure~\ref{fig:newton}, and the Newton polytope is the pink shaded area in the figure. Similarly, the right panel of Figure~\ref{fig:newton} shows the Newton polytope of $1+S^2+2ST+T^2$, which is also a symbol letter at six points.
The normal fans of the Newton polytopes of $1+ST+T^2$ and $1+S^2+2ST+T^2$ are shown in Figure~\ref{fig:fan}. The two fans are actually isomorphic to each other, under the \textbf{flip} along the red dashed line shown in the figure. Interestingly, the letter map (\ref{eq:R62map}) generates exactly this isomorphism; that is, the normal fans of the letters $1+ST+T^2$ and $1+S^2+2ST+T^2$ are mapped into each other under the letter map\footnote{On the right-hand side of (\ref{eq:R62map}), the normal fan of $\frac{\left(1+S T+T^2\right)^2}{S^2 T^2}$ is just that of $1+S T+T^2$. To see this, note that the overall square in the numerator of $\frac{\left(1+S T+T^2\right)^2}{S^2 T^2}$ simply rescales the Newton polytope of $1+S T+T^2$, and the $T$ and $S$ in the denominator is simply a translation; scaling and translation have no effect on the normal fan of the polytope. Similarly, the normal fan of $\frac{1+S^2+2 S T+T^2}{S^2}$ is the same as $1+S^2+2 S T+T^2$.}. Furthermore, the letter map generates an isomorphism between the normal fans of $S+T$ and $1+T^2$ as well, and the isomorphism is again given by the same \textbf{flip} above, see Figure~\ref{fig:fan2}. Including the trivial cases of $T$ and $S$, the letter map acts as an isomorphism on the normal fans of the symbol letters; and the isomorphism is always given by the \textbf{flip}.
This observation generalizes to higher multiplicities (at least through eight points where we have explicit letter maps to check against) that the letter map acts as an isomorphism on the normal fans of letters, with the isomorphism given by,
\begin{equation}
\textbf{flip}_1 \otimes \textbf{flip}_2 \otimes \cdots \otimes \textbf{flip}_{n-5} \,,
\end{equation}
where $\textbf{flip}_i$ is the same \textbf{flip} as in the six-point case but in the $T_i$-$S_i$ plane.
\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\filldraw[pink] (0,0) -- (4,0) -- (2,2) -- cycle;
\filldraw[black] (0,0) circle (2.5pt);
\filldraw[black] (4,0) circle (2.5pt);
\filldraw[black] (2,2) circle (2.5pt);
\draw[thick] (-1,0) -- (4.5,0) node[anchor=west]{$T$};
\draw[thick] (0,-1) -- (0,4.5) node[anchor=south]{$S$};
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\filldraw[pink] (0,0) -- (4,0) -- (0,4) -- cycle;
\filldraw[black] (0,0) circle (2.5pt);
\filldraw[black] (4,0) circle (2.5pt);
\filldraw[black] (0,4) circle (2.5pt);
\filldraw[black] (2,2) circle (2.5pt);
\draw[thick] (-1,0) -- (4.5,0) node[anchor=west]{$T$};
\draw[thick] (0,-1) -- (0,4.5) node[anchor=south]{$S$};
\end{tikzpicture}
\end{subfigure}
\caption{The Newton polytopes of the polynomials $1+ST+T^2$ (left) and $1+S^2+2ST+T^2$ (right). Each black dot corresponds to a monomial, and the pink shaded area is their convex hull.}
\label{fig:newton}
\end{figure}
\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\draw[red, thick, dashed] (-2.4,1) -- (2.4,-1);
\draw[thick] (-3,0) -- (3,0) node[anchor=west]{$T$};
\draw[thick] (0,-3) -- (0,3) node[anchor=south]{$S$};
\draw[ultra thick, blue, -latex] (0,0) -- (-1.77,-1.77) node[anchor=north]{1};
\draw[ultra thick, blue, -latex] (0,0) -- (1.77,-1.77) node[anchor=north]{2};
\draw[ultra thick, blue, -latex] (0,0) -- (0,2.5) node[anchor=west]{3};
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\draw[red, thick, dashed] (-2.4,1) -- (2.4,-1);
\draw[thick] (-3,0) -- (3,0) node[anchor=west]{$T$};
\draw[thick] (0,-3) -- (0,3) node[anchor=south]{$S$};
\draw[ultra thick, blue, -latex] (0,0) -- (0,2.5) node[anchor=west]{$1'$};
\draw[ultra thick, blue, -latex] (0,0) -- (2.5,0) node[anchor=north]{$2'$};
\draw[ultra thick, blue, -latex] (0,0) -- (-1.77,-1.77) node[anchor=north]{$3'$};
\end{tikzpicture}
\end{subfigure}
\caption{The normal fans of the Newton polytopes of $1+ST+T^2$ (left) and $1+S^2+2ST+T^2$ (right). The rays are drawn in blue arrows; they are the inner normal vectors of the facets of Newton polytopes in Figure~\ref{fig:newton}. The cones are $\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{3,1\}\}$ and $\{\{1'\},\{2'\},\{3'\},\{1',2'\},\{2',3'\},\{3',1'\}\}$ respectively. The fan on the left is isomorphic to the fan on the right by the \textbf{flip} along the red dashed line, the line at an angle $-\pi/8$ with the $T$-axis.}
\label{fig:fan}
\end{figure}
\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\draw[red, thick, dashed] (-2.4,1) -- (2.4,-1);
\draw[thick] (-3,0) -- (3,0) node[anchor=west]{$T$};
\draw[thick] (0,-3) -- (0,3) node[anchor=south]{$S$};
\draw[ultra thick, blue, -latex] (0,0) -- (-1.77,1.77) node[anchor=east]{1};
\draw[ultra thick, blue, -latex] (0,0) -- (1.77,-1.77) node[anchor=west]{2};
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\begin{tikzpicture}
\draw[red, thick, dashed] (-2.4,1) -- (2.4,-1);
\draw[thick] (-3,0) -- (3,0) node[anchor=west]{$T$};
\draw[thick] (0,-3) -- (0,3) node[anchor=south]{$S$};
\draw[ultra thick, blue, -latex] (0,0) -- (-2.5,0) node[anchor=north]{$1'$};
\draw[ultra thick, blue, -latex] (0,0) -- (2.5,0) node[anchor=north]{$2'$};
\end{tikzpicture}
\end{subfigure}
\caption{The normal fans of the Newton polytopes $S+T$ (left) and $1+T^2$ (right). The rays are drawn in blue arrows; the cones are $\{\{1\},\{2\}\}$ and $\{\{1'\},\{2'\}\}$ respectively. The two fans are isomorphic to each other, by the \textbf{flip} along the same red dashed line as in Figure~\ref{fig:fan}.}
\label{fig:fan2}
\end{figure}
\section{Evidence at All Multiplicities} \label{sec:number}
Since the equality (\ref{eq:main}) involves an antipode map, the order of the symbol entries are reversed on the right-hand side; the first entries of the symbol become the last entries, and vice versa. The symbol letters that can appear in first entries and last entries are heavily constrained by physics. These are often called the \emph{first-entry conditions} and the \emph{final-entry conditions} in the literature. For Eq.~(\ref{eq:main}) to hold, the number of first entries and the number of final entries must be the same. As we will see in this section, this is indeed the case in parity-even kinematics.
The first entries of a symbol encode its branch points. For a physical amplitude, its branch points correspond to vanishing Mandelstam variables,
\begin{equation} \label{eq:num_first}
s_{i,i+1,\ldots,j} = (p_i+p_{i+1}+\cdots+p_j)^2 \propto \left<i-1,i,j-1,j\right> \,.
\end{equation}
where $\left<i,j,k,l\right> \equiv \det(Z_i Z_j Z_k Z_l)$ and $Z_1,\ldots,Z_n$ are momentum twistors \cite{Hodges:2009hk}; all first entries are therefore of the form $\left<i-1,i,j-1,j\right>$. These letters come in two different cases. The first case is of the form $\left<i-1,i,i+1,i+2\right>$; there are $n$ such letters for $i=1,\ldots,n$. The second case is when $(i-1,i)$ and $(j-1,j)$ are \emph{not} adjacent to each other, e.g. $\left<i-1,i,j-1,j\right>=\left<1,2,4,5\right>$; there are $n(n-5)/2$ such letters.\footnote{The 2 in the denominator comes from the symmetry $\left<i-1,i,j-1,j\right> = \left<j-1,j,i-1,i\right>$.} Since the remainder function is conformal invariant, only conformal invariant combinations of such letters can show up, i.e. the little group weight of every $Z_i$ has to drop out; this take $n$ out of the total number of allowed letters. Overall, the number of allowed first entries is,
\begin{equation}
\text{\# of first entries } = n + \frac{n(n-5)}{2} - n = \frac{n(n-5)}{2} \,.
\end{equation}
On the other hand, the allowed final entries are constrained to be of the form $\left<i-1,i,i+1,j\right>$ \cite{Caron-Huot:2011dec}. Let us count the number of allowed letters. First, again there are $n$ letters of the form $\left<i-1,i,i+1,i+2\right>$. Next, when $(i-1,i,i+1)$ are not next to $j$, there are $n(n-5)$ such letters. And again we have to take $n$ out of these for conformal invariance. Overall,
\begin{equation}
\text{\# of final entries } = n + n(n-5) - n = n(n-5) \,.
\end{equation}
Now, the parity operation preserves letters of the form $\left<i-1,i,j-1,j\right>$, and maps \cite{Golden:2013xva}
\begin{equation}
\left<i-1,i,i+1,j\right> \leftrightarrow \left<i,j-1,j,j+1\right> \,.
\end{equation}
So the number of parity-even final entries is,
\begin{equation}
\text{\# of parity-even final entries } = n + \frac{n(n-5)}{2} - n = \frac{n(n-5)}{2} \,,
\end{equation}
which is exactly the same as the number of parity-even first entries. This therefore supports a symmetry of $R_n$ in parity-even kinematics that involves the antipode map, such as (\ref{eq:main}).
\section{Discussion} \label{sec:discuss}
In this paper we conjecture an antipodal symmetry (\ref{eq:main}) for all two-loop remainder functions in parity-even kinematics. We explicitly check the existence of such a symmetry at symbol level through eight points, and provide some all-multiplicity evidence.
First, let us emphasize the presence of the antipode map in this symmetry. The antipode map arises naturally as part of the Hopf algebra structure of multiple polylogarithms, and has a long history in the study of such functions in the mathematical literature. However, its importance in physics has been under-explored. One of the rare examples where the antipode map shows up is in the ``antipodal duality'' of ref.~\cite{Dixon:2021tdw}. In this paper we give another such example. Given the ubiquity of MPLs in physical quantities, we should expect the antipode map to play a much bigger role than we have currently understood. It is also interesting that our symmetry map at six points (\ref{eq:R62mapOld}) bears resemblance to the map in six-particle amplitude/three-particle form factor duality \cite{Dixon:2021tdw}. One might wonder whether the antipodal symmetry for higher-point amplitudes could potentially hint at a general higher-point ampitude/form factor duality.
Further, the antipodal symmetry seems to point to some deep structures of the amplitudes, especially in connection with tropical geometry. Indeed, it has been known for a while that some information of the amplitudes is encoded in certain tropical polytopes (or their normal fans) \cite{Arkani-Hamed:2019rds,Drummond:2019cxm,Henke:2019hve}. These are polytopes that are associated with the amplitude as a whole; specifically, they are Minkowski sums of the Newton polytopes of subsets of symbol letters. However, the letter map in (\ref{eq:main}) preserves the tropical fan structures \emph{letter by letter}. This suggests there might be information contained in the tropical geometry of individual symbol letters, that has yet to be studied.
Many things are still unsatisfactory. For example, the symmetry involves a non-trivial letter map, which is a change of basis of the symbol alphabet, but the map cannot be obtained from a change of variables of the underlying kinematics; this makes it difficult to lift the symmetry to the full function level. It might be possible to shed some light on this problem by looking at the full functional form of some two-loop remainder functions \cite{Goncharov:2010jf,Golden:2013xva,Golden:2018gtk,Golden:2021ggj} and the cluster algebraic structures associated with them, which had played an important role in either simplifying or constructing these full functions. Also, the naive generalization of (\ref{eq:main}) does not seem to work beyond two loops, starting at three loops and six points. It would be interesting to see if there exists a clever normalization of amplitudes\footnote{One guess would be the BDS-like normalization \cite{Alday:2009dv,Yang:2010as}. The antipodal symmetry holds for one- and two-loop BDS-like-normalized amplitudes at six and seven points, but still fails at three loops. Also, the BDS-like normalization is not defined for $n$ a multiple of four (see also \cite{Golden:2019kks}).} (or maybe a more complicated transformation) that generalizes the antipodal symmetry to an all-loop statement.
\acknowledgments
I would like to thank Lance Dixon for stimulating discussions and comments on the draft, and Lance Dixon, {\"O}mer G{\"u}rdo{\u{g}}an, Andrew McLeod and Matthias Wilhelm for collaboration on related projects. This work was supported by the US Department of Energy under contract DE--AC02--76SF00515, and in part by the National Science Foundation under Grant No. PHY-1748958. I am also grateful to the Kavli Institute for Theoretical Physics, TASI 2022 and the University of Colorado Boulder for hospitality.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 637 |
\section{Introduction}
The Rayleigh-Jeans portion of the dust spectral energy distribution (SED) of
IR-luminous galaxies produces a strong negative $K$-correction
and makes the observed submillimeter flux of such galaxies
almost invariant at $z>1$ to $z\sim10$ \citep{blain93}.
Thus if there are IR-luminous galaxies at high redshift, observations at
submillimeter wavelengths are a powerful way to find them.
However, to date, all but one of the identified submillimeter galaxies (SMGs),
other than those around luminous quasars, are at redshifts lower than 4,
likely because of the limited resolution of the
current submillimeter instruments and the limited sensitivity
of the current radio instruments, which are needed to locate the sources.
The Submillimeter Common-User Bolometer Array (SCUBA) on the
single-dish James Clerk Maxwell Telescope (JCMT) resolved 20\%--30\%
of the submillimeter extragalactic background light (EBL) into point
sources brighter than $\sim2$ mJy at 850 $\mu$m
(\citealp{smail97,barger98,hughes98,barger99,eales99}). Because of the low resolution
of JCMT ($\sim15\arcsec$ at 850 $\mu$m), identifications of the submillimeter
sources have to assume the radio--far-infrared (FIR) correlation in local galaxies
\citep[see, e.g.,][]{condon92} and rely on radio interferometry
to pinpoint the location of the submillimeter emission.
Optical spectroscopy of radio identified SMGs shows that they are
ultraluminous sources ($>10^{12}$ $L_{\sun}$, corresponding to a star formation rate
of $10^2$--$10^3$ $M_{\sun}$ yr$^{-1}$) peaking at $z\sim2$--3 and that
they dominate the total star formation in this redshift range \citep{chapman03,chapman05}.
However, the positive $K$-correction of the radio synchrotron emission
makes the radio wavelength insensitive to high-redshift galaxies, and radio
observations can only identify 60\%--70\% of the blank-field
submillimeter sources \citep{barger00,ivison02}.
The radio unidentified SMGs are commonly thought
to be at redshifts higher than the radio detection limit
(typically $z\gtrsim3$--4) but there has been a lack of direct
evidence for such a high-redshift radio-faint submillimeter population.
To date, there is only one spectroscopically confirmed SMG at $z>4$
\citep[$z=4.547$]{capak08}. Luminous radio-faint SMGs contribute $\sim10\%$ to
the submillimeter EBL measured by the \emph{COBE} satellite \citep{puget96,fixsen98}.
If most of these galaxies are indeed at high redshifts, then this implies a
large amount of star formation in massive high-redshift systems.
\begin{figure*}[ht!]
\epsscale{1.16}
\plotone{fig1.eps}
\caption{Our new ultradeep
NIR $J$, F160W, and $K_s$ band images of GOODS 850-5.
Each panel has a $24\arcsec$ width. North is up.
The three images have identical surface brightness scales (in $f_\nu$).
The $2\arcsec$ diameter circles
mark the SMA position from W07, which has an uncertainty of $0\farcs2$.
\label{pic_nicmos_moircs}}
\end{figure*}
With recent developments in submillimeter interferometry,
it is now possible to directly locate submillimeter sources
without relying on radio interferometers. \citet{younger07}
observed a sample of SMGs with the Submillimeter Array (SMA).
Several of their SMA detections are radio faint and consistent with
being at redshifts higher than the radio identified SMGS.
We have also been carrying out a program specifically targeting radio-faint submillimeter
sources with the SMA. In \citet[][hereafter W07]{wang07} we reported our
first identification in this program, GOODS 850-5.
GOODS 850-5 was detected in our JCMT
SCUBA jiggle-map survey of the Great Observatories Origins
Deep Survey-North (GOODS-N, \citealp{giavalisco04a}) with an
850 $\mu$m flux of $12.9\pm2.1$ mJy \citep{wang04}.
It was also detected in the combined jiggle and scan map
of the GOODS-N (GN 10, see \citealp{pope06} and references therein).
It is the second brightest submillimeter
source in our jiggle-map catalog of the GOODS-N and has a total IR
luminosity of $\sim2\times10^{13}$ $L_{\sun}$. It did not have a
$5 \sigma$ radio counterpart in the deep Very Large Array (VLA)
1.4 GHz catalogs of \citet{richards00} and \citet{biggs06}.
In W07, an SMA 880 $\mu$m detection of GOODS 850-5 was obtained
and its counterpart was found to be extremely faint in the optical and
near-IR (NIR). \citet[][hereafter D08]{dannerbauer08} soon followed up with new 1.25 mm
and 20 cm detections of GOODS 850-5. It turns out that the correct \emph{Spitzer}
counterpart had already been suggested by \citet{pope06} but this identification
was not confirmed and the high redshift of the source was not realized
until the accurate position was obtained by W07.
However, the exact redshift of GOODS 850-5 is
unclear. Both W07 and D08 found that the NIR SED is consistent
with a galaxy at $z\sim6$, and the submillimeter and radio SED is consistent with
$z\sim4$. While both groups agree that this is a high-redshift SMG,
W07 favors $z\sim6$ but D08 favors $z\sim4$ based on the radio--FIR correlation.
The $z\sim6$ redshift suggested
by W07 is based on a featureless power-law continuum in the IRAC bands
and a non-detection in a relatively shallow $K_s$ band image. However,
it is fair to say that
this photometric redshift is not a secure one, which would require the detection of
at least one prominent spectral feature. To obtain a better constraint on the
redshift, we carried out ultradeep NIR imaging in the $J$, $H$, and $K_s$ bands,
hoping to detect the stellar continuum at $\lesssim2$ $\mu$m.
However, we found instead that the SED of GOODS 850-5 at 1--3 $\mu$m
is quite weird---it is not detected even at a 5 nJy sensitivity at 1.6 $\mu$m
(compared with its $\sim1$ $\mu$Jy flux at 3.6 $\mu$m). Explaining this is challenging,
but our analyses show that it greatly strengthens the previous W07 suggestion of $z\gtrsim6$.
In this paper we present our new ultradeep NIR imaging observations of
GOODS 850-5 and complete analyses of the likelihood of its redshift.
The new observations and data used in this paper are described in \S~\ref{sec_obs}.
Photometric redshift analyses in the optical, NIR, FIR, and radio are described in \S~\ref{sec_photoz}.
The mass and age of the stellar population and the star formation rate of GOODS 850-5
are estimated, and the nature of this galaxy is discussed in \S~\ref{sec_property}.
The implications for galaxy evolution are
discussed in \S~\ref{sec_discussion}. We summarize and make some final
remarks in \S~\ref{sec_summary}.
The cosmological parameters adopted in this paper are $H_0=71$ km s$^{-1}$ Mpc$^{-1}$,
$\Omega_{\Lambda}=0.73$, and $\Omega_M=0.27$.
\begin{figure*}
\epsscale{1.17}
\plotone{fig2.eps}
\caption{Multiwavelength images of GOODS 850-5.
Each panel has a $24\arcsec$ width. North is up.
The SMA position is labeled with $2\arcsec$ diameter circles and has an
uncertainty of $0\farcs2$.
The color codes in the optical, NIR, and IRAC images are
labeled in the pictures. Grayscale images have negative color scales.
The MOIRCS and NICMOS images are from
this work and the VLA image is from D08 and G.\ Morrison (in preparation).
The rest are from W07.
\label{thumbnails}}
\end{figure*}
\section{New NIR Imaging and Existing Data}\label{sec_obs}
\subsection{\emph{HST} NICMOS F160W Imaging}
We observed GOODS 850-5 with the Near Infrared Camera and Multi-Object
Spectrometer (NICMOS) on the \emph{Hubble Space Telescope} (\emph{HST})
in Cycle 16. We used the NIC3 camera and the F160W filter to obtain the highest
sensitivity. The observations were made in November and December 2007,
containing a total of 16 \emph{HST} orbits in four visits. In each orbit, we made only one
exposure that is as long as possible ($\sim47$ minutes) in order to minimize the
contribution of read noise. The exposures were dithered to provide
0.5 pixel sampling.
The data reduction was carried out in the Interactive Data Language (IDL)
environment. We started the reduction with the standard pipeline-calibrated images
provided by the \emph{HST} Archive, which are flux calibrated and have instrumental
singatures removed. We applied a background subtraction to each exposure
by fitting the object masked image with a smooth polynomial surface. The brightest
cosmic ray hits and hot pixels were removed in each image with a spatial sigma
filtering. The offsets between images were determined by measuring the positions
of the detected objects in the central part of the field of view.
We did not attempt to correct for the optical distortion of NIC3.
The 16 images were then drizzled to a common grid of $0\farcs1$ per pixel
(two times finer than the original pixel of NIC3) and combined to form a deep image.
The absolute astrometry was matched to the GOODS-N ACS catalog \citep{giavalisco04a}
but corrected for the $0\farcs4$ offset in the ACS frame to match the radio frame
\citep{richards00}.
Before the images were combined, another sigma filtering was applied to drizzled
pixels in the same coordinate grids to remove fainter cosmic ray hits.
When the images were combined, each pixel was weighted by its exposure time
and each image was inverse-weighted by the background brightness to achieve
the highest S/N. The reduced image is shown in Figure~\ref{pic_nicmos_moircs}.
The final image has an extremely deep 1 $\sigma$ sensitivity of 4.9 nJy.
Surprisingly, even with such a great depth, we are not able to detect
GOODS 850-5.
\subsection{Subaru MOIRCS $J$ and $K_s$ Imaging}
We obtained extremely deep $J$ and $K_s$ images of GOODS 850-5
with the Multi-Object InfraRed Camera and Spectrograph \citep[MOIRCS, ][]{ichikawa06}
on the 8 m Subaru Telescope on Mauna Kea. MOIRCS contains two 2k~$\times~$2k
HAWAII2 detectors, covering a field of view of $\sim 4\arcmin \times 7\arcmin$ with a
pixel scale of $0\farcs117$. The images used here were taken in two large MOIRCS
imaging campaigns in the GOODS-N led by Japanese groups and our group based in
Hawaii between December 2005 and January 2008. All the $J$ band exposures and
approximately half of the
$K_s$ band exposures were made by Japanese investigators and were obtained
from the Subaru public archive. Parts of the Japanese data were published in
\citet{kajisawa06}. The rest of the $K_s$ exposures were made by our group and will
be published elsewhere. The majority of the observations were performed under photometric
conditions with excellent seeing
of $0\farcs25-0\farcs6$. A very small fraction of the data has a large extinction
of $>0.5$ magnitude or poor seeing of $>0\farcs7$, and they were excluded in
this work.
The MOIRCS images were reduced with the IDL based
SIMPLE Imaging and Mosaicking Pipeline (SIMPLE, W.-H.\ Wang 2008,
in preparation\footnotemark[8]).
Images within a dither set (typically 20--30 minutes in length) were flattened
using an iterative median sky flat in which a simple median sky was first
derived to flat the images and then a second median sky was derived by masking
all the detected objects using the flattened images. After the images were flattened,
the residual sky background was subtracted with a
smooth polynomial surface. The brightest cosmic ray
hits were removed by a spatial sigma filtering in each flattened image.
MOIRCS produces almost nearly circular fringes in
roughly half of the images. The fringes were modeled in polar coordinates
where they are nearly perfect straight lines and were subtracted from the images in
the original Cartesian coordinates.
\footnotetext[8]{also see http://www.aoc.nrao.edu/$\sim$whwang/idl/SIMPLE/index.htm}
The package SExtractor \citep{bertin96} was used to measure object positions
and fluxes in each flattened, sky subtracted, and fringe removed image in
a dither set. The first-order derivative of the optical distortion function was
derived by measuring the offsets of each object in the dither sequence as a
function of location in the images. Absolute astrometry was obtained by matching
the detected objects to a reference catalog constructed with brighter and compact
objects in the GOODS-N ACS catalog
(matched to the radio frame) and the GOODS-N SuprimeCam catalog \citep{capak04}.
The images were then warped directly from the raw frames to a
common tangential sky plane with a sub-pixel accuracy.
All projected images were weighted by their sky transparencies, exposure
times, inverse background brightnesses, and pixel-to-pixel efficiencies
(i.e., flat field) to obtain optimal S/N.
The weighted images were then combined to form a large mosaic. When images
from a dither set were combined, a sigma filter was applied to pixels that have the
same sky positions to further remove fainter cosmic rays.
The images were calibrated by observing various UKIRT Faint Standards \citep{hawarden01}
at least every half night on each detector.
Data taken under nonphotometric conditions and poorly calibrated archive data
were recalibrated with photometric data taken by our group.
The final mosaics have weighted exposure times (relative to
median sky brightness and transparency) of 13.2 and 23.7 hours in the
$J$ and $K_s$ bands, respectively, at the location of GOODS 850-5.
The image qualities are very good, with FWHMs of $0\farcs46$ at $J$ and
$0\farcs42$ at $K_s$. The rms astrometry error between the MOIRCS source positions
and the ACS/SuprimeCam reference catalog is $0\farcs08$.
The reduced MOIRCS images in the region around GOODS 850-5
are shown in Figure~\ref{pic_nicmos_moircs}.
The $1 \sigma$ sensitivity limits at $J$ and $K_s$ are both 14 nJy.
As with the F160W image, GOODS 850-5 is not detected in the
$J$ and $K_s$ images.
\subsection{NIR Photometry}
Although GOODS 850-5 is not detected in any of the above ultradeep NIR
images, it is useful to measure its fluxes at the SMA position (J2000 = 12:36:33.45,
+62:14:08.65, W07) to determine whether there is any low level flux recorded.
It is also important to obtain realistic flux limits/errors in these bands for the use of
photometric redshift fitting. We placed apertures with diameters that are 1.5 times the
FWHMs of compact objects in the fields ($0\farcs46$, $0\farcs3$, and $0\farcs42$ in $J$, F160W,
and $K_s$, respectively) at the SMA position to measure the fluxes of GOODS 850-5.
For each image, the background was estimated in a $3\arcsec$ area around GOODS 850-5 after
detected objects and GOODS 850-5 itself were masked.
The measured $J$ and F160W fluxes are negative, and the $K_s$ flux is 0.65 $\sigma$.
To estimate flux uncertainties, we first masked all detected objects in the images
and then randomly placed the apertures in the neighborhood of GOODS 850-5.
The dispersions in the random aperture fluxes are considered as flux uncertainties
in these three bands. This procedure is necessary especially for the two MOIRCS bands,
where the images were not drizzle combined and therefore the noise is more correlated
between pixels. Such flux errors also include the uncertainties in background subtraction
and confusion noise from faint undetected sources in the field.
We summarize the measurements in Table~\ref{tab1}.
\begin{deluxetable}{lc}[hb!]
\tablecaption{NIR Photometry of GOODS 850-5\label{tab1}}
\tablehead{\colhead{Wave Band} & \colhead{Flux (nJy)}}
\startdata
MOIRCS $J$ & $-0.8\pm13.8$ \\
NICMOS F160W & $-2.75\pm4.90$\\
MOIRCS $K_s$ & $9.1\pm13.9$
\enddata
\end{deluxetable}
\subsection{Existing Data}
GOODS 850-5 is not detected in the GOODS-N \emph{HST} ACS images but is clearly
detected in the \emph{Spitzer} IRAC 3.6--8 $\mu$m and MIPS 24 $\mu$m images
(GOODS \emph{Spitzer} Legacy Program, M.\ Dickinson et al.\ 2008, in preparation).
Its IRAC and MIPS fluxes were first measured by \citet{pope06}. Because GOODS 850-5
is blended with brighter nearby IRAC and MIPS sources, in W07 we remeasured the IRAC
and MIPS fluxes with a PSF fitting method to better isolate its flux.
In this work, we adopt the W07 results. (The flux errors in W07 and here include the
uncertainties in the deblending processes.)
GOODS 850-5 is not detected in
the MIPS 70 $\mu$m and 160 $\mu$m bands \citep{frayer06,huynh07}.
We adopt the 1 $\sigma$ limits of 2 mJy at 70 $\mu$m and 30 mJy at 160 $\mu$m.
In the submillimeter, we adopt the SCUBA 850 $\mu$m jiggle map flux of Wang et al.\ (2004)
and the SMA 880 $\mu$m flux of W07. GOODS 850-5 was detected by D08 with
the IRAM Plateau de Bure Interferometer at 1.25 mm with a flux of $5.0\pm1.0$ mJy.
In W07, we used the older Very Large Array (VLA) 20 cm image of
\citet[also see \citealt{richards00}]{biggs06}, in which GOODS 850-5 has a flux of
$18.7\pm8.0$ $\mu$Jy\footnotemark[9]. G.\ Morrison et al.\ (in preparation) obtained a deeper
VLA 20 cm image of GOODS-N. GOODS 850-5 was detected with a highly significant
flux of $34.4 \pm 4.2$ in the new VLA image (D08). Here we adopt the latest D08 values at
1.25 mm and 20 cm. The photometry used in this work is summarized in Table~\ref{tab2}.
Figure~\ref{thumbnails} show the multicolor images of GOODS 850-5.
Comparing to the figure shown in W07, the main difference is the new Subaru
and \emph{HST} NIR images and the new VLA 20 cm image.
\footnotetext[9]{This flux was measured at the location of the SMA position, which is
$\sim0\farcs3$ away from the best-fit VLA position. G.\ Morrison (2008, personal
communication) and R. Ivison (2008, personal communication) found best-fit VLA fluxes
of $24.1\pm5.8$ and $23.8\pm5.9$ $\mu$Jy, respectively, from the image of \citet{biggs06}.}
\begin{deluxetable}{lcc}
\tablecaption{Photometric Data of GOODS 850-5\label{tab2}}
\tablewidth{220pt}
\tablehead{\colhead{Wave Band} & \colhead{Flux ($\mu$Jy)} & \colhead{Reference}}
\startdata
ACS F435W & $-0.013\pm 0.004$ & 2 \\
ACS F606W & $-0.004 \pm 0.003$ & 2 \\
ACS F775W & $0.001 \pm 0.006$ & 2 \\
ACS F850LP & $-0.009 \pm 0.009$ & 2 \\
MOIRCS $J$ & $-0.0008\pm0.014$ & 1\\
NICMOS F160W & $-0.002.8\pm0.0049$ & 1\\
MOIRCS $K_s$ & $0.0091\pm0.014$ & 1\\
IRAC 3.6 $\mu$m & $1.14 \pm 0.14$ & 2 \\
IRAC 4.5 $\mu$m & $1.64 \pm 0.13$ & 2 \\
IRAC 5.8 $\mu$m & $2.33 \pm 0.24$ & 2 \\
IRAC 8.0 $\mu$m & $5.37 \pm 0.37$ & 2 \\
MIPS 24 $\mu$m & $46.3 \pm 9.2$ & 2 \\
MIPS 70 $\mu$m & $<2000$ & 3 \\
MIPS 160 $\mu$m & $<30000$ & 4 \\
SCUBA 850 $\mu$m & $12900 \pm 2100$ & 5 \\
SMA 880 $\mu$m & $12000 \pm 1400$ & 2 \\
IRAM 1.25 mm & $5000\pm1000$ & 6 \\
VLA 20 cm & $34.4\pm4.2$ & 6
\enddata
\tablerefs{(1) this work; (2) \citet[W07]{wang07}; (3) \citet{frayer06}; (4) \citet{huynh07};
(5) \citet{wang04}; (6) \citet[D08]{dannerbauer08}.}
\end{deluxetable}
\section{SED and Redshift of GOODS 850-5}\label{sec_photoz}
\subsection{Non-detections in $J$, F160W, and $K_s$}
The results of our new NIR imaging are surprising and not particularly
easy to understand.
GOODS 850-5 is clearly detected in the IRAC 3.6 $\mu$m band at 8 $\sigma$.
Despite the great sensitivities at F160W and $K_s$, which are respectively
29 and 10 times higher than those at 3.6 $\mu$m, GOODS 850-5 is not detected.
This reveals an extraordinarily red SED in the NIR. The 1 $\sigma$ $K_s$ flux
upper limit (23 nJy) implies a $S_{3.6\mu\rm m}/S_{Ks}$ flux ratio of $>50$ or
a spectral slope of $\alpha < -7.8$. For comparison, objects with $R-K>5$,
corresponding to $\alpha < -2.5$, are called ``extremely red objects''
\citep[EROs, e.g.,][]{mccarthy04}. The spectral slope of GOODS 850-5 is
more than three times that of EROs, and the $K_s-3.6$ $\mu$m color
(in AB scale) of GOODS 850-5 is $\gtrsim2$ higher than the $I-J$ color
of Hu-Ridgway 10, which is a prototype dusty starburst ERO \citep{hr10}.
High redshift UV emitting objects are commonly selected with red
colors of $\gtrsim2$ (in the AB system) between two adjacent optical filter bands
(the ``Lyman-break'' technique, e.g., \citealt{steidel99,fan01}),
corresponding to $\alpha \lesssim -10$. The flux ratio of GOODS 850-5
between $K_s$ and 3.6 $\mu$m is $\sim8$ times that of a minimal
Lyman break.
(A Lyman break between $K_s$ and 3.6 $\mu$m would imply $z>17$.)
\begin{figure*}
\epsscale{0.85}
\plottwo{fig3a.eps}{fig3b.eps}
\caption{NIR colors of GOODS 850-5 and galaxies between $z=0$ and 10:
(a) colors between the F160W and the 3.6 $\mu$m bands; (b) colors
between the $K_s$ and 3.6 $\mu$m bands. Solid curves are colors of E, Sbc, and Im
galaxy types in \citet{cww80}, without reddening. Dotted curves are the
same types of galaxies with $A_V=2.0$ (a) and $A_V=5.0$ (b).
We adopt the extinction law in \citet{calzetti00}. Horizontal dashed lines are the
lower limits for the colors of GOODS 850-5, derived from the observed $1\sigma$
upper limits of its F160W and $K_s$ fluxes.
\label{nir_colors}}
\end{figure*}
The extremely red color of GOODS 850-5 sets a strong constraint on its redshift.
It can be seen in Figure~\ref{nir_colors} that it is very difficult to explain the
observed colors between 1.6 and 3.6 $\mu$m with galaxies at $z<5$ even with a
considerable amount of reddening. Simply based on the two colors
in Figure~\ref{nir_colors}, $z\sim6$ appears to be the most likely
redshift for GOODS 850-5. However, we can use all the filter bands simultaneously
to obtain better redshift estimates with the photometric redshift technique.
\subsection{NIR Photometric Redshift}\label{sec_nir_photoz}
In W07 we derived photometric redshifts for GOODS 850-5 and found that
galaxies at $z\sim6$ provided the best fits. Here we used our new photometric
data in the NIR to improve our photometric redshift estimates.
We used all the \emph{Spitzer} IRAC,
\emph{HST} ACS and NICMOS, and Subaru MOIRCS data and the latest version of the
Hyperz package \citep{bolzonella00}\footnotemark[10] for this calculation.
In each band, we added a 5\% of zeropoint error quadratically.
We used zeros to replace negative fluxes in the four ACS bands and the $J$ and F160W
bands, since negative fluxes are not physical. (The F775W flux is consistent
with zero.) We used the non-zero $K_s$ flux.
The photometric redshifts were derived independently
with two SED template sets. The first is the latest stellar population synthesis
model of \citet[BC03]{bc03}. The second contains the empirically observed galaxy
spectra of \citet{cww80} from E to Im types and the starburst spectra of \citet{kinney96}.
Both SED sets are widely used by the optical extragalactic community for
photometric redshift estimates. The combination of the latest Hyperz with the
BC03 model has the nice feature of estimating ages and stellar masses of galaxies.
Hyperz also limits the age of the galaxies not to exceed the age of the universe.
\footnotetext[10]{also see http://www.ast.obs-mip.fr/users/roser/hyperz/}
We adopted the standard extinction law of \citet{calzetti00}.
There is a subtlety in the maximum extinction to be allowed.
Explaining the extremely red color of GOODS 850-5 between
F160W/$K_s$ and 3.6 $\mu$m requires a strong Balmer break
(rest wavelength 4000 \AA) at a high redshift of $z>5$ (or a Lyman break at $z>16$).
However, if we allow very large extinctions, this red color might be
reproduced with a highly reddened low-redshift galaxy.
For example, \citet{mobasher05} suggest that the very red $z-J$ color of the object
HUDF-JD2 may come from a Lyman break at $z\sim6.5$.
On the other hand, the mid-IR (MIR) observations of
\citet{chary07} suggest that HUDF-JD2 is at $z\sim2$ and the red $z-J$ color comes
from an extinction of $A_V\sim4.9$ (also see \citealp{fontana06,dunlop07}).
Spectroscopically confirmed SMGs have
a typical extinction of $A_V\sim1$--3 \citep{smail04,swinbank03}, but the
spectroscopic sample may be biased toward less reddened systems.
Photometric redshift fitting to SMGs without spectroscopy generally gives
$A_V\sim0$--5. From these examples, $A_V<6$ seems to be a reasonable limit
for high-redshift SMGs. (For heavier extinctions, the shape of the extinction curve becomes a
much more important uncertainty than the $A_V$ value itself.)
We adopted this limit for our photometric redshift analyses.
Later we will show that, unlike in the case of HUDF-JD2, a $z<3$ redshift with
a large extinction is ruled out for GOODS 850-5 by its radio and FIR SED.
\begin{figure}
\epsscale{1.1}
\plotone{fig4.eps}
\caption{NIR photometric redshift results. The top panel shows the observed
SED between 0.4 and 8 $\mu$m, and the best-fit SEDs from the two SED template
sets. Filled symbols are detections and the error bars are smaller than the symbols.
Open symbols with arrows are 1 $\sigma$ upper limits. The open symbol with
an error bar is the 0.65 $\sigma$ $K_s$-band measurement. Note the vertical
scale of the top panel. The bottom panel
shows the distributions of the $\chi_\nu^2$, $A_V$, and probability of the fits.
\label{nir_photoz}}
\end{figure}
Figure~\ref{nir_photoz} shows the results of our photometric redshift analyses.
Both SED template sets give best fits at redshifts between 6 and 7.
The BC03 best-fit is a galaxy at $z=6.9$ that formed in a single burst of star formation
and has now aged 0.7 Gy with an extinction of $A_V=1.8$. The empirical SED set
provides a best fit at $z=6.4$, with an elliptical type and $A_V=0.5$.
The BC03 set systematically gives higher extinctions at all redshifts,
likely because of the limit in the galaxy age, i.e., an unreddened galaxy
that is sufficiently red to fit the observed SED may be too old for the age of
the universe. Nevertheless, the $\chi_\nu^2$ distributions from the two
SED sets are consistent with each other. In our subsequent analyses, we
adopt the BC03 result as it includes an age limit for the galaxy and provides
physical quantities such as the age and stellar mass. The 68\% confidence range (1 $\sigma$
for a Gaussian distribution) from the integration of the BC03 probability function
is $z=6.0$--7.4. We note that redshifts between 4 and 5 are
not entirely ruled out, although the fit is poorer here. On the other hand,
$z<3$ is safely ruled out by the strong limits from the NICMOS and MOIRCS
non-detections.
Comparing to the photometric redshift analyses in W07, the probability distribution
now becomes narrower, which is an important improvement of this work.
On the other hand, the minimum $\chi_\nu^2$ increases, indicating that it is
generally difficult to explain the red color between 1.6 and 3.6 $\mu$m.
We tested photometric redshifts at $z>10$ and obtained nearly perfect fits at $z>17$
(i.e., a Lyman break between $K_s$ and 3.6 $\mu$m). However, this high redshift
cannot explain the observed SED in the FIR and radio. We do not consider
$z>10$ in this paper.
\subsection{Millimetric Redshift}\label{sec_radio_ir_photoz}
We used the radio and FIR portion of the SED for another photometric
redshift estimate (the millimetric redshift estimate).
Because there is not a clear correlation between the
radio/dust SEDs and the stellar SEDs of galaxies, this millimetric
redshift estimate was carried out independently of the above NIR
photometric redshift. The earliest version of such millimetric
redshift estimates was carried out using the spectral index between two wavebands
in the radio and submillimeter \citep{carilli99, barger00, yun02} based on
the well known radio--FIR correlation in the local universe.
This method has larger errors caused by the uncertainty in the dust temperature.
Advanced versions utilize full SED fitting in the radio and FIR
as the amount of available data increases.
Here we used all the data listed in Table~\ref{tab2} between
24 $\mu$m and 20 cm, including the two non-detections at 70 and
160 $\mu$m. These two non-detections play a key role in ruling out low redshifts of
$z<3$ but do not provide strong constraints at $z>4$.
We used two sets of radio and FIR SED templates.
The first set contains model SEDs of Arp 220 (ultraluminous starburst
with cooler dust emission), NGC 6090 (luminous starburst),
M 82 (low luminosity starburst with warm dust) and M 100
(normal spiral) adopted from \citet{silva98}, and Mrk 231 (ultraluminous dusty
active galactic nucleus, AGN, with warm dust) derived from the photometry in
NED\footnotemark[11]. The second SED set includes all of the 105 SED
models in \citet[CE01]{chary01}.
\footnotetext[11]{The NASA/IPAC Extragalactic Database
(NED) is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space Administration.}
\begin{figure}
\epsscale{1.1}
\plotone{fig5.eps}
\caption{Radio and FIR photometric redshift results. The top panel shows the observed SED of
GOODS 850-5 and the best-fit SEDs. The bottom panel shows the reduced
$\chi^2$ of the fits vs.\ redshift. In the bottom panel, the black curve only shows the
best-fit among all 105 model SEDs in CE01 at each redshift.
\label{fig_radio_ir_photoz}}
\end{figure}
In Figure~\ref{fig_radio_ir_photoz}, we present the best-fit SEDs and
the redshift distribution of $\chi_\nu^2$ for each SED template. A low redshift
of $z<3$ is ruled out and this agrees with the NIR photometric redshift.
The best-fit with the Silva et al.\ models
comes from Arp 220 at $z=3.9$. This is perhaps not too surprising
since Arp 220 has been commonly considered as the local analogue to
high-redshift SMGs. The next best fit comes from M 82. It is at $z=6.2$ and
the fit is slightly poorer than that with Arp 220. The fit with the spiral template M 100 has
$\chi_\nu^2>10$ everywhere and is not shown in the figure.
The best-fit AGN template has a redshift of 9.1, which was discussed in
W07. The best-fit of the CE01 templates is at $z=4.5$ and has a
$\chi_\nu^2$ ($\sim0$) lower than that from the Silva et al.\ models. This unreasonably
low $\chi_\nu^2$ is perhaps a result of the very wide range of models (105 of them).
Generally speaking, the CE01 models provide good fits in the
$z\sim3.5$--5.5 range. D08 used both the two-waveband spectral index method and
the SED fitting method (also with the CE01 templates, but only with
the 24 $\mu$m, 850 $\mu$m, 1.25 mm, and 20 cm bands) and obtained $z\sim3.7$
and $z\sim3.3$ respectively. Our photometric redshift results are slightly higher
than those in D07, likely due to the introduction of the two upper limits at
70 and 160 $\mu$m, which strongly disfavor $z<3.5$. Nevertheless, the results here and
those in D08 are still broadly consistent.
\subsection{$z\sim4$ or $z\gtrsim6$?}\label{4_or_6}
We are now faced with two different photometric redshift results.
The NIR photometric redshift suggests $z\sim6$--7, while the millimetric
redshift suggests $z\sim4$. We favor the NIR redshift of $z>6$ because we
believe that our knowledge about the stellar photospheric emission of galaxies is more
robust than that about the dust and radio emission. This is not only because the stellar
spectral synthesis models (such as the Bruzual \& Charlot models) and the empirical
spectra (such as the Coleman, Wu, \& Weedman templates) have been widely adopted
and tested in numerous extragalactic studies from $z\sim0$ to $z>7$,
but also because of the various uncertainties in the radio/IR photometric redshift.
Below we discuss each of the uncertainties.
First, the two-waveband spectral index method suffers from the uncertainty
in dust temperature. Second, the stellar contamination in the 24 $\mu$m band
is quite uncertain at $z>4$. All NIR photometric redshift models for GOODS 850-5
at $z>4$ predict observed 24 $\mu$m stellar photospheric fluxes that are much weaker
than those in the Silva et al.\ and CE01 models. If we decrease the stellar emission in
the Silva et al.\ models to match the observations, all fits become better than those in
Figure~\ref{fig_radio_ir_photoz}. In particular, the M 82 fit at $z\sim6$ slightly shifts to
$z\sim5.5$ and becomes better than the Arp 220 fit at $z\sim4$.
Third, and most importantly, there is a fundamental problem in the millimetric redshift,
which is the uncertainty in the radio--FIR correlation.
The radio--FIR correlation is a tight correlation observed for local
galaxies \citep{condon92}, but the detailed physical mechanism of this correlation
is still unclear. Several groups have attempted to measure this correlation
on different samples of high-redshift galaxies with various methods,
and the results remain controversial \citep{appleton04,boyle07,vlahakis07}.
If high-redshift SMGs are brighter in the radio, as suggested by Vlahakis et al.,
then the redshift of GOODS 850-5 should be higher than that inferred from the local
radio--FIR correlation. In addition to the uncertainties in observational
determinations of the correlation, there are also physical reasons
to suspect that the radio--FIR correlation at high redshift would be different than
that in the local universe. For example, the higher energy density of the cosmic
microwave background radiation at high redshift leads to a stronger inverse-Compton
cooling for the cosmic-ray electrons \citep{condon92},
which suppresses the radio emission. If this is the dominant effect at high redshift,
the correct redshift of GOODS 850-5 should be lower than that inferred from the
local correlation. On the other hand, galaxies tend to have smaller sizes at high redshift.
This may enhance the interstellar magnetic field and the synchrotron radiation,
but the total radio emission can be further complicated by ionization and bremsstrahlung
loss \citep{thompson06}.
Unfortunately, at least in the case of GOODS 850-5, the result of the
millimetric redshift highly relies on the validity of the radio--FIR
correlation. To show this, we repeated the same photometric redshift fitting
but did not use the 20 cm data point. The resultant $\chi_\nu^2$ distributions are shown in
Figure~\ref{fig_ir_photoz}. In this experiment, all the fits have $\chi_\nu^2$
around 1.0 (the expectation value) between $z\sim3$ and $z>10$.
This shows that the redshift of GOODS 850-5 highly depends on its dust temperature
when the radio--FIR correlation is not assumed.
While it is established that many high-redshift SMGs have cool dust SEDs
similar to that of Arp 220, a small number of SMGs have warmer, M 82-like SEDs
(e.g., \citealt{clements08}, also see Figure~2 in W07).
Because of this, we cannot rule out higher redshifts of $z>6$ and an M 82-like
SED for GOODS 850-5.
\begin{figure}[h!]
\epsscale{1.1}
\plotone{fig6.eps}
\caption{FIR-only photometric redshift results. Description the curves is as in
Figure~\ref{fig_radio_ir_photoz}. The plot legend (except for CE01)
are ordered by dust temperature: Arp 220 has the coolest dust and Mrk 231 has the
warmest dust. Note that the vertical scale is different than that in Figure~\ref{fig_radio_ir_photoz}.
This plot shows that the observed FIR SED of GOODS 850-5 can be reasonably
fitted with various templates from $z\sim4$ to 10.
\label{fig_ir_photoz}}
\end{figure}
\subsection{Summary on the Redshift}
A low redshift of $z<3$ is ruled out by both the NIR and millimetric
redshifts for GOODS 850-5. The NIR photometric redshift suggests
$z\sim6$--7, but the millimetric redshift suggests $z\sim4$.
The later strongly relies on the validity of the local radio--FIR correlation.
If we do not assume the local radio--FIR correlation, redshifts between 3 and 10
seem equally possible for the observed FIR SED of GOODS 850-5.
We consider the $z\gtrsim6$ NIR photometric redshift as a more likely one,
but $z\sim4$ is still a possibility for GOODS 850-5.
\section{Properties of GOODS 850-5}\label{sec_property}
\subsection{Active Galactic Nucleus?}
The possibility of the existence of an AGN was discussed by W07.
The lack of an X-ray counterpart in the 2 Ms \emph{Chandra} image \citep{alexander03}
rules out a Compton-thin AGN at $z<3$. An AGN with an X-ray luminosity
$\sim10^{42}$ erg s$^{-1}$ at $z>3$ cannot be detected by \emph{Chandra} and
therefore is still possible. If the radio and FIR emission of GOODS 850-5
is entirely powered by a Compton-thick dusty AGN like Mrk 231, its redshift would be $z\sim9$
(\S~\ref{sec_radio_ir_photoz}, W07), which is less likely. Furthermore, our new observation
in the NIR shows a strong spectral break around 2--3 $\mu$m, which is
an important feature of old stellar populations. On the other hand, QSOs typically
have power-law SEDs across a wide wavelength range from the UV to the NIR and
do not show strong spectral breaks like this. We conclude that there is no evidence
for an AGN in any of the observations.
\subsection{Stellar Population}
The use of the BC03 models and the latest Hyperz allows us to estimate the stellar
masses and ages. The SED of GOODS 850-5
is nearly a power law in the IRAC bands but shows a clear spectral break between
1.6 and 3.6 $\mu$m. This can hardly be explained by a normal extinction curve.
A spectral break in the unreddened stellar SED is required, and the most natural
spectral break is the 4000 \AA\, Balmer break, which is a signature of older stars.
Hyperz agreed with this interpretation and all the best-fit models between $z=4$ and 10 have
nearly the age of the universe. For example, at the best fit of $z=6.9$, the age of the universe is
0.79 Gyr and the best-fit model has an age of 0.7 Gyr . The minimum age
with good fits ($\chi_\nu^2$ probability $>0.5\times$ the best-fit one) is 0.5 Gyr,
corresponding to a formation redshift of 10--14. Because there are no detections in
the $K_s$ and F160W bands, the actual strength of the Balmer break in GOODS 850-5 is
unknown. Therefore we do not think we can constrain its age with sufficient accuracy.
Nevertheless, we can conclude that it requires a well established stellar population
with an age that is significantly large compared to the cosmic time to explain
the observed SED at $<10$ $\mu$m.
Since the observed IRAC fluxes are likely dominated by old stars with little
AGN contributions, they can be used to make a measurement of the stellar masses.
Figure~\ref{fig_mass} shows the Hyperz/BC03 best-fit masses as a function of
redshift, and the uncertainty range. At $z=4$ and 6.0, the best-fit masses are
3 and $5\times10^{11}$ $M_{\sun}$, respectively. The range allowed by the
photometric redshift fitting is quite narrow at $z>4$. We therefore believe that the true
uncertainty in the mass estimate is more likely from the stellar population
model. In BC03, we can see that the mass-to-light ($M/L$) ratio is a strong function of
initial mass function (IMF) and a weaker function of metallicity. The adopted IMF in
BC03 is the \citet{chabrier03} IMF, which is more top-heavy than the standard
\citet{salpeter55} IMF. If the IMF is Salpeter, the stellar mass of GOODS 850-5 would
be approximately 2 times larger than the above values. Another important factor
is the likely existence of the thermally pulsing asymptotic giant branch
\citep[TP-AGB,][]{maraston05} phase, which is not included in the BC03 models.
Such a phase decreases the $M/L$ ratio for a young stellar population in the rest-frame
NIR. \citet{maraston06} found that the inclusion of the TP-AGB stars decreases the
mass by $\sim60\%$ on average, compared to the BC03 models.
This would decrease the stellar mass of GOODS850-5 to $3\times10^{11}$ $M_{\sun}$
at $z\sim6$, but this is still a very large mass.
\begin{figure}
\epsscale{1.1}
\plotone{fig7.eps}
\caption{Stellar mass vs.\ redshift, derived from Hyperz with the
BC03 models. Solid curve is the best-fit result. Dotted curves
are the maximum and minimum stellar masses when the
$\chi_\nu^2$ is larger than the minimum $\chi_\nu^2$ by 1.0
(see Figure~\ref{nir_photoz} for the distribution of minimum $\chi_\nu^2$),
showing how the stellar mass changes when the fit is perturbed
around the best-fit. At $z<4$, the fits become very poor although the best-fit
mass still follows the trend at $z>4$. At $z\sim6.9$, the minimum and
maximum range is approximately $\pm40\%$ around the best fit.
There is another factor of 2 uncertainty caused by the uncertainty
in the stellar population model (see text).
\label{fig_mass}}
\end{figure}
\subsection{IR Luminosity and Star Formation Rate}
The data at $>24$ $\mu$m allow us to robustly determine the total IR luminosity
(integrated from 8--1000 $\mu$m, e.g., \citealt{sanders96}).
Figure~\ref{fig_lir} shows the IR luminosity of the best-fit \citet{silva98} models
(FIR only, \S~\ref{4_or_6} and Figure~\ref{fig_ir_photoz}) as functions of redshift and
SED type. We consider two redshifts here: $z\sim4.0$, as suggested by the millimetric
redshift estimate, and $z\sim6.9$, as suggested by the NIR photometric redshift.
At $z\sim4$ the best-fit model is Arp 220 and the best-fit IR luminosity is
$1.4\times10^{13}$ $L_{\sun}$. At $z\sim6.9$ the best-fit model is M 82
and the luminosity is $2.6\times10^{13}$ $L_{\sun}$. These are all comparable to
the result directly inferred from the submillimeter flux using the standard formulas
(e.g., \citealt{blain02}, $L_{\rm IR} = 1.9 \times 10^{12} S_{850 \mu \rm m} L_{\sun}/$mJy,
which is $2.5\times10^{13}$ $L_{\sun}$ for GOODS 850-5).
\begin{figure}
\epsscale{1.1}
\plotone{fig8.eps}
\caption{Total IR luminosity for the best-fit models vs.\ redshift. The IR luminosity is
derived from SED fitting to data between 24 $\mu$m and 1.25 mm. The result
from the M 100 SED is not shown since M 100 does not fit the observations
($\chi_\nu^2 > 10$ for all redshifts).
\label{fig_lir}}
\end{figure}
The star formation rate for GOODS 850-5 can be estimated with
the relation $\dot{M}=1.7\times10^{-10} L_{\rm IR}/L_{\sun}$ \citep[e.g.,][]{kennicutt98}.
The results are 2400 $M_{\sun}$ yr$^{-1}$ for $z\sim4$ and 4400 $M_{\sun}$ yr$^{-1}$
for $z\sim6.9$. This assumes the standard Salpeter IMF.
It is interesting to compare the above star formation rates with the non-detections in the
NIR. Using the UV luminosity versus star formation rate conversion in \citet{kennicutt98},
$\dot{M}=1.4\times10^{-28}L_{\rm UV}/$(\rm erg s$^{-1}$ Hz$^{-1}$),
and a minimum IR star formation rate of 2000 $M_{\sun}$ yr$^{-1}$, we found an
unattenuated UV (1500--2800 \AA) luminosity of $1.4\times10^{31}$ erg s$^{-1}$ Hz$^{-1}$.
At $z=4$--6.9, this corresponds to approximately 40--20 $\mu$Jy at 1.6 to 2.1 $\mu$m,
assuming a young burst SED from BC03. The observed F160W and $K_s$ flux upper limits then imply
the extinctions for the starburst component to be at least $A_V\sim6.7$--4.6.
Figure~\ref{fig_hidden_burst} shows the SEDs of such hidden starbursts at $z=4$ and 6.9.
The lower limits for the extinctions of the starburst component are significantly higher
than those derived from the optical/NIR SED.
We will come back to this in \S~\ref{sec_coexist}.
\begin{figure}
\epsscale{1.1}
\plotone{fig9.eps}
\caption{NIR SEDs of ongoing starbursts suggested by the IR luminosity of GOODS 850-5
at $z=4.0$ and 6.9. Two curves are shown for each redshift: one with no extinction,
derived from a BC03 instantaneous burst model with an age of 5 Myr and a star formation
rate of 2000 $M_{\sun}$ yr$^{-1}$; the other with an extinction that is required to minimally
hide the burst in the F160W and $K_s$ bands such that the hidden burst contributes
$<50\%$ to the observed 1 $\sigma$ upper limits. The squares are the observed SED of
GOODS 850-5 at $>1$ $\mu$m. \label{fig_hidden_burst}}
\end{figure}
\subsection{Nature of the 24 $\mu$m Emission}
All the above discussions on the existence of an AGN,
the stellar mass, and the IR luminosity depend (to various degrees)
on whether the observed 24 $\mu$m emission is dominated by
emission from stellar photosphere or from dust (see the extensive discussion on the 24
$\mu$m emission from HUDF-JD2 in \citealp{mobasher05}).
Roughly speaking, the observed 24 $\mu$m flux of GOODS 850-5 is
unlikely to be dominated by stellar emission, as this would
require $A_V>8$ at $z\sim6$ to explain the observed red 24-to-8.0 $\mu$m
color without dust emission. If we include the 24 $\mu$m band in the
optical/NIR photometric redshift analyses, the fit becomes poor
at all redshifts.
We can quantify the strength of stellar radiation
at 24 $\mu$m using photometric redshift fits
without including the 24 $\mu$m data. The best-fit BC03 models
in \S~\ref{sec_nir_photoz} imply 24 $\mu$m stellar emission of
4.6 and 2.9 $\mu$Jy at $z\sim4$ and 6.9, respectively, corresponding
to 10\% and 6\% of the observed 24 $\mu$m flux. These seem
unusually small, especially at $z\sim6.9$, where 24 $\mu$m
corresponds to rest-frame 3 $\mu$m. We can compare these with
local templates in \citet{silva98} to see if they make any sense.
Table~\ref{tab3} compares the ratios of $L_{\rm IR}/M_{\star}$ and
the ratios of dust-to-stellar radiation at 24 $\mu$m for Arp 220, NGC 6090,
M 82, and GOODS 850-5, at $z\sim4$ and $z\sim6.9$.
We found that once the dust-to-star ratios in the observed 24 $\mu$m emission
is normalized by the $L_{\rm IR}/M_{\star}$ ratios, GOODS 850-5
is not different from local galaxies. It is also interesting to note that
at $z\sim4$ and 6.9 the normalized dust-to-star ratios are similar to
Arp 220 and M 82, respectively. This is consistent with what we found
in the photometric redshift analyses: GOODS 850-5 is similar to Arp 220
if it is at $z\sim4$ but more similar to M 82 if $z>6$.
\begin{deluxetable}{llccc}[h!]
\tablecaption{Dust and Stellar Contributions at 24 $\mu$m\label{tab3}}
\tablehead{\colhead{$z$} & \colhead{Galaxy}
& \colhead{$L_{\rm IR}/M_{\star}$} &
\colhead{$S_{24}^{\rm dust}/S_{24}^{\star}$} &
\colhead{$(S_{24}^{\rm dust}/S_{24}^{\star})^{\prime}$} }
\startdata
4 \\
\hline
& Arp 220 & 14 & 4.8 & 0.34\\
& NGC 6090 & 1.5 & 1.6 & 1.1 \\
& M 82 & 0.2 & 2.7 & 13 \\
& GOODS 850-5 & 46 & 15 & 0.33\\
\hline
6.9 \\
\hline
& Arp 220 & 14 & 0.08 & 0.0057\\
& NGC 6090 & 1.5 & 0.07 & 0.047\\
& M 82 & 0.2 & 0.08 & 0.40\\
& GOODS 850-5 & 52 & 9.1 &0.18
\enddata
\tablecomments{$L_{\rm IR}/M_{\star}$ is in solar unit.
$(S_{24}^{\rm dust}/S_{24}^{\star})^{\prime} =
(S_{24}^{\rm dust}/S_{24}^{\star}) / (L_{\rm IR}/M_{\star})$.}
\end{deluxetable}
There are two subtle issues in the above comparison. First, the
underlying assumption for the normalization is that the $M/L$ ratios
are similar for the galaxies at 24 $\mu$m (rest-frame 3.0--4.8 $\mu$m
for the redshifts of interest). Given the long wavelengths, this is
acceptable. Second, the estimates of the IR luminosity for GOODS 850-5
slightly depends on how much of the observed 24 $\mu$m emission
comes from the warmer dust components. However, from
Figure~\ref{fig_lir} we can see that the uncertainty is well within
a factor of 2 and the IR luminosity is mostly determined by the
three measurements in the millimeter and submillimeter.
Therefore the 24 $\mu$m measurement can still be considered
as a semi-independent check for all the analyses in longer and
shorter wavelengths. The fact that the extrapolations of the optical/NIR
photometric redshift analyses agree well with the millimetric one at
24 $\mu$m (in terms of dust-to-star ratio) does show an excellent self
consistency. It is fair to conclude that the unusually large dust-to-star
ratio in the observed 24 $\mu$m emission of GOODS 850-5 is simply
a consequence of its extremely large IR luminosity.
Two question arises once we establish that $\gtrsim90\%$ of the
observed 24 $\mu$m emission comes from dust: (1) Does this affect
our millimetric redshift analyses? (2) Does the dust
emission extend to the IRAC bands and bias our stellar mass
estimates? To examine (1), we decreased the stellar contribution
in the Silva et al.\ templates by factors of 2 to a few tens and repeated the
photometric redshift fitting. We found that the fits improve slightly.
The best-fit redshift with Arp 220 does not change significantly and
the best-fit redshift with M 82 decreases from $\sim6$--7 to $\sim5.5$--6.
Thus, the general conclusions on the millimetric redshift
and the IR luminosity are both fairly insensitive to the assumed stellar
contribution to the 24 $\mu$m flux.
The answer to (2) is negative as well. The dust spectral slope for
starbursts is extremely steep at $<1.6$ $\mu$m (IRAC bands for $z>4$),
and the blue end of the dust emission does not contaminate the IRAC fluxes.
This is still true even if the observed 24 $\mu$m emission comes from a
dusty AGN with dust much warmer than starbursts. \citet{mobasher05}
compared the observed 24 $\mu$m emission from HUDF-JD2 with dusty AGN
templates Mrk 231 and NGC 1068. We follow the procedure in Mobasher et al.\
and compare the SEDs of GOODS 850-5 and NGC 1068. We found that
the dust emission only contributes to the IRAC fluxes at $z<4$.
At $z>6$, similar to what Mobasher et al.\ concluded for HUDF-JD2 at $z=6.4$,
the dusty AGN does not bias the IRAC stellar mass estimates.
We do not consider Mrk 231 here as its SED in the literature is contaminated by
stellar emission at rest-frame $<3$ $\mu$m (as pointed out by Mobasher et al.)
and hence it does not provide a definite answer. Moreover, as shown in
our photometric redshift analyses, to explain the large millimeter fluxes of GOODS 850-5
with a Mrk 231-like warm dust, it would require unlikely high redshifts of
$z\sim8$--10. In short, we conclude that the dust emission at 24 $\mu$m
is unlikely to bias our IRAC-based stellar mass estimates regardless of the existence
of a dusty AGN.
\subsection{Coexistence of Old Stars and a Starburst}\label{sec_coexist}
The properties of GOODS 850-5 are puzzling.
There appear to be two inconsistencies that are related to each other.
(1) While the IR luminosity implies an intensive ongoing starburst and the formation
of a young galaxy, the NIR SED suggests that most of the observed stellar
radiation comes from old stars without any detectable rest-frame UV radiation.
(2) Our interpretation of the NIR SED suggests $A_V\lesssim2$, but $A_V>4.6$ is
required to hide all the young stars.
One conceivable way to solve this is to see if $A_V>4.6$ with young stellar populations
could provide reasonably good fits in the photometric redshift. If yes, we may argue
that the entire young galaxy is behind an $A_V>4.6$ dust screen
and that the observed IRAC fluxes indeed mostly come from young
stars. To investigate this possibility, we limited the
photometric redshift fitting to the BC03 models with $A_V=4.6$--10, $z>4$, and ongoing
star formation. At $z\sim4$, fits with such highly extinguished stellar populations
all have $\chi_\nu^2>2.7$, which is significantly poorer than the fits with $A_V\lesssim2$.
At $z>5$, the $\chi_\nu^2$ becomes larger than 5 and the models fail to fit the data.
We conclude that $A_V>4.6$ for a young galaxy is a less likely case, although not fully
ruled out for $z\sim4$. It is more likely that the observed IRAC fluxes mostly come from
a relatively old stellar population with a moderate extinction of $A_V\lesssim2$.
Another possible and indeed sensible scenario to explain the above inconsistencies
is that the star forming region (with $A_V>4.6$) in GOODS 850-5 is different from the
region that produces most of the observed IRAC fluxes ($A_V<2$).
GOODS 850-5 is not resolved by the SMA ($\sim2\farcs2$ beam FWHM, S/N=8.6)
and by the VLA ($\sim1\farcs7$, S/N=8.2). These imply an uncertainty of
$\sim0\farcs2$ (beam FWHM divided by S/N) for its beam-convolved size
and $\sim0\farcs8$ for the upper limit of its intrinsic size.
The upper limit corresponds to $\sim4.5$ kpc at $z=6.9$ and is consistent
with the CO sizes of low-redshift SMGs \citep{tacconi06}.
On the other hand, the IRAC fluxes suggest a massive stellar population of
$M_{\star} \sim10^{11.5} M_{\sun}$, which may be spatially offset from the
starburst region and may have less extinction.
An interesting object for comparison is the prototypical dusty starburst ERO,
Hu-Ridgway 10.
\citet{stern06} found that its 10 $\mu$m silicate feature implies $A_V\sim11$
but its optical/NIR SED implies a much smaller $A_V\sim2.4$.
They ascribed this to a heavily obscured starburst in Hu-Ridgway 10, similar to our hypothesis
for GOODS 850-5. Unfortunately, the current data of GOODS 850-5 do not allow
us to use the silicate feature for an extinction measurement.
Another possibility to test the above two-component hypothesis for GOODS 850-5 is to see if
we can fit the observed NIR SED with two components. We found that the photometric
redshift fitting becomes poorer at all redshifts if we include hidden bursts shown
in Figure~\ref{fig_hidden_burst}. The increase in $\chi_\nu^2$ is from $\sim0.2$
at $z=4$ to $\sim0.5$ at $z=6.9$. However, we note that the actual extinction for
the starburst component can be higher than the minimally hidden burst shown in
Figure~\ref{fig_hidden_burst}. Once we increase the extinction for the burst component
by $\sim2$--3 (still less than the extinction in Hu-Ridgway 10), the photometric redshifts
reduce to the ones shown in \S~\ref{sec_nir_photoz} at all redshifts.
In other words, the current data are insufficient for further testing the two-component
hypothesis.
We conclude that a massive galaxy with an old stellar population but with
some dust screening hosting a much more dusty nuclear starburst can explain
the observations of GOODS 850-5.
\section{Discussion}\label{sec_discussion}
For the stellar masses and star formation rates
estimated at $z\sim4$ and 6.9, the mass build-up times are both
$\sim110$ Myr, respectively corresponding to 7\% and 14\% of their cosmic times.
It is clearly more difficult to produce the stellar mass for the $z\sim6.9$ case,
since there is less time available for the formation of the galaxy.
Given that all the stars we see are old, it requires an intensive but short burst of
$>10^3 M_{\sun}$ yr$^{-1}$ at $z>10$. This also suggests that GOODS 850-5
has undergone two distinct bursts of star formation: one
produced the old stars observed in the IRAC bands, and the other is producing the
observed dust emission in the MIPS and (sub)millimeter bands.
The estimated mass of $M_{\star}\sim10^{11.5}M_{\sun}$ is similar to typical SMGs
at $z<4$ \citep[e.g.,][]{smail04,dye08} but is more massive than other galaxies
observed at $z>6$. The most massive Lyman-break (UV luminous) selected galaxies at
$z\sim6$--7 have stellar masses of $\sim10^{10}$--$10^{10.5}$ $M_{\sun}$
\citep{labbe06,yan06,eyles07}.
GOODS 850-5 is 10 times more massive than these optically selected galaxies,
suggesting that the very massive galaxies at these redshifts may be still in dusty
starbursting phases and therefore may be missed by deep optical surveys.
We can estimate the limit for the number of massive, dust-hidden galaxies at $z>6$.
The photometric redshift uncertainty range for GOODS 850-5 is $z\sim6.0$--7.4
and our GOODS-N SCUBA survey area for sources brighter than GOODS 850-5 is
$\sim100$ arcmin$^2$. Therefore the comoving volume of our survey for
sources similar to GOODS 850-5 is $\sim3\times10^5$ Mpc$^3$. This implies
a number density of $\sim3\times10^{-6}$ Mpc$^{-3}$ for
massive objects of $M_{\star}\sim10^{11.5}$ $M_{\sun}$. It is unclear how a
radio-faint submillimeter selection is related to stellar masses. However, the above
density is probably a lower limit, since GOODS 850-5
is the first radio-faint SMG studied in our SMA survey and the survey is
not yet complete. This lower limit is slightly larger than the maximum
value suggested by \citet[see their Figure~11]{yan06} for optically selected
galaxies at $z\sim6$, and is significantly larger than the values from the $\Lambda$CDM
hydrodynamic simulations of \citet{nagamine04} and \citet{night06}.
This emphasizes the importance of dusty galaxies at high redshift, but we
clearly need a much larger high-redshift SMG sample to reach cosmologically
meaningful conclusions.
It is interesting to ask whether current $\Lambda$CDM models can produce
$10^{11.5}$ $M_{\sun}$ galaxies at all at $z>6$. In standard $\Lambda$CDM
models, massive galaxies form in later cosmic times (aka.\ ``hierarchical'' galaxy formation).
On the other hand, evidence of ``cosmic downsizing''
\citep{cowie96} has been observed at many different wavelengths over
a broad range of redshifts, and this has been considered by some as
anti-hierarchical. The existence of ultraluminous SMGs at $z>2$ once put a strain on the
$\Lambda$CDM models \citep{baugh05}. Moreover, the stacking detection of a
population of faint SMGs at low redshifts of $z\sim1$, which dominates the
submillimeter EBL, further demonstrated the downsizing behavior in the SMG
population from $z\sim4$ to $z=0$ \citep{wang06,serjeant08}.
Recent semi-analytic and hydrodynamic models are able to reproduce
galaxies with $M_{\star}>10^{10}$ $M_{\sun}$ up to $z\sim5$--6
under the $\Lambda$CDM framework \citep{bower06,night06}.
At slightly higher redshifts of $z=6$--7, the $\Lambda$CDM N-body simulations of
\citet{lukic07} provide a halo density of $\sim10^{-5}$ (Mpc/h)$^{-3}$ for the
mass range of $10^{12}$--$10^{13}M_{\sun}$. The density rapidly drops to
$10^{-9}$--$10^{-11}$ (Mpc/h)$^{-3}$ for the mass range of $10^{13}$--$10^{14}M_{\sun}$.
If we assume a matter-to-star mass ratio of $\gtrsim10$, the density of
$\sim3\times10^{-6}$ Mpc$^{-3}$ that we derived for high-redshift SMGs seems to
match the upper bound in the simulations. This suggests that it is plausible
to find approximately one $z>6$ dark halo in our survey area that is massive enough to host a
hyperluminous SMG. However, whether the simulations can reproduce the observed
intensive starburst and the large stellar mass within such a halo in a short cosmic time
remains to be seen.
Lastly, can objects like GOODS 850-5 play a role in the reionization of the universe?
The latest 5 yr \emph{WMAP} result \citep{dunkley08} implies a reionization
redshift of $11.3\pm1.4$ (for instantaneous reionization) and the massive
old stellar population in GOODS 850-5 implies intensive star formation at
$z>10$. We use the standard formulation in \citet{madau99},
$\dot{\rho} = 0.013 \times f_{\rm esc}^{-1} \times [(1+z)/6]^3$ $M_{\sun}$ yr$^{-1}$ Mpc$^{-3}$,
where $\dot{\rho}$ is the minimum star formation rate density required
for reionization, $f_{\rm esc}$ is the escaping fraction of ionizing photons, and a
Salpeter IMF is assumed. With the number density estimated above,
$f_{\rm esc}\sim0.1$, and assuming that the initial burst in GOODS 850-5 is as intensive
as the current one, we found that the formation of GOODS 850-5 at $z>10$
contributed $<10^{-2}$ to the ionizing photons that are required for reioinization.
The fraction would be even smaller if the initial burst of GOODS 850-5 were dusty
(i.e., a smaller $f_{\rm esc}$). This is consistent with the current picture that the universe
is ionized by a large amount of low luminosity objects.
\section{Summary and Final Remarks}\label{sec_summary}
Our new ultradeep NIR observations reveal many unusual properties of GOODS 850-5.
It is detected by the SMA, IRAM, and VLA from the submillimeter to centimeter
wavelengths, and by \emph{Spitzer} between 3.6 and 24 $\mu$m, all with high
significance. On the other hand, it is not detected in the $J$, F160W, and $K_s$ bands even
with the nano-Jansky sensitivities. We analyzed the
photometric redshifts of GOODS 850-5. The NIR photometric redshift suggests
a high redshift of $z\sim6.9$ and rules out $z<3$. The millimetric redshift
also rules out $z<3$ and suggests $z\sim4$ if we assume the local radio--FIR correlation.
Without this assumption, $z=4$ to 10 is equally possible for the observed IR SED.
We conclude that $z\gtrsim6$ is more likely for GOODS 850-5
but $z\sim4$ remains a possibility.
To explain the observed NIR SED of GOODS 850-5 and the IR luminosity requires
an established stellar population that formed at a large look-back time
coexisting with an intensive ongoing starburst that is completely invisible in the
rest-frame UV. The old stars observed in the IRAC bands have a large mass of
$M_{\star}\sim10^{11.5}M_{\sun}$
and are $\sim10$ times more massive than optically selected massive galaxies at $z>6$.
The current burst of star formation seems to be compact with a total IR luminosity
of 1.4--$2.6\times10^{13}$ $L_{\sun}$ and a star formation rate of 2400--4400
$M_{\sun}$ yr$^{-1}$.
It is deeply enshrouded by an $A_V>4.6$ dust screen so its UV radiation is
not detected by NICMOS at 1.6 $\mu$m and by Subaru at 2.1 $\mu$m.
This large extinction required for the starburst component also makes it
difficult to directly confirm its existence in the IRAC bands and the
two-population hypothesis remains to be tested.
The high redshift of GOODS 850-5, if confirmed, will have important implications for
galaxy formation and evolution, as discussed in this paper and in W07.
Its extreme faintness at $<2$ $\mu$m prevents any optical and NIR spectroscopy
with current space and ground-based instruments. A precise measurement
of its redshift will most likely come from millimeter ``redshift machines'' on large
telescopes such as the 110 m Green Bank Telescope. Spectroscopy in the
MIR and FIR with the \emph{Herschel Space Observatory} is another possibility.
In general, the observations of GOODS 850-5 remind us that there is a
class of unexpected objects that are not included in the traditional picture of
radio identified SMGs. The SMA is likely to reveal more examples in the near future.
Studies of cosmologically large samples of such high-redshift SMGs, however, will
require combinations of next-generation instruments, such
as the Expanded VLA, Atacama Large Millimeter/Submillimeter Array,
\emph{James Webb Space Telescope}, and large bolometer arrays on
single-dish millimeter telescopes.
\acknowledgments
We thank L.\ Silva and R.\ Chary for providing the SED templates,
the Subaru and \emph{HST} staff for help with observations and data reduction,
G.\ Morrison for providing us with the latest VLA image, C.\ Carilli, F.\ Owen,
and H.\ Hirashita for very helpful discussions, and the referee for comments that
greatly improved the manuscript.
This work is supported by \emph{HST} grant HST-GO-11191,
the NRAO Jansky Fellowship program (W.-H.W.), NSF grants
AST 0239425 and 0708793 (A.J.B.) and AST 0407374 and 0709356 (L.L.C.), the
University of Wisconsin Research Committee
with funds granted by the Wisconsin Alumni Research Foundation,
and the David and Lucile Packard Foundation (A.J.B.).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 101 |
Q: i am getting "segmentation fault" while executing this program This is a code for implementation of singly linked list in c++.
Three basic operations i.e.,insertion, deletion and display is being performed. The data is passed from the main function. The code getting complied but during execution it shows "Segmentation Fault".
Please tell me what is wrong with the code.
#include<iostream>
using namespace std;
class list
{
private:
typedef struct node
{
int data;`
node* next;
}* nodeptr;
nodeptr head;
nodeptr curr;
nodeptr temp;
public:
list();
void insert(int addData);
void delet(int delData);
void display();
};
list::list()
{
head=NULL;
curr=NULL;
temp=NULL;
}
void list::insert(int addData)
{
nodeptr n=new node;
n->next=NULL;
n->data=addData;
if(head!=NULL)
{
curr=head;
while(curr!=NULL)
{
curr=curr->next;
}
curr->next=n;
}
else
{
head=n;
}
}
void list::delet(int delData)
{
nodeptr delptr=NULL;
temp=head;
curr=head;
while(curr!=NULL&&curr->data!=delData)
{
temp=curr;
curr=curr->next;
}
if(curr==NULL)
{
cout<<"Data not found.";
delete delptr;
}
else
{
delptr=curr;
curr=curr->next;
temp->next=curr;
delete delptr;
}
}
void list::display()
{
curr=head;
while(curr!=NULL)
{
cout<<"\n"<<curr->data;
curr=curr->next;
}
}
int main()
{
list l;
l.insert(1);
l.insert(2);
l.insert(3);
l.insert(4);
l.insert(5);
l.display();
cout<<"\nDeleting:";
l.delet(3);
l.display();
l.delet(5);
l.display();
return 0;
}
A: The problem is with insert function:
curr=head;
while(curr!=NULL)
{
curr=curr->next;
}
curr->next=n;
After reaching the curr to null you are accessing it. It should be:
curr=head;
while(curr->next !=NULL)
{
curr=curr->next;
}
curr->next=n;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,147 |
Q: ExecuteNonQuery() returning -1 I am calling a SQL Server stored procedure from C#, consisting of a straightforward INSERT statement.
The ExecuteNonQuery() returns the number of rows affected by the statement OK unless there is no update, in which case I'm getting -1.
This doesn't make much sense to me; how can -1 rows be affected?
A:
The @@ROWCOUNT function is updated even when SET NOCOUNT is ON.
Source
So, since you are at a dead-end with turning NOCOUNT off, try this:
Use @@ROWCOUNT with ExecuteNonQuery
At the end of your SP, execute SELECT @@ROWCOUNT and use ExecuteScalar instead of ExecuteNonQuery. This will return the result that you desire.
A: You can get -1 after calling ExecuteNonQuery() with stored procedure having insert/delete/update query. This happens if stored procedure do not call any query(Insert/Update/Delete). Here is one example which will help you to understand this.
StoredProcedure- MySQL
DELIMITER $$
DROP PROCEDURE IF EXISTS sp_DML $$
CREATE PROCEDURE sp_DML
(
P_Operation varchar(50)
)
BEGIN
if(P_Operation='Insert')
/* Insert Statment*/
end if;
if(P_Operation='Update')
/* Update Statment*/
end if;
if(P_Operation='Delete')
/*Delete Statment*/
end if;
END$$
If u call this SP using ExecuteNonQuery(); and passing parameter P_Operation as 'NoOpeartion' which does not satisfy any condition written into Stored procedure then it will always return -1.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,332 |
ALSO BY STEPHEN HOLMES
Passions and Constraint
Anatomy of Antiliberalism
Benjamin Constant and the Making of Modern Liberalism
ALSO BY CASS R. SUNSTEIN
Free Markets and Social Justice
Legal Reasoning and Political Conflict
The Partial Constitution
After the Rights Revolution
Democracy and the Problem of Free Speech
One Case at a Time: Judicial Minimalism on the Supreme Court
THE COST
OF RIGHTS
Why Liberty Depends on Taxes
W. W. Norton & Company
New York • London
TO GEOFFREY STONE
CONTENTS
ACKNOWLEDGMENTS
INTRODUCTION:
COMMON SENSE ABOUT RIGHTS
PART I:
WHY A PENNILESS STATE CANNOT PROTECT RIGHTS
Chapter One: All Rights Are Positive
Chapter Two: The Necessity of Government Performance
Chapter Three: No Property without Taxation
Chapter Four: Watchdogs Must Be Paid
PART II:
WHY RIGHTS CANNOT BE ABSOLUTES
Chapter Five: How Scarcity Affects Liberty
Chapter Six: How Rights Differ from Interests
Chapter Seven: Enforcing Rights Means Distributing Resources
Chapter Eight: Why Tradeoffs Are Inescapable
PART III:
WHY RIGHTS ENTAIL RESPONSIBILITIES
Chapter Nine: Have Rights Gone Too Far?
Chapter Ten: The Unselfishness of Rights
Chapter Eleven: Rights as a Response to Moral Breakdown
PART IV:
UNDERSTANDING RIGHTS AS BARGAINS
Chapter Twelve: How Religious Liberty Promotes Stability
Chapter Thirteen: Rightsholders as Stakeholders
Chapter Fourteen: Welfare Rights and the Politics of Inclusion
CONCLUSION:
THE PUBLIC CHARACTER OF PRIVATE FREEDOMS
APPENDIX:
SOME NUMBERS ON RIGHTS AND THEIR COSTS
NOTES
INDEX
ACKNOWLEDGMENTS
IT IS A KEEN PLEASURE to thank the many friends and colleagues who helped us with this book. We first stumbled across the cost of rights, as a subject for inquiry and analysis, in discussions at the Center on Constitutionalism in Eastern Europe at the University of Chicago. A healthy respect for the fiscal preconditions of effective rights enforcement arose naturally from observing the underprotection of basic liberties in the insolvent states of Eastern Europe and the former Soviet Union. Russia's great experiment with the jury trial, to choose a typical example, went awry when it began to consume 25 percent of already inadequate local court budgets. One of our principal goals here has been to apply what we learned in this eye-opening context to the debate about rights underway in the United States. We thank Dwight Semler, the coordinator of the Center, and our co-directors, Jon Elster, Larry Lessig, and Wiktor Osiatynski, as well as Andras Sajo, for many challenging discussions. We are also grateful for searching criticisms and wily suggestions on the manuscript from Elster, Bruce Ackerman, Samuel Beer, Martin Krygier, Martha Nussbaum, Richard Posner, and Bernard Yack. Sophie Clark, Keith Sharfman, Matthew Utterbeck, and Christian Lucky provided invaluable research assistance. Appreciation also goes to our editor, Alane Salierno Mason, for her incisive comments and steady encouragement.
Introduction:
COMMON SENSE ABOUT RIGHTS
ON AUGUST 26, 1995, a fire broke out in Westhampton, on the westernmost edge of the celebrated Long Island Hamptons, one of the most beautiful areas in the United States. This fire was the worst experienced by New York in the past half-century. For thirty-six hours it raged uncontrollably, at one point measuring six miles by twelve.
But this story has a happy ending. In a remarkably short time, local, state, and federal forces moved in to quell the blaze. Officials and employees from all levels of government descended upon the scene. More than fifteen hundred local volunteer firefighters joined with military and civilian teams from across the state and country. Eventually, the fire was brought under control. Astonishingly, no one was killed. Equally remarkably, destruction of property was minimal. Volunteerism helped, but in the end, public resources made this rescue possible. Ultimate costs to American taxpayers, local and national, originally estimated at $1.1 million, may have been as high as $2.9 million.
Opposition to government has been a defining theme of American populism in the late twentieth century. Its slogan is, Don't tread on me! Or as Ronald Reagan put it, "Government isn't the solution; it's the problem." More recently, critics of all things governmental, such as Charles Murray and David Boaz, have claimed that an "adult making an honest living and minding his own business deserves to be left alone," and that the "real problem in the United States is the same one being recognized all over the world: too much government."
Yet in Westhampton, on the spur of the moment, public officials were able to organize and direct a costly and collective effort to defend private property, drawing liberally on public resources contributed by the citizenry at large, for the emergency rescue of real estate owned by a relatively small number of wealthy families.
There is nothing exceptional about this story. In 1996, American taxpayers devoted at least $11.6 billion to protecting private property by means of disaster relief and disaster insurance. Every day, every hour, private catastrophes are averted or mitigated by public expenditures that are sometimes large, even massive, but that often go unrecognized. Americans simply assume that our public officials—national, state, and local—will routinely lay hold of public resources and expend them to salvage, or boost the value of, private rights. Despite the undesirably high incidence of crime in the United States, for instance, a majority of citizens feel relatively secure most of the time, in good measure due to the efforts of the police, publicly salaried protectors of one of our most basic liberties: personal or physical security.
Public support for the kind of "safety net" that benefited the home owners of Westhampton is broad and deep, but at the same time, Americans seem easily to forget that individual rights and freedoms depend fundamentally on vigorous state action. Without effective government, American citizens would not be able to enjoy their private property in the way they do. Indeed, they would enjoy few or none of their constitutionally guaranteed individual rights. Personal liberty, as Americans value and experience it, presupposes social cooperation managed by government officials. The private realm we rightly prize is sustained, indeed created, by public action. Not even the most self-reliant citizen is asked to look after his or her material welfare autonomously, without any support from fellow citizens or public officials.
The story of the Westhampton fire is the story of property ownership across America and, in truth, throughout the world. Indeed, it is the story of all liberal rights. When structured constitutionally and made (relatively speaking) democratically responsive, government is an indispensable device for mobilizing and channeling effectively the diffuse resources of the community, bringing them to bear on problems, in pinpoint operations, whenever these unexpectedly flare up.
The Declaration of Independence states that "to secure these rights, Governments are established among men." To the obvious truth that rights depend on government must be added a logical corollary, one rich with implications: rights cost money. Rights cannot be protected or enforced without public funding and support. This is just as true of old rights as of new rights, of the rights of Americans before as well as after Franklin Delano Roosevelt's New Deal. Both the right to welfare and the right to private property have public costs. The right to freedom of contract has public costs no less than the right to health care, the right to freedom of speech no less than the right to decent housing. All rights make claims upon the public treasury.
The "cost of rights" is a richly ambiguous phrase because both words have multiple and inevitably controversial meanings. To keep the analysis as focused and, along this dimension, as uncontentious as possible, "costs" will be understood here to mean budgetary costs and "rights" will be defined as important interests that can be reliably protected by individuals or groups using the instrumentalities of government. Both definitions require elaboration.
DEFINING RIGHTS
The term "rights" has many referents and shades of meaning. There are, broadly speaking, two distinct ways to approach the subject: moral and descriptive. The first associates rights with moral principles or ideals. It identifies rights not by consulting statutes and case law, but by asking what human beings are morally entitled to. While no single agreed-upon theory of such moral rights exists, some of the most interesting philosophical work on rights involves an ethical inquiry, evaluative in nature, of this general kind. Moral philosophy conceives of nonlegal rights as moral claims of the strongest sort, enjoyed perhaps by virtue of one's status or capacity as a moral agent, not as a result of one's membership in, or legal relationship to, a particular political society. The moral account of rights tries to identify those human interests that may not, before the tribunal of conscience, ever be neglected or intruded upon without special justification.
A second approach to rights—with roots in the writings of the British philosopher Jeremy Bentham, American Supreme Court Justice Oliver Wendell Holmes, Jr., and legal philosophers Hans Kelsen and H. L. A. Hart—is more descriptive and less evaluative. It is more interested in explaining how legal systems actually function and less oriented toward justification. It is not a moral account. It takes no stand on which human interests are, from a philosophical perspective, the most important and worthy. It neither affirms nor denies ethical skepticism and moral relativism. Instead it is an empirical inquiry into the kinds of interests that a particular politically organized society actually protects. Within this framework, an interest qualifies as a right when an effective legal system treats it as such by using collective resources to defend it. As a capacity created and maintained by the state to restrain or redress harm, a right in the legal sense is, by definition, a "child of the law."
Rights in the legal sense have "teeth." They are therefore anything but harmless or innocent. Under American law, rights are powers granted by the political community. And like the wielder of any other power, an individual who exercises his or her rights may be tempted to use it badly. The right of one individual to sue another is the classic example. Because a right implies a power that can be wielded, for good or ill, over others, it must be guarded against and restricted, even while being scrupulously protected. Freedom of speech itself must be trimmed when its misuse (such as shouting "Fire!" in a crowded theater) endangers public safety. A rights-based political regime would dissolve into mutually destructive and self-defeating chaos without well-designed and carefully upheld protections against the misuse of basic rights.
When they are not backed by legal force, by contrast, moral rights are toothless by definition. Unenforced moral rights are aspirations binding on conscience, not powers binding on officials. They impose moral duties on all mankind, not legal obligations on the inhabitants of a territorially bounded nation-state. Because legally unrecognized moral rights are untainted by power, they can be advocated freely without much worry about malicious misuse, perverse incentives, and unintended side effects. Rights under law invariably raise such misgivings and concerns.
For most purposes, moral and positive accounts of rights need not be at odds. Advocates of moral rights and describers of legal rights simply have different agendas. The moral theorist might reasonably say that, in the abstract, there is no "right to pollute." But the positivist knows that, in American jurisdictions, an upstream landowner can acquire a right to pollute a river from a downstream landowner. The points are not contradictory, but simply pass each other in the night. Those who offer moral accounts and those who offer positive accounts are asking and answering different questions. So students of collectively enforceable rights have no quarrel with those who offer moral arguments on behalf of one or another right or understanding of rights. Legal reformers should obviously strive to align politically enforceable rights with what seems to them to be morally right. And those charged with enforcing legal rights would do well to convince the public that these rights are morally well founded.
But the cost of rights is in the first instance a descriptive, not a moral, theme. Moral rights have budgetary costs only if their precise nature and scope are politically stipulated and interpreted—that is, only if they are recognizable under law. True, the cost of rights can be morally relevant, for a theory of rights that never descends from the heights of morality into the world of scarce resources will be sorely incomplete, even from a moral perspective. Since "ought implies can," and lack of resources implies cannot, moral theorists should probably pay more attention than they usually do to taxing and spending. And they cannot fully explore the moral dimensions of rights protection if they fail to consider the question of distributive justice. After all, collectively provided resources are often, for no good reason, channeled to secure the rights of some citizens rather than the rights of others.
Rights are ordinarily enforced in functioning and adequately funded courts of law. Not included among the rights discussed in this book, therefore, are rights such as those of women raped in war zones of Bosnia or Rwanda. Existing political authorities have in effect turned their backs on the sickeningly brutal wrongs perpetrated under such conditions, claiming that such crimes do not fall under their jurisdictions. Precisely because remedial authorities universally shrug them off, such miserably neglected "rights" have no direct budgetary costs. In the absence of a political authority that is willing and able to intervene, rights remain a hollow promise and, at present, place no burdens on any public treasury.
Not even the ostensibly legal rights guaranteed under international human rights declarations and covenants will be discussed here unless subscribing national states—capable of taxing and spending—reliably support international tribunals, such as those in Strasbourg or the Hague, where genuine redress can be sought when such rights are violated. In practice, rights become more than mere declarations only if they confer power on bodies whose decisions are legally binding (as the moral rights announced in the United Nations Declaration of Human Rights of 1948, for example, do not). As a general rule, unfortunate individuals who do not live under a government capable of taxing and delivering an effective remedy have no legal rights. Statelessness spells rightslessness. A legal right exists, in reality, only when and if it has budgetary costs.
Because this book focuses exclusively on rights that are enforceable by politically organized communities, it pays no attention to many moral claims of great importance to the liberal tradition. This regrettable loss of scope can be justified by an enhanced clarity of focus. Even if legally unenforceable rights are put to one side, enough difficult problems remain to occupy our attention.
Philosophers also distinguish between liberty and the value of liberty. Liberty has little value if those who ostensibly possess it lack the resources to make their rights effective. Freedom to hire a lawyer means little if all lawyers charge fees, if the state will not help, and if you have no money. The right to private property, an important part of liberty, means little if you lack the resources to protect what you own and the police are unavailable. Only liberties that are valuable in practice lend legitimacy to a liberal political order. This book does not focus exclusively on the budgetary costs of rights that are enforceable in courts of law, therefore, but also on the budgetary costs of making those rights exercisable or useful in daily life. The public costs of police and fire departments contribute essentially to the "protective perimeter" that makes it possible to enjoy and exercise our basic constitutional and other rights.
DEFINING COSTS
American law draws an important distinction between a "tax" and a "fee." Taxes are levied on the community as a whole, regardless of who captures the benefits of the public services funded thereby. Fees, by contrast, are charged to specific beneficiaries in proportion to the services they personally receive. The individual rights of Americans, including the right to private property, are generally funded by taxes, not by fees. This all-important funding formula signals that, under American law, individual rights are public not private goods.
Admittedly, the quality and extent of rights protection depends on private expenditures as well as on public outlays. Because rights impose costs on private parties as well as on the public budget, they are necessarily worth more to some people than to others. The right to choose one's own defense lawyer is certainly worth more to a wealthy individual than to a poor one. Freedom of the press is more valuable to someone who can afford to purchase dozens of news organizations than to someone who sleeps under one newspaper at a time. Those who can afford to litigate obtain more value from their rights than those who cannot.
But the dependency of rights protection on private resources is well understood and has traditionally attracted greater attention than the dependency of rights protection on public resources. Lawyers who work for the American Civil Liberties Union (ACLU) voluntarily accept a cut in personal income in order to defend what they see as fundamental rights. That is a private cost. But the ACLU is also a tax-exempt organization, which means that its activities are partly financed by the public. And this, as we shall see, is only the most trivial way in which rights protection is funded by the ordinary taxpayer.
Rights have social costs as well as budgetary costs. For instance, the harms to private individuals that are sometimes inflicted by criminal suspects released on their own recognizance can reasonably be classed among the social costs of a system that takes serious measures to protect the rights of the accused. A comprehensive study of the costs of rights, therefore, would necessarily devote considerable attention to such nonmonetary costs. But the budgetary costs of rights, treated in isolation from both social costs and private costs, provides an ample and important domain for exploration and analysis. Focusing exclusively on the budget is also the simplest way to draw attention to the fundamental dependence of individual freedoms on collective contributions managed by public officials.
Unlike social costs, "net costs" (and benefits) cannot be temporarily put to one side. Some rights, although costly up front, increase taxable social wealth to such an extent that they can reasonably be considered self-financing. The right to private property is an obvious example. The right to education is another. Even protecting women from domestic violence may be viewed in this way, if it helps bring once-battered wives back into the productive workforce. Public investment in the protection of such rights helps swell the tax base upon which active rights protection, in other areas as well, depends. Obviously enough, the value of a right cannot be assessed by looking solely at its positive contribution to the gross national product (GNP). (While the right of prisoners to minimal medical care is not self-financing, it is no less obligatory than freedom of contract.) But the long-term budgetary impact of expenditures on rights cannot be left out of the picture.
Rights, it should also be noted, may impose a burden on the public fisc beyond their direct costs. A foreign example will help drive this point home. Freedom of movement was created in South Africa by the abolition of the notorious pass laws. But the public costs of building urban infrastructure—water supply, sewage systems, schools, hospitals, and so forth—for the millions who, using their newly won freedom of movement, have flooded from the countryside into cities, is proving astronomically high. (Since the abolition of South Africa's pass laws was one of the most indisputably just acts of recent times, it should not be necessary to prevaricate about its indirect financial costs in order to defend it.) On a more modest scale, here at home, the Third Amendment freedom from having troops quartered in private homes requires that taxpayers fund the construction and maintenance of military barracks. Similarly, a system that scrupulously protects the rights of criminal suspects will make it more costly to apprehend criminals and prevent crimes. And so on.
Such indirect costs or compensatory expenditures, because they directly involve budgetary outlays, fall within the "costs of rights" as narrowly defined in this book. They are especially important because, in some cases, they have led to the curtailment of the rights of Americans. For example, Congress has instructed the secretary of transportation to withhold funding from those states that have not yet abolished the right to ride a motorcycle without a helmet. This decision was based in part on a study made at Congress's request of medical costs associated with motorcycle accidents, including the extent to which private accident insurance fails to cover actual costs. If concern for indirect public costs plays such an important role in the legislative restriction of what some consider our freedoms, the theory of rights obviously cannot leave such costs out.
Finally, this is a book about the nature of legal rights, not a detailed study of public finance. It asks what we can learn about rights by reflecting on their budgetary costs. The rough dollar amounts cited here are therefore meant to be illustrative only. They are certainly not the product of an exhaustive and precise inquiry into the budgetary costs of various rights. To calculate accurately the costs of protecting any given right is immensely complicated, for bookkeeping reasons alone. In 1992, judicial and legal services in the United States cost the taxpayer roughly $21 billion. But joint costs and multi-use facilities make it difficult to specify what portion of this $21 billion was spent on the protection of rights. Similarly, police training presumably improves the humane treatment of suspects and detainees. But while it helps protect their rights, training is primarily intended to increase the capacity of police officers to apprehend criminals and prevent crimes, and in that way to protect the rights of law-abiding citizens. So how could we possibly calculate the exact percentage of the police training budget that is earmarked for the protection of the rights of suspects and detainees?
Empirical research along these lines is certainly desirable. But before such research can be sensibly undertaken, certain conceptual foundations must be laid. To lay such foundations is one of the principal purposes of this book. Once the costs of rights becomes an accepted topic of research, students of public finance will have ample incentive to provide a more precise and thorough account of the dollar amounts devoted to the protection of our basic liberties.
WHY THIS TOPIC HAS BEEN IGNORED
Although the costliness of rights should be a truism, it sounds instead like a paradox, an offense to polite manners, or perhaps even a threat to the preservation of rights. To ascertain that a right has costs is to confess that we have to give something up in order to acquire or secure it. To ignore costs is to leave painful tradeoffs conveniently out of the picture. Disappointed by the way recent conservative majorities on the Supreme Court have limited various rights first granted during the tenure of Chief Justice Earl Warren, liberals may hesitate to throw a spotlight on the public burdens attached to civil liberties. Conservatives, for their part, may prefer to keep quiet about—or, as their rhetoric suggests, may be oblivious to—the way that the taxes of the whole community are used to protect the property rights of wealthy individuals. The widespread desire to portray rights in an unqualifiedly positive light may help explain why a cost-blind approach to the subject has proved congenial to all sides. Indeed, we might even speak here of a cultural taboo—grounded in perhaps realistic worries—against the "costing out" of rights enforcement.
The widespread but obviously mistaken premise that our most fundamental rights are essentially costless cannot be plausibly traced to a failure to detect hidden costs. For one thing, the costs in question are not so terribly hidden. It is self-evident, for instance, that the right to a jury trial entails public costs. A 1989 study provides a dollar amount: the average jury trial in the United States costs the taxpayer roughly $13 thousand. Just as plainly, the right to reasonable compensation for property confiscated under the power of eminent domain has substantial budgetary costs. And the right of appeal in criminal cases clearly assumes that appellate tribunals are publicly funded. And that is not all.
American taxpayers have a serious financial interest in damage suits against local governments involving hundreds of millions of dollars every year in monetary claims. In 1987 alone, New York City paid out $120 million in tort expenses; in 1996, this figure had risen to $282 million. Understandably, every large city in the country is trying to implement tort liability reform, for the right of individuals to sue municipal governments is placing an increasingly intolerable drain on local budgets. Why should judges, narrowly focused on the case before them, have the power to decide that taxpayers' money must be spent on tort remedies rather than, for instance, on schoolbooks or police or child nutrition programs?
Legal professionals understand perfectly well the budgetary implications of the right to sue local governments for damages. They also know that taxpayer money can be saved by openly or surreptitiously curtailing other sorts of rights. The taxpayer's interest in lower taxes can be accommodated, for instance, by de-funding defense services for the poor.
Public savings can be achieved just as effectively by tightening standing requirements for civil actions (by curtailing classical rights), as by tightening eligibility requirements for food stamps (by curtailing welfare rights). When judges hold pretrial conferences to encourage out-of-court settlements in order to reduce delay and congestion in court, they implicitly acknowledge that time is money—more specifically, that court time is taxpayers' money. Under the due process clause, government agencies must provide some sort of hearing in connection with taking away a person's liberty or property (including driver's licenses and welfare benefits), but courts routinely take budgetary expenses into account when deciding how elaborate a hearing to hold. In 1976, discussing the procedural safeguards required by a due process guarantee, the Supreme Court said that
the Government's interest, and hence that of the public, in conserving scarce fiscal and administrative resources is a factor that must be weighed. At some point the benefit of an additional safeguard to the individual affected by the administrative action and to society, in terms of increased assurance that the action is just, may be outweighed by the cost. Significantly, the cost of protecting those whom the preliminary administrative process has identified as likely to be found undeserving may in the end come out of the pockets of the deserving since resources available for any particular program of social welfare are not unlimited.
Statements of this sort, which have become central to the particular legal question of "how much process is due?" may seem like common sense, but their implications have not yet been fully spelled out or thought through.
In interpreting statutes and precedents, and in deciding who may sue whom, courts of appeal as a matter of course take account of the risk of being overwhelmed by costly suits. More generally, courts are given discretion over their own caseloads because, among other things, public expenditures earmarked for the system of justice are limited. Rules such as the Eleventh Amendment (which bans suits against states for money damages in federal court) suggest that American public officials have always understood the costs to the taxpayer of unrestricted individual rights to sue the government. Today, the nationwide move toward no-fault auto insurance, which restricts the rights of individuals to sue other individuals for personal injury, reflects a growing concern over the exorbitant costs, including public costs, of certain private rights. The rise of medical malpractice tribunals has similar sources. Everyone knows that it is very expensive to make existing facilities readily accessible to people with disabilities as mandated by the Americans with Disabilities Act of 1990. But should it not be just as obvious that taxpayers (who else?) must foot the bill when judges hold that compensation is to be paid for a taking of private property or interpret overcrowding in prison as a violation of the Eighth Amendment prohibition on cruel and unusual punishment?
LIBERALS MAY BE SKEPTICAL INITIALLY about the very subject of this book. But why should cost consciousness diminish our commitment to the protection of basic rights? To ask what rights cost, first of all, is not to ask what they are worth. If we could establish to the last penny what it would cost to enforce, say, the right of equal access to justice in a given budgetary year, we would still not know how much we, as a nation, should spend on it. That is a question for political and moral evaluation, and it cannot be settled by accounting alone.
Such considerations are unlikely to assuage liberal apprehensions, however, given the current and apparently bipartisan crusade to cut public expenditures. Fearing that short-sighted voters may respond all too eagerly to "we cannot afford it" arguments put forward by conservatives, liberals may worry, reasonably enough, that cost-benefit analysis will be misused by powerful private interests. They may fear that inevitable disclosures of waste, inefficiencies, and cost overruns—while good in principle—will eventually lead to a further slashing of budgetary allocations for the protection of even our most precious rights. This fear is not wholly unjustifiable. But its appropriateness depends a good deal on what cost-benefit analysis actually entails.
Conservative anxieties are equally acute, but assume a different hue. Many conservatives cling instinctively to a cost-blind approach to the protection of the so-called negative rights of property and contract, because staring hard at costs would shatter the libertarian fiction that individuals who exercise their rights, in the classic or eighteenth-century sense, are just going about their own business, immaculately independent of the government and the taxpaying community. The public costs of nonwelfare rights show, among other things, that "private wealth," as we know it, exists only because of governmental institutions. Those who attack all welfare programs on principle should be encouraged to contemplate the obvious—namely, that the definition, assignment, interpretation, and protection of property rights is a government service that is delivered to those who currently own property, while being funded out of general revenues extracted from the public at large.
So neither liberals nor conservatives, at the outset, are likely to welcome an inquiry into the costs of rights. And a third obstacle to such a study stems from the distinctive sensibility, and perhaps the vested interests, of the legal profession itself. The judiciary prides itself on being insulated from the political process, following the dictates of reason rather than expediency and commonly deferring to the legislature and executive in fiscal matters. But in practice, judges defer much less in fiscal matters than they appear to, simply because the rights that judges help protect have costs.
That rights are financed by the extractive efforts of the other branches does not mesh smoothly with judicial self-images. The problem is serious. Are judges, though nominally independent, actually dangling on purse strings? Does justice itself hinge on riders attached to spending bills? And how can a judge, given the meager information at his or her disposal (for information too has costs) and his or her immunity to electoral accountability, reasonably and responsibly decide about an optimal allocation of scarce public resources? A judge may compel a street to remain open for expressive activity or a prison to improve living conditions for prisoners, but can that judge be sure that the money he or she commandeers for such ends would not have been used more effectively by inoculating ghetto children against diphtheria?
This dilemma does not affect judges alone. Take civil liberties litigators: because they conceive of rights as weapons with which to confront and attack government, they may be uncomfortable with an inquiry into the budgetary cost of rights that focuses attention on a very simple and concrete way in which rights are "creatures" of government. Generally speaking, the costliness of rights protection explodes a powerful illusion about the relationship between law and politics. If rights depend in practice on the going rate of taxation, then does not the rule of law hinge upon the vagaries of political choice? And is it not demeaning to understand rights, which after all protect human dignity, as grants awarded by the public power (even if the power in question is democratically accountable)? As guardians of priceless values, must not judges, especially, rise above the daily compromises of power-wielders and power-seekers?
Whatever the merits of the "should" in this case, it has little relevance to what "is." To imagine that American law is or can be untouched by the tradeoffs familiar to public finance can only blind us to the political realities of rights protection. For the cost of rights implies, painfully but realistically, that the political branches, which extract and re-allocate public resources, substantially affect the value, scope, and predictability of our rights. If the government does not invest considerable resources to ensure against police abuse, there will be a great deal of police abuse, whatever the law on the books may say. The amount the community chooses to expend decisively affects the extent to which the fundamental rights of Americans are protected and enforced.
ATTENTION TO THE COST OF RIGHTS raises a flurry of additional questions, not just about how much various rights actually cost, but also about who decides how to allocate our scarce public resources for the protection of which rights, and for whom. What principles are commonly invoked to guide these allocations? And can those principles be defended?
Finally, the simple insight that rights have costs points the way toward an appreciation of the inevitability of government and of the various good things that government does, many of which are taken so much for granted that, to the casual observer, they do not appear to involve government at all. Attention to the public costs of individual rights can shed new light upon old questions such as the appropriate dimensions of the regulatory-welfare state and the relationship between modern government and classical liberal rights. Public policy decisions should not be made on the basis of some imaginary hostility between freedom and the tax collector, for if these two were genuinely at odds, all of our basic liberties would be candidates for abolition.
PART I:
WHY A PENNILESS
STATE CANNOT
PROTECT RIGHTS
Chapter One
ALL RIGHTS ARE POSITIVE
IN ROE V. WADE, the Supreme Court ruled that the U.S. Constitution protects a woman's right to have an abortion. A few years later, complications arose: does the Constitution also mandate public funding of abortions? Does it require the government to defray the costs of nontherapeutic abortions if the government is already subsidizing childbirths? In Maher v. Roe, the Court concluded that the Constitution does no such thing. A denial of Medicaid payments, it explained, "places no obstacles—absolute or otherwise—in the pregnant woman's path to an abortion." This is because "an indigent woman who desires an abortion suffers no disadvantage as a consequence of the state's decision to fund childbirth," for the government is in no way responsible for her penury. According to the Court, a state legislature's refusal to foot this particular bill, while it may effectively deny safe abortion to a penniless woman, in no way violates that woman's "right" to choose.
To reconcile its holding in Roe with that in Maher, the Court drew a crucial distinction. It said that "there is a basic difference between direct state interference with a protected activity and state encouragement of an alternative." Apparently, the Constitution can, with unimpeachable consistency, first prohibit the government from intruding and afterward permit the government to withhold support. A woman is constitutionally protected from impermissible restrictions by state agencies, the Court went on to explain. But her freedom of choice does not carry with it "a constitutional entitlement to the financial resources to avail herself of the full range of protected choices." Protection from a burden is one thing, entitlement to a benefit is another. And indeed such a distinction between a liberty and a subsidy sounds like common sense. But is it supportable? On what grounds?
Behind the distinction adduced by the Court lies an unspoken premise: immunity from invasion by the state involves no significant entitlement to financial resources. Theorists who share this assumption see constitutional rights as shields established solely to protect vulnerable individuals from arbitrary imprisonment, intrusions on contractual freedom, takings of property, and other forms of governmental abuse. Personal liberty can be secured, they typically argue, simply by limiting the government's interference with freedom of action and association. Individual freedom requires not governmental performance but only governmental forbearance. Construed along these lines, rights resemble "walls against the state," embodying the assurance that Congress "shall make no laws" restricting private liberty or imposing excessive burdens. By dividing government against itself, the Constitution prevents public authorities from intruding or abridging or infringing. The limited government that results leaves plenty of room for private individuals to mind their own business, to breathe and act freely in unregulated social realms. Such immunity from government meddling is even said to be the essence of constitutionalism. And while action is costly, inaction is relatively cheap, or perhaps free. How could anyone confuse the right to noninterference by public authorities with monetary claims upon the public treasury?
THE FUTILITY OF DICHOTOMY
The opposition between two fundamentally different sorts of claim—between "negative rights" such as those granted in Roe and "positive rights" such as those denied in Maher—is quite familiar. But it is anything but self-evident. It does not appear anywhere in the Constitution, for one thing. It was wholly unknown to the American framers. So how does it arise? It has profoundly shaped the legal landscape of the United States, but does it provide a cogent classification of different kinds of rights? Does it make sense?
Without some simplifying scheme, admittedly, the plethora of rights entrenched in American law are hard to think about in an orderly fashion. U.S. citizens successfully claim such a cornucopia of rights, and these rights are so palpably diverse, that generalization about them sometimes seems beyond our reach. How should we think systematically about rights so disparate as the right to strike and freedom of conscience, the right to sue journalists for libel and the right to be free from unreasonable searches and seizures? And how should the right to vote be compared to the right to bequeath one's property, or the right of self-defense to freedom of the press? What do these highly variegated claims have in common? And how can we classify or subdivide, in a rational way, the rights protected and enforced in the United States today?
Even a selective list of the everyday rights of ordinary Americans will make our embarrassment of riches clear. It is not easy to arrange in useful categories such strikingly diverse claims as the right to an abortion, the right to practice one's profession, the right to terminate an agreement, the right to be considered for parole, consumer rights, parental rights, the right to submit evidence before a review board, the right to testify in court, and the right against self-incrimination. Under what basic headings should we classify the right to change one's name, the right of private security guards to make arrests, the exclusive right to decide who publishes (copyright), stock-purchase rights, the right to recover money damages for defamation, tenants' and landlords' rights, the right to smoke the dried leaves of some (but not all) plants, and the right to judicial review of the rulings of administrative agencies? Are there purposes for which it is helpful to sort into two basic groupings—say, the positive and the negative—the right of legislative initiative, the right not to be denied a job because of sexual preference, the right to return to a job after taking unpaid maternity leave, the right to interstate travel, freedom of testation, and the right to inform authorities of a violation of the law? And what about hunting and fishing rights, the right to keep and bear arms, a landowner's right to abate nuisances upon his land, mineral rights, the right to present testimony about the victim of a crime in order to influence the sentencing of a perpetrator, pension rights, the right to give to charity tax-free, the right to recover a debt, the right to run for office, the right to use extrajudicial arbitration methods, and the right to view obscene materials at home? And how should we classify visitation rights in prison, the right to dispose of one's property as one wishes, the right of an expelled high school student to a hearing, the right to marry and divorce, the right of first refusal, the right to be reimbursed for overpayments, the right to the presence and advice of an attorney before custodial interrogation by law enforcement authorities, the right to emigrate, the right to receive counseling about birth control, and the right to use contraceptives?
This ramshackle inventory of only some of the everyday rights of ordinary Americans suggests the magnitude of the challenge facing anyone who wants to map the sprawling terrain of our individual liberties. Even if we set aside archaic-sounding anomalies, such as the "right of rebellion," we will have a tough time organizing into two mutually exclusive and jointly exhaustive groups the swarm of claims and counterclaims that help structure the commonplace expectations and routine behavior of U.S. citizens today.
THE LURE OF DICHOTOMY
True, grand efforts at simplification cannot be impeded. For some purposes, moreover, simplification can be useful; the question is whether the relevant simplification helps illuminate reality. Among recent attempts to impose an easily grasped order on the multiplicity of basic rights invoked and enforced in this country, the one to which the Supreme Court, for good or ill, has lent the weight of its authority has been far and away the most influential. In classrooms and on editorial pages, in judicial opinions and before congressional committees, a distinction is routinely drawn between negative rights and positive rights, or (what is often perceived to be the same thing) between liberties and subsidies. The distinction gains its initial plausibility, perhaps, because it seems to track the politically more familiar contrast between small government and big government.
This dichotomy has taken deep root in common thought and expression. Those Americans who wish to be left alone prize their immunities from public interference, it is said, while those who wish to be taken care of seek entitlements to public aid. Negative rights ban and exclude government; positive ones invite and demand government. The former require the hobbling of public officials, while the latter require their affirmative intervention. Negative rights typically protect liberty; positive rights typically promote equality. The former shield a private realm, whereas the latter reallocate tax dollars. The former are privative or obstructionist, while the latter are charitable and contributory. If negative rights shelter us from the government, then positive rights grant us services by the government. The former rights include the rights of property and contract and, of course, freedom from being tortured by the police; the latter encompass rights to food stamps, subsidized housing, and minimal welfare payments.
This storybook distinction between immunities and entitlements has become so influential, even authoritative, that the Supreme Court was able to assume its validity without serious examination or even argument. Neither its relative historical novelty nor its palpable inadequacy has weakened its hold on academic analysis or the public imagination. But wherein lies its seemingly irresistible appeal?
The attraction of this categorization stems partly from the moral warning or moral promise it is believed to convey. Conservative devotees of the positive/negative rights distinction routinely urge, for instance, that welfare rights are potentially infantilizing and exercised on the basis of resources forked out free of charge by the government. Classical liberal rights, they add by way of contrast, are exercised autonomously, American-style, by hardy and self-sufficient individuals who spurn paternalism and government handouts.
Critics of the regulatory-welfare state also interpret the immunities/entitlements dichotomy in the light of a simplified narrative of historical betrayal or decline. Negative rights, they say, were the first liberties to be established, having been wisely institutionalized at the Founding, if not earlier, whereas positive rights were added afterward, in an ill-considered twentieth-century deviation from the original understanding. When the United States was first created, the protection and enforcement of basic rights was limited to guarantees against tyrannical and corrupt government. Only much later—with the New Deal, the Great Society, and the Warren Court—were supererogatory entitlements to public assistance introduced. Instead of protecting us from government, this conservative story continues, welfare rights make people dependent on government, thus eroding "real freedom" in two different ways: by unfairly confiscating the private assets of the wealthy and imprudently weakening the self-sufficiency of the poor. By profligately adding new positive rights to old negative ones, modern liberals such as Franklin Delano Roosevelt and Lyndon Johnson not only betrayed the Founders' conception of freedom, but also summoned into existence a whole flock of impoverished and dependent citizens who now, alas, must be elbowed off the government soup wagon.
This narrative of decline is recounted with palpable earnestness by political conservatives. American progressives could not disagree more. Nevertheless, they too frequently assume that there are basically two kinds of rights, the positive and the negative. They merely redescribe the shift from immunities to entitlements as a progressive tale of evolutionary improvement and moral growth. While conservatives deplore the emergence of taxpayer-subsidized welfare rights, progressives applaud the rise of positive guarantees—interpreting this as a sign of political learning and an improved understanding of the requirements of justice. Charitable impulses have finally come to the fore and been codified into law. New Deal and Great Society America broke with the narrow principles that served the interests of property holders and big business to the detriment of the majority. Viewed with hindsight, negative rights were limited or perhaps even cruel. The eventual rise of positive rights registered a novel appreciation of the need to supplement non-interference with public provision.
One and the same distinction, in effect, obligingly serves two contrary outlooks. While American liberals typically associate rights of property and contract with immoral egoism, American conservatives link private liberties to moral autonomy. Progressives trace entitlements to generous solidarity, while libertarian conservatives connect welfare handouts to sickly dependency. Opposite evaluations are attached, but the conceptual skeleton is the same. Although politically nonpartisan, the negative/positive rights dichotomy is by no means politically innocent, shaping as it does some of our most important debates. It provides the theoretical underpinnings for both attacks on and defenses of the regulatory-welfare state. The negative/positive polarity, we might even say, furnishes a common language within which welfare-state liberals and libertarian conservatives can understand each other and trade abuse.
But who is correct? Are property rights instruments of selfish egoism or sources of personal autonomy? Do welfare rights (including those to medical care or employment training) express solidarity and fellow-feeling or erode initiative and inculcate dependency? Should individuals be protected only from government or also by government? These questions encapsulate much of the American rights debate today. Naturally, any dichotomy that appeals simultaneously to both the Left and the Right is likely to be hard to criticize and immensely difficult to slough off. Taken-for-grantedness, however, does not mean that the distinction is justifiable either descriptively or normatively. Upon inspection, the contrast between two fundamental sorts of rights is more elusive than we might have expected, and much less clear and simple than our Supreme Court has assumed. In fact, it turns out to be based on fundamental confusions, both theoretical and empirical. But if the distinction itself is flawed, then perhaps neither side of the American rights debate is on solid ground.
THE COST OF REMEDIES
"Where there is a right, there is a remedy" is a classical legal maxim. Individuals enjoy rights, in a legal as opposed to a moral sense, only if the wrongs they suffer are fairly and predictably redressed by their government. This simple point goes a long way toward disclosing the inadequacy of the negative rights/positive rights distinction. What it shows is that all legally enforced rights are necessarily positive rights.
Rights are costly because remedies are costly. Enforcement is expensive, especially uniform and fair enforcement; and legal rights are hollow to the extent that they remain unenforced. Formulated differently, almost every right implies a correlative duty, and duties are taken seriously only when dereliction is punished by the public power drawing on the public purse. There are no legally enforceable rights in the absence of legally enforceable duties, which is why law can be permissive only by being simultaneously obligatory. That is to say, personal liberty cannot be secured merely by limiting government interference with freedom of action and association. No right is simply a right to be left alone by public officials. All rights are claims to an affirmative governmental response. All rights, descriptively speaking, amount to entitlements defined and safeguarded by law. A cease-and-desist order handed down by a judge whose injunctions are regularly obeyed is a good example of government "intrusion" for the sake of individual liberty. But government is involved at an even more fundamental level when legislatures and courts define the rights that such judges protect. Every thou-shalt-not, to whomever it is addressed, implies both an affirmative grant of right by the state and a legitimate request for assistance addressed to an agent of the state.
If rights were merely immunities from public interference, the highest virtue of government (so far as the exercise of rights was concerned) would be paralysis or disability. But a disabled state cannot protect personal liberties, even those that seem wholly "negative," such as the right against being tortured by police officers and prison guards. A state that cannot arrange prompt visits to jails and prisons by taxpayer-salaried doctors, prepared to submit credible evidence at trial, cannot effectively protect the incarcerated against tortures and beatings. All rights are costly because all rights presuppose taxpayer funding of effective supervisory machinery for monitoring and enforcement.
The most familiar government monitors of wrongs and enforcers of rights are the courts themselves. Indeed, the notion that rights are basically "walls against the state" often rests upon the confused belief that the judiciary is not a branch of government at all, that judges (who exercise jurisdiction over policeofficers, executive agencies, legislatures, and other judges) are not civil servants living off government salaries. But American courts are "ordained and established" by government; they are part and parcel of the state. Judicial accessibility and openness to appeal are crowning achievements of liberal state-building. And their operating expenses are paid by tax revenues funneled successfully to the court and its officers; the judiciary on its own is helpless to extract those revenues. Federal judges in the United States have lifetime tenure, and they are quite free from the supervisory authority of the public prosecutor. But no well-functioning judiciary is financially independent. No court system can operate in a budgetary vacuum. No court can function without receiving regular injections of taxpayers' dollars to finance its efforts to discipline public or private violators of rights, and when those dollars are not forthcoming, rights cannot be vindicated. To the extent that rights enforcement depends upon judicial vigilance, rights cost, at a minimum, whatever it costs to recruit, train, supply, pay, and (in turn) monitor the judicial custodians of our basic rights.
When the holder of a legal right is wronged, he may usually petition a taxpayer-salaried judge for relief. To obtain a remedy, which is a form of government action, the wronged party exercises his right to use the publicly financed system of litigation, which must be kept readily available for this purpose. To have a right, it has been said, is always to be a potential plaintiff or appellant. Rights can be retrenched, as a consequence, by making it harder for complainants to seek vindication before a judge. One way to do this is to deprive courts of their operating funds. To claim a right successfully, by contrast, is to set in motion the coercive and corrective machinery of public authority. This machinery is expensive to operate, and the taxpayer must defray the costs. That is one of the senses in which even apparently negative rights are, in actuality, state-provided benefits.
To protect rights, judges exact obedience. Courts issue injunctions to restrain the unlawful infringement of patents or to force realty companies to rent to African Americans under the Fair Housing Act of 1968. To insure freedom of information, courts order federal agencies to provide information requested by the public. Liberty, in such cases, hinges upon authority. When judicial oversight is lax, rights are correspondingly flimsy or elusive. American immigration authorities routinely discriminate on the basis of physical disability, political opinion, and national origin. To remark that aliens trying to enter the United States have few legal rights is to observe that, under American law, they have little access to publicly funded judicial remedies.
But courts are not the only tax-funded government bodies to deliver remedies. For instance, consumer protection bureaus in various states regularly receive complaints and act to protect consumers' rights by penalizing the unfair and deceptive practices of retailers. At the federal level, the Consumer Product Safety Commission spent $41 million in 1996 identifying and analyzing hazardous products and enforcing manufacturer compliance with federal standards. Many other government agencies serve similar rights-enforcing functions. The Department of Justice itself spent $64 million on "civil rights matters" in 1996. The National Labor Relations Board (NLRB), which cost the taxpayer $170 million in 1996, protects the rights of workers by imposing obligations on management. The Occupational Safety and Health Administration (OSHA)—$306 million expended in 1996—defends the rights of workers by obliging employers to provide a safe and healthy workplace. The Equal Employment Opportunity Commission (EEOC), with a 1996 budget of $233 million, safeguards the rights of employees and job seekers by obliging employers not to discriminate in hiring, firing, promotion, and transfers. In every one of these cases, the cost of enforcing rights can be chalked up to the price of enforcing their correlative duties.
To be sure, it is possible to complain that several or all of these agencies are wasteful or too expensive, or even that some of them should be abolished. But while no particular set of institutions is ideal, some substantial governmental machinery for providing remedies must remain, for rights have nothing to do with autonomy from public authority. Because the wholly private and self-sufficient individual has no rights, it is implausible to be "for rights" and "against government."
A few more examples will help clarify this point. The right to bequeath one's private property to heirs of one's choice—"the right to speak after death"—is obviously a power that no individual testator can exercise autonomously, without the active assistance of state agencies. (Proceedings for construing and establishing the validity of wills, and arbitrating the disputes to which wills sometimes give rise, are managed by probate courts, which are funded by taxpayers, not merely by user fees.) And the right to make an enforceable will is perfectly typical, for no rightsholder is autonomous. What would the right to marry mean without public institutions, which must spend taxpayers' money to define and create the institution of marriage? What would the right to child support mean in practice if state agencies could not successfully fulfill requests to locate parents or deduct unpaid support from federal and state tax refunds? What would the copyrights owned by private American entertainment industries be worth in, say, China, if the U.S. government did not put its official weight behind their enforcement?
Something similar can be said about the right to private property. American law protects the property rights of owners not by leaving them alone but by coercively excluding nonowners (say, the homeless) who might otherwise be sorely tempted to trespass. Every creditor has a right to demand that the debtor repay his debt; in practice, this means that the creditor can instigate a two-party judicial procedure against a defaulting debtor in which a delict is ascertained and a sanction imposed. And he can also count on the sheriff to "levy upon" the personal property of the debtor, to sell it, and then to pay the delinquent's debts from the proceeds. The property rights of creditors, like the property rights of landowners, would be empty words without such positive actions by publicly salaried officials.
The financing of basic rights through tax revenues helps us see clearly that rights are public goods: taxpayer-funded and government-managed social services designed to improve collective and individual well-being. All rights are positive rights.
Chapter Two
THE NECESSITY OF
GOVERNMENT PERFORMANCE
THE IDEA THAT RIGHTS ARE ESSENTIALLY AIMED "against" government, rather than calling on government, is patently wrong when applied to what is sometimes called "private law." Rights in contract law and tort law are not only enforced but also created, interpreted, and revised by public agencies. At both federal and state levels, courts and legislatures are constantly creating and readjusting the legal rules that give meaning to rights, as well as specifying and respecifying the various exceptions to these rules. By adjudication and legislation, public authorities not only enforce contracts but also decide which contracts are enforceable and which are unenforceable, unconscionable, or otherwise meaningless pieces of paper. Judges and legislators not only award damages to the victims of negligence but also identify which excuses are legally acceptable for what might otherwise be classified as negligent behavior. The right of American citizens to sue an FBI agent for violating their rights under color of law is wholly defined by statutes and statutory and constitutional interpretation. The rights of out-of-state recreational and commercial fishers owe much of their content to judicial interpretation of the privileges and immunities clause and all of their content to positive law.
The rules defining ordinary rights of this sort are intricate, technical, and full of highly subtle qualifications. In American jurisdictions, for instance, contract law generally stipulates that an injured party cannot collect damages for a loss that he could have avoided after he learned of the breach of contract. An individual who asserts his rights under contract law or tort law must therefore master, or submit to, a complex tissue of rules and exceptions that are, in turn, administered by state officials. He must avail himself of the public power first for the specification of these rules (and exceptions), then for their interpretation, and finally for their enforcement.
The plaintiff's right to bring an action at law against a defendant is not adequately described as a right "against" the state. It is neither a right to be independent of the state nor a right that protects the rightsholder from the state, but rather a right to use state power to give legal effect to a private agreement, to enjoin trespassers from entering private property, to collect compensatory or punitive damages from someone who has negligently or recklessly caused an injury, and so forth. When I sue someone under contract or tort law, I am not trying to get the government "off my back"; I am trying to get it "on my case." In private law, the rightsholder does not need government forbearance; he needs government performance.
To draw attention to the positive role of government in the protection of each and every American liberty is not to deny that, for very limited purposes, some versions of the negative/positive dichotomy may be usefully applied to the analysis of rights. It is perfectly plausible to distinguish between performances and forbearances. The landowner has a legal right that passersby refrain from trespassing on his land. A contract holder may have a right to ensure that third parties do not interfere with an ongoing contractual relationship. In each case, to have a right is to have a legal power to prevent others from acting in a harmful way. Such a right to the self-restraint of others can be usefully contrasted to rights to compel the desirable actions of others, such as the right of a creditor legally to coerce a debtor to repay a debt, or a right of a contracting party to compel his contractual partner to perform.
Because American law recognizes wrongful omissions as well as wrongful commissions, the distinction between rights to require action and rights to prohibit action is useful and important. But it should not be confused with the much less plausible distinction between negative and positive rights, as these concepts are commonly deployed, not only by the Supreme Court. The wholly reasonable distinction between forbearance and performance lends no credence to the opposition between immunity against government interference and entitlement to government service. For the forbearance/performance dichotomy, as just described, does not, in the first instance, refer to government action at all. One private individual has a right either to force another private individual to act or to preclude another private individual from acting. In both cases, obviously, enforcement of a right requires decisive government performance. To protect myself from trespassers and to collect from a delinquent debtor, I have a right to set in motion a tax-funded system of litigation, devoted to accurate fact-finding (which is far from easy) and operated by government bodies—namely, the courts.
HOW EXCEPTIONAL ARE CONSTITUTIONAL RIGHTS?
But are not private-law rights (such as the right to sue for breach of contract) quite unlike constitutional rights (such as freedom of speech)? It makes little sense to distinguish between property rights and welfare rights by calling the former negative and the latter positive. Is it more plausible to label private-law rights as positive (requiring government action), and constitutional rights as negative (requiring governmental self-restraint)? When speaking of rights against state action, after all, the Supreme Court was referring exclusively to constitutional rights. So this question arises: are the liberties protected under the Bill of Rights wholly negative? Do they require the state to refrain from acting without requiring the state to act?
Some constitutional rights depend for their existence on positive acts by the state, and the government is therefore under a constitutional duty to perform, not to forbear, under the Constitution as it stands. If it allows one person to enslave another, by doing nothing to disrupt an arrangement that amounts to involuntary servitude, the state has violated the Thirteenth Amendment. Under the First Amendment's protection of freedom of speech, states must keep streets and parks open for expressive activity, even though it is expensive to do this, and to do it requires an affirmative act. Under the protection against "takings" of private property without just compensation, the government is probably under an obligation to create trespass law and to make it available to property owners, and a partial or complete repeal of the law of trespass—a failure to act, in other words, to protect private property—would likely be unconstitutional. If a judge accepts a bribe offered by a defendant and therefore does nothing to protect the plaintiff's rights, the judge has violated the due process clause. If a state declines to make its courts available to enforce certain contract rights, it has probably impaired the obligations of contracts, in violation of the contracts clause. In all these cases, the government is obliged, by the Constitution, to protect and to perform.
Practically speaking, the government "enfranchises" citizens by providing the legal facilities, such as polling stations, without which they could not exercise their rights. The right to vote is meaningless if polling place officials fail to show up for work. The right to just compensation for confiscated property is a mockery if the Treasury fails to disburse. The First Amendment right to petition for a redress of grievances is a right of access to government institutions and a right, incidentally, that assumes that the government can perform for the benefit of aggrieved citizens. Nor is this all.
If an agency of the American government tries to deprive anyone of life, liberty, or property, it is required to give that person timely notice and provide an opportunity to be heard before an impartial body. The right to subpoena witnesses in one's own defense is useless if the court's solemn writs and summonses are greeted with laughter. And what does it mean to say that state and federal governments are prohibited from denying equal protection before the law if not that they are required to provide it? Protection against unequal treatment by government officials requires other government officials to receive and resolve complaints. The constitutional right to due process—like the private right to bring an action in contract or tort—presupposes that, at the taxpayers' expense, the state maintains and makes accessible complex and relatively transparent legal institutions within which the cumbersome formalities of fair, public, and understandable adjudication occur.
Admittedly, some important constitutional rights are plausibly styled as duties of the government to forbear rather than to perform. But even those "negative rights"—such as prohibitions on double jeopardy and excessive fines—will be protected only if they find a protector, only if there exists a supervisory state body, usually a court of some kind, able to force its will upon the violators or potential violators of the rights at issue. Even rights reasonably described as operating "against" the state require the (affirmative) creation and strengthening of relations of oversight, command, and obedience so that rogue officials (including police officers and prison guards) do not behave cruelly or discriminatorily. In some cases, public officials must indeed be kept out of protected zones. But those zones qualify as protected only because of affirmative government, and to achieve the desired protection, vulnerable individuals must have relatively easy access to a second, higher-level set of government actors whose decisions are deemed authoritative.
Nonperforming public officials—whether apathetic or bribe-taking or remissly supervised—will not enforce constitutional rights any more effectively than they enforce rights held under statutes and common law. The very idea that a certain kind of process is "due" demonstrates that constitutional rights impose affirmative obligations on the state. Giving citizens access to courts and other adjudicative forums is not like giving them access to natural harbors and navigable waters, because the government must not only brush aside hindrances to access, but must actually create the institutions to which access is being granted. "Avenues of relief" are maintained in passable condition by government officials. The operating expenses of American courts alone run in the billions of dollars every year, and the American taxpayer picks up the tab.
RIGHTS AND POWERS
Invariably, rights pit power against power. Under tort law, rights enlist the power of government to extract compensatory or punitive damages from private wrongdoers. Under constitutional law, rights bring the power of one branch of government to bear upon wrongdoers from other government agencies. For instance, in the late 1960s, the Supreme Court protected the right of students to wear black armbands to school (in a protest against the Vietnam War) by overruling public high school authorities. Protection "against" government is therefore unthinkable without protection "by" government. This is exactly what Montesquieu had in mind when he asserted that freedom can be protected only if power checks power. No legal system can defend people against public officials without defending people by means of public officials.
When a right is enforced, moreover, somebody wins and somebody loses. The enforcement of a right (whether it is a right against racial discrimination or a right to collect compensatory damages) is "accepted" by the losing party because that party has no choice, that is, because the full power of the state has come down on the side of the rightsholder, and thus against the losing party. Conversely, curtailing a right often involves curtailing the power of the government agency that enforces it in the face of serious resistance. For instance, if a political pressure group wants to cut back the existing rights of American workers, it will try to diminish the authority of OSHA, the EEOC, or the NLRB. This is strong evidence that rights depend essentially on power.
The dependency of liberty on authority should be especially obvious in the United States, where rights against abuse by state and local officials have long been enforced by federal officials. The "incorporation doctrine," which extends most of the Bill of Rights to the states, protects individual liberties not by removing government from the scene, but by giving national authorities the power to overrule state authorities. The Fourteenth Amendment prohibits the states from denying anyone equal protection of the law or depriving them of life, liberty, or property without due process of law. Such a prohibition would be hollow if the federal government did not have the power to bear down on recalcitrant states.
"Congress shall have power to enforce this article by appropriate legislation." All three Civil War amendments contain such enforcement clauses. So the amended Constitution explicitly vests the federal government with the capacity to realize in practice the individual rights it proclaims in principle. Without such governmental powers, rights would have no "bite." To protect the rights of southern blacks, more than once in our history the national government has dispatched federal troops to the South. Without such a show of force, the individual rights of a large group of Americans would have remained a cruel charade. To prevent racial segregation in education, national involvement was necessary, sometimes including the threat to meet violence with violence. Until Congress and the former Department of Health, Education, and Welfare applied irresistible financial pressure, in any case, school districts in the deep South simply ignored the Supreme Court's desegregation orders. When state government is discriminating, the right to be free from racial discrimination, like the right to property, requires affirmative assistance from government, in this case the nation itself.
In the area of voting rights, the same pattern has prevailed. The Voting Rights Act of 1964—designed to vindicate constitutional rights—called for more involvement by the national government, not less. Until Congress legally prohibited the use of literacy tests, states contrived to disenfranchise black Americans for reasons of race. This is just a further illustration of a general truth: individual rights are invariably an expression of governmental power and authority.
Not included in the original Constitution, the Bill of Rights was added to the Constitution two years after its ratification partly to appease those who desired a weaker and more constrained national government. But that was not its only purpose, and that has not been its effect in practice. By extending the scope of the Bill of Rights, the Supreme Court, a national institution, has steadily encroached on preserves of state power. State autonomy has been whittled away and federal power correspondingly enhanced in the name of individual rights. (Admittedly, the opposite has also occasionally occurred.) Indeed, one of the consequences of the enhancement of federal power has been to apply the prohibition on uncompensated takings of private property to the states—requiring state governments, for instance, to compensate people, as a matter of federal constitutional law, when regulation has rendered their beachfront property valueless.
Decentralizing government has no logical connection with limiting the encroachment of government into society. Many of the original limits on Congress's authority were not meant to preserve immunity from government, but simply to clear a space for unsupervised state regulation, as opposed to federal regulation, of private economic behavior. To create a national market, against the protectionist impulses of local authorities, the federal government had no choice but to erode state regulatory autonomy. And this is perfectly normal: a lower authority will usually retreat only when a higher authority steps forward.
The framers of the American Constitution sought to establish a strong and effective government armed with capacities that the anemic government created under the Articles of Confederation notoriously lacked. A constitution that does not organize effective and publicly supported government, capable of taxing and spending, will necessarily fail to protect rights in practice. This has been a lesson long in learning, and not only for libertarians and free-market economists, but also for some human-rights advocates who have selflessly devoted their careers to a militant campaign against brutal and over-mighty states. All-out adversaries of state power cannot be consistent defenders of individual rights, for rights are an enforced uniformity, imposed by the government and funded by the public. Equal treatment before the law cannot be secured over a vast territory without relatively effective, honest, centralized bureaucratic agencies capable of creating and enforcing rights.
Chapter Three
NO PROPERTY WITHOUT TAXATION
ACCORDING TO THE BRITISH PHILOSOPHER JEREMY BENTHAM, "property and law are born together and die together. Before the laws there was no property; take away the laws, all property ceases." Every first-year law student learns that private property is not an "object" or a "thing" but a complex bundle of rights. Property is a legally constructed social relation, a cluster of legislatively and judicially created and judicially enforceable rules of access and exclusion. Without government, capable of laying down and enforcing compliance with such rules, there would be no right to use, enjoy, destroy, or dispose of the things we own. This is obviously true for rights to intangible property (such as bank accounts, stocks, or trademarks), for the right to such property cannot be asserted by taking physical possession, only by an action at law. But it is equally true of tangible property. If the wielders of the police power are not on your side, you will not successfully "assert your right" to enter your own home and make use of its contents. Property rights are meaningful only if public authorities use coercion to exclude nonowners, who, in the absence of law, might well trespass on property that owners wish to maintain as an inviolable sanctuary. Moreover, to the extent that markets presuppose a reliable system of recordation, protecting title from never-ending challenge, property rights simultaneously presuppose the existence of many competent and honest and adequately paid civil servants outside the police force. My rights to enter, use, exclude from, sell, bequeath, mortgage, and abate nuisances threatening "my" property palpably presuppose a well-organized and well-funded court system.
A liberal government must refrain from violating rights. It must "respect" rights. But this way of speaking is misleading because it reduces the government's role to that of a nonparticipant observer. A liberal legal system does not merely protect and defend property. It defines and thus creates property. Without legislation and adjudication there can be no property rights in the way Americans understand that term. Government lays down the rules of ownership specifying who owns what and how particular individuals acquire specific ownership rights. It identifies, for instance, the maintenance and repair obligations of landlords and how jointly owned property is to be sold. It therefore makes no more sense to associate property rights with "freedom from government" than to associate the right to play chess with freedom from the rules of chess. Property rights exist because possession and use are created and regulated by law.
Government must obviously help maintain owner control over resources, predictably penalizing force and fraud and other infractions of the rules of the game. Much of the civil law of property and tort is designed to carry out this business. And the criminal justice system channels considerable public resources to the deterrence of crimes against property: larceny, burglary, shoplifting, embezzlement, extortion, the forging of wills, receiving stolen goods, blackmail, arson, and so forth. The criminal law (inflicting punishments) and the civil law (exacting restitution or compensation) conduct a permanent, two-front, and publicly financed war on those who offend against the rights of owners.
David Hume, the Scottish philosopher, liked to point out that private property is a monopoly granted and maintained by public authority at the public's expense. As the English jurist William Blackstone, following Hume, also explained, property is "a political establishment."2 In drawing attention to the relation between property and law—which is to say, between property and government—Bentham was making the very same point. The private sphere of property relations takes its present form thanks to the political organization of society. Private property depends for its very existence on the quality of public institutions and on state action, including credible threats of prosecution and civil action.
What needs to be added to these observations is the correlative proposition that property rights depend on a state that is willing to tax and spend. Property rights are costly to enforce. To identify the precise monetary sum devoted to the protection of property rights, of course, raises difficult issues of accounting. But this much is clear: a state that could not, under specified conditions, "take" private assets could not protect them effectively, either. The security of acquisitions and transactions depend, in a rudimentary sense, on the government's ability to extract resources from private citizens and apply them to public purposes. On balance, property rights may even place a charge upon the public treasury that vies with the burden of our massive entitlement programs.
None of this denies that protection of property rights can be a valuable investment that increases aggregate wealth over time. On the contrary, the extraction and redistribution of resources necessary to protect property rights is relatively easy to justify. Indeed, American liberalism, like its counterparts elsewhere in the world, is based on the reasonable premise that public investment in the creation and maintenance of a system of private property is richly repaid, not least of all because reliably enforced property rights help increase social wealth and therefore, among other benefits, swell the tax base upon which government can draw to protect other kinds of rights. But the strategic wisdom of an initial investment does not undo the fact that it is an investment.
The immense up-front costs of protecting private property mount even higher if we include, as we surely must, protection from foreign looters and marauders. The thousands of civilians expelled from their homes in Abkhazia or Bosnia—like other forced migrants throughout the world—know that property rights are a mirage without military forces trained and equipped to protect owners from forcible seizures by invading armies or drunken paramilitary gangs. The defense budget in a free-market society is a widely shared public contribution to, among other ends, the protection of private property. Americans spent $265 billion in 1996 on defense and an additional $20 billion on veterans' benefits and services. Military expenditures must unquestionably be counted among the public costs of the property rights that many Americans peaceably exercise and enjoy.
Conscription of low-income youth represents an important way in which property holders may benefit directly from the "civic contributions" of the propertyless. Individual property holders are fundamentally dependent on collective efforts, both diplomatic and military, organized by the government, to protect their land and housing stock from seizure by property-grabbing adjacent states. Montana "Freemen," citizens of the Republic of Texas, and other self-styled government-bashers who pretend they can defend their autonomy with mail-order shotguns and hunting rifles would, in reality, be wholly unable to prevent their private property from being gobbled up even by relatively weak foreign powers if most of their fellow citizens did not regularly submit themselves to taxation and conscription by the national political community.
Where real estate is involved, in fact, ownership becomes quickly enmeshed with sovereignty (or with aspirations to sovereignty, as Palestinians caught selling land to Israelis find out). Defense spending is surely the most dramatic example of the dependency of private rights on public resources. It reveals the statist preconditions of laissez-faire, the authority that underwrites liberty. At common law, only the sovereign is said to have an absolute interest in land: ordinary landowners "hold of the sovereign." This quaint legalism expresses a deep truth. An autonomous individual, in a liberal society, cannot create the conditions of his own autonomy autonomously, but only collectively.
The most ardent antigovernment libertarian tacitly accepts his own dependency on government, even while rhetorically denouncing signs of dependency in others. This double-think is the core of the American libertarian stance. Those who propagate a libertarian philosophy—such as Robert Nozick, Charles Murray, and Richard Epstein—speak fondly of the "minimal state." But describing a political system that is genuinely capable of repressing force and fraud as "minimal" is to suggest, against all historical evidence, that such a system is easy to achieve and maintain. It is not. One piece of evidence to the contrary is the amount we spend, as a nation, to protect private property by punishing and deterring acquisitive crimes. In 1992, for instance, direct expenditures in the United States for police protection and criminal corrections ran to some $73 billion—an amount that exceeds the entire GDP of more than half of the countries in the world. Much of this public expenditure, naturally, was devoted to protecting private property. Even a purportedly hands-off state, if it wants to be serious about encouraging economic activity, must reliably protect homeowners and shopkeepers from burglars, arsonists, and other threats.
An effective liberal government, designed to repress force and fraud, must avoid arbitrary and authoritarian tactics. Those who wield the tools of coercion must be institutionally disciplined into using it for public, not private, purposes. Ideally conceived, a liberal government extracts resources from society fairly and efficiently and redeploys them skillfully and responsibly to produce socially useful public goods and services, such as the deterrence of theft. A successful liberal state must be politically well organized in precisely this sense. Its government must be capable of creating a favorable business climate in which investors are confident that they will reap rewards tomorrow for efforts made today. Without such a state, well-functioning markets, capable of producing prosperity, are very unlikely to emerge or survive. A state capable of reliably repressing force and fraud and enforcing property rights is a cooperative achievement of the first magnitude, and the world is unfortunately filled with negative examples. But if private rights depend essentially on public resources, there can be no fundamental opposition between "government" and "free markets," no contradiction between politically orchestrated social cooperation and footloose individual liberty.
Property owners are far from being self-reliant. They depend on social cooperation orchestrated by government officials. Defense against land-grabbing foreign predators is only one example of the way liberal individualism depends on effective collective action. Recordation is another. American taxpayers expended $203 million for general property and records management in 1997. Sunk costs in our recordation system are much larger. For real estate markets to operate effectively, a reliable system of titles, deeds, and land surveys must be in place. Land registries and offices of public records require skilled and honest staffs. The "free market" is unlikely to put roofs on public buildings where records are stored or establish criminal penalties to deter bribery of officials in charge of registering titles to real or personal property. Surveyors, too, must be paid and monitored. The bare unobstructed latitude to buy and sell private property will not produce an explosion of mutually beneficial private exchanges unless potential buyers receive some sort of guarantee that the putative owner is selling something he (and he alone) actually owns. Without clearly defined, unambiguously assigned, and legally enforceable property rights, ownership does not encourage stewardship. Title holders will neither cultivate their fields nor repair their homes if their rights are not reliably protected by the public power.
Additional examples of government expenditures for the sake of private property owners are legion; it is unnecessary to think that all or even most are defensible in order to see the basic pattern. The American taxpayer spent almost $10 billion in 1996 for agricultural subsidies designed to increase the value of the private property rights of American farmers. The Army Corps of Engineers expended around $1.5 billion in 1996 on flood-plain management and other forms of flood control. The Coast Guard spent $1.26 billion in the same year in search and rescue missions, aids to navigation, marine safety (including the removal of dangerous wrecks and derelicts at sea), ice breaking, and so forth, all of which helps protect the private property of American shippers and boat owners. Copyright, which is a form of property, also involves public expenditure. The Copyright Office and Copyright Royalty Tribunal, taken together, cost $28 million in 1996; $18 million of this amount was covered by user fees, leaving roughly $10 million on the tab of the ordinary taxpayer.
The relatively high rate of owner occupancy in the United States is a creation not only of governmentally conferred rights but also of American mortgage, insurance, and tax law. It is certainly not a product of government disengagement or laissez-faire. Some property owners would be forced to liquidate their holdings if they were not allowed to deduct the depreciation of their assets from their taxable income. And a tax deduction is a form of public subsidy. This is just one more example of the way private property is affirmatively sustained by public subsidies.
Private property is not only protected by government agencies, such as the fire department. It is, more generally, a creation of state action. Legislators and judges define the rules of ownership, just as they establish and interpret the regulations governing all of our basic rights. Does the accidental finder of goods have a legal right to judicial protection? Does a purchaser acquire an ownership right to property bought for value and in good faith from a thief? What rights against a present occupant belong to the owner of a future interest in real property? How many years of wrongful possession destroy the title of the original owner? Can an illegitimate child inherit from its natural parents by intestate succession? What happens if one joint owner sells his portion of jointly owned property? Can I, without notice, cut off branches from my neighbor's tree if they overhang my land? Do I have a right to pile a mountain of garbage in my front yard? Can I build an electrical fence around my land with voltage high enough to kill trespassers? Can I erect a building that cuts off my neighbor's vista? Can I advertise the free viewing of pornographic videos in my front window? Can I stick posters on my neighbor's fence? Under what conditions is copyright assignable? How much do which creditors collect in case of bankruptcy? What rights do pawnbrokers have over goods left to them upon pledge?
Thousands of questions of this sort are continuously asked by those who have property rights and regularly answered by legislatures and courts, that is, by state agencies. The answers given shift over time. In the United States, answers also vary from one jurisdiction to another. For instance, spouses have a right to income from each others' property in Idaho, Louisiana, Texas, and Wisconsin. In most of the rest of the country, they have no such rights. The state cannot "leave the owner alone," therefore, because an owner is an owner only on the precise terms laid down at particular times by specific legislatures and courts.
To protect our property rights, American courts must administer a technically intricate and changing body of rules. These rules are especially vital when two or more individuals have overlapping claims to the same piece of property. Private property as we know it exists only because legislation and adjudication has specified the respective ownership rights of rival claimants—for instance, the property rights of authors and publishers in a book or the property rights of employers and employees in the invention of employees. Upon the death of a co-owner of real property, the law must decide if ownership rights are to be transferred to the living co-owner(s) or to the heirs of the deceased co-owner. The law assigns property rights by creating and enforcing rules for authoritatively settling disputes among rival claimants. To perform this function, judges must be trained, equipped, paid, protected from extortion, and provided with a technical and clerical staff. This is what it means to call the right to property a privately enjoyed public service.
Along the same lines, the basic ingredients of the law of tort—for example, my right to demand compensatory damages from those who have negligently or willfully damaged my property—strongly suggest that property rights are less like latitudes and more like entitlements than American public rhetoric commonly allows. Those who demand greater rights to compensation from government for public "takings"—through regulation or otherwise—are in reality seeking entitlements. They want to be protected publicly and through law. This is not an argument against their claim of right. The regulatory state might well work better if government had to pay property owners for the diminished value of land whenever, for example, new environmental regulations have impeded development. But arguments to this effect should not be based on undiscriminating protests against public invasions of autonomously held rights.
Many political conservatives, but not they alone, urge government to "get out of the marketplace." For their part, some liberals counter that government quite legitimately interferes with, or "steps into," the market whenever and wherever disadvantaged Americans are at risk. But this familiar debate is built on sand. No sharp line can be drawn between markets and government: the two entities have no existence detached from one another. Markets do not create prosperity beyond the "protective perimeter" of the law; they function well only with reliable legislative and judicial assistance.
Of course, inept governments can and do commit economic blunders. Without doubt, ill-devised and poorly timed policies can and do make markets function poorly. The question is not free markets or government but what kind of markets and what kind of government. Governments not only have to lay the essential legislative and administrative foundations for a functioning market economy, they can also act to make market systems more productive. They do so, for example, by adjusting the exchange rate of the national tender against foreign currencies, by disrupting anticompetitive monopolies, by building bridges and railroads, and by financing the vocational training of the future workforce. As even Friedrich Hayek, the great critic of socialism, remarked, "The question whether the state should or should not 'act' or 'interfere' poses an altogether false alternative, and the term 'laissez-faire' is a highly ambiguous and misleading description of the principles on which a liberal policy is based."
A liberal economy cannot function unless people are willing to rely on each other's word. For a market to be national, and not merely local, reliance must extend beyond a small circle of mutual acquaintances. In such a system, reliance on the word of relative strangers cannot arise from personal reputations for fairness alone. It must be cultivated and reinforced by public institutions. For one thing, the government must make courts and other institutions available to enforce contracts. Public authorities cultivate the "reliance interest" by attaching property and foreclosing liens. Judges can send an individual to jail for contempt of court if he fails to comply with an order to carry out a contract lawfully entered into. Likewise, laws against defamation, geared to the protection of business and financial reputations, help foster economically beneficial social trust. If contracts were not reliably enforced, it would be more difficult and perhaps even impossible to buy goods on credit or by installment. Without the active help of a sheriff, authorized by a court writ, a seller could not easily repossess consumer goods from a defaulting installment purchaser. More generally, payment by the installment plan, beneficial for the economy as a whole, would be shunned if contracts were not reliably enforced.
In the truly autonomous realm, beyond the reach of government, property is not well protected. (In the abandoned warehouse at the edge of town where you lost your wallet, your right to your property is not worth much.) Where the public power cannot effectively intrude, moreover, extortion is rampant and borrowers are unable to obtain long-term loans, for one function of the liberal state is to lengthen the time horizons of private actors by predictably enforcing known and stable rules. Property is worth little if you, and potential purchasers, do not believe in the future. Confidence in long-term stability is partly a product of reliable law enforcement, that is, of forceful and decisive state action.
But the first thing a government must do to make a market system work is to overcome the age-old rule of force and threats of force. Free markets do not function properly if profit-seekers uninhibitedly engage in criminal violence. Libertarians recognize this fact, but they fail to appreciate the extent to which it undermines their boasted opposition to "government" as well as to taxing and spending. Long-gestation investment in productive facilities, which creates jobs, is unlikely where assets are indefensible against private extortionists. Neoclassical economics supposes that private competitors will not resort to violent crime in the pursuit of gain. Within its own framework, laissez-faire theory is helpless to explain the basis of civilization, the general renunciation of violence by advantage-seeking individuals and groups. Why do most American entrepreneurs hesitate to threaten and kill their competitors? The theory of free markets, as it is currently taught in American universities, tacitly assumes that the problems of short time horizons and violent competition, characterizing the state of nature, have already been solved. For the most part, in other words, the science of economics (unlike, say, the science of anthropology) tacitly presupposes the existence of an active and reliable system of criminal justice.
Even on their own terms, doctrinaire libertarians must acknowledge that government cannot "pull out" of the economy without leaving private individuals helplessly vulnerable to ruthless predators. The relatively peaceful exchange of goods and services, as we know it, is a product of civilized self-restraint and therefore should be understood as a historically improbable and fragile achievement. In the state of nature, a handful of killers and thieves willing to employ deadly force and hazard their lives on a dare can cow a large civilian population. They can establish anticompetitive monopolies, for instance, and dramatically shrink the sphere of voluntary exchange. Only a reliable public power can break such an anarchical reign of fear and legal uncertainty. Only a state can create a vibrant market. Furthermore, only a national government can weave together disconnected local markets into a single national market. For why would a wholesaler in New Jersey sell to a retailer in California if contracts could not be reliably enforced across state lines?
If the government wholly disengages from the economy, the economy will not be free in the sense we admire, and it will certainly not produce the historically unprecedented prosperity to which many Americans have grown accustomed. Voluntary exchanges will occur, as they do even in the poorest of countries, and we may see inchoate versions of well-functioning markets. But government inaction creates an economic system vexed by force, monopoly, intimidation, and narrow localisms. The individual's freedom, his "right to be left alone" by thugs and thieves, cannot be separated from his entitlement to state help—that is, his claim to a range of public services (basic legal provisions and protections) from the government. The effort of social coordination it takes to build even a "minimal" state, capable of repressing force and threats of force, is truly massive and should not be taken for granted.
Capitalists certainly know this and tend not to invest where political risk is excessive, as in some of the emerging Eastern European democracies. Their problem is not too much government but too little government. When government is incoherent, incompetent, and unpredictable, economic actors do not think very far into the future. Not free-enterprise but robber capitalism—the rule of the violent and the unscrupulous—thrives in the absence of law and order.
Swindling is nearly as great a threat to free markets as force, and enforceable antifraud law also presupposes a well-organized and effective system of governance. To some extent markets themselves will deter fraud; people who lie and cheat at the drop of a hat tend not to compete well. But without effective antifraud legislation, private parties will often hesitate to undertake what both sides nevertheless anticipate would be a mutually advantageous voluntary exchange. Antifraud legislation, in turn, costs taxpayer money to enforce. The Federal Trade Commission (FTC) spent $31 million in 1996 investigating unfair and deceptive practices and removing other obstacles to market performance. Perhaps this is too much, perhaps the case for an FTC is weak, but any market requires governmental assistance in protecting against fraud, and that assistance is likely to be costly.
The Securities and Exchange Commission (SEC), through its "full disclosure" program (which cost the taxpayer $58 million in 1996), requires publicly traded companies to furnish management, financial, and business information on a regular basis so that investors will be able to make informed decisions. The SEC spent an additional $101 million in 1996 on the prevention and suppression of fraud in the securities market. Oversight of the stock market and commodity futures market cost the American taxpayer $355 million in 1996.
In the absence of government machinery capable of detecting and remedying misrepresentation and false dealing, free exchange would be an even more risky business than it is. The act of buying and selling is often worrisome in the absence of reliable means to counteract the asymmetry of knowledge between buyer and seller. The seller frequently knows something the buyer needs to know. That is one reason why the risk-averse fear commercial exchanges as possible scams, why they cling to suppliers they know personally rather than shopping around for bargains. Public officials can discourage this kind of clinging, promote market ordering, and discourage swindlers by insuring against any damage arising from the asymmetry of information between buyers and sellers. To help consumers make rational choices about where to obtain credit, for instance, the Consumer Credit Protection Act forces any organization that extends credit to disclose its finance charges and annual percentage rate. Just so, consumers benefit from competitive markets in restaurants because, as voters and taxpayers, they have created and funded sanitation boards that allow them to range adventurously beyond a restricted circle of personally known and trusted establishments. The enforcement of disclosure rules or antifraud statutes is no less a taxpayer-funded spur to market behavior than government inspection of food handlers.
The appropriate level of federal spending and government oversight will remain controversial. Nothing said above is intended as a defense of any particular program; some existing programs should undoubtedly be scaled down. What cannot be denied is that enforceable antifraud legislation is a common good, embodying biblically simple moral principles (keep your promises, tell the truth, cheating is wrong). Moreover, the benefits of antifraud law cannot be captured by a few but are diffused widely throughout society. It is a public service, collectively provided, and serving to reduce transaction costs and promote a free-wheeling atmosphere of buying and selling that would be very unlikely to arise if "caveat emptor!" were the sole rule.
Admittedly, the current economic boom in China suggests that, when suitably integrated into the world economy, a society without a strong court system can use kinship and other informal networks to breed credible commitments even in the absence of reliable judicial enforcement of property rights. In most industrialized societies and as a general rule, however, free markets depend on enforceable contract law and a liberal style of governance. To deter fraud, a government must be interventionist and well funded. American taxpayers have proven willing to foot the bill partly because they see the obvious advantages in the monitoring of private exchanges by politically accountable officials.
Government must not only repress force and fraud, invest in infrastructure and skills, enforce stockholders' rights, and provide securities exchange oversight and patent and trademark protection. It must legally clarify the status of collateral. And it must regulate the banking sector and credit markets to prevent pyramid schemes and ensure a steady flow of credit to businesses rather than cronies. The enforcement of antitrust law is also crucial. For the reliable delivery of these public services, markets require government. At the taxpayer's expense, the state must foster innovation, encourage investment, boost worker productivity, raise production standards, or stimulate the efficient use of scarce resources. It can do this, among other ways, by defining property and contract rights clearly, assigning them unambiguously, and protecting them impartially and reliably. The job is neither easy nor cheap.
To do all this, governments need first to collect money through taxation and then to channel it intelligently and responsibly. Rights enforcement of the sort presupposed by well-functioning markets always involves "taxing and spending." Needless to say, the inevitable dependency of markets on law, bureaucracy, and public policy does not imply that government initiatives are always wise or beneficial. As a political community, we have choices—but only among competing regulatory regimes.
Chapter Four
WATCHDOGS MUST BE PAID
IN 1992, THE ADMINISTRATION OF JUSTICE in the United States—including enforcement, litigation, adjudication, and correction—cost the taxpayer around $94 billion. Included in this allocation were funds earmarked for the protection of the basic rights of suspects and detainees. Because it always presupposes the creation and maintenance of relations of authority, the protection of individual rights is never free. True of the rights of property and contract, this also applies to the rights protected within our system of criminal justice, including of course the rights of people who are not in fact criminals. Here again, rights enforcers must be in a position to tell potential rights violators what to do and what not to do. The history of habeas corpus confirms the validity of the thesis that an abusive power can be successfully counterattacked only by another power. Classical liberal rights necessarily depend on relations of command and obedience that, in turn, are expensive to create and maintain. This can be observed clearly in the case of prisoners, whose rights cannot be even minimally protected unless their custodians are monitored from above and penalized for abuses. Although sometimes denounced as a hindrance to law enforcement, protecting the rights of prisoners means nothing more than forcing correctional officers to obey the law. These rights are sometimes controversial, but the basic point—the need to monitor public officials who exercise coercion—is quite general and applies, in different forms, to the rights of the law-abiding as well as of those convicted of crimes.
Protecting prisoners' rights, even quite modestly, is costly. To avoid degrading treatment, prison cells must be ventilated, heated, lit, and cleaned. Prison food must provide minimal nutrition. The Eighth Amendment demands that prison wardens and guards provide minimally humane conditions of confinement. A prison official violates a constitutional right where the deprivation alleged is, objectively, "sufficiently serious" and if he acts with "deliberate indifference" to inmate health and safety. In the federal prison system alone, medical care costs ran to $53 million in 1996. Authorities cannot segregate inmates from the general prison population without using fair procedures. Officials institutionally positioned to penalize flagrant abuses (such as murder or torture) must "monitor the monitors." And to assure access to the appeals process, prison authorities must provide prisoners with "adequate law libraries or adequate assistance to persons trained in the law."
In other words, the right to be treated decently in the system of criminal justice—by police, prosecutors, judges, prison guards, and probation officers—presupposes the power of bureaucratic superiors to punish and deter misconduct by subordinates. Procedures must be established and responsibility assigned for determining the legality or illegality of detention. The enforceable rights of the interrogated are the enforceable duties of the interrogators. The rights of prisoners are the duties of wardens and guards. Protecting rights within the American criminal justice system requires oversight of the law-enforcement apparatus. Whatever their attitude toward red tape, defenders of rights cannot be consistently antibureaucratic, for police and prison guards behave more decently when monitored than when unwatched. And second-level supervisory personnel must be given adequate training and paid a living wage.
The cost of training and monitoring correctional officers is a concrete illustration of the indispensable contribution of the taxpaying community to the protection of individual liberties. True, it is more familiar to style the rights protected within our criminal justice system as purely negative, as rights against the government, as shields from police and prosecutorial and custodial abuse. But attention to the cost of rights should help us focus attention on the other side of the coin, namely on the forms of state action required for rights of suspects and detainees to be a palpable reality rather than a mere paper promise. Nor, it is important to emphasize, are the rights protected by the criminal justice system solely protections of criminals, or even of the wrongly accused. Ordinary citizens depend, for their protection against the state and thus for their so-called negative liberties, on the taxpayer-funded training and monitoring of the police.
Because it involves federal supremacy, the extension of most Fourth, Fifth, and Sixth Amendment protections to individuals suspected, accused, or convicted of crimes within the states nicely exemplifies the positive side of ostensibly negative rights. The government, as the agent of American taxpayers, provides the accused with certain weapons (rights) which, it is expected or hoped, will help reduce improper conduct by officials and even the odds against the occasionally overwhelming power of the prosecution. Thus, the right to a speedy, fair, and public jury trial is an entitlement to a taxpayer-funded benefit or service.
Needless to say, the rights of accused Americans—rich and poor, black and white—are not protected equally. But our criminal justice system would be even more grossly unfair if the community as a whole did not subsidize some basic protections. In the 1996 U.S. budget, covering only federal trials, $81 million went to fees and expenses for obtaining witnesses. The accused does not have to rely on his own resources to compel witnesses to testify in his favor; he is legally entitled to employ resources drawn from the community as a whole. Ability to pay bears no rational relation to innocence or guilt. This, at least, is the Supreme Court's explicit rationale for the right of the indigent accused, even on appeal, to a lawyer whose salary will be paid by the public. Equal protection implies a constitutional right of access to whatever appellate process a state makes generally available. Under existing law, American taxpayers must pay for blood grouping tests for indigent defendants in paternity cases and for psychiatric assistance for indigent defendants in some criminal cases. And to ensure that court-appointed attorneys are not in the pocket of the prosecutor, some sort of independent supervision is obviously required.
Even the right of the accused to be free pending trial presupposes the bureaucratic capacity to set up and administer systems of bail and release on recognizance. Such a right would be unavailable if the state could not perform—that is, if the criminal justice system could not, with relative accuracy, distinguish defendants who will show up for trial from those likely to jump bail, or train its police well enough to conduct a competent investigation without keeping suspects uninterruptedly behind bars.
The duty of the police to refrain from unreasonable searches and seizures is meaningless unless the courts have the capacity to compel the police to comply with the Constitution. This capacity depends importantly on social norms and expectations and on the training and norms of the police, but it also depends on the fiscal wherewithal of the judiciary. Searches must be authorized in advance by warrants issued by neutral and detached magistrates upon proof of probable cause, and the salaries of these nonpartisan judges cannot be manipulated in an ad hoc manner by officials in the other branches of government. The exclusionary rule, barring from trial any evidence gathered illegally, is one way the American judiciary has tried to enforce police compliance or at least to offer constitutional instructions to officers engaged in crime prevention. The exclusionary rule has been gradually softened by exceptions, to be sure. But why has this tendency to diminish the pre-existing rights of suspects and defendants been supported by those who want to be tough on crime? Only because such a rule represents a form of supervisory interference thought to handcuff the police and weaken the fight against crime by permitting police illegality to taint otherwise solid evidence. To erode a right—whether desirable or not—often means impairing a publicly funded supervisory power.
In effect, the rights of the accused and the incarcerated contract and expand as the American judiciary is sometimes more, sometimes less deferential toward the executive branch's war on crime. This oscillation shows, yet again, that the breadth of our liberties depends upon the resolve of our authorities. But it is worth stressing that rights cannot be based on government forbearance, for an even more basic reason. Rights come into being only after a government agency, often a court, makes the effort to define such basic terms as "excessive," "reasonable," and "cruel." The precise scope of our rights changes over time as the courts decide. The court's job is not simply to prevent the executive branch from acting abusively (taking that term as a rough placeholder for what the Constitution forbids). It also has to set down the criteria for distinguishing abusive from nonabusive action. This is an affirmative task it cannot avoid. When is a search or seizure unreasonable? At what point in time does a suspect have a right to counsel—already at the line-up, or only at the preliminary hearing? Under what conditions can officers initiate interrogation? In the criminal justice system, rights always presuppose at least one form of state action because they always assume that the court has given answers, for better or worse, to these and other similar questions. Judicial inaction, a refusal to answer, is not an option.
The Rehnquist Court has reinterpreted and thus reduced many of the rights in criminal procedure established by the Warren Court. It has achieved this end not by flat prohibitions but by its own readings—namely, by drawing distinctions and redefining a handful of essential terms. Even under Warren-era rules, the prosecution was able to introduce at trial evidence that the police, in the absence of a warrant, had found "in plain view." But the Rehnquist Court has enlarged this category by admitting, for example, evidence detected by aerial surveillance using sophisticated cameras. By distinguishing between a mere "stop" and a genuine "arrest" the current Court has also permitted the use of evidence disclosed by police friskings, such as weapons or contraband, that would otherwise have been excluded. It has similarly declared that the "reasonable expectation of privacy" does not cover sealed garbage bags deposited in a dumpster. The Sixth Amendment guarantees an accused person the right "to be confronted with the witnesses against him," but the Court has decided that this right can be waived in cases involving the sexual abuse of children who would be psychologically harmed by having to sit face to face with their presumed victimizer.
Some of these new lines drawn by the Court are quite reasonable, while others seem less so. But this is a side issue; what matters here is that the rights of Americans are creatures of state action. The very scope of our rights against police, prosecutorial, and custodial abuse is established by judicial interpretation, that is, by government performance. The enforcement of these rights by judicial authority over executive-branch officials is merely a secondary illustration of the dependence of individual liberty on state action. The first and most basic way in which publicly funded authorities affect liberty is by defining its scope. The community does not protect any imagined freedoms, but only those which, at any given historical moment, its government, largely through its judiciary, identifies as enforceable rights, and is willing to protect, which is to say fund, as such.
The American system of criminal justice is expensive, in part, because it is designed both to avoid falsely convicting innocent defendants and to prevent lethally armed police officers and prison guards from mistreating even those who are declared guilty. That the costs of these arrangements, indispensable for the protection of basic rights, must be publicly defrayed has theoretical as well as financial significance. Such costs bring into sharp relief the essential dependency of rights-based individualism on state action and social cooperation.
PART II:
WHY RIGHTS CANNOT
BE ABSOLUTES
Chapter Five
HOW SCARCITY AFFECTS LIBERTY
JOSHUA DESHANEY WAS BORN IN 1979. His parents were divorced a year later and his father, Randy DeShaney, remarried soon after he was awarded legal custody of the infant. In January 1982, Randy DeShaney's second wife charged her husband with child abuse, alerting the Winnebago County (Wisconsin) Department of Social Services (DSS) that Joshua's father was beating the boy. Officials from DSS interviewed the father, who denied the charges. In January 1983, Joshua was admitted to a local hospital with multiple bruises and abrasions. Suspecting child abuse, the examining doctor notified DSS. Joshua was placed in the temporary custody of the hospital.
Three days later, after conducting an exam, a team of public officials concluded that the evidence of abuse did not warrant keeping Joshua in public custody. A month later, Joshua was again treated for injuries. A DSS caseworker made monthly home visits during which she observed more head wounds. In March 1984, Randy DeShaney beat his four-year-old son so cruelly that the boy lapsed into a coma. Emergency surgery disclosed hemorrhages caused by recurrent blows to the head. Joshua survived but with severe brain damage, and he is expected to spend the rest of his life in an institution for the severely retarded.
Joshua's mother brought suit on his behalf against DSS, claiming that its failure to provide protection against this sickening brutality constituted a violation of Joshua's fundamental rights under the Constitution. The Supreme Court rejected this claim, asserting that although Joshua's case was undoubtedly tragic, he had suffered no constitutional wrong.
While widely criticized, the DeShaney decision has also found powerful defenders within the American legal community. These defenders divide into two camps. Some echo the Court's own reasoning, alleging that Joshua possessed no constitutional right to state protection. His constitutional rights were not violated because such rights safeguard private individuals exclusively from public officials; they do not entitle people to state protection from their fellow citizens. The Constitution protects individuals from private action only if the government has somehow authorized or encouraged or sponsored the action, or was significantly involved in bringing it about. Since there is no right to affirmative government assistance, and since DSS oversight of child custody cases did not seriously implicate the state in the abusive behavior, no constitutional protection came into play.
Other defenders of this disputed decision take a different line, arguing more pragmatically and not relying on a sharp distinction between negative and positive liberties. Instead of headlining the Constitution's chilly indifference to Joshua's fate, they argue that American courts, for various reasons, cannot effectively manage scarce resources. Instead of alleging that people have no right to affirmative assistance from the state, or that no "state action" was involved, they claim that courts are poorly positioned to make rational decisions about how executive agencies should allocate their budgets and their time. By attending to the difference between these two quite distinct rationales for the controversial DeShaney decision, we can deepen our understanding of the issues raised by the budgetary cost of rights.
DOES THE CONSTITUTION PROTECT AGAINST
PRIVATELY INFLICTED HARMS?
The first line of reasoning, articulated by the Court itself, ignores the issue of costs. The due process clause, the Court declared, operates as "a limitation on the State's power to act, not as a guarantee of certain minimal levels of safety and security." The Court added that "its language cannot fairly be extended to impose an affirmative obligation on the State to ensure that [people's] interests do not come to harm. . . . Its purpose was to protect the people from the State, not to ensure that the State protected them from each other." These few words are rich with implications. Behind this grand pronouncement, in fact, lies a comprehensive theory of negative constitutionalism, which implies the following: the Constitution is designed principally to prevent action by federal authorities. It is a giant restraining order imposed by citizens upon their government. Not the First and Fourteenth Amendments alone, but the Constitution as a whole ties the hands of public officials in order to protect the population from tyrannical rule. That is not only its overriding purpose, but also its almost exclusive purpose.
While constitutional rights hamstring public officials, according to this widely accepted view, they place no constraints whatsoever on miscreants out of office. As a result, the Constitution does not oblige public officials to protect individuals from private force and fraud, and the government's failure to prevent private wrongs is not a form of state action for which officials could be held judicially accountable.
But the text of the Constitution hardly settles the issue. True, the due process clause bans the state from "depriving" people of life, liberty, or property, but to know whether the state has "deprived" anyone of anything, we need to know what people are entitled to have. If "liberty" includes a right to police protection, then the state deprives people of "liberty" when it fails to provide police protection. If "liberty" includes freedom from private brutality or intrusion, then the state deprives people of "liberty" when it allows people to be subject to private brutality or intrusion. The text is therefore inconclusive. Or suppose it is agreed that the Constitution does not protect people from private action; how much follows from this? Even if the Constitution does not protect people against private acts, it may impose a duty on the state to protect private people against private intrusions. The fact that the Constitution applies largely or even exclusively to "the state" does not eliminate this possibility.
Indeed, it is not hard to think of constitutional rights that oblige state action to protect individuals from privately inflicted harms. If a state decided not to protect your property against private trespassers—if, in other words, the state repealed, selectively or entirely, the laws of trespass—a serious question would arise whether the state had "taken" your property by failing to protect you from private trespassers. To "own" property is to have a right to exclude others, and if a state will not affirmatively help you to exclude others, it may well, under existing law, have taken away what you own. Thus, the right to private property may entail a right to government protection via the trespass laws. Or consider the right to contractual liberty. The Constitution protects people against state impairments of contractual obligations. If a state refused to make its courts available to enforce certain contracts, it would probably be taken to be "impairing" contractual rights. The contracts clause therefore has a positive dimension too, insofar as it guarantees an affirmative right to the use of courts (and government resources) to protect contractual guarantees.
Even those who insist that constitutional rights protect citizens exclusively from public authorities—and not at all from each other—are likely to admit that the Thirteenth Amendment is a graphic exception. "Neither slavery nor involuntary servitude . . . shall exist within the United States" prohibits a form of traditionally private behavior. In a way, the ban on slavery can be read as a straightforward ban on private enslavement; it can also be read as a directive to the government, ensuring that government will not permit involuntary servitude.
Other examples of such an obligation are legion. What if Jones sues Smith to enjoin a threatened assault and then Smith bribes the judge, who accordingly rules for Smith? In this case, under existing law, Jones's rights have been violated because public officials failed to protect him. And that is only the beginning: in many cases, the government's involvement with private actions has been deemed sufficient to trigger constitutional constraints, even if it appears that private individuals are asking for state help against other private individuals.
Racially restrictive covenants between private buyers and sellers can be challenged under the equal protection clause because private contracts are hollow unless the government makes its full coercive powers available to enforce them. The use of those coercive powers raises a serious constitutional problem, even in the context of an apparently private real estate deal. The Fourteenth Amendment prohibits a private lawyer from using his peremptory challenges to eliminate jurors on the basis of race; the involvement of the justice system brings the Constitution into play. Political parties, which belong to civil society not to the state, are constitutionally banned from conducting primaries in a racially discriminatory fashion. Because government is so directly involved in its operation, the First Amendment limits the freedom of Amtrak, a nominally private corporation, to quash artistic expression at Penn Station. The Fourteenth Amendment prohibits racial discrimination by a private restaurant that rents space in a municipal parking garage. Prison authorities can be sued under the Constitution for injuries inflicted on one prisoner by another if these authorities demonstrated serious indifference to the inmates' well-being.
Outside of the constitutional context, an affirmative obligation of government to protect private citizens from each other is a logical consequence of ordinary rights enforcement. Union members have a right to report the unscrupulous conduct of union officials. But this right is effectively meaningless unless the government visibly protects whistle-blowers from violent reprisals. Indeed, since the enforcement of rights always creates "losers," the affirmative duty of the government to protect "winners" from acts of private retaliation is a necessary correlate of every right. A battered wife has a perfectly well established legal right to report abuse. But what if her husband carries a firearm? In that case, her right will be a cruel sham unless the city government has spent tax dollars on such protective measures as shelters for battered women. The individual's right to testify is likewise hollow unless the government takes upon itself the (costly) obligation of protecting witnesses from retaliation. The $23 million that the Department of Justice spent in 1996 on witness protection programs can be understood in this light. To enforce rights consistently, public authorities must also bring the full force of the law down upon private individuals who try to inflict physical injury upon other private individuals simply because the latter are exercising their rights. This is yet another way in which personal liberty presupposes active government performance—and yet another reason why rights have costs.
Thus, it does not suffice to declare, in a blanket fashion, that American governments, federal and state, are under no "affirmative obligation" to protect American citizens. The Constitution was not designed to wash the government's hands; nor is that an appropriate role for the Supreme Court. It certainly seems reasonable to say that once welfare officials became aware of the abusive behavior of Joshua's father, they were legally obliged to do something about it. If such an obligation existed, then the boy's rights were violated by the state's action and inaction. At the very least, this sort of ruling cannot be precluded by the curious claim that the American government is never under any legal obligation to protect American citizens. A Supreme Court ruling, after all, is not only the disposition of a particular case; it also broadcasts a message to the public about the basic purpose and meaning of the American social contract. Evaluated in this light, the line of reasoning in DeShaney is seriously flawed.
The theoretical importance of the case, however, lies in its lessons for the "absoluteness" of rights. Might the Supreme Court have been arguing more narrowly that Joshua's rights were not absolute, because they were subject to budget constraints?
AN ARGUMENT FROM SCARCITY
The second, more pragmatic argument does not reject the view that Joshua had some sort of right to state protection, but simply takes costs, in the sense of competing goods, into account. Although this reasoning was not emphasized in the majority opinion in DeShaney, it almost certainly influenced the outcome in the case, because it supplies the simplest and surest route to that outcome. Rights enforcement often does not depend on courts alone. To remedy past rights violations and deter future rights violations, courts must rely on willing cooperation from government agencies, which, in turn, necessarily operate within stringent fiscal and other constraints. In the context of social services, the problem is clear. To deal with potentially boundless problems, departments of social services are endowed with embarrassingly bounded resources, and they must allocate the scant means at their disposal, using their detailed knowledge of the situation on the ground, as they judge most effective. Hard budget constraints imply that some potential victims of child abuse will become actual victims of child abuse, and the state will have done little or nothing about it. This is deplorable, but in an imperfect world of limited resources, it is also inevitable. Taking rights seriously means taking scarcity seriously.
Courts are not well positioned to oversee the tricky process of efficient resource allocation conducted, with more or less skill, by executive agencies, nor are they readily able to rectify past misallocations. Judges do not have the proper training to perform such functions and they necessarily operate with inadequate and biased sources of information. This is why, under American law, Federal Aviation Administration (FAA) agents generally cannot be sued for their unlucky choice of which civilian aircraft to inspect in which sequence, for the courts obviously cannot take upon themselves responsibility for planning the work schedules of government personnel. Faced with a particularly pressing problem, how can a judge measure its urgency compared to that of other social problems competing for governmental attention, and about which he knows virtually nothing? How can judges, in deciding a single case, take account of annual ceilings on government spending? Unlike a legislature, a court is riveted at any one time to a particular case. Because they cannot survey a broad spectrum of conflicting social needs and then decide how much to allocate to each, judges are institutionally obstructed from considering the potentially serious distributive consequences of their decisions. And they cannot easily decide if the state made an error when concluding, before the fact, that its limited resources were most effectively devoted to cases A, B, and C, rather than to case D—even if it turns out that case D involved a calamity like Joshua DeShaney's. (Perhaps cases A, B, and C were also disasters.)
While judges may be perfectly competent to spot egregious violations of rights and even to invalidate egregious misallocations of resources, they cannot intelligently decide, in most such cases, when imaginable remedies are better channeled to other pressing needs. From this perspective, the DeShaney case is most charitably understood not as a dramatic pronouncement that the American government owes no protection to American citizens, but rather as a sober recognition that rights have costs, that funds for the protection of the entire array of legal rights must be drawn from the same inevitably limited budgets. In cases of this sort, courts should be very hesitant to substitute their own judgment for that of executive agencies. Courts cannot easily participate in the job of priority-setting and the optimal distribution of scarce resources that the plaintiff in DeShaney called upon them to undertake.
This is a fairly plausible defense of the general approach in DeShaney, although not a convincing justification of the particular outcome. The evidence of prior knowledge by state authorities was sufficient to implicate them in the brutal deed, and the abuse was so grave, and so likely, that the modest expenditure that would have been required to prevent it could have been constitutionally mandated without creating an imperialistic judiciary liable to substitute its judgment everywhere for that of the executive branch. But the real importance of the case lies in the opposition it raises between a (false) claim that the Constitution creates only negative rights, and a (true) claim that courts are not in a good position to assess claims that involve resource allocation.
What the two rival rationales for the decision show is that the understanding of basic rights, and therefore of the relation of the judiciary to the other branches of government, depends on a prior choice either to ignore costs or to take them into account. In its opinion, the Court paid no heed to the question of scarce public resources. It could justify the state "inaction" it wished to defend as such only by declaring that a child beaten horribly after having been consigned to his cruel father's custody by court order and while under the government's custodial supervision suffered no violation of his basic rights. The result was one of the most shockingly brutal opinions of modern Supreme Court history. Shockingly brutal and altogether unnecessary. For a narrower and more reasonable justification, based partly on cost, was readily at hand, involving the nonabsolute character of rights that depend on expenditures. The DeShaney decision thus provides a powerful incitement to explore more deeply the limits that fiscal constraints necessarily impose, and should impose, upon the proper sphere of judicial decision-making.
RHETORIC AND REALITY
Rights are familiarly described as inviolable, preemptory, and conclusive. But these are plainly rhetorical flourishes. Nothing that costs money can be an absolute. No right whose enforcement presupposes a selective expenditure of taxpayer contributions can, at the end of the day, be protected unilaterally by the judiciary without regard to budgetary consequences for which other branches of government bear the ultimate responsibility. Since protection against private violence is not cheap and necessarily draws on scarce resources, the right to such protection, presuming it exists, cannot possibly be uncompromisable or complete. The very same is true of more familiar individual rights to protection against government abuse. For instance, my right to compensation for the taking of my property under the eminent domain power is worthless if the Treasury is empty and unable to pay. If rights have costs, then the enforcement of rights will always be sensitive to the taxpayer's interest in saving money. Rights will regularly be curtailed when available resources dry up, just as they will become susceptible to expansion whenever public resources expand.
Rights are relative, not absolute claims. Attention to cost is simply another pathway, parallel to more heavily traveled routes, to a better understanding of the qualified nature of all rights, including constitutional rights. It should be a useful supplement to more familiar approaches, not least of all because the conventional cost-blind theory of rights has reinforced a widespread misunderstanding of their social function or purpose. Attention to the costs of rights reveals the extent to which rights enforcement, as actually carried out in the United States (and elsewhere), is shot through with trade-offs, including monetary trade-offs. This does not mean that decisions should be made by accountants, only that public officials and democratic citizens must take budgetary costs into account.
Public finance is an ethical science because it forces us to provide a public accounting for the sacrifices that we, as a community, decide to make, to explain what we are willing to relinquish in pursuit of our more important aims. The theory of rights, if it hopes to capture the way a rights regime structures and governs actual behavior, should take this reality into account. Courts that decide on the enforceability of rights claims in specific cases will also reason more intelligently and transparently if they candidly acknowledge the way costs affect the scope, intensity, and consistency of rights enforcement. And legal theory would be more realistic if it examined openly the competition for scarce resources that necessarily goes on among diverse basic rights and also between basic rights and other social values.
Chapter Six
HOW RIGHTS DIFFER FROM INTERESTS
RIGHTS ARE SOMETIMES DESCRIBED AS MORALLY CHARGED and almost irrebuttable claims, to be sharply distinguished from everyday assertions of interest. Whereas interests are always a matter of more or less, thereby implying trade-offs and compromises, rights are a matter of principle, demanding a kind of clinched, unblinking intransigence. At least that is the way many legal theorists and human-rights advocates tend to speak. A similar viewpoint has been memorably articulated by Ronald Dworkin—a leading American theorist of rights—who, in an evocative phrase, portrays rights as "trumps" that can be played in court against government officials.
This metaphor captures an important aspect of American legal reality. Although no right can flatly override all other considerations, rights can nevertheless qualify as "absolute" in a limited sense. When basic rights are at stake, the government cannot casually invoke mundane considerations as justification for non-enforcement. Legal theorists are only following popular preconceptions and ordinary language, then, when they conceptualize rights as claims qualitatively distinct from mere assertions of interest. Extenuating circumstances (such as exorbitant costs or scarce administrative capacities) may easily excuse the government from protecting a mere interest. But these same considerations will excuse the failure to protect a right only under special and highly restricted conditions.
Dworkin has frequently acknowledged the need to balance one right against another and also the occasional necessity of curtailing otherwise important rights in the name of competing social values of sufficient urgency. Rights cannot be overridden by invoking general utility, he writes, but "a state may be justified in overriding or limiting rights on other grounds" and "the most important . . . of these other grounds invokes the notion of competing rights that would be jeopardized if the right in question were not limited." Freedom of the press may perhaps be restricted by the right to privacy or freedom from malicious libel. Contrariwise, freedom of the press can be expanded by contracting the right to sue for libel. The right to engage in collective bargaining requires the legal abolition of the right to make yellow-dog contracts, whereby workers once "voluntarily" agreed not to join a union. And so forth.
The curtailing of civil liberties to combat terrorism is unquestionably lamentable, but such trade-offs have been made in the past and will no doubt be made again. Although it should have done so, "strict scrutiny" did not in fact prevent the Court from giving its blessing to the flagrantly discriminatory internment of Japanese-Americans in World War II. And there is little guarantee that similar infringements will not occur when pertinent reasons arise that again seem convincing to judges.
The need for swift governmental action is a commonly accepted rationale for overriding important rights. For example, property can be seized without prior notice (an action that would ordinarily violate due process of law) if a shipment of pharmaceuticals has been dangerously adulterated or if a vehicle transporting contraband is about to escape the grasp of the police. Freedom of information can be restricted, or defined in a limited way, on the grounds not only of national security but also to protect sensitive data about government personnel. Under emergency conditions, freedom of movement can be legally curtailed to prevent the spread of highly contagious fatal diseases. And the right to ride a motorcycle without a helmet can be abolished, partly because of the medical and rehabilitation costs such activity imposes on the community as a whole.
A large part of lawyering involves discovering judicially acceptable excuses for actions or omissions that would otherwise be deemed unlawful or unacceptable. As the category "excusable homicide" suggests, even the most socially unacceptable behavior can be justified, as a matter of law, in special circumstances (such as self-defense). Mitigating factors can be invoked to justify governmental as well as private action. What the rights-as-trumps view implies is only that a government that curtails civil liberties must persuasively invoke important public interests. To violate central constitutional values, the state should have even weightier values on its side.
But while the rights-as-trumps view is perfectly at home with the notion that rights occasionally clash with other rights and with other public interests as well, so that judicial balancing is often required, it neglects the idea that rights cannot be absolute because their enforcement depends on the timely delivery of limited public money to the agents charged with enforcing them. Some conflicts among rights stem from a common dependency of all rights on limited budgetary outlays. Financial limits alone exclude the possibility of all basic rights being enforced maximally at the same time. Rights invariably demand or imply trade-offs of a financial sort. And expenditure patterns will to some extent be determined politically. Attending to costs helps explain why property rights clash with property rights, why the local police department cannot protect Jones's dilapidated home adequately if it has already committed its sole stake-out team to guard Smith's luxurious estate.
To be sure, some basic rights, such as freedom of speech or the right to vote, may not be bought and sold on the open market; the ban on trading political rights is designed partly to ensure that political power is not concentrated in any individual or group. So rights are not commodities in a simple sense. But when the price soars, rights enforcement necessarily becomes more selective. We can obtain costly goods and services only by relinquishing something else of value. The world of value is complex and the world of available possibilities is larger than the world of co-available possibilities. There is nothing cynical or degrading about admitting as much or acknowledging that this pattern applies to basic rights as well as to ordinary commodities. Of course, it does not follow that rights must be tossed along with everything else into a gigantic cost-benefit calculating machine created and operated by economists.
Although it is theoretically misleading to portray rights as absolutes, such a description can be defended as psychologically and rhetorically useful. Civil libertarians, like politicians, used-car salesmen, and advertising executives, are keenly aware that exaggeration has a mnemonic function, and they know by experience that their uncompromising phraseology often pays off. Hyperbole can draw special attention to what they see as crying needs, thereby increasing the chance that citizens and representatives will treat certain interests with exceptional sensitivity and seriousness. Perhaps a (misleading) emphasis on the absolute character of free speech will stiffen the spine of citizens and representatives when the pressure for (unjustified) censorship is especially great. But overstatement can create problems too, and an insistence that rights are absolute may lead to the over-protection of some rights to the detriment of others that have an even greater claim. And since political attention, too, is a scarce resource, the more time officials lavish on one claim, the less time they have for another.
Defeasibility is an inescapable characteristic of all legal rights, including constitutional rights. Another important reason, apart from costs, why legal rights must always be subject to curtailment or limitation is also worth revisiting: rights are, in reality, legal powers that can be exercised over others. Powers can always be misused. Rights must be subject to restrictions in order to prevent their exploitation for wrongful ends. For instance, the right to self-defense is well established in American law, but it is justifiable only because, or to the extent that, courts keep an eye out for its abuse. You cannot claim to have acted in self-defense, for example, if you were not seriously endangered. Similarly, the rights of a stockholder to sue a company's management can be used to harass and eventually to obtain a handsome bribe for dropping the case. The possibility of abusive suits must be taken into account by legislators and judges who determine the conditions under which the right to sue fails. The American legal system makes continuous remedial and compensatory adjustments to handle the unintended side effects that necessarily occur whenever the government hands individuals the discretionary right to wield the public power, to dip into the public purse.
But—it will be asked—are not some human interests intrinsic and not merely instrumental goods? While some things are valuable merely as means, are not other things good in themselves, because of the good things that they, on balance, bring into being? True, freedom of speech serves to improve the quality of public decision-making and to reduce the level of government corruption. But is it not also valued for its own sake, simply because censorship is an indignity, an insult to human autonomy? The answer is yes: some interests do have intrinsic value. But even intrinsic goods have costs; they cannot exist without public effort and a substantial expenditure of resources. Protecting rights that are valued for their own sake will entail dangers, downsides, unintended side effects, opportunity costs, and other troubles, for there are few gains without losses. Thus, the right to a hearing serves dignitary functions and is not designed just to ensure accurate fact-finding. But if it is very expensive to hold elaborate hearings, government may not be required to hold elaborate hearings. And the visitation rights of grandparents on the side of the noncustodial parent may seem "sacred" in a way, and certainly such rights are not of merely instrumental value; but such rights are regularly obliterated in American jurisdictions in cases of adoption, out of concern for the countervailing interests of the child.
Indeed, the rights of Americans are constantly expanding and contracting under the impact of legislative and adjudicative action. Rights are interests that, politically and judicially, are highly valued at that moment; but they are not merely that. Within American legal culture, rights are interests of a special kind. Attention to the cost of rights does not render meaningless the fundamental liberal distinction between interests and rights. "Rights talk" is essential because it raises the threshold of justification for interfering with interests deemed especially important.
When rights are at issue, some arguments are not merely insufficiently weighty but altogether inadmissible. This is true in private law as well as constitutional law. The debtor cannot legally refuse to pay his debt because his creditor is an apostate, although he can refuse to pay, under certain conditions, if the product he received proves defective. Analogously, our system of religious liberty does not allow government to suppress a minority's religious practices because the minority's god is not the true God, although it can ban the consumption of hallucinogens in specific contexts. Our system of political liberty does not deprive people of the vote because incumbents fear how people will vote. Our system of free expression does not allow government to regulate ideas simply because officials or citizens think those ideas are wrong or dangerous, but it can regulate them for other reasons. And once we identify the category of permissible and impermissible reasons for action in any particular system, we are well on our way toward understanding what rights, as interests of a special type, mean in practice.
For example, the Winnebago County DSS could not justify its failure to protect a child from his father's brutality by invoking racial or religious considerations. It could not say, "We protect white children but not black children." Whether or not the Constitution obliges the government to protect individuals from private harms, its use of such a justification would have been, without any question, absolutely forbidden. Similarly, a court cannot deny child custody to a divorced white mother simply because she is now cohabiting with a black man. That justification for state action is blocked. America's rights regime is "absolutist" in this sense: it rules out certain reasons unconditionally while proscribing actions and inactions only conditionally.
Formulated differently, rights are regulatory, not prohibitive. American courts do not ordinarily defend constitutional rights simply by barring government actions as unlawful. What courts do, rather, is to require that the level or branch of government involved provide legitimate and substantial grounds for restrictions imposed and actions undertaken or omitted. This is one way that the American judiciary contributes to democratic accountability—compelling legislative and executive authorities, whenever they infringe upon the interests currently denominated as rights, publicly to articulate the legitimacy and importance of the goals they are pursuing and the appropriateness of the means they select. Rights rule off-limits certain justifications for action or inaction.
To avoid misinterpreting rights as un-overrideable vetoes blocking the path of policy, we could choose to emphasize the perennial need to balance among conflicting interests. But the "balancing" metaphor is just as misleading as the vague notion that rights are absolutes. If all rival claims must be weighed against one another, then claims of right are not essentially different from claims of interest. But this is a simplification, for when a right is in play, government cannot justify non-enforcement simply by claiming that some discernible interests lie on the other side.
This is a familiar phenomenon in daily life. If a friend tells you something in confidence, you may breach the confidence if doing so is necessary to save that friend's life; you may not blab simply because it is fun to gossip about your friend's problems. If a friend is getting married, you may regretfully decide not to attend the wedding, perhaps, if your child is sick and you cannot find substitute care. But you may not decline simply because there is a swell sitcom on television at the hour of the ceremony. Our ordinary decision-making is routinely based on the exclusion of certain reasons as utterly irrelevant, rather than merely unimportant. So outside the law, decision-making is touched by "absoluteness" in this sense and is not merely a matter of balancing.
The same is true of decisions made in the legal sphere. The law's elevation of a certain subset of interests into legally enforceable rights usually deletes, for the time being, certain justifications from the menu of acceptable reasons for interfering with them. To the extent that certain justifications are inadmissible, the right does indeed work, for restricted purposes, in absolutist fashion. But because more persuasive justifications always remain admissible, rights never qualify as uncompromisable when the would-be rights violator comes up with legitimate and sufficiently weighty grounds for neglecting them. Scarcity of resources is a legitimate, however regrettable, reason for failure to protect rights. The two rationales for DeShaney, even though neither is convincing in the end, provide a useful illustration of this powerful truth.
AMONG CONSTITUTIONAL RIGHTS, freedom of speech is one of the most precious. It is worth protecting even, or rather especially, in extreme circumstances, for free speech makes it much more likely that the violation of other rights will be reported. Alongside its many psychological, moral, artistic, religious, and economic functions, liberty of expression is an essential precondition for democratic self-government. It helps ensure political accountability, mop up governmental corruption, drag into the daylight abuses of power, and improve the quality of policy-making by enlisting suggestions and criticisms from specialists out of office as well as from the public at large. In less-developed countries, freedom of speech can even help prevent famines. This is why freedom of expression and communication is sometimes described as the liberty on which all other liberties depend. No surprise that free speech has a special place in American legal culture and has been frequently styled as uninfringeable.
Nevertheless, like other forms of public behavior—which always entails the risk of mutual harm among private individuals and groups—speech is regulated every day, and with good reason. A right is a power, and any power can be misused. Americans would certainly be worse off if the U. S. government dealt with free speech as if it were untouchable. There are (reasonable) laws on the books restricting perjury, attempted bribery, price-fixing, fraudulent and misleading commercial advertising, child pornography, conspiracy, threats to assassinate the president, and many other forms of speech. Not even free-speech purists favor abolishing all such restrictions in the name of individual freedom and autonomy. In practice, doctrinaire extremists in this area are merely trying to shift, usually relatively slightly, the line that political and judicial authorities have drawn when regulating communication and expression. Those who claim that they are "free-speech absolutists" do not really mean it. Some constraints on speech are merely common sense, even in a nation strongly committed to freedom of expression. We would be less free if freedom of speech were treated as a preemptory claim immune to regulation, even when other important interests or rights are in jeopardy.
But what principles help us separate constitutionally protected speech from constitutionally unprotected speech? Constitutional lawyers have been extraordinarily creative in elaborating such principles. But in the United States, whenever the right to free speech is widely perceived to have socially unacceptable consequences (including the undesirable social costs of perjury and the other illicit speech acts listed above), this right is abridged without much embarrassment. Freedom of expression can and will be compromised when the side effects of the unlimited exercise of this right are perceived to be exceptionally harmful. Some such infringements are morally dismaying, but others are not, and in any case they are politically unavoidable. Freedom of speech will be intruded on when, in the eyes of the judiciary, the reasons for doing so have sufficient legitimacy and weight, and less drastic means are not readily (which may mean inexpensively) available. Conversely, a constitutional right prevails when publicly and judicially acceptable justifications for intruding upon it cannot be found.
The controversial issue of flag burning illustrates the point. The government cannot regulate flag burning on the ground that public officials hate protesters, or believe that this is an especially heinous and unpatriotic act, or fear that many people will be upset by an expression of outrage against the symbol of the country's nationhood. But government can regulate flag burning on the neutral grounds of protecting private property from destruction. Freedom of speech is classified as a precious right rather than an ordinary interest because of the sharply restricted conditions under which it can be compromised.
At the time when the First Amendment was adopted, relatively few of its framers had a particularly radical idea of free speech. Most of them agreed that orderly government was an inherently fragile and vulnerable creation that must be protected, in certain circumstances, from the potentially corrosive power of words. Certainly the framers did not intend to ban regulation of anything that could come from a mouth or a pen. There is much dispute about what the framers particularly believed, but no one can deny that the current conception of the free speech principle is far broader than the understanding held by its authors. The meaning of free speech in the United States began to evolve in the 1790s and has been developing ever since. Its scope, at any given time, has always depended upon changing interpretations by a changing Court. Today, spending money to elect a candidate is a form of constitutionally protected free speech, whereas burning one's draft card is not. There is nothing inevitable about this dispensation; right or wrong, it is literally a matter of interpretation.
Today, the government generally may not punish speech because people are offended by the ideas that it contains. Some individuals and groups may be grievously offended by the ideas expressed in a communist tract. But even if the moral injury is large—even if people become suicidally depressed from prolonged exposure to offensive ideas—offense ordinarily does not count as a legitimate basis for public action, at least not in the United States. In the context of speech, outrage at the content of expressed ideas is flatly excluded as a ground for governmental regulation. Whatever the consequences, offense is usually an unacceptable reason for restricting speech. Even the controversial restrictions on sexual harassment in the workplace are justified as a way of preventing employment discrimination, not offense.
Freedom of speech implies far more than a right against direct censorship of disfavored opinions. Every tyrant knows that he can effectively stifle annoying public protests, even without explicitly banning expression as such, simply by cordoning off arenas where demonstrations and rallies are likely to be staged. Hence the right to free speech, protected under American law, includes a right of access to public forums and, as a logical consequence, a right to ensure that certain public places—such as public streets and parks—are kept open and available for expressive activity.
In this particular way, freedom of speech does not simply require that the government adopt a hands-off approach, for maintaining open public spaces will ordinarily entail nontrivial public expenses, presupposing a degree of compulsory taxing and spending. The right to set up a soapbox and enter a publicly subsidized space where listeners can gather and supporters parade imposes costs on some citizens for the benefit of others. Indeed, the Supreme Court has strongly suggested that the government cannot charge the immediate users of freedom of speech, such as protesters in a public park, for the expenses for speech-related activities. All taxpayers, including those who are not especially free-speaking or interested in protest, must pay. Strollers do not need to purchase tickets to walk around in most public parks. Similarly, legal rights are subsidized by taxes levied on the community at large, not by fees paid by the individuals who happen to be exercising them at the moment. Because this is a necessary, not an accidental arrangement, redistribution in the field of rights protection seems to be inevitable.
The implications may be profound, for in stark contrast to its reasoning in the DeShaney case, the Court has indicated that government subsidies, in the free speech context, may well be constitutionally required. How could the Court distinguish the cases? Perhaps what it aims to say is that freedom of speech, properly understood, means that publicly subsidized expressive arenas must be assigned a high budgetary priority no matter what other claims are competing for community resources. That may be implied by classifying freedom of speech as a right rather than merely an interest of American citizens. But if this is the Court's point, its cost-free style of argumentation prevents it from formulating its conclusion in terms clear enough to invite constructive criticism, or from elucidating its deeper assumptions and spelling out the wider implications of its approach.
Chapter Seven
ENFORCING RIGHTS MEANS
DISTRIBUTING RESOURCES
THE RIGHT TO VOTE is no more costless than any other right. Putting aside all private expenditures for political campaigns, the 1996 elections probably cost the American taxpayer somewhere between $300 million and $400 million. Of course, accurate nationwide statistics are difficult to come by. This is partly because almost all of the public costs of running elections are borne by states and municipalities. Federal spending is minimal. State taxpayers pay the costs of printing ballots, registration materials, and voter guides, while municipal taxpayers defray the expenses of staffing and maintaining polling stations. Voting booths must be kept in working order, bans on advertising near the polling stations must be enforced, and vote fraud must be deterred and detected. (Running a mayoral election, it should be noted, costs a city no less than running a senatorial or presidential election. Once the initial investment in holding an election has been made, the additional costs of adding more candidates and ballot initiatives is minimal.)
As the legal philosopher Hans Kelsen once remarked, "to the citizen's right of vote corresponds the duty of the election officer." And that election officer, he might have added, will ordinarily be paid. Polling stations must be opened in various locations, geographically distributed to give approximately equal access to all voters. Under certain conditions, states are constitutionally obliged to make absentee voting procedures available to inmates awaiting trial or convicted of misdemeanors. And local and state governments must use general tax revenues to put into place all the preconditions for fair elections, since they cannot condition the right to vote on the payment of an individualized poll tax or user fee. Such a governmentally managed subsidy is necessarily redistributive.
Perhaps because the costs of elections vary so greatly from city to city, state officials seem strangely reluctant to engage in a thorough accounting. Available figures are nonetheless suggestive. In Massachusetts, a state law passed prior to the 1996 presidential elections mandated longer hours for polling stations. Implementing this tiny amendment to the law cost Massachusetts taxpayers $800,000. In California, where a study of electoral expenses was commissioned by the state government, the cost of any statewide election (whether presidential, senatorial, gubernatorial, etc.) runs around $45–50 million. This is also true for any referendum requiring a separate ballot. Printing and mailing costs for voter guides alone, including those printed in Spanish as well as English, can range from $3 million to $12 million. In California, the cost per voter is estimated to run from $2 to $5, depending on each municipality's voting system.
Today, the right to vote would be unconstitutionally infringed if courts were not permitted to outlaw impermissible racial gerrymandering. The money for such remedial activities and, more generally, for organizing and carrying out free and fair elections is extracted from both willing and unwilling taxpayers, from voters and nonvoters alike. Voting would be a very different act, would bear a very different social meaning, if voters alone had to pay a fee to defray the public costs of conducting an election, instead of all taxpayers having to pay. That a modest form of compulsory redistribution is involved is obviously not an argument against the right to vote. Indeed, we are so used to the taxing and spending presupposed by representative government that we simply take it for granted.
If both the right to free speech and the right to vote require public expenditures, presuppose redistributive decisions, and are relative rather than absolute goods, the same is likely to be true of other rights as well. The Fourth Amendment confers protection against unreasonable searches and seizures. It obliges the government to perform a service that can, under some conditions, be extremely expensive—namely, to monitor police behavior accurately and to deter misbehavior by a fair but also swift and reliable system of punishment. And if citizens are to hold police officers accountable for their actions, they must also finance the procedural protections that accused officers, too, deserve. As a practical matter, resources extracted from the taxpayer will have to be targeted to ensure that lethally armed police officers neither behave unlawfully nor are falsely convicted of behaving unlawfully. Private liberty depends on the quality of public institutions.
THOSE WHO ACCLAIM RIGHTS as trumps sometimes also construe them as barriers defending the most cherished individual interests against a repressive or meddlesome community. Individuals invoke their rights to fend off the majority. Rights protect individuals from mob rule. There is some truth to this antimajoritarian idea. We are all familiar with the lone dissident fighting for his freedom to engage in nonconformist speech and the religious outsider seeking to practice her religion despite majority bigotry and intolerance. But are rights adequately described as claims that the solitary individual raises against the community in which he or she was born and bred? The idea that rights are erected against the community is obviously too simple, for rights are interests on which we, as a community, have bestowed special protection, usually because they touch upon "the public interest"—that is, because they involve either the interests of the collectivity as a whole or the fair treatment of various members of the community. By recognizing, protecting, and financing rights, the collectivity fosters what are widely construed to be the deeper interests of its members.
Property rights encourage individuals to improve their property by allowing owners to capture the benefits of improvement. This arrangement is a social one created for social purposes; it has a perceptibly positive effect on a nation's real estate and capital stock. Other seemingly individual rights are likewise collectively conferred, designed, reshaped, interpreted, adjusted, and enforced to promote what are widely seen as collective interests. They are protected by public institutions, including legislatures and courts, for collective reasons. Admittedly, and importantly, rights may operate in some sense "against" the collectivity once they are vested in individuals. Government may not confiscate property simply because a majority wants to do so. But even in such cases, rights are guaranteed in the first instance both by and for the collectivity. Since it has no existence apart from the individuals who compose it, a collectivity can define, confer, interpret, and protect rights only if it is politically well organized and only if it can act in a coherent manner through the instrumentality of an accountable government.
Arguing that rights serve collective purposes, the philosopher Joseph Raz remarks, "If I were to choose between living in a society which enjoys freedom of expression, but not having the right myself, or enjoying the right in a society which does not have it, I would have no hesitation in judging that my own personal interest is better served by the first option." The right to free expression benefits individuals largely because of its social consequences: diminishing the risk of ill-considered government action, promoting scientific progress, encouraging the dissemination of knowledge, and ensuring that government oppression or abuse will sometimes be met by clamorous protest. Individuals in a society without free speech suffer most from what the lack of freedom does to that society. So, too, are both individual and social welfare promoted by the rights to a fair trial, freedom from unreasonable searches and seizures, and freedom of religion. In all these cases, the relevant right helps secure goods for many people beyond those who personally assert it at the moment. This is one reason why most rights are funded out of general revenues rather than by narrowly targeted user fees.
Chapter Eight
WHY TRADEOFFS ARE INESCAPABLE
WITH THESE WORDS, President Franklin D. Roosevelt proposed a Second Bill of Rights in 1944:
We have accepted, so to speak, a second Bill of Rights under which a new basis of security and prosperity can be established for all—regardless of station, race, or creed.
The right to a useful and remunerative job in the industries or shops or farms or mines of the Nation;
The right to earn enough to provide adequate food and clothing and recreation;
The right of every farmer to raise and sell his products at a return which will give him and his family a decent living; . . .
The right of every family to a decent home;
The right to adequate medical care and the opportunity to achieve and enjoy good health;
The right to adequate protection against the economic fears of old age, sickness, accident, and unemployment;
The right to a good education.
HALF A CENTURY LATER, people all over the world are still debating what rights belong in the Constitution. Should a Constitution, for example, protect the right to social security? How should we understand the rights to housing, welfare, and food? Should there be a constitutional right to employment? Roosevelt's detractors scoff at his attempt to put such "rights" on the same footing as the classical freedoms from government interference. They strenuously object to the very idea of constitutionalizing such rights, even though the International Covenant on Economic, Social, and Cultural Rights (adopted by the United Nations in 1966), which has been copied verbatim into many new post-communist constitutions, does treat minimal social and economic guarantees as if they were equivalent to civil liberties and political rights.
It is familiarly said that welfare rights and other social and economic guarantees are aspirational or open-ended. There is never a point at which they are completely protected. This characterization is correct, but it should not be made with the assumption that old-fashioned rights, such as freedom from unreasonable searches and seizures or police brutality, are fully enforceable. Those who object to welfare rights because they cost money should not assume that property rights can be fully safeguarded, for the conventional contrast between aspirational welfare rights and limited property rights does not survive scrutiny. Our freedom from government interference is no less budget-dependent than our entitlement to public assistance. Both freedoms must be interpreted. Both are implemented by public officials who, drawing on the public purse, have a good deal of discretion in construing and protecting them.
The argument that poor nations can afford the first generation but not the second generation of rights is not wholly misdirected, but, as stated, it is far too simple. If first-generation rights are taken seriously, and if they turn out to be quite expensive, truly poor nations cannot afford them either. They cannot ensure that a right to a fair trial is always respected in practice, just as it is not always respected in poor neighborhoods in the United States, notwithstanding this country's historically unprecedented wealth. All rights are open-ended for the simple reason that rights have costs and hence can never be perfectly or completely protected. All rights are aspirational.
Should nations—whether poor or rich—constitutionalize social and economic guarantees? This is not only a philosophical question about the essential nature of rights as such, but also an acutely pragmatic one, raising issues of institutional competence and also of public finance that should be decided by considering available resources, predictable side effects, and competing goals. Philosophical arguments may show that minimal guarantees deserve to be classed among basic human interests. It is perfectly obvious that people cannot lead decent lives without certain minimal levels of food, shelter, and health care. But calling the crying need for public assistance "basic" may not get us very far. A just society would ensure that its citizens have food and shelter; it would try to guarantee adequate medical care; it would strive to offer good education, good jobs, and a clean environment. But which of these goals should it pursue by creating rights, legal or even constitutional? This is a question that cannot be answered by abstract theory alone; everything depends on context.
Those opposed to constitutionalizing welfare rights usually argue along the following lines. A constitution is a legal document with limited tasks. If a country tries to make legally binding and judicially enforceable everything that a decent society requires, its constitution risks losing its coherence. If Americans created expensive constitutional rights to housing and health care, which depend on the state of the economy, we would overload our Bill of Rights. Indeed, by labeling as "constitutional rights" valuable services that we sometimes cannot afford to deliver, we may even cheapen traditional American liberties in the eyes of citizens, who will begin to see constitutional rights as claims to be honored or not, depending on resources available at the time.
These points have some force. But since all rights depend on the state of the economy and public finances, the decision to constitutionalize or not to constitutionalize welfare rights cannot be made on such grounds alone. Not a single right valued by Americans can be reliably enforced if the Treasury is empty. All rights are protected only to a degree, and this degree depends partly on budgetary decisions about how to allocate scarce public resources. If rights have costs then, like it or not, "politics is trumps," to use political scientist B. Guy Peters's aphoristic reminder of the inevitable role of political choice in the creation of public budgets.
Some countries (Germany is an example) have constitutionalized certain kinds of welfare rights without noticeably cheapening freedom of the press or procedural guarantees. By contrast, the American welfare state relies almost entirely on statute, not the Constitution. But there is less to this distinction than meets the eye. The demand for welfare rights arises forcefully out of modern economies and societies. For the most part, the level of protection welfare rights receive is determined politically, not judicially, whether such rights are officially constitutionalized or not.
One might think that in developing nations second-generation constitutional rights to minimum welfare guarantees are not desirable, because they would cost a great deal more than first-generation rights to more familiar liberties (a distinction of degree), because they would give the wrong kind of power to the judiciary, because they would not produce adequate social returns, or because they would send the wrong signal about the basic point of government. These are practical issues. But to consider first-generation rights "priceless" and second-generation rights "costly" is not only imprecise, it also encourages the fantasy that the courts can generate their own power and impose their own solutions, whether or not the legislative or executive branches happen to support them. The American judiciary may or may not be the forum of basic principle, but it is certainly constructed and buttressed by the extractive branches of government, which provide the fiscal wherewithal to nourish and house the judiciary and, generally, to keep it alive and functioning. To focus on the cost of rights is therefore to shed light on an important and poorly understood aspect of the American separation of powers.
While many rights appear in the American Constitution, it is a mistake to think that their specific content is chiseled in constitutional granite. During no thirty-year period is the concrete meaning of our basic constitutional rights likely to remain constant. As old social problems fade away and new social problems spring up, the way rights are construed naturally evolves. To draw attention to how the rights of Americans are ceaselessly changing is emphatically not to defend relativism, to say that basic human interests differ wildly across cultures, or to imply that governments should define rights however they wish. But as a descriptive matter, rights are in important respects context-dependent. The way they are interpreted and applied shifts with changing circumstances and with advances or retreats in knowledge. Freedom of speech is a revealing example. What freedom of speech means in contemporary American constitutional jurisprudence is not what it meant fifty or one hundred years ago. The significance and implications of First Amendment rights have not stood still in the past and will surely continue to change in the future.
Many reasons account for this ceaseless and unpredictable evolution. Judgments about issues of value, fact, and harm change with time and place. But another source of variation is more mundane, for rights are rooted in the most shifting of all political soils, that of the annual budgetary process, a process thick with ad hoc political compromises. Built on such shifting terrain, rights are bound to be less indefeasible than the desire for legal certainty might lead us to wish. To take account of this unstable reality, therefore, we ought not to conceive of rights as floating above time and place, or as absolute in character. It is more realistic and more productive to define rights as individual powers deriving from membership in, or affiliation with, a political community, and as selective investments of scarce collective resources, made to achieve common aims and to resolve what are generally perceived to be urgent common problems.
The constitutions of Germany, Mexico, Brazil, Hungary, and Russia include, in various forms, a right to a safe and healthy environment. (The extent to which these rights can be enforced through the court systems in these countries is debatable, but it is modest even in the best of cases.) In the United States, too, people have argued vigorously on behalf of such a legally entrenched third-generation right at the national level. They urge that the interest in environmental protection is systematically undervalued in ordinary political processes, and that future generations deserve protection against environmental degradation perpetrated by those now living, who, being myopic and self-interested, are all too likely to act as faithless trustees. As theoretical arguments, these claims have considerable force.
Yet even if the interest in environmental protection were promoted to the status of a judicially enforceable right, it would still be protected only to some degree, and its public costs would grow in direct proportion to the degree of protection afforded. Environmental protection is a very costly business. Not even the Superfund (designed to ensure clean-up of abandoned toxic-waste dumps) is unlimited. Rescuing endangered species—poached and poisoned to the point of extinction—can be expensive. And these are only two examples. In the United States, more than fifty million people continue to live in areas that fail to meet national ambient-air-quality standards. Although the nation already expends more than $130 billion per year on environmental regulation, it is not clear if our environmental regulations represent, in their current form, the most intelligent uses of limited resources.
In environmental protection, increasing attention is being paid to the phenomenon of "health-health trade-offs," which occur when regulation of one risk simultaneously increases another risk. An absolutist or single-minded approach to specific risks may well increase overall or aggregate risks. If the personal interest in being free from sulfur dioxide, which is certainly not trivial, were treated as an absolute right, the result would be a range of additional social problems, including new environmental problems; perhaps elimination of sulfur dioxide would lead to more dangerous replacements, or create serious waste-disposal problems. Inevitably, resources devoted to some problems will draw away resources from others; a government that channels the lion's share of its environmental resources to clean up hazardous-waste dumps will find itself penniless to protect clean air or clean water. Single-minded protection against highly salient environmental risks may compromise larger and longer-run environmental interests. Aggressive protection against dangers from nuclear power accidents may increase the price, and decrease the supply, of nuclear power, and in that way increase dependence on fossil fuels, which create environmental problems of their own. A no-compromise attitude will therefore produce confusion and arbitrariness and may, on balance, disserve the very rights it intends to promote.
To be enacted and implemented sensibly, enforceable rights to a safe and healthy environment would have to channel limited resources to the highest priority problems. Supreme Court Justice Stephen Breyer has vigorously argued that poor priority-setting is a central obstacle to good regulation. This suggests that anyone entrusted with respecting environmental rights will have to make hard decisions about what problems and which groups have an overriding claim on collective resources. A key goal of the legal system ought to be to overcome the problem of selective attention, a general problem that emerges whenever participants focus on one aspect of an issue to the exclusion of other aspects. In a way, an emphasis on the cost of rights can be understood as a response to the problem of selective attention. "Health-health" trade-offs are paralleled by "environment-environment" trade-offs, as when protection of clean air increases solid-waste disposal problems, and the "rights-rights" trade-offs that arise when, for example, use of the legal system to protect environmental quality makes fewer resources available to protect, say, against criminal violence.
The environment is for the most part collectively enjoyed and, if the air gets a good deal cleaner or less clean, many or most of us will be positively or negatively affected. This point is important, for any general "right" to environmental quality could entail an individual plaintiff's capacity to dictate at least minimal levels of water and air quality for thousands or even millions of people. Environmental interests, recast as judicially enforceable rights, could have serious collective consequences on both the cost and benefit sides. They would certainly involve a redistribution of resources from some people to others in the form of taxation, and an additional redistribution at the point of expenditure.
So what would be the effect, in the United States, of creating a constitutional right to environmental protection? Some environmentalists say that a safe environment is an absolute good and should be provided "whatever it takes." But safety is a relative, not an absolute, concept. The question is "How safe?" rather than "Safe or not?" Achieving higher levels of safety requires both private and public expenditures, and perhaps those expenditures are best made elsewhere. If it were enforceable in court, a constitutional right to a safe environment could entrust judges with the job of identifying the point at which such a right has been adequately respected. Are courts better equipped to carry out this task than they are to micromanage the Winnebago County DSS? For one thing, they lack the fact-finding capacity in the environmental arena that would justify their making particular allocative decisions. For another, they are not politically accountable. Equally important, they lack the overview of the tangle of economic and environmental issues that would be necessary, at a minimum, for deciding rationally that one policy should be chosen over an alternative.
The professional incapacity of judges does not, by itself, establish that the palpable interest of a nation's citizens in environmental quality has no conceivable place in a constitution. Perhaps such a "right" should be created and simply construed as a directive to the legislature, not to the courts. Perhaps such a "right" would not be judicially enforceable at all but would instead be useful as a weapon in political debate. Perhaps such a quasi-right or symbolic right could be designed not to ensure any particular outcome but instead to flag the importance and denounce the government's disregard of environmental interests. Perhaps courts could play a modest and appropriate role by calling public attention to cases in which political actors have conspicuously defaulted on their responsibilities—as the Supreme Court should have done in DeShaney itself.
Whether a particular nation should enshrine a right to environmental quality in its constitution remains debatable. Under current conditions, with an active, vigorous, and frequently successful environmental movement, a constitutional amendment of this kind probably would not make sense for the United States. But if third-generation rights ever become judicially enforceable, they will be less distinctive than both critics and proponents seem to expect. From the perspective of public finance, the three generations of rights occupy a continuum, rather than being radically distinct kinds of claims. Extending the insight of Justice Breyer, we might even say that poor priority-setting afflicts the entire domain of rights enforcement. The question is always "How well protected?" rather than "Protected or not?" Anyone entrusted with enforcing legal rights will have to make hard decisions about what problems, and which groups, have an overriding claim on collective resources in particular circumstances.
Those charged with monitoring child custody cases are not the only ones who must bear this burden. Are citizens who are subjected to police abuse in a position altogether different from that in which Joshua DeShaney found himself? Consider their right to be free from unreasonable searches and seizures. Although constitutionally entrenched and undoubtedly a right, this right cannot be absolute in the sense of uncompromisable. No right can be uncompromisable if its scope hinges upon the shifting judicial interpretation of a word as vague and indeterminate as "unreasonable." Even more importantly, the Fourth Amendment right cannot be absolute unless the public is willing to invest the enormous amounts necessary to ensure that it is seldom violated in practice. The fact that the Fourth Amendment is violated so regularly shows that the public is not willing to make that investment.
A police officer told one of the authors that the Fourth Amendment does not give him "much trouble," because "I don't violate the Fourth Amendment unless I say I violated the Fourth Amendment, and I never say I violated the Fourth Amendment." Monitoring officials cannot do their job effectively unless they can obtain reliable information about misbehavior from sources independent of the officials suspected of abuse: officers on duty have a palpable incentive to knead and color the facts when crafting reports for higher officials, including the judiciary. Exorbitant information costs sometimes make the price of protecting even the most precious rights prohibitively high. Although the right to be free from unreasonable searches and seizures is constitutionally guaranteed, it is violated every day in practice. The politics of budget-making is one reason why.
Not only is the right to private property financed by the community, but the indisputably nonabsolute character of that right is a function of, among other factors, cost. What would it take to ensure that the rights of owners were never violated? The degree to which property rights are actually enforced varies with historical circumstances, political resolve, and state capacities, including meager or bountiful tax revenues. In protecting private property, a liberal polity (even one free of corruption and racial bias) necessarily husbands its scarce resources with an eye to competing social purposes. Some funds must be held in reserve, for instance, for the enforcement and protection of other kinds of rights. To enforce rights fairly, the government cannot expend its entire annual budget on protecting the property rights of a few individuals during the first few months of the budgetary year. Nor would any property owner be willing to hand over 100 percent of his income and wealth to have 100 percent perfect police protection of his (thereby non-existent) estate.
The decision about how thoroughly to protect property rights overloads the fact-finding and accounting capacities of police departments, administrative agencies, and courts of law. True, property rights are protected selectively rather than fairly for other, less palatable reasons as well. To the extent that publicly salaried officers devote more time to deterring and punishing acquisitive crimes in rich white neighborhoods than in poor black or Latino neighborhoods, property rights resemble the legally camouflaged interests of the strong. Such a lopsided enforcement of rights is surely a violation of equality before the law. But even if law enforcement officers did not favor some groups over others, they would still be selective in the delivery of protection from assault and theft.
Rights remain rights even though they will not always be enforced to the hilt, or even as thoroughly as would be possible were resources more plentiful or taxpayers more open-handed. Trade-offs in rights enforcement must and will be made. Scarce resources will be allocated between monitoring the police and (for example) paying and training the police, between monitoring the police and monitoring electoral officers, between monitoring the police and providing legal aid to the poor, providing food stamps to the poor, educating the young, nursing the elderly, financing national defense, or protecting the environment.
Morally speaking, incomplete protection of property rights is far easier to swallow than half-hearted protection of the helpless from beatings and killings. We accord property rights special, but not the greatest possible, protection. But are the interests of some Americans in not being brutalized or murdered given the same level of regard as the interests of other Americans in protection of their property rights? Was the palpable benefit to Joshua DeShaney of retaining his normal brain functions given the highest imaginable level of administrative protection? Was it accorded a level of protection greater or less than that received by the homeowners of Westhampton? There seems to be something obscene about the very comparison, not to mention the distressing answers such questions may elicit. But they do suggest that, in reality, no right can be uncompromisable, for rights enforcement, like everything costly, is inevitably incomplete.
Those who describe rights as absolutes make it impossible to ask an important factual question: Who decides at what level to fund which cluster of basic rights for whom? How fair, as well as how prudent, is our current system of allocating scarce resources among competing rights, including constitutional rights? And who exactly is empowered to make such allocative decisions? Attention to the costs of rights leads us not only into problems of budgetary calculation, as a consequence, but also into basic philosophical issues of distributive justice and democratic accountability. Indeed, it leads us to the edge of what is perhaps the outstanding philosophical dilemma of American political theory: What is the relationship between democracy and justice, between principles of collective decision-making, applicable to all important choices, and norms of fairness that we consider valid regardless of deliberative decisions or majority will?
In the DeShaney case, the Court was simply wrong to conclude that constitutional rights never include a right to state help. But it was right to the extent that it implicitly acknowledged a severe problem, for the protection of human lives always involves allocative decisions, and judges are not always in a good position to determine if one set of allocations is better or worse than the realistic alternatives. The cost of rights does not justify the Court's decision in the DeShaney case itself. More generally, however, scarcity is an entirely legitimate reason for the government's failure to protect rights absolutely. This insight draws out the commonalities between the first, second, and third generations of rights. All depend on collective contributions. All can be seen as selective investments of scarce resources. All are in an important sense aspirational, for none can ever be perfectly or completely enforced. Of course, there are differences as well. But the similarities are strong enough to belie the view that those rights that have been proposed and introduced more recently betray the basic spirit of the American Constitution.
PART III:
WHY RIGHTS ENTAIL
RESPONSIBILITIES
Chapter Nine
HAVE RIGHTS GONE TOO FAR?
WHILE STILL A HIGH SCHOOL STUDENT IN WISCONSIN, and only a minor, John Redhail became a father. The child's mother filed a successful paternity action against him, and the court ordered Redhail to pay $109 per month until the child reached the age of eighteen. Indigent and unemployed, Redhail did not make the payments. Two years later his application to marry Mary Zablocki was denied on the grounds that Redhail had failed to pay child support and, under Wisconsin law at the time, those who had not met their child support obligations could be deprived of the right to marry.
The Supreme Court of the United States held that the Wisconsin law in question was unconstitutional. The right to marry, the Court explained, is "fundamental," and a state cannot enforce a support order through the unusual means of denying marriage licenses. Such license denials do not deliver money into any child's hands, and other available collection strategies would not intrude on constitutionally protected rights.
Should a deadbeat father's right to marry trump his moral responsibilities toward his child? This fundamental liberty obviously could not exist in the absence of governmentally created and managed procedures. In its current form, it is a product of government, not of nature. Should it not be abridged when doing so can "send a message" and perhaps help ensure that men fulfill their most basic social duty? Since the community defrays the costs whenever children become public charges, can it not restrict the freedom of those who are morally and legally obliged to provide support? Do private rights, when interpreted as preemptory claims, operate as excuses for moral shirking? Must our responsibilities to family and community wither and fade as the domain of our individual liberties expands?
Beneath these legal questions lurk even deeper worries. Has America recently witnessed an explosion of rights at the expense of traditional moral duties? Does our political culture now entice individuals to act however they please, without heeding the consequences, especially the consequences for others? Should it be obligatory for Americans—John Redhail and those in similar positions—to forgo their ephemeral and egotistical wants, pull up their socks, and act responsibly? And what is the relevance of the fact that private rights, such as those asserted by John Redhail, have public costs?
The idea that rights have "gone too far" while responsibilities have correspondingly shriveled has become something of a platitude. In the 1950s, according to a familiar tale, Americans enjoyed fewer rights, insisted much less on their personal freedoms, and (it supposedly follows) took their responsibilities to both self and others most earnestly. Since the 1960s, by glaring contrast, licentiousness has swept the land. Americans now think that it is a glorious idea to do whatever they have a right to do—to receive a paycheck while refusing to work, to abuse drugs and alcohol, to behave promiscuously, or to have children out of wedlock. Nor does this fable neglect the government's noxious role in promoting cultural decay. After the Supreme Court under Chief Justice Earl Warren and other government agencies started lavishing rights on nonconformists, ordinary citizens began to disregard their traditional duties. The government's irresponsible overprotection of rights helped breed the population's irresponsible neglect of obligations.
Claims to this effect issue from a dazzling variety of sources: President Clinton, Robert Dole, Supreme Court Justice Clarence Thomas, General Colin Powell, George Will, many members of the U.S. Senate, and a wide range of academics, including Mary Ann Glendon, Amitai Etzioni, William Galston, and Gertrude Himmelfarb. Glendon fears that "rights talk" has drawn Americans into greater selfishness and atomism, that a culture of rights has politically devalued altruism, mutual concern, and assistance to one another. Will, Galston, and Powell plead for a resurrection of "shame" as a means of inculcating sobriety and discipline. Himmelfarb speaks of the "demoralization of society," meaning a wholesale retreat of morality from our social world, and draws unflattering comparisons between a degenerate America today and Victorian England, where pervasive respect for moral virtues purportedly ensured a greater role for social responsibility. Many critics complain that during the 1960s and 1970s the U.S. Supreme Court was seized by the promiscuous counterculture. Thereafter, it lavished rights unstintingly on the rebellious, the untrustworthy, and the deviant. This, they say, is how America began its current downward slide.
The notion that rights are intrinsically corrosive of duties is especially appealing to conservative critics of social programs designed to help the poor. But such apprehensions are also shared by liberals. Both ends of the current political spectrum identify rights with irresponsibility and an attenuated sense of duty, although they have different forms of moral laxity in mind. The Right belabors the licentiousness of the poor, while the Left laments the licentiousness of the rich. Conservatives typically decry the wanton behavior of young uneducated black mothers hooked on public aid. They claim that welfare entitlements undercut responsibility by delivering paychecks to those who refuse to get out of bed in the morning, dress themselves, and show up punctually for work. For their part, liberals deplore the reckless conduct of junk-bond dealers, overpaid CEOs, industrial polluters, and companies that relocate for a small profit regardless of how plant closings affect aging workforces and abandoned communities. They accuse the privileged of displaying a devil-take-the-hindmost fondness for their own property and privileges. One side is obsessed by the want of responsibility toward oneself, while the other focuses its outrage on irresponsibility toward others. But both aspire to restrict the liberties of those who fail to comply with basic moral rules. In this sense, John Redhail—brassily asserting his rights while furtively ducking his responsibilities—epitomizes what each camp believes to have gone wrong with America.
But is the United States today really suffering from a culture of "anything goes"? Do most Americans inconsiderately pursue immediate interests or impulses with little thought for social consequences? And can this cult of heedlessness, assuming it exists, be causally traced to an "explosion of rights"? In what sense, if at all, has the entitlement mentality caused family breakdown, sexual permissiveness, and a wasting of the work ethic? We are frequently asked to believe that individuals throughout the land have been shedding their responsibilities while scrambling greedily for their rights and that morality has been rinsed out of law. Since rights are ultimately latitudes or exemptions from control, we are told, irresponsible behavior is programmed into the genetic code of America's rights-based regime. In this view, after rights to get divorced and live off welfare began to be accepted without embarrassment in the United States, the country's citizens started thinking that there is nothing—however selfish or self-destructive or antisocial—that they are not licensed to do. To arrest the ongoing social decay, Americans of all classes must be weaned from their pathological attachment to personal liberties.
RESPONSIBILITY TALK
We should think more responsibly about responsibility. Have increases in criminal behavior resulted from the enforcement of rights or from, say, demographic, technological, economic, educational, and cultural changes largely independent of rights? Even if certain rights have, on balance, increased irresponsible behavior in some domains, sweeping causal generalizations are dubious. "Responsible behavior" may be defined as conduct that reduces harm to both self and others. Can we plausibly claim that there has been a general reorientation of American society from responsibilities (thus understood) to rights?
In many spheres of social life today, people shirk their duties, behave inconsiderately, ignore the serious problems of others, and ought, in general, to behave more responsibly. But this is no innovation of the last thirty years; in one form or another, it has always been the case. It is true today even in countries where individual rights are uniformly disrespected or wholly unknown. So what has the culture of rights added to mankind's enduring proclivity to recklessness, insensitivity, and short-term thinking?
Two possibilities have already been discussed at length. When interpreted either as negative immunities from government influence or as non-negotiable claims, rights may indeed become formulas for irresponsibility. If property owners are persuaded that their ownership rights are perfectly secured when their government simply steps out of the picture, they may also underestimate how thoroughly their individual freedoms depend on community contributions. When civil libertarians style a small number of rights as absolute, they may neglect the distributional consequences of expending scarce resources on a limited set of what they have identified as the most urgent social interests. Those who believe that they have a right to engage in certain behavior may not understand that it is not right to do what they have a right to do. So, yes, where rights are poorly understood, they can encourage irresponsible conduct.
Nevertheless, rights and responsibilities can hardly be separated; they are correlative. The mutual dependence of rights and responsibilities, their essential inextricability, makes it implausible to say that responsibilities are being "ignored" because rights have "gone too far." Add to this the fact that rights are immensely heterogeneous. Is the right to engage in collective bargaining, on balance, de-responsibilizing? What about habeas corpus? The right to a fair trial? The right to self-defense? The right to vote? Rights to due process and equal treatment do not tell government officials, at least, that "anything goes."
Ordinary contract law prohibits American courts from enforcing irresponsible debts, such as those contracted among gamblers. Such interdictions are natural, for contract law as a whole is a system for enforcing social responsibilities. The right of a promisee to sue a promisor for breach of promise is the classical illustration of the thesis that rights and duties are correlative. And the pattern is general. If Smith has a right to his property, then Jones has a duty not to trespass upon it. If Jones has a right to a percentage of the proceeds from his bestseller, the publisher has a duty to send him what he is due. To protect the rights of Smith the nonsmoker, the government must increase the responsibilities of Jones the smoker. If Jones's freedom of religion is constitutionally protected, public officials have toward him a duty of toleration. If Smith has a right to be free from racial discrimination in employment, employers have a duty to ignore the color of Smith's skin. If Jones has a right in a criminal trial to exclude evidence gathered illegally against him, the police have a duty to get a valid warrant before they search his house. If Smith has a right to sue a newspaper for libel, the newspaper has a duty to check its facts.
The United States once denied enslaved African Americans the rights to own property and to make contracts, to take care of their children and to vote. These denials did not inculcate habits of responsibility. Societies where liberal rights are weakly enforced—that is, where predatory behavior among strangers abounds—do not witness a flourishing of social responsibility. Historical evidence suggests, on the contrary, that rightslessness is the richest possible breeding ground for individual and social irresponsibility. In this more sociological sense, too, rights and responsibilities are far from opposites.
Contrary to the critique of those who seek more responsibility, the current American legal system, rather than reflecting the anarcho-libertarian principle of "anything goes," publicly articulates and coercively enforces reams of legal prohibitions. And many of these coercive constraints were created in the supposedly responsibility-phobic 1960s and 1970s—including rules against environmental degradation, against dangerous workplaces, and against the sexual harassment of working women. Some important constraints are much older, such as rules against unlicensed amateurs setting themselves up in private practice as eye surgeons. Today, the national government limits the right of tobacco companies to advertise their products on the grounds that such otherwise protected commercial speech decreases responsible behavior among the young. (Addiction means precisely this: addicted individuals cannot, in any simple sense, "freely choose" not to smoke; as a consequence, the government cannot foster individual liberty, where addictive substances are concerned, simply by assuming a posture of laissez-faire.) Social responsibility is far from neglected by American law. While amply supported by colorful anecdotes, the report of an across-the-board decrease in the social responsibility of Americans since the 1960s is scantily corroborated by reliable evidence.
Rights and responsibilities are routinely reconfigured as time passes; individuals now act responsibly in realms where they once behaved irresponsibly and vice versa. In some cases, at least, they have relinquished rights they once enjoyed. Here are a few examples:
• Social norms, and sometimes law, now discourage environmentally destructive behavior. In many circles, littering invites social disapproval. Recycling is common; people willingly recycle. Companies engage in a wide range of activities designed to reduce pollution, presumably to escape social disapproval and to act responsibly. One of the most effective environmental programs simply requires companies to make available to the public information about their toxic releases. Responding to public pressure, companies have substantially reduced their emissions. A more trivial but in its way remarkable example: In big cities, people clean up after their dogs.
• In general smoking has declined. From 1978 to 1990, a steep drop in cigarette smoking took place. The decline was especially pronounced among young African Americans, who have been exercising responsibility where they once indulged their liberties. The smoking rate among blacks between the ages of 18 and 24 fell from 37.1 percent in 1965, to 31.8 percent in 1979, to 20.4 percent in 1987, to 11.8 percent in 1991, to 4.4 percent in 1993. (There has been a disappointing rise since that time, but rates remain low by earlier standards.) Part of the decline stems from the fact that smokers no longer enjoy the legal rights they once took for granted: in many places, smoking is now illegal. Part of the drop-off also reflects a growing perception that smoking is harmful to both self and others.
• Whereas employers could once fire employees at will, they no longer have this right, at least not in its 1950s form. As a result of the Occupational Safety and Health Act, civil rights laws, workers' compensation laws in their modern guise, and common law developments, employers are now constrained in their authority to dismiss employees. Employers now labor under a legal duty to provide a safe workplace, and they can no longer discharge employees on discriminatory grounds. Social norms also discourage irresponsible (which is to say arbitrary) discharges.
• Employers and teachers were once free to engage in sexual harassment. Indeed, the very category of "sexual harassment" did not exist until recently, and both social norms and law authorized teachers and employers to seek sexual favors from those over whom they exercised power. Employers and teachers were essentially licensed to indulge in what is now punishable as harassing behavior. A traditional right has therefore been legally extinguished. Responsible behavior in this area is increasingly widespread, partly because of new law, and partly because of patterns of social disapproval that are inducing men to behave more responsibly.
• In many states, men no longer have a legal right to rape their wives. As a result of new legislation, husbands must act more responsibly. Sexual intercourse must be consensual even within marriage.
• Until recently, racist and anti-Semitic statements were common fare even in relatively public places. Such statements are still largely uncontrolled by law, and bigots have a legal right to utter racial slurs if they are so inclined. But many Americans shun talking in such irresponsible ways or at least do so less often than they once did. On this count, at least, civility has increased.
Even though socially and personally responsible behavior has fallen off in some areas, in other words, talk of a wholesale decline of responsibility is overblown. Indeed, it would not be especially difficult to concoct a self-congratulatory report on a whole new wave of responsibility in America: whereas they used to cling pertinaciously to their selfish rights, it could be said, Americans have now learned to act with generosity, social conscience, and concern for others. But why answer one half-baked narrative with another? What has happened in the last twenty years is a perfectly ordinary process of legal evolution, in which both responsibilities and rights have been redefined. The law has recognized some new rights while disestablishing some old ones.
Whether all of the recent developments are welcome is entirely beside the point. This kind of modification is only to be expected. In the relevant period both law and social norms have changed, as they never cease to do. Who knows what kinds of responsibilities and irresponsibilities will be produced by new law and new norms thirty years from now?
The dichotomy between rights and responsibilities is especially misleading because many rights are specifically created in order to make government more responsible. The right to exclude testimony extracted under duress is designed to prevent arresting and interrogating officers from beating confessions out of detainees. Most constitutional rights, in fact, are crafted to induce responsible conduct among agents of the state. They are incentives to self-discipline, partly but not only because rights imply duties. The right to vote and freedom of the press, especially, are meant to have, and sometimes do have, a responsibilizing effect on officials who can be ousted from office or held up to public ridicule.
When American law enforces social responsibility, it does not ordinarily do so in the name of an ideal code of conduct. Instead, American law usually imposes responsibilities as the counterparts, or preconditions, of rights. The responsibilities of polluters are the mirror image of the rights of the public to a nontoxic environment. Smokers and employers have duties because nonsmokers and employees have rights. The crime of marital rape, quite obviously, imposes a duty in the name of a right. The rights of stockholders are the duties of company directors and managers.
For a debtor to respect the rights of his creditor, he must act responsibly. So must a government that respects the contractually attained rights of all parties subject to its jurisdiction. Property rights inhibit both private theft and the confiscatory whims of public officials, thereby making both ordinary citizens and public officials behave more responsibly than they might otherwise tend to do. A government that enforces and protects rights, moreover, cannot do so unless it channels scarce tax revenues to public uses, rather than into the pockets of corrupt officials. Full and fair compensation for any property seized for public purposes requires a well-functioning system of public finance. The simple fact that rights have costs, therefore, already demonstrates why rights entail responsibilities.
Indeed, the cost of rights allows us to slip into the rights/responsibilities debate by a side door. Property rights have costs because, to protect them, the government must hire police officers. Responsibility is involved here, first, in the honest routing of taxpayers' dollars into the salaries of the police. It is involved a second time when, at considerable expense, the government trains police officers to respect the rights of suspects. And responsibility comes in a third time when the government, again at the taxpayers' expense, monitors police behavior and disciplines abuses to prevent officers from abridging civil rights and civil liberties by, for example, breaking into people's homes, manufacturing evidence, and beating up suspects. Attention to the cost of rights, in other words, heightens our understanding of the mutually supportive relationship between rights and responsibilities. And the same holds true when we turn from classical rights to the rights characteristic of the modern regulatory state.
The "social disintegration" litany of Left and Right will no doubt remain a staple of American political debates, for it apparently serves subcognitive needs. At least, laments of this sort cannot be easily quieted by evidence or argument. But such complaints are based on a serious misconceptualization of rights, and showing this may still be useful.
MORALITY IN LAW
American law, admittedly, vests individuals with the right to do things that are widely considered to be morally wrong. This is not an accidental but an essential feature of any liberal regime or indeed of any free country. Americans have the legal right to engage in conduct that responsible and even moderately sane people will scrupulously avoid. So while American law has moral sources, it is not coextensive with the moral sensibility of the community.
The indifference of law to morality, however, should not be exaggerated. The moral codes that impinge on law have changed somewhat, it is true, but they have not vanished, and it is not even clear that they have been reduced on balance. American tort law, for instance, remains shot through with morally laden categories such as "negligence" and "recklessness," and these categories routinely guide the way state power is used. In the past few decades, morally charged legal constraints on harmful behavior have increased, not decreased, in such areas as product liability and consumer protection. In criminal law, the perception that the accused has acted with "an abandoned and malignant heart" or "a culpable state of mind" continues to influence the decisions of prosecutors and judges alike. And in the United States, unlike in other Western countries, anyone who causes a death, however accidentally, while committing a felony, may be charged with murder—a perhaps futile attempt to make felons behave more responsibly while committing their lesser crimes.
Along the same lines, the list of crimes against morality that are still punished in America is quite impressive: statutory rape, incest, indecent exposure, prostitution, child pornography, and lewd and lascivious conduct. Habitual drunkenness provides grounds for divorce in most states. Adultery remains illegal under the law of many states, as well as under American military law. And American law recognizes morality in another sense as well: to write or say that a person is immoral—that he is a womanizer or watches filthy movies or is a miser or would commit crimes if he were not afraid of getting caught—all this is, in some states, actionable per se and does not require the plaintiff to prove special damages. In other words, morality has hardly disappeared from our courtrooms or our streets.
Responsibility, moreover, is frequently a product of law. The right to drive a car does not include a right to drive it irresponsibly. In fact, since 1960 the law has imposed more, rather than fewer, constraints on both manufacturers and drivers, designed to increase safety. Spouses are still legally responsible for one another's debts. In most states, it remains very difficult to disinherit a spouse. Americans are also remarkably compliant with the tax laws (well over 90 percent of the public complies fully); indeed, Americans are far more compliant than citizens in some countries where individualism and individual rights play a less conspicuous role in social self-understanding. Massive tax evasion in, say, Russia today does not stem from a culturally ingrained attachment to individual rights. Yet observable increases in responsibility do not result only from fear of criminal and civil sanctions: without some element of "civic virtue," bolstered no doubt by the public perception that the government spends tax revenues more or less responsibly, that most people do their fair share, and that rich Americans, in particular, are not wholly exempt from taxation, the costs of running the Internal Revenue Service would be much steeper.
RIGHTS AS LATENT RESPONSIBILITIES
An accused party has a right to get out of jail on (not excessive) bail before trial in order to prepare a better defense. In this case, the rightsholder himself has a right to act responsibly. Not only do rights typically entail responsibilities for others vis-à-vis rightsholders, but rightsholders themselves are sometimes made more responsible by virtue of exercising their rights. This is another reason why the clarion call for fewer rights and more responsibilities is ultimately incoherent.
Aristotle objected to Plato's enthusiasm for collective child-rearing on the grounds that if everyone is responsible for every child, and if particular individuals are not denominated "parents," children will not receive decent care. The very same logic justifies the right to private property. If everyone owns everything then, in a sense, no one owns anything. One of the problems with this sad state of affairs is that in a system of collective ownership, the costs of dilapidation are spread thinly, and thus catastrophically, across society. Each individual in a position to maintain and repair property loses little by decay and gains next to nothing by maintenance. In a system without private ownership or coercive organization, the costs of maintenance are borne by each person, while the benefits of maintenance are widely shared. Hence individuals have scant incentive to engage in timely and arduous repairs. If rewards for upkeep and improvement cannot be captured by owners, houses and farms and factories are very unlikely to be kept up and improved. Acting with an eye to tomorrow, individuals deprived of enforceable property rights are likely to engage in uncoordinated inaction, or acts of negligence that produce massive collective harms. As Aristotle objected to Plato, private rights can be a spur to action that is socially beneficial and, from society's point of view, highly responsible.
Any farmer toiling to repay a bank loan can explain that the right to private property is both an onerous burden and an incitement to effort. Not only do property rights compel owners to pay the costs of their own property's dilapidation, but well-defined and unambiguously assigned property rights nourish responsibility by allowing individuals to capture the returns on their investments. They also help lengthen the time horizon of owners, who can thereby hope to benefit tomorrow from exertions made today.
Property rights also play an essential role in systems of political accountability, giving taxpayers a material incentive to monitor the way governments put reluctantly shelled-out tax revenues to use. So multiple linkages between private ownership and social responsibility are clear even before we look at the ways the American legal system uses rights to layer social responsibilities on top of ownership rights—imposing zoning restrictions on the sale of pornography, using the tax code to prod homeowners to safeguard and improve their assets, preventing factory owners and landowners from polluting the aquifers, and discouraging restaurant owners from shutting their doors to racial minorities.
Pleas to downplay rights and inculcate responsibilities are less helpful than their authors intend because they convey the impression that rights-and-responsibilities is a zero-sum game: any increase in one automatically decreases the other. They obscure the essential fact that, in the American legal system, rights are public services that the government must answerably deliver in exchange for tax revenues responsibly paid by ordinary citizens. Rights would go miserably unprotected if these mutual accountabilities failed. The exchange of equal rights for social cooperation lies at the heart of liberal-democratic politics. Rights are what responsive government and informed citizenship are all about. That rights have costs demonstrates their dependence on what we might as well call "civic virtue." Americans possess rights only to the extent that, on the whole, they behave as responsible citizens.
None of this is meant to deny the urgency of various issues touched upon by the advocates of "more responsibility." But drug use, AIDS, divorce, out-of-wedlock births, welfare as a way of life, single-parent families, children in poverty, and violent crime cannot be so casually traced to an alleged "culture of rights." The terrible social pathology of our public housing projects should be addressed in more concrete and less exalted terms. None of these problems can be solved by diminishing the respect for rights in American legal culture. Nor are useful solutions likely to arise from sweeping claims about the acids of modernity.
Chapter Ten
THE UNSELFISHNESS OF RIGHTS
POLITICAL THEORY KNOWS OF RIGHTS WITHOUT RESPONSIBILITIES, namely the pre-legal rights of individuals in philosopher Thomas Hobbes's "state of nature," where individuals have "even a right to one another's body." To protect this sort of "right," though we should probably not call it that, every individual is a freelancer, forced to shift for himself. Truculent males are more likely than females to succeed at such a brutal game. To escape the state of nature means to obtain a wholly new kind of interest: a legal right, that is, a claim that carries with it serious responsibilities. All legally enforceable rights are "artificial" in the sense that they presuppose the existence of that imposing human artifice, the public power, designed to promote social cooperation and inhibit mutual harm.
To enjoy such rights, an individual must renounce his "natural right" to punish unilaterally all those who, in his subjective opinion, have injured him. This renunciation is the germ of liberal responsibility. That legally enforceable rights entail responsibilities, even in the rightsholder himself, is also apparent from the difference between seeking a remedy at law and paying a Mafia hit man to wreak private revenge. Indeed, the right to litigate, even if grossly overused, helps induce those whose interests have been grievously harmed to seek redress "responsibly," that is, within legal channels, rather than in vigilante fashion. When an injured party seeks remedy in court (instead of the back alley), she must make an effort to prove her case. To obtain a writ of sequestration or an order to garnish a debtor's wages, a creditor must bear a considerable burden of proof and face rebuttal in an open procedure. That is, the rightsholder herself must behave responsibly, in a public setting, if she wants government help in enforcing her claim.
When they work well, liberal rights shrewdly deploy incentives to induce responsible behavior and self-discipline among private citizens as well as among public officials. One individual's rights to sue another for plagiarism, abuse of trademark, or commercial fraud—all of which entail a taxpayer-subsidized right of access to a public system of litigation—probably make people act on balance more "responsibly" (however that elusive term is defined) than they otherwise would. To inhibit irresponsible speech, the state provides a forum for vindicating interests in reputation. Legislators jack up damages to make sure homeowners responsibly shovel their sidewalks. And so forth.
Admittedly, my right to sue you for negligent behavior can be used frivolously or irresponsibly. But so can my right to vote or, for that matter, a bottle of tranquilizers. Because they place a legal power over others into the hands of individuals, legal rights can always be abused. The personal ordeal of being sued in court includes the costs of defending oneself at trial and of submitting to the disagreeable burdens of discovery. But to safeguard against a misuse of the power to bring an action for damages, liberal systems do not abolish the power (a cure worse than the disease), but instead create countervailing powers, by establishing, for instance, rules that throw financial hardship upon parties who lodge insubstantial or frivolous or fraudulent claims. These rules themselves take the form of rights—rights to be free from abuse of judicial process—which embed in American law a standard of responsible behavior.
The costs of rights include the cost of imposing sanctions for noncompliance. This explains why societies where rights are systematically ignored are anything but preserves of moral responsibility. Rights enforcement means that a politically organized society consistently and fairly punishes those who trample lawlessly upon the most important interests of others. To inhibit the abusive behavior of those who stand to gain from violating rights is impossible without dipping into public funds. Remedies for past rights violations and disincentives for future rights violations are costly because they always involve enforced responsibilities. The debtor must repay. The promisor must perform. And the judges who enforce contracts and punish law-breakers must abstain from taking bribes.
As systems of incentives inducing self-limiting—and that means responsible—behavior, rights should be associated not with a hands-off but with a liberal, as opposed to authoritarian, regulatory style. Rights, from this perspective, should be described neither as latitudes nor as entitlements, but rather as consciously designed or historically evolved techniques for inducing sober, decent, and mutually respectful behavior. Rights compel both those who can exercise them and those who must respect them to internalize the harms that may result from their own laxities and misbehaviors.
Some theorists draw a historical distinction between individual rights, purportedly invented in modern times, and a "right order," allegedly embraced in antiquity and the Middle Ages (when "right conduct" supposedly flourished). But the contrast is misleading. Historically, no such age of untrammeled civic virtue and responsibility ever existed. And liberal rights are today integral to our conception of a "right order." They encourage right conduct. While the results are certainly imperfect, and sometimes worse than that, rights in America have helped build a social constellation in which private individuals usually refrain from harming one another and where citizens more or less responsibly contribute to the Treasury while officials use these funds more or less responsibly to defray the costs of rights. This is probably the only sort of order possible in a large, heterogeneous society such as the United States, where people from widely different backgrounds and with diverse beliefs are asked to cooperate in a common life.
Because rights are costly, they could never be protected or enforced if citizens, on average, were not responsible enough to pay their taxes and public officials were not, on the whole, responsible enough to use extracted revenues for public purposes rather than pocketing them for private enrichment. The sad tale of America's decaying social fabric and failing civic virtues would be more persuasive, and the situation of the country more desperate, if citizens routinely refused to pay their taxes. One reason they do not resist more universally is that, by and large, their rights are enforced. That is to say, they see that their taxes are used at least in part to protect what they understand to be their basic liberties.
THE ENTITLEMENT MENTALITY?
Since rights and responsibilities, far from being mutually exclusive, are corollaries, to depict the evolution of the American rights culture as a dramatic eclipse of dutifulness by libertinage is to make a hash of social and legal reality. For the culture of rights is always also a culture of responsibility. Legal permissions logically imply legal obligations, and rights always restrict even as they permit. Formulated differently, to make the enforcement of rights into the principal goal of public policy, the United States has developed a regulatory style that necessarily emphasizes duties, prohibitions, obligations, and restraints. To potential violators, every right "just says no."
Even where there has been a perceptible decline in social responsibility, it is intellectually irresponsible to trace this downward slide to the growing appeal of individual rights. To be sure, people may well be engaging in more irresponsible sexual behavior than they did, say, in 1955. But this is itself a crude thought (are sexual harassment and marital rape, both newly against the law, responsible?), and in what sense has this trend been unleashed or accelerated by an expansion of legal rights? Are not levels of promiscuity far better understood as a product of changing social norms and technologies? Not only does evidence suggest that new contraceptive options have been an important contributor, but changing sexual mores reflect changing relations between men and women. One of the roots of the "sexual revolution" is the refusal of women to be held to different standards than men. Was American society more responsible when male promiscuity was admired and female promiscuity reviled? Once the playing field was leveled, so to speak, traditional prohibitions and inhibitions began to crumble. Explaining why such moral norms and social expectations change when they do is far from easy. But there were undoubtedly multiple causal factors at work, the massive entry of women into the workforce and the greater availability of contraception being two. It would certainly be adventurous to suppose that the lengthening list of legally recognized rights was the principal moving force.
But what about the commonplace that a "right to welfare" discourages productive labor? This sounds rather plausible at first hearing. But an argument on the other side, formulated by Adam Smith, also has a certain weight: "That men in general should work better when they are ill fed than when they are well fed, when they are disheartened than when they are in good spirits, when they are frequently sick than when they are generally in good health, seems not very probable."
Even if we accept Smith's speculations, should we not be worried by the growth of the entitlement mentality? Has not the overextension of welfare rights encouraged dependency, unwanted children, or other social ills? Can we not encourage responsible conduct by cutting back on public aid? These are legitimate questions, and there is some evidence that welfare rights have produced dependency and associated social ills. But the data are mixed. This is especially true of the claim that rising illegitimacy rates are a product of public largesse. Evidence collected on the most important (and now repealed) welfare program, Aid to Families with Dependent Children (AFDC), does not unambiguously demonstrate that welfare fosters illegitimacy. Under the AFDC program, welfare rates were actually decreasing in the period when illegitimacy rates were increasing. Reductions in AFDC benefits do not correlate with decreases in illegitimate births, and increases in AFDC benefits do not correlate with increases in illegitimate births. Thus the data create a serious question of interpretation.
Food stamps may lessen the ordeal of being poor, and whenever it becomes especially horrible to be poor, fewer people may be poor, but this does not by itself demonstrate that food stamps breed sloth or multiply illegitimate births. A great deal depends on prevailing social norms associated with work, welfare, and family. In many sectors of society, these norms powerfully encourage steady employment, stigmatize public aid, and condemn out-of-wedlock pregnancy. When such norms lose their sway over hearts and minds, it is difficult to determine the precise extent to which welfare rights play a supplementary role in discouraging work or increasing rates of childbirth among poor unmarried women. As a result, it is far from obvious that penalizing single mothers financially will substantially reduce the frequency of single motherhood. To say so is not to defend existing welfare programs; whatever the shortcomings of such programs, the blanket claim that rights breed irresponsibility simply should be greeted with skepticism and tested by evidence.
THE QUESTION OF SELFISHNESS
A prominent and especially evenhanded critic of rights, Professor Mary Ann Glendon echoes widespread worries about the way rights undermine responsibilities and political culture in general. "Our current American rights talk," she claims, stands out for "its prodigality in bestowing the rights label, its legalistic character, its exaggerated absoluteness, its hyperindividualism, its insularity, and its silence with respect to personal, civic, and collective responsibilities." With this alleged lopsidedness of our legal culture in mind, Glendon devotes considerable attention to the "duty to rescue," a duty not recognized by American law. If a bystander ignores someone who is drowning, he will not be held accountable, even if the rescue could have been accomplished with little effort. Glendon deplores this result, arguing at a minimum for a statement, in law, that such a duty exists. Such a duty might initially seem to be far removed from the legalistic world of individual rights. But appearances can be misleading. Implicitly, Glendon's argument for a new duty is a plea for a new right: a right to assistance, to be granted to vulnerable people and held by them against other individuals and the government. With a similar logic, antiabortion activists aim to discourage what they consider to be immoral and irresponsible behavior by creating a constitutional right to life and vesting it in the fetus. Not only do rights create duties, but the imposition of a duty often serves to create a right.
The culture of rights is simultaneously a culture of liabilities and hence of responsibilities. So why should rights in general be accused of promoting selfishness? The right to vote gives public officials an incentive to put their self-interest to one side, or rather to identify their personal interest (in being re-elected) with the public's interest in good governance. The rights to equal protection and to fair hearings do not seem especially amoral or antisocial. The rights protected under the Fourteenth Amendment are aimed at eliminating the immoral and antisocial effects of racial discrimination by public officials within the states. Far from being antisocial, such rights promote communal decency by protecting against the exclusion of subordinated groups. Many rights reflect some degree of altruism on the part of ordinary citizens and most, when reliably protected, can help increase altruism and habits of responsibility.
Some of the core liberal rights—such as freedom of speech and association—are designed to encourage forms of deliberation and communal interaction, practices that the critics of "rights talk" otherwise seem to favor. Freedom of association conspicuously protects collective action. So does the right to preach or to put out a newspaper. These freedoms are meant to stimulate social communication, not to protect isolated individuals in a presocial order or to promote back-turning on others and hedonistic self-involvement. Although the right to freedom of speech may be owned and used by individuals, it is also the precondition of an eminently social process, namely, democratic deliberation. Free speech fosters liberal sociality, the opportunity for people to communicate and disagree and bargain freewheelingly with one other in public arenas. Freedom of the press, which props open public channels of communication, is emphatically communal in character. Indeed, anyone who owns a speech right may, by using it, conceivably contribute to the collectivity and its goals. This is why the government cannot "buy" speech rights, even if speakers privately wish to sell them.
The right to receive a jury trial and the right to serve on a jury (regardless of race) are two additional time-honored American liberties that are far from atomistic. In these cases, the community purchases a right that ensures an important role for ordinary citizens in adjudicative proceedings. To say that Americans live in a "procedural republic" is to acknowledge that individuals are not judges in their own cases and that citizens create and maintain (among other things) common institutions through which they can solve some of their common problems. Part of the goal of a fair trial is to ensure that diverse citizens can work together to decide accurately questions of guilt or innocence. The constitutional right to due process—like the private right to bring an action in contract or tort—presupposes that, at the taxpayers' expense, the government makes fact-finding institutions accessible to those whose interests are at stake. The right to a fair trial is eminently social. It provides an important mechanism for community self-governance.
As mentioned earlier, the rights created under contract law and tort law can be just as accurately described as legal powers. The right to sue for negligence or breach of contract implies the power to impose a severe, even debilitating, financial burden on another human being. Since our legal system creates and maintains such dangerous instrumentalities, which may occasionally be exploited for private advantage, it must also make an effort to ensure that they are wielded responsibly. It must provide relief from wrongful findings of liability and mistaken repossession. And this it does, although imperfectly. Irresponsible and frivolous suits are no doubt a curse, but the American legal system nevertheless devotes considerable resources to discouraging the misuse of well-protected rights, including the right to sue.
Blackstone put the case for the procedural republic this way: "if individuals were once allowed to use private force as a remedy for private injuries, all social justice must cease, the strong would give law to the weak, and every man would revert to a state of nature."7 The culture of rights encourages people to settle their conflicts juridically, to seek redress for their grievances through legal channels, without resort to violence and threats of violence. This is no small contribution to peaceful social coexistence and cooperation.
Chapter Eleven
RIGHTS AS A RESPONSE TO MORAL BREAKDOWN
WE HAVE A RIGHT TO SPEAK OFFENSIVELY, even abhorrently, but most people do not and should not exercise this right very often. A lawyer has a right to refuse to do pro bono work, but lawyers should generally do pro bono work. An extremely wealthy person has a right to hoard all his money (after taxes) and to give none of it to charity, but miserliness is not to be encouraged. The right to make an enforceable will can be used by a multimillionaire lavishly to appoint his favorite cat cemetery, but he should support more deserving social causes.
Some Americans may think that, because they have a right to do something, they cannot be criticized or blamed for doing it. Some extremists, including broadcasters and Hollywood producers and owners of music companies, interpret any objection to their offensive or degrading talk as an unwarrantable violation of their freedom of speech. But a well-functioning liberal culture distinguishes legal sanction from moral censure.
Philosophers distinguish between "the right" and "the good," that is, between the uniform rules of justice Americans are jointly compelled by law to obey, and the various personal ideals they severally choose to embrace. In the same spirit, a distinction can be drawn between the legally wrong and the personally immoral. We avail ourselves of the public power to deter unlawful breaches of promise, torts, and crimes. To discourage behavior that is immoral but not illegal, by contrast, we deploy private persuasion and disapproval, but not public coercion.
Under the U.S. Constitution as currently interpreted, women have a right to have abortions. For many Americans, this right epitomizes the way individual liberty promotes personal irresponsibility. It is certainly inadequate to justify abortion simply by invoking the right to "privacy"—a word that does not appear in the Constitution and that, in any case, fails to do justice to the issue. To oppose the abstract "right to life" to the equally abstract "right to choose" is also of little help. Rather than protecting an abstract right to choose, we should probably be focusing on the most effective way to provide young women with decent opportunities and prospects. But lawful efforts to discourage pregnancies that are likely to result in abortions, and even to dissuade pregnant women from undergoing abortions, should not be uniformly banned as interferences with a "right." The right to abortion imposes correlative responsibilities on public officials, and some of these, such as police protection for employees at abortion clinics, require the expenditure of taxpayers' money. Nonviolent efforts to reduce the staggering number of abortions per year (a grim 1.5 million), at least if accompanied by affirmative help for those with few options, may well be excellent investments.
Law should be and is shaped by moral aspirations, and it is perfectly consistent, from a moral standpoint, to insist that the right to an abortion ought to be exercised, or should have to be exercised, only very rarely. Steps can be taken to make abortions less common, as through education about contraception, prevention of coercive intercourse, and better opportunities for young women.
However we may think about this vexing subject, it demonstrates that while individuals may have a perfect legal right to do something, others can have an equal right to complain nonviolently about their doing it. Indeed, a large part of moral education consists of the inculcation of norms and values that discourage behavior that is harmful or offensive but not unlawful. Behavior of questionable morality can be effectively discouraged by informal social opprobrium, without turning it into a crime or a tort.
According to some self-styled advocates of responsibility, a new emphasis on what people have a right to do has produced a culture of moral relativism and standardlessness in which Americans typically insist on their rights without giving a second thought to whether their conduct is valuable to themselves or society. These cultural critics are particularly worried that grants of rights lead people, and especially disadvantaged people, to think of themselves as victims, and to specialize in seeking redress and protection from government. The recognition of rights, they conclude, can fuel dependency, self-pity, and lack of initiative.
Similar worries about rampant "victimology" have recently surfaced in arguments about sex equality. Some critics of feminism—and some feminists—argue that an overemphasis on rights against sexual harassment, date rape, and pornography has abraded the sense of personal agency and responsibility and encouraged women to enroll in a cult of victimhood, thereby making it all the more difficult for them to obtain equality and self-respect. Admittedly, people who regard themselves as victims, who do not perceive their own capacities for self-help, and who think that the world somehow owes them a living may fail to engage in activity that is ultimately rewarding to themselves or society. But there is little historical evidence to support the speculation that people who gain legal rights uniformly begin to see themselves as passive victims who need assume no responsibility for their own fate. Everything depends on the particular legal rights in question, for those who enjoy particular rights often have more agency as a result.
Many critics of the regulatory-welfare states are enthusiastic about rights of contract and property and would like to strengthen the legal protection given to those rights by, for example, requiring government to compensate people whose land loses value as a result of government regulation. Such critics are asking, in effect, for new or strengthened rights. But would it be plausible to allege that stronger rights of property and contract would debase their beneficiaries into victims, or that people who urge such rights are promoters of "victimology"? This is an odd claim, even though rights of property and contract do erode certain archaic habits of self-help—even though those who purchase private security or fire prevention services are likely thereafter to refrain from honing their traditional capacities for self-defense.
In fact, people whose rights are reliably enforced—including their rights to make contracts, to own property, to be free from segregation, and to be free from sexual harassment—may also be likely to be more secure actors in society and to cooperate more actively with a system that grants them equal respect. Some rights are even a precondition for individual agency: individuals who lose control of their person and property become far more likely to see themselves as victims. Perhaps those who until now have been neglected by their government would stop playing "victim" and become agents and citizens if their rights were reliably protected. Indeed, a crucial social purpose of rights protection is to do precisely this. Whether rights increase or decrease self-reliance depends on their content, context, and effects. To suggest that rights as such reduce those who enjoy them into helpless supplicants and victims—replacing family with promiscuity and work with dependency—is not plausible.
Martin Luther King, Jr., called on the state for protection against private racial discrimination. As a litigator, Thurgood Marshall helped establish a right against racial discrimination by the state. Neither King nor Marshall can be plausibly accused of promoting a cult of victimhood. On the contrary, they are generally thought to have helped establish greater independence for African Americans. (King, incidentally, although often described as an enthusiast for racial neutrality and color-blindness, was a resolute supporter of race-conscious affirmative action programs.) Their advocacy of rights was part and parcel of their reformist dynamism and refusal to assume a passive stance. What would someone who advocates responsibility while denegrating rights say about their assertion of rights? Would he really describe it as a fatal first step toward the cult of victimization among African-Americans?
RIGHTS ARISE WHERE NORMS AND DUTIES FAIL
So how, if at all, can the abortion right be best justified? The answer has everything to do with social context, not to mention with pervasive failures of social responsibility that extend far beyond the parties most immediately involved. Under conditions of equality on the basis of sex, the argument for a constitutionally protected abortion right in the United States would be far weaker. Without widespread poverty, the right to have an abortion within the United States—that is, the right to have an abortion without undertaking an expensive trip abroad—would raise less serious questions of basic fairness. In a society in which duties to the vulnerable were taken very seriously, the case for a right to an abortion would be less plausible than it now is. In such a society, women who need help would get it—before, during, and after pregnancy. The availability of social assistance would argue against the right to abortion, by making childbearing and child-rearing less difficult and less a source of inequality than they now are for many women. In such a society, men as well as women would be required (by social norms, if not by law) to dedicate their bodies to the protection of their children. (It is noteworthy that nowhere in American law are men required to devote their bodies to the protection of third parties, even when their own children, for example, need a blood transfusion or a bone-marrow transplant.) Most important, in such a society, restrictions on abortion would be based on a general and neutral form of compassion for the vulnerable, rather than the now-pervasive desire—prominent, though by no means universal, in the pro-life movement—to control women's sexual and reproductive capacities.
In other words, abortion rights would be harder to justify if restrictions on abortion did not allow male legislators, administrators, and judges to impose traditional gender roles through law, and in that way to continue a system of discrimination based on sex. In a society free from sex discrimination, any right to abortion might seem puzzling or even unnecessary. But this is not the society in which Americans now live.
Under conditions of pervasive gender inequality, the protection of abortion rights can best be understood as a social answer, as responsible as any alternative in light of those conditions, to a tragic initial failure of social responsibility. The right to have an abortion is harder to justify in the abstract. Here lies a general lesson: rights often emerge when private and public institutions falter and individuals fail to carry out their duties responsibly. When the environment is severely degraded, when the vulnerable are abandoned to fend for themselves, or when children are at risk, "rights claims" are commonly raised. When individuals engage in criminal behavior because bad social conditions have loosened moral inhibitions (thou shalt not steal, thou shalt not kill), the costs of providing police protection are sharply increased, and new claims of rights arise from the victims of crime. Hence the proposed Victims' Rights Amendment to the Constitution is a response to both individual and social dereliction. The claims of rights to clean air and water, to food, to a decent place to live, to a safe workplace, to children's rights, or to "free reproductive choice"—all these must be understood in context as compensatory responses to an original dereliction of social responsibility.
When beneficial social norms are working well, legal regulations often prove unnecessary. When beneficial norms break down, rights claims become increasingly insistent. Social norms and legal rules solve similar problems by different means. A strong social norm against littering would have the same effect as a well-enforced law against littering. Both would help society avoid behavior that is in each case not terribly bothersome, but that is in the aggregate highly undesirable. Like laws, social norms help coordinate social behavior. To the extent that Americans live in the grip of norms of cooperation, which pervasively encourage people to do their share by contributing small amounts of time or labor to projects that can succeed only when a large majority makes such contributions, rights claims do not even arise.
Informal social disapproval is often more powerful and effective than legal rules enforced by courts of law and can provide a less expensive and more efficient way to achieve widely desired social aims. If companies are polluting too much, if smokers are irritating or endangering nonsmokers, or if poor people are using drugs, public education campaigns to promote community norms might be able to improve the situation at a relatively low cost. Admittedly, the American government does not have a particularly sterling record as the moral preceptor of the nation. And preaching at people does not always make them good. (The audience may tune out or even rebel.) But government can and often does use the spread of information or the colorful depiction of the benefits of cooperation to society's advantage: increases in recycling and decreases in cigarette smoking are two recent examples.
When efforts at moral persuasion fail, rights are likely to be asserted instead. Arguments "against rights," therefore, may make more sense if reinterpreted as complaints about inadequate social norms and our need to respond to their defects. The right to be free from certain kinds of pollution ("nonsmokers' rights") and the right to be free from racial hate speech (a right vindicated by many campus speech codes) are regularly advanced when social norms falter. And once such rights are legally recognized, the costs to the taxpayer may be high.
Not all social norms are good; some are evil. A social norm against allowing black Americans to vote thwarted the purposes of the Fifteenth Amendment for almost a century. Indeed, the cost of rights enforcement is sometimes prohibitively high because pernicious social norms often cannot be broken without resorting to unconscionable force. The difficulty of enforcing civil rights in the teeth of racist habits and beliefs illustrates the point. The right to be protected from racial discrimination is better enforced in the United States military than in our civil society partly because civilians in the grip of racist norms have a greater capacity to resist the commands of authority than racists in uniform.
Rights enforcement is dependent both on coercive authority and on social norms, good or bad. It is limited because coercive authority is limited by scarce resources and because, whereas socially beneficial norms can make rights and coercion unnecessary, socially divisive norms can cripple them both or even render them futile. Legal rights may arise in response to faltering norms, but they will not be respected or enforced in the absence of at least some normative appeal. Prohibition—enacted as a kind of victims' rights amendment designed to protect the families of heavy drinkers—is probably the most vivid example. Likewise, such "welfare rights" as AFDC were ultimately doomed because they came to lack widespread public support.
None of this suggests that there has been a general shift in America away from responsibilities and toward rights. In 1995, despite the many John Redhails among us, more than ten billion child support dollars were collected in the United States, a historic high. In countless other areas, too, people are now much more responsible than they used to be. And their enhanced responsibility often seems connected to a broadening and deepening of rights.
Yet when the state's capacity to protect its citizens is limited, it should be careful about mechanically enforcing moral responsibilities. As one commentator has recently remarked of coarse-grained legal attempts to compel deadbeat fathers to take responsibility for their children, "Poor mothers often break up with the father of their children because he is physically abusive. Once the break comes, they want him out of their lives. If a state agency forces an angry, abusive man to start paying child support, he may reassert his parental rights and begin harassing the mother again. Fanning these embers may not, in fact, be such a good policy." If authorities drag a deadbeat father back into the life of his abandoned child, then they owe the child, and its mother, protection from the physical abuse that may ensue. A responsible government will not declare and enforce the rights of child support recipients if it is not prepared on the spot to pay the costs of such protection. The exercise of legal rights often invites a violent response and the cost of protecting rightsholders from retaliation should certainly be counted among the cost of rights themselves. No responsible political community will extend rights to its citizens unless it is willing to pay these subsidiary costs as well.
By their nature, in sum, rights impose responsibilities, just as responsibilities give birth to rights. To protect rights, a responsible state must responsibly expend resources collected from responsible citizens. Instead of lamenting a fictional sacrifice of responsibilities to rights, one should ask which concrete package of complementary rights and responsibilities is likely to confer the most benefits on the society that funds them.
PART IV:
UNDERSTANDING RIGHTS
AS BARGAINS
Chapter Twelve
HOW RELIGIOUS LIBERTY
PROMOTES STABILITY
WHY DO AMERICANS OBEY THE LAW? Why do most U.S. citizens, most of the time, voluntarily adjust their behavior to intricate legal rules, pay their taxes, show up for jury duty, and go along with the occasionally unreasonable decisions of sundry political and judicial officials? A complete answer to "the compliance question" would no doubt allude to habit, imitation, deference, respect for norms, social solidarity, and the coercive power of the state. But ordinary citizens will not routinely respect the law if they do not also perceive it as legitimate. And that means they must see the burdens imposed by law as more or less fairly shared.
Compliance with law partly derives from a social understanding that the government safeguards and promotes fundamental human interests, including basic individual liberties. That is to say, the enforcement of rights not only presupposes the power to tax and spend, it also helps create popular acceptance of the power to tax and spend. Incumbents create political goodwill by funding rights that citizens want. While the protection of basic rights depends fundamentally upon the actions of the coercive and extractive state, the coercive and extractive activity of the state can be justified in the eyes of its citizens by its contribution to protecting, in an even-handed way, their most cherished interests.
One reason why citizens feel morally obliged to fulfill elementary civic duties is that the system they thereby uphold adequately, though by no means perfectly, defends their fundamental freedoms. Legally and politically protected rights are among the most widely prized public services delivered by the liberal-democratic state. And American citizens willingly shoulder the not-inconsiderable burdens imposed upon them by their national, state, and local governments in part because these governments distribute with a modicum of fairness a whole range of precious public goods, such as firefighting and enforceable private rights. While the state musters protection, citizens reciprocate with cooperation. Cooperation is much less likely to be forthcoming when rights protection is diluted, erratic, or absent, or when government creates and protects rights that should not qualify as such. Incumbents also have a deplorable tendency to stint on the protections they offer to politically weak individuals and groups whose cooperation they do not especially need.
The rights-for-cooperation swap is a perennial theme of liberal political theory, usually evoked with the celebrated metaphor of a "social contract." The government agrees to protect citizens both from each other and from its own rogue officials; in exchange, citizens lend the government their support. Liberal societies are held together not only by habit, authority, shared culture, feelings of belonging, and fear of the police, but also by a widespread perception of mutual advantage. This is one reason why people feel reluctant to contribute their share if others are shirking. Societies flourish when individuals exercise mutual self-restraint, submit themselves to clear rules binding equally on all, and pull their weight in common endeavors. When these pieces are in place, individual burdens are perceived to be outweighed by individual and collective gains.
An empirically oriented theory of rights must consider how individual liberties create and sustain cooperative relations both among groups of citizens and between citizens and their government. Why should citizens willingly defray the costs of rights enforcement? They may disburse from fear, of course, or from habit, without asking why. But they may also perceive these rights to be worth the price. This is what it means to call rights and especially basic rights the cornerstone of the liberal social contract, the source of the legitimacy of liberal political authority. Granted by governments and accepted by citizens in a trading of concessions, rights may even, at a stretch, be deemed bargains. This is not the whole story, but it is a helpful metaphor. Indeed, it is more than metaphor; as a purely descriptive or historical matter, many rights owe their origin to bargains among diverse people seeking to cooperate or at least to coexist peaceably with one another. By attempting to construe them in this way, we can clarify how rights can be regularly safeguarded by government action, even if they cannot, for practical reasons, always be enforced in courts of law.
Some of those who wince at the very mention of the cost of rights may distrust close inquiry into the trade-offs that rights enforcement inevitably entails. They may also object on moral or theoretical grounds to the notion that rights are granted in exchange for civil cooperation, of which tax "contributions" are but the most easily metered example. Neglect of the question of the cost of rights may reflect an even deeper discomfort with the conception of rights as bargains. If rights are unassailable and universal, based on impartial reason and claimable by all rational beings, how can they be reduced to a matter of "you scratch my back while I scratch yours"? And indeed this way of formulating the matter does seem crass and simple-minded. If many rights cannot be bought and sold on the marketplace of commodities—if you cannot sell your right to speak freely or your vote—how can we assimilate rights generally to items in a barter exchange?
The answer is that bargains come in different forms and many of them are neither petty nor ignominious. Mutually beneficial and perfectly honorable trades occur every day—in workplaces, families, and statehouses as well as in supermarkets—and they are not objectionable simply because they are perceived to be advantageous to everyone involved. For instance, to lower the intensity of industrial strife, which might well disrupt interstate commerce, the American government assigns some rights to employers and some rights to employees and then tries to enforce the assembled package of rights and duties against the parties. Rights enforcement, in this case, is actually a strategy of conflict management and a form of bargain enforcement. Indeed, the use of rights to enhance mutually beneficial social cooperation is perfectly routine, as the very idea of a social contract should suggest. Laws themselves, creating and embodying rights, are sometimes best understood as bargains. To say that rights can be seen as services delivered to fulfill the terms of a bargain is simply to affirm that rights-based political systems are stabilized in part—though only in part—by a widespread perception of mutual gain. By setting forth clear rules for resolving disputes without violence and stabilizing social expectations in a heterogeneous society, rights create a particularly stable style of social coexistence and cooperation.
As instruments for improving individual and collective welfare, rights naturally require renunciations of various kinds from all members of the community (not only from public officials), renunciations that ought to be abundantly repaid by the benefits that ensue from reciprocity, specialization, and the pooling of efforts. So the American social contract should not be described simply as a rights-for-cooperation exchange, with government serving up rights and citizens answering with cooperation. The American social contract involves a more deliberative and reflective deal among rights-respecting citizens themselves, between the rich and poor, for example, and among members of discordant religious sects.
Roughly speaking, theorists of democracy fall into two camps: those who see politics as a matter of interest-group deals among self-interested private groups and those who see politics as a process of deliberation and reason-giving. But dealing and deliberating both play roles in the creation of rights, and they are not easy to disentangle in practice. "Bargains" (or agreements) can and do emerge from processes of discussion and reflection, with self-interest, narrowly understood, not being the principal or only moving force. Indeed, social pacts codified into legally enforceable rights often reflect judgments about what is ultimately right and fair, not merely about what is convenient at the moment. But even when politics is dominated by deliberative democracy, so that narrow interests do not overwhelm the process of reason-giving, the quest for compromise and mutual adjustment often play a significant and even constructive role.
Religious liberty can be usefully examined along these lines. A classical liberal right with many abstract or philosophical rationales, freedom of conscience originated and evolved in large measure as a bargain among private social groups meant to ensure intersectarian comity, mutual forbearance, and social cooperation.
"CONGRESS SHALL MAKE NO LAW RESPECTING an establishment of religion, or prohibiting the free exercise thereof." These two rules have proved surprisingly difficult to reconcile. Conflicts between the free exercise clause and the no-establishment clause continue to raise some of the thorniest problems in American constitutional law. When protecting free exercise, by making civil halls available for religious meetings, does the state help establish a religion? When banning religious organizations from receiving generally available funds, does the state obstruct free exercise? Attempts to comply strictly with any interdiction of political support for religion—such as the ban on school-ordered prayer or the refusal to allow religious groups to use school facilities after hours—might be thought to place illicit burdens on free exercise. Similarly, scrupulous attempts to secure free exercise—by paying welfare benefits, for example, to those who refuse to accept employment because it would require them to work on the Sabbath—might seem to violate the prohibition on public support for private belief.
The practical dilemma here, whatever else it implies, reinforces one of the central claims of this book. Even those who seek neutrality about religion insist that the state must provide its standard services—above all police and fire protection—to religious organizations as well as to everyone else. This costs a great deal of money, especially when tensions among religious groups, or between the religious and the nonreligious, are running feverishly high. Because it regularly provides benefits to many nonreligious groups in society, the state cannot achieve a semblance of neutrality without subsidizing religion in numerous contexts. A government policy of hands off (no establishment) does not guarantee individual liberty (free exercise). Laissez-faire provides distinctly inadequate protection of the right to free exercise of religion. Advocates of school prayer must surely acknowledge as much. And those who reject school prayer, too, should be able to see that constitutional rights to free exercise of religion cannot be upheld by a state policy of noninterference, but instead require government performance and taxpayer support.
Yet the costs that religious liberty regularly impose upon taxpayers are poorly understood. One reason is that freedom of religion is routinely perceived as an exemplary "negative" liberty, designed to stay the government's hand, to curtail the power of Congress and state legislatures to interfere with the freedom of individuals to worship, believe, and live according to their consciences. In this conventional view, freedom of religion is designed to shut the government out of a walled-off private space in order to preserve the moral autonomy of individuals dealing with their hopes and fears about (among other things) their own and their loved ones' mortality.
In this context, the doctrine of non-entanglement and the image of a "wall of separation" between church and state are typically invoked to demonstrate that freedom of religion, like all other constitutional rights, requires the state to withdraw rather than to intrude, to refrain from acting rather than to act, to disengage rather than to engage. Freedom of religion, it is said, shelters vulnerable individuals—whether in solitary moments of prayerful reverie or unorthodox dissent or simple nonbelief—from the intrusive and conformist pressures of a potentially bigoted state.
Yet the contribution of religious liberty to individual autonomy should not be allowed to obscure its origin in, and contribution to, peaceable social coexistence. While permitting us to be autonomous in our deepest convictions, religious liberty depends essentially on the smooth functioning of a certain kind of legitimate political authority. It also enables our religiously heterogeneous society to operate passably well, playing a stabilizing role and encouraging social concert. Its obvious utility in a religiously heterogeneous society helps account for the original recognition of the right, for its enormous contemporary importance in America and elsewhere, and for the evident willingness of taxpayers to defray its costs.
Religious liberty is certainly no more costless than other legal rights. American citizens are more or less free to worship or not, as they wish, but their freedom in this respect makes a claim upon the public fisc, even when it is not subsidized out of public budgets (through, for example, police and fire protection of churches and other religious institutions). For instance, religious liberty entitles citizens to judicial remedies (funded by taxpayers) whenever religious freedoms are infringed by public officials. Although the law on this point is shifting and complex, religious liberty may on occasion require exemptions from general law, and exemptions can prove expensive.
Religious liberty is costly, to an even greater degree, because it implies government readiness to intervene impartially whenever serious tensions arise among sects. Religious conflict in America presents a legitimate arena for government regulation, and therefore imposes all the ordinary public costs of monitoring, deterrence, and access to relief. Able to organize themselves and express their views in public, Jehovah's Witnesses can play anti-Catholic records in Catholic neighborhoods. Such freedom will certainly make claims upon the Treasury when, for example, it requires police supervision of potentially provocative expressions of religious fervor. Perhaps more pertinent today, religious liberty requires the government to supply relief if a private sect or church illicitly uses threats of coercion to prevent individual members from exit or apostasy.
Controversies continue to rage about the degree to which secular norms may be imposed on religious communities. Should antidiscrimination principles be applied against churches that, for example, discriminate on the basis of sex? In general American law answers that they should not be, although this answer has not gone uncontested. By contrast, the ban on ordinary civil wrongs—assault, battery, trespass, and so forth—is fully applied to religious organizations. However one judges the Court's 1990 decision to uphold the criminalization of peyote use in American Indian religious ceremonies (on the ground that the law was neutral and did not discriminate against religion), freedom of religion in the United States would soon become illusory if religious cults operated freely behind a wall impenetrable to government officials, where no coercive public authority could be brought to bear, for example, on unscrupulous or mentally unhinged cult leaders. Because rights are potentially dangerous powers, a government committed to enforcing rights must also work to ensure that they are not seriously abused.
Religious liberty is tolerable only when it is practiced according to certain rules of civilized self-restraint, rules that must be coercively enforced. Religious groups in America are forbidden from using coercion or threats of coercion in pursuing their religious convictions, for the right to do so belongs exclusively to the government. The government alone wields this right because it is, or claims to be, the only agent that represents the interests not of one sector, region, class, or sect in society, but rather the public interest, that is, the broadly shared interests of all citizens without exception, whatever their religious beliefs.
Superficial appearances aside, freedom of religion has nothing to do with state inaction or paralysis. It does not even imply that government must refrain from action (refrain, say, from enforcing laws against murder, or rape, or possessing hallucinogens) within the confines of church property. Rather than prohibiting state interference in sectarian affairs, the First Amendment simply regulates the manner and scope of such action. More profoundly, religious liberty provides a script for publicly subsidized cooperation and mutuality. It stipulates, for example, that citizens in a multidenominational America, when acting through the shared instrumentalities of their government, cannot publicly proclaim that adherents of minority sects and nonbelievers are second-class citizens or somehow unworthy or unwelcome members of the community.
The phrase "when acting through . . . government" deserves further commentary. The principal threat to religious liberty is not the government as such, but rather private religious sects that might, if the opportunity arose, employ the instrumentalities of government to enforce their sectarian beliefs on unwilling fellow citizens. To protect religious liberty from "government interference," as a consequence, is actually a roundabout way of protecting religious liberty from infringement by private parties.
In reality, freedom of religion needs to be protected less against the government than against the intolerant and domineering arrogance of private sects. This indirect effect of religious liberty clearly bears upon the thesis that constitutional rights protect individuals only against state action, not against private parties. Even if technically true, this legalistic claim needs to be interpreted realistically and in context: the First Amendment actually protects individual liberty from private interference through sectarian capture of government or "under color of law." A system of religious liberty must always include rules of mutual restraint that aim to prevent self-righteous citizens from making certain divisive, humiliating, and community-embittering gestures, not in all contexts, but while occupying specific roles—for instance, when acting as public-school officials. Laws that lack a secular purpose, such as those requiring the teaching of creationism in school, are unconstitutional because they give public power to private sects in a context crucial for the future peaceful coexistence of America's many religious and nonreligious groups.
As a matter of history, one of the original goals of the bar on sectarian teaching in public schools, entrenched in American constitutional law, was to shield such schools from being rent by denominational conflict. Separation between church and state helped create a common institution where socially dominant Protestants would have to learn that they could not exploit their majority status or sneer at Catholics and others as un-American. This required affirmative government efforts, in less than totally favorable circumstances, and not a policy of hands-off. It was also, incidentally, an unembarrassed attempt to legislate morality or at least to inculcate a form of moral self-restraint, which is an important aspect of morality.
The publicly funded schoolhouse was consciously fabricated as neutral territory—neutral not in the sense of free of values, but in the sense of protective of multiple convictions, remaining neutral among them. It was meant to provide an apprenticeship in coexistence, tolerance, and common action—not, of course, to teach moral skepticism or relativism. The channeling of ostensibly public resources for the educational benefit on one sect alone, by contrast, would not merely have fostered political divisions along religious lines, but would also have spoiled the implicit social pact that allows members of rival confessions in America to feel that, differences aside, they are all in the same boat and that they share a common government designed to seek out and pursue common purposes. This is the sense in which religious freedom can be seen as a social pact among churches and sects in which the government serves to some extent as a broker rather than a partner. (This is not to deny that government officials themselves benefit from cooperative relations among rival denominations.)
Religious liberty is one of the central means by which the multidenominational United States handles its inner diversity. Our pluralistic society, we might say, is held together by a division. The "barrier" between church and state has a positive, not merely a negative, function. It permits and encourages common citizenship despite religious pluralism, allowing citizens to disagree about ultimate matters while concurring on penultimate ones. Americans can disagree about "the good" (that is, the personal and religious ideals they deem worth pursuing), while agreeing about "the right" (the rules of justice that govern nonviolent coexistence and cooperation in a world of scarce resources).
Social cooperation in a heterogeneous society, including the ability to display a degree of tolerance and mutual respect, presupposes that people can put to one side their most fundamental disagreements and concur instead on more abstract or more particular matters. Citizens with different religious backgrounds and beliefs can commit themselves to religious liberty, or to the Constitution as a whole, even though the grounds supporting that commitment are quite diverse. And citizens with different religious convictions can agree on a number of particular practices from their diverse starting points. The fundamental rights of Americans can be agreed to by a heterogeneous citizenry whose adherence to common rules is supported by a wide array of attitudes and beliefs.
When acting through the state, which exercises a rough monopoly over the legitimate means of coercion, Americans are asked to put aside for the moment their conflicting religious convictions. But when acting outside state channels—through nongovernmental groups and in ordinary social contexts—they can freely act upon, or act out, their religious beliefs. Freedom of religion is far from exclusively individualistic, in other words. It necessarily includes the highly social freedoms to worship together, to preach and proselytize, and to found new churches and sects. To the extent that it involves social organization and public interaction, freedom of religion, like any other permission to act, raises the possibility of conflict among individuals and groups. And it is here that most of the public costs of maintaining religious liberty arise.
As the case of abortion reveals, religious disagreements can sometimes burst irrepressibly into public debate. But even such a seemingly irreconcilable conflict as the American abortion debate, where ultimate values are at stake, has not poisoned all social communications in America or rendered impossible the democratic resolution of other problems. The abortion controversy has been largely kept within relatively moderate bounds, for religious and unreligious Americans alike understand the delicate pact of mutual tolerance on which their polity is based. That truce or process of mutual adjustment and self-restraint, far from being demeaning, is a premise of our common life. Those committed to it are following principle as well as expediency.
Bargains are moral relationships, as well as strategic ones. That is because bargains implicitly encourage each party to see itself as a part of the whole, as only one legitimate claimant among others. According to the American social contract, when I assert my freedom of conscience, I am simultaneously affirming that all other citizens, whatever their private creed, enjoy one and the same right. This reference to reciprocity and fairness across individuals—which naturally restricts what any single individual can justifiably do or claim—is implicit in every assertion of a constitutional right under American law. Impartiality and fairness in this area not only help sustain social comity but also illustrate why it is so misleading to construe eighteenth-century rights as intrinsically selfish and antisocial.
Chapter Thirteen
RIGHTSHOLDERS AS STAKEHOLDERS
IT IS IMPOSSIBLE TO UNDERSTAND THE PLACE OF PROPERTY RIGHTS in the American social contract without asking how such rights affect those members of society who possess little or no property. How can the propertyless—to ask about the most basic form of social cooperation—be deterred from looting and burning? The criminal justice system helps safeguard accumulations of private wealth from the indignation of the poor as well as from the greed of the unscrupulous. But to protect property rights by coercive means alone, the state would have to wield frighteningly massive amounts of lethal force. Such vast and discretionary police powers would not only cost property owners dearly; it would also make them feel constantly vulnerable to rogue authorities. So the practical question, for would-be accumulators of private wealth, is how to deter theft and arson without relying exclusively on coercion.
How can the state be made strong enough to protect property rights, but not so strong that its lethally armed officers will be tempted to violate property rights for their personal enrichment? This puzzle, which touches the essence of liberalism, can be answered best by posing a second question: how can wealth be made legitimate in the eyes of poverty? Alternatively, how should the government treat the poor if one of its principal goals is to protect effectively, but with a minimal amount of abusable coercion, the property rights of the rich?
A full answer to this question would refer to publicly financed education, decent opportunities to enter the job market, the widest possible diffusion of private property, and many other state-managed allocations of collective resources. The disadvantaged will be far likelier to contribute to a common good if they believe that the privileged, too, are contributing their fair share. And a prudential approach to poverty will surely include giving the propertyless enough food from the table to prevent them from falling into rage or despair. The most ardent advocates of private property might try to ensure that everybody has some basic nourishment and shelter. Alleviating extremes of desperation among the poor may also stem from moral principles, sheer compassion, or fellow feeling, but since the castle is not safe if the cottage is starving, poverty relief sometimes emerges, perhaps most reliably, as a rich man's strategy of self-defense.
Because welfare transfers from the rich to the poor have been traditionally motivated by fear of worker radicalism, they have tended to lose middle-class support after the shrinkage of the industrial workforce and the disappearance of communism as a seemingly viable alternative to capitalism. But for owners to receive reliable protection for their property while the government obtains a steady flow of revenue, both incumbents and owners still need the cooperative self-restraint of indigent people, especially indigent young men. The underlying motivation here is easy to grasp, for, as Judge Richard Posner has remarked, "poverty in the midst of plenty is likely to increase the incidence of crime." True, the wealthy can respond to this problem in a purely private manner. The moderately well-to-do can retreat into gated communities where they can effectively insulate themselves from the consequences of lower-class despair. But this is not a happy strategy even for people with money: insulation is costly, and not only in terms of dollars. If this becomes a trend, of course, social cohesion will be at risk, and it is safe to say that middle-class support for welfare programs will continue to wane.
WELFARE RIGHTS AS A SOCIAL BARGAIN
"Rights talk" in the United States is exuberantly partisan. Indeed, the political affiliations of Americans are good predictors of which rights they favor and which they disfavor. Economic conservatives want to beef up property rights while watering down welfare rights. Religious conservatives praise the right to life and condemn total separation of church and state. American Civil Liberties Union (ACLU) liberals support freedom of speech and censure the right to school prayer. Welfare-state liberals favor entitlements to public assistance and disfavor the right of companies to close plants whenever they wish.
We might even say that political positions in the United States are largely defined by the decision to propose or cherish some rights and censure others. And often arguments for or against certain rights are supported by careful arguments. But the advocate of any given right has a vested interest in making it seem that his favored right inhabits a pristine, extrahuman orbit of "law," or "the Constitution," into which bargains and clashing political values never intrude. This conceit cannot survive examination. The American rights debate is a debate about appropriate bargains and values; it is fueled by partisan passions and conflicting moral judgments and commitments. So how has American politics nevertheless achieved the relatively consensual character upon which foreign observers so frequently remark? Could it be that the American consensus, to the extent that it exists, will survive if and only if all important social groups feel they have something to gain from mutual forbearance, that is, to the extent that each is granted some important cluster of worthwhile rights?
Even in the absence of any kind of poor relief, private ownership may engender spillover effects beneficial to the poor. Job creation is one of the most persuasive arguments in favor of private ownership publicly guaranteed. The taxation-for-protection contract, encapsulated in reliably enforced property rights, is often and accurately said to confer many palpable side benefits upon the nonrich, not only new jobs but also economic growth in general, diminished costs of subsistence goods relative to wages, and an economically vibrant counterweight to tyranny (which would inevitably harm everyone, including the poor). Moreover, the provision of opportunity and assistance to poor people always touches upon publicly shared conceptions of justice. A fair society tries to guarantee reasonable opportunities for all and also to ensure that no one drops beneath a decent floor. This is part of what is meant by the central liberal idea of society as a cooperative venture.
Unless society is organized as a cooperative venture, private property as we know it cannot be created and maintained. Large American corporations could never have developed their current wealth and power without many kinds of government support. Similarly, wealthy and successful individuals owe their riches and success to social institutions that, while demanding cooperation from all, distribute rewards selectively and unequally. A capitalist economy provides the legal preconditions for the unequal accumulation of wealth. Such unequal accumulations do not fall from the sky. However hard people work, it is always an oversimplification to attribute differences of acquired wealth solely to the wealthy's "own efforts." People begin from massively different starting points, and someone born on one street in, say, Chicago, New York, or Los Angeles may have much worse life prospects than someone else born a mile away. In any case, private exertions take the form they do, and are rewarded as they are, only because of institutional arrangements that are politically chosen, administered by government and enforced through law. Arrangements that spawn unequal accumulations of wealth can certainly be justified on liberal principles, at least if they generate advantages for most. They can also be adjusted—without any offense to these principles—to ensure that some of this amassed fortune is diverted to provide minimally decent opportunities and well-being for ordinary citizens. Indeed, the very objective that justifies those arrangements—the promotion of human well-being—also argues for adjustments designed to help those who are otherwise disadvantaged. Those adjustments are part of a social bargain that, if it works well, works to the benefit of all.
Without such modest assistance, Americans born into poverty through no fault of their own might begin to interpret our social contract, whose rules they are in any case forced to obey, as a giant swindle perpetrated by the well-to-do. Something of this sort has happened before in our history. It may be happening again today.
That the rich—owing their wealth, in part, to cooperatively maintained law and government—should pay for the voluntary self-restraint and cooperation of the impoverished, rather than trying to cow them into a facsimile of self-restraint, is forcefully asserted by even the most impeccably liberal theorists. For instance, John Stuart Mill wrote that "since the state must necessarily provide subsistence for the criminal poor while undergoing punishment, not to do the same for the poor who have not offended is to give a premium on crime." The right to a bottom floor of subsistence may well provide an incentive to self-discipline and cooperative behavior. There is nothing particularly eleemosynary about poor relief, when viewed from the property owner's perspective. Some forms of poor relief are compelled by abstract principles of justice. Much of it is supported by fellow feeling. But welfare benefits can also be understood as a tactical side payment to the poor attached to the original taxation-for-protection deal struck between property owners and their government.
PROPERTY RIGHTS AS A SOCIAL BARGAIN
Political scientist Theda Skocpol has argued persuasively that the American welfare state originated in the extensive system of veterans benefits that grew out of the American Civil War. That welfare rights were first introduced as veterans benefits not only helps explain taxpayer willingness—at least in wartime or in the wake of war—to defray their costs. It also lends credence to the more general thesis that rights are politically stabilized as part of social bargains. It is therefore striking that, in the common-law tradition, property rights themselves originated as veterans benefits.
To simplify a complex story, William the Conqueror created property rights, as they eventually came to exist under the common law, when he distributed plots of seized lands to the Norman noblemen who had helped him overrun England. Common-law property rights, as enforceable in court, did not descend from high principle but were rather rough-hewn in a process of social give-and-take. This historical curiosity fits well with the fact that, as a matter of current legal reality, property rights, far from being rigidly fixed, remain subject to considerable renegotiation.
The enforcement of property rights in the United States is sustained partly by a mutually beneficial taxation-for-protection exchange between owners and government. Owners are willing to be taxed, to some degree, in order to have their property reliably protected against roughneck vandals and roving bandits—not to mention against deliberately kindled or accidental fire. For its part, the government is willing to refrain from imposing confiscatory tax rates, not only because of political incentives, but also because public officials understand that reliable long-term revenues will be augmented if citizens are encouraged to accumulate private wealth, keep honest books, and bank and invest their earnings inside the country, or at least within the purview and reach of the IRS. This cooperative relationship increases the security of both parties, extending their time horizons and permitting both to undertake long-term planning and long-gestation investments.
In this sense, property rights represent a selective application of public resources not only in order to encourage self-restraint on all sides—the government must refrain from confiscation and private owners must refrain from hiding their assets and acquiring property by means of force or fraud—but also to elicit new forms of creative activity, from both government and private individuals. Such socially beneficial inventiveness is unlikely to emerge under conditions where transactions and acquisitions are nerve-wrackingly insecure. Because both sides profit, the bargain can be self-enforcing and stable over time. Although the government ordinarily cannot be sued in court for failing to ensure the property rights of particular people against burglars and incendiaries, public officials who are "soft on crime" can be ousted from office.
The right to property should also be understood as an indispensable condition for democratic citizenship. The latitude, more or less broad, to accumulate private wealth is justifiable, despite the considerable inequalities it necessarily entails, partly because a decentralized and unplanned economy helps provide a reliable material basis for an unintimidated political opposition. If property can be confiscated at whim, people are not likely to have the independence and the security that will permit them to criticize the government openly. The high status of the right to property in the American system of governance reflects a general understanding that citizens can best deliberate together if their property is shielded from public officials. This is yet another way in which the right to private property serves the common good.
RIGHTS AS STRATEGIES OF INCLUSION
The prudential, as opposed to moral or humanitarian, origins of public assistance have been copiously documented. Modern public health and sanitation programs were launched in burgeoning cities because the wealthy, although they could afford the best doctors that money could buy, could not thereby protect themselves from contagious diseases ravaging the poor. Similarly, health care for workers serves the needs of employers. Regular employment and homeownership reduce the level of social instability and violent crime. Effective consumer protection can stimulate consumer demand. But by far the best reason to think of public provision as part of a social bargain is the above-mentioned origins of welfare in warfare. War in general accustoms citizens to higher tax rates, the revenues from which are then used, in peacetime, for social programs of various kinds. This development will be understandable only if we interpret welfare rights in part as bargains, as concessions granted to groups whose cooperation is necessary or desirable. In wartime, especially, property owners are confronted with their radical dependency on cooperation from the citizenry at large, especially the poor.
When those with little or no property are reluctant to fight fiercely against foreign looters and conquerors, the property rights of the rich are of little worth. For prudential reasons alone, property owners have an incentive to prevent the impoverished from feeling alienated from the polity. For their own purposes, moreover, they need to mobilize the poor, not merely sedate or placate them. To enlist the active support of the indigent, rather than merely their inert acquiescence, governments need to make palpable gestures of inclusion. Far from being negative protections from governmental interference, civil rights—such as the right to vote, the right to a fair trial, and the right to publicly subsidized education—are ways of pulling excluded individuals into the community.
A foreign example, once again, may help illustrate the way in which legal rights function to promote civic inclusion. To the great surprise and frustration of human rights workers from the West, gypsies in Eastern Europe, whose basic survival strategy involves a scrupulous avoidance of direct contact with political authorities, often refuse to protect their rights by going to court. People who go to court, after all, must register their names, occupations, and whereabouts and provide other sensitive information to public officials. To assert one's rights is to enroll in the state's decision-making apparatus, and that is exactly what many of Eastern Europe's gypsies refuse to do. To avoid a potentially dangerous form of co-optation by public authorities, they willingly forgo their rights. They perfectly understand that constitutional rights, far from walling off a zone of private liberty beyond the reach of the state, are integral parts of a social contract on the basis of which government agencies extend their authority to virtually all sectors of social life.
Both property rights and welfare rights represent efforts to integrate differently situated citizens into a common social life. Far from eschewing all contact with government, the holders of property rights are indispensable partners of the modern liberal state. Institutionalized in memory of the last war and partly in anticipation of the next, welfare rights—involving cash transfers, medical care, food, housing, jobs, job training, or some combination of these—are one means among many to make the disadvantaged feel they, too, are participants in a shared national venture. Because all parties benefit, such a conjunction of property rights and welfare rights can be self-enforcing and stable over time.
Like wealth, poverty in America is in important ways a product of political and legal choices. Our law of property—which includes rules governing inheritance—determines who "lacks resources." Without government and law, some of the propertyless would quickly be able to procure considerable resources by private violence or stealth. That they do not do so more than they do is partly a product of legal coercion and social norms, but also of perceived mutual advantage. None of this is meant to deny that personal initiative, industriousness, thrift, and self-reliance are important virtues. Some people are poor because they lack such qualities. But if existing distributions of resources are a function of law, then a sensibly designed welfare program is a coherent part of a liberal-democratic polity, rather than an inexplicable departure from its core assumptions.
BARGAINS AND EQUALITY
To conceive rights as benefits funneled to citizens in exchange for political support may seem to violate the principle that rights must be impartially enforced. Do not all American citizens, even those with little of value to offer in an exchange, deserve to have, and have in fact, equal rights? After all, we do not reserve the right to a fair trial only to those who make tangible social contributions—to the healthy, say, but not to the chronically ill. And the right to vote is not restricted to the prime "stakeholders" in the country, that is to say, to property owners or those who pay hefty taxes.
Admittedly, the metaphor of bargains may seem to conflict with the moral promise of human equality. Bargaining suggests that our public authorities will prove most eager to guarantee valuable rights to those capable of rendering the government (or its incumbents) a needed service in return. To construe rights as legally enforced social bargains implies that the rich and powerful, for no convincing moral reasons, are likely to get better value from one and the same set of rights than the poor and the powerless receive. To think of rights as bargains is to expect that more prominent stakeholders will, in fact, reap larger dividends. For example, if welfare benefits represent a quid pro quo, then welfare cuts will fall most heavily on those with little political clout. In times of fiscal austerity, if rights are bargains, those who do not vote or make campaign contributions (say, recipients of food stamps) will suffer a greater loss of rights than the more influential beneficiaries of Social Security and Medicare.
While this picture is morally unappealing, it has a good deal of descriptive power. It is certainly not remote from the actual state of affairs. In societies such as the United States, which are generally and correctly judged to be free, the rich and the powerful enjoy many advantages not shared by the impoverished and the feeble, including advantages associated with the enforcement of their rights. While the rich use their private wealth to buy exquisite or tasteless luxuries, they also spend it to leverage better results from their civil liberties and basic rights than the moneyless can hope to achieve. They can hire private security personnel to improve the protection of their persons and property. They can exercise their constitutional right to have an abortion, even without government financial assistance. They can send their children to religious schools, which the indigent cannot always manage, even though the latter, too, are supposed to have their freedom of religious conscience constitutionally guaranteed. To exercise their freedom of speech the wealthiest citizens can purchase access to the mass media. To exercise their freedom to choose their public officials, they can make massive campaign contributions. And the well-off can notoriously hire the shrewdest lawyers and thus take disproportionate advantage of rights constitutionally assigned to everyone, but in a way that their less well situated fellow citizens cannot conceivably afford.
Imposing private costs—in the form, say, of user fees—is a standard way of conserving scarce resources, such as access to an institution that resolves conflicts. But screening techniques that impose private costs prevent only the poor, not the rich, from instigating frivolous appeals. True, the "contingency fee" system (whereby an attorney agrees to collect a fee only if a suit for damages turns out to be successful) is available for some cases, and it provides some poor people with the key to the courthouse. Judges also occasionally help poor litigants by awarding costs. Nevertheless, it is generally true that wealthy people derive far more than fair value from their supposedly equal rights. It would certainly be implausible to suggest that legal aid to the poor fully redresses the imbalance of resources between indigent and wealthy defendants.
This partiality of supposedly impartial rights to those endowed, for whatever reasons, with private resources is troubling. Certainly a good deal can and should be done to improve the situation, including, for example, better campaign-finance laws, better public monitoring of police abuse, and better legal services for the indigent. But a society in which private wealth could never be used to boost the use-value of "equally protected rights" would not be a free society in the way Americans use this term. To level the playing field so that all criminal defendants received roughly the same quality of legal counsel, for example, regardless of their personal assets, would require an unacceptable degree of governmental supervision and discretionary coercive control. A government capable of entirely neutralizing the influence of private resources on the value of individual rights would have to be so immensely powerful, in fact, that even the trivial misuses of its powers that would be bound to occur would probably be worse for most citizens (including the poor) than the inequalities it was ostensibly established to abolish.
The American social contract is a swindle to the extent that it leaves poor citizens below a decent floor. But helping the poor does not entail abolishing inequality. What the poor want most, after all, is not equality but help, and this they sometimes can and do receive (we continue to argue about how much and in what form) under various welfare, education, and employment programs. The effort to counteract desperate conditions, and to ensure that everyone has minimally decent life prospects, should not be confused with egalitarianism as a political creed.
Inequality of results will always be inescapable so long as rights impose private as well as public costs. Every American citizen has the right to sue the police for civil damages, but only a party with substantial private resources has a fighting chance to do so successfully. Those most likely to suffer police abuse ordinarily have no such resources and hence, in practice, enjoy no such right. Freedom of speech and the press, the right to legal counsel, the right to choose one's public officials, and even freedom of conscience are all enhanced by the superaddition of private resources to those already provided from the public budget. That the supposedly equal right to acquire private property is taken advantage of by some individuals more fully than by others does not, presumably, require extensive commentary.
But individual purchasing power is not the only source of bias in the allocation of private-law and constitutional rights in the United States. Vital public services are allocated unequally because the weak and the poor, being relatively disorganized, have too little political leverage to obtain their share of public resources. Unfortunately but inevitably, whenever money is distributed, power is likely to have some influence on who loses and who gains. Politically untouchable expenditures are usually those that provide special benefits to well-organized social groups. As government-managed services, rights are no more likely to be showered upon all citizens equally than public works are to be divvied up fairly among diverse localities with unequal lobbying power or skill. This observation is not meant to be casually cynical, however, or complacent or resigned. The dependency of rights on power does not spell cynicism because power itself has various sources. It arises not from money or office or social status alone. It also comes from moral ideas capable of rallying organized social support. Civil rights activists worked hard to mobilize support for their ideas because they unsqueamishly acknowledged that rights depend on social organization and political power. And the unquestionable contribution of the civil rights movement to the protection of civil rights for African Americans corroborates the thesis that rights reflect effective politics, and not merely the dictates of moral conscience.
"Equal protection" under a liberal regime, or at least the moral obligation to protect the rights of the weak, can have a serious and palpable meaning. But this meaning will not be discovered or made plain if we shut our eyes to the powerful inequalities of influence that pervade all societies, even liberal ones.
Chapter Fourteen
WELFARE RIGHTS AND THE POLITICS OF INCLUSION
INDIVIDUAL FREEDOM, however defined, cannot mean freedom from all forms of dependency. No human actor can single-handedly create all of the preconditions for his own action. A free citizen is especially dependent. He may feel "independent" when he goes into a do-it-yourself store and buys a do-it-yourself kit. But his autonomy is an illusion. Liberal theory should therefore distinguish freedom, which is desirable, from nondependence, which is impossible. Liberty, rightly conceived, does not require a lack of dependence on government; on the contrary, affirmative government provides the preconditions for liberty. The Bill of Rights is a do-it-yourself kit that citizens can obtain only at taxpayer-funded outlets.
On the basis of a democratically enacted statute, such as the Federal Tort Claims Act, an individual citizen may enter a court and sue the government for violating her rights. When so doing, she is acting as a free citizen even though her individual action presupposes prior state action. So a liberal nation cannot obliterate the dependency of individuals and subgroups on the state. But why should it try? Dependency of certain sorts is facilitative, not debilitating, especially if laws are democratically revisable and politicians are democratically removable from office. My right to vote or make a will depends on government provision of legal facilities serving these ends. When government declines to furnish such legal facilities (as it does, say, when it denies marriage licenses to gay couples), it is, rightly or wrongly, denying individual rights. What advances individual liberty is not nondependence on law and government, but a certain style of dependency, one that encourages personal initiative, social cooperation, and self-improvement.
Public education, provided to all, and not only on the basis of ability to pay, is only the most obvious example of affirmative state assistance, funded collectively and designed to foster individual and group self-help. Property rights have a similar purpose and result. This insight should also encourage us to redesign our regulatory and welfare programs, not in order to eliminate dependency (which is futile), but to create the kind of dependency that fosters self-help and makes it possible for most people to lead decent lives.
What Americans cherish as "independence" is actually dependence on a certain set of (liberal) institutions. I can escape the corkscrew of the local strongman—that is, be independent—only if I have the public power on my side. Empty treasuries and debilitated administrations make a mockery of paper rights. We do not have to look to foreign shores to see this point. What do we observe when we look into our poorest neighborhoods? Do indigent Americans really have the very same rights as the rest of us (freedom from unreasonable searches and seizures, protection against police abuse, the right to a fair trial), plus a wide array of welfare rights delivered at no expense to themselves? In reality, many inner-city Americans live without enforceable rights because, having been virtually abandoned by their government, they are virtually stateless.
Wealthy Americans are seldom neglected in this way. The Americans who most genuinely "shift for themselves" are neither wealthy homeowners nor recipients of public aid, but rather those among the homeless who eschew shelters and soup kitchens in favor of garbage cans, subway grates, and spare change. To say that such individuals shift for themselves is to say that they have little access to the legal machinery that could protect them from undeserved institutionalization or from assault by teenagers with baseball bats and gasoline cans.
Liberal governments must also prevent the disparity between luxury and misery from growing so glaring that class hatreds begin to threaten social stability and the regime of private property itself. One way to head off such dangers is through publicly funded education, designed to provide the means for individual self-development and, when necessary, for escaping desperate family conditions. But government can also respond to the threat of tensions between haves and have-nots through various antipoverty and job-training programs. The United States's largely successful earned income tax credit (EITC) is a good example. In addition, government can support a mortgage system with the tax code and by legally backing up the power of private banks to evict defaulting borrowers. A well-organized mortgage system, in turn, can spur construction and allow more and more moderate-income families to become owner-occupants and thereby to join the politically reliable middle class, widely defined.
So prudence has no fundamental quarrel with morality. Welfare rights can be fair as well as expedient. To some extent, they can be expedient only because they are perceived as basically fair. And as anyone who has ever run an office knows, fairness is not merely a moral norm; it is also a powerful management tool. Without it, group morale and the inclination to pitch in will dwindle or collapse. That the same is true on a national scale is powerfully suggested by the efficiency gains in revenue collection when taxes are perceived to be roughly equitable.
The obvious partiality of supposedly impartial rights toward those with private resources raises a problem of political legitimacy. Marxist writers (among others) direct our attention to this difficulty by deprecating basic rights as "merely formal," as scams perpetrated on the many, of genuine value solely to the few. All the poor person receives from capitalist democracy, allegedly, is "the right to sleep under a bridge at night." This is a gross exaggeration, but not one to be breezily dismissed. Indeed, if supposedly impartial rights accrued solely to the advantage of the rich, the American government's vital claim to represent society as a whole, rather than being a tool of special interests, would not only be tarnished. It would crumble.
The American social contract can hold only to the extent that all influential economic, racial, and religious groups believe that they are being treated with respect and rough fairness or, at least, that they are receiving a palpable return for their cooperation, collaboration, and self-restraint. Hence if one powerful sect captured the government and used it exclusively for partial or sectarian purposes, other citizens in a multidenominational country would correctly infer that an underlying social contract had been breached. And if glaring discrepancies between luxury and misery destroy the sense that all citizens are somehow in the same boat, as they threaten to do in the United States today, the government's capacity to enlist necessary social cooperation for its policies will founder.
The state's concern for political stability occasionally leads it to infringe upon otherwise constitutionally protected rights, as when the FBI employs wiretaps in response to an alleged terrorist threat. But the principal expression of the state's overriding interest in political stability is the precise balance of rights it positively protects. In an endeavor to stabilize a system of private property, the American system provides, or at least attempts to provide, the propertyless with a form of compensatory "security" that operates as a psychological equivalent to reliably enforced property rights. A democratic government cannot possibly equalize the capacity to take advantage of all the rights that it claims to guarantee. But it can modify the corrosive impression that the reliable rights "of all Americans" belong exclusively to the rich. It can do this, for example, by providing legal counsel free of charge to the indigent, by providing education for all children, and by ensuring that poor people receive food, shelter, decent health care, and employment opportunities.
At the risk of oversimplification, the public protection of the private rights of property owners can be understood as the following sort of bargain: the government first lays down, interprets, and enforces the rules that assign property to particular individuals, and then it provides security of possession to owners in exchange for political support and a steady flow of revenue. The delivery of welfare rights (understood capaciously to include more than cash transfers) is part of an ancillary exchange by which the government and the taxpaying citizens recompense the poor, or at least give them symbolic recognition, for their cooperative behavior in both war and peace. Most importantly, welfare rights compensate the indigent for receiving less value than the rich from the rights ostensibly guaranteed equally to all Americans.
Entitlement programs cost the American taxpayer $700 billion in 1996. This astronomical expenditure, which accounted for 30 percent of the budget, was not simply an expression of fellow feeling or a logical corollary of principles of justice. Rather, entitlements can be shaved back but not eliminated entirely because they lend legitimacy both to the property rights of the wealthy and to the state apparatus that enforces them. In this sense, they are a bargain among social groups in which the government of the day acts as a go-between.
Seen in this way, such rights represent an unsentimental politics of inclusion, slightly mitigating, not abolishing, the disparities of wealth incident to a liberal economy. One might even say that social welfare programs create a modern version of the ancient "mixed regime," a system that gives both poor and rich a stake. The contemporary mixed regime, however, is inscribed not in the organization of powers, as it was in ancient Rome (the Senate representing the patricians and the Tribunes representing the plebes), but rather in the expanded list of basic rights. The modern class compromise is reflected in the combination of property rights and welfare rights now characteristic not only of the United States but of all liberal-democratic regimes. Whether these rights are constitutionally entrenched, as in most European countries, or left to public policy, as in the United States, is of no particular import for the perceived value and stabilizing effect of the modern exchange of property rights for welfare entitlements.
If welfare rights in America really are granted in exchange for social cooperation, then one should expect benefits to flow to the best-organized groups among the relatively disadvantaged. One might even expect the most successful welfare programs to be those beneficial to the "middle classes." This is in fact the case. The most successful American welfare programs are organized not as bargains between classes, in fact, but as parts of an intergenerational contract among members of the middle class, broadly defined.
Most Americans spend two-thirds of their lives working. The earning cohort, through the government accountable to it, voluntarily devotes a substantial percentage of its income to supporting both the young, through publicly funded education (costing many millions of dollars; precise figures are hard to obtain), and the old, through Medicare ($130 billion in 1996) and Social Security ($375 billion in 1996), programs that consume a large and growing share of federal revenues. This intergenerational redistribution, or system of rights, is sometimes advertised as a payback scheme, but it was never designed so that individual contributors would take out what they originally put in. Instead it is a transfer plan that presupposes that the donor cohort imaginatively identifies with previous and subsequent generations. To keep the country going, working taxpayers swallow genuine losses in exchange for gains by the young and the old. Of course, debates go on about the appropriate content of the Social Security system, and serious changes are currently afoot. But in its broad contours the system is stable and widely accepted, and the public support it receives is an important commentary on the moral economy of the nation.
The bargain between rich and poor in the United States, not surprisingly, is less robust. The relevant sums are far lower when poor people alone are the recipients; thus, for example, $82 billion was allocated to Medicaid in 1996 and $27 billion for food stamps. Some conservatives argue that programs designed to help the poor are objectionable in principle—simply because they are funded by "takings." Others say that welfare benefits are counterproductive in practice. While the first objection makes no sense, the second must be tested empirically. Lukewarm public support for programs that target the poor alone has a further implication. While it may sound fair or prudent to restrict entitlements to the most desperately impoverished, programs that prove to be of no benefit to members of the middle class or others with political clout risk becoming extremely attractive candidates for the next round of budget cuts.
MAKING SENSE OF WELFARE RIGHTS
That free governments regularly provide public services, make selective investments, design incentives for self-discipline, and broker bargains for improving social cooperation should not be controversial. What needs stressing is that governments do all these things when enforcing rights. All governments develop techniques for handling social conflicts and eliciting social cooperation. Liberal governments typically do so by creating, assigning, and enforcing rights. As a historical matter, many basic rights enjoyed by Americans today grew out of social bargains ensuring fruitful collaboration on a national scale among highly diverse groups. This is true for religious liberty, private property, and social welfare guarantees.
Some European constitutions guarantee all citizens a right to publicly financed education up to a certain age. In practice, Americans have a similar system of guarantees, even though access here to free education is provided not by the federal government under the national Constitution but by the states. Whether or not the right to an education is guaranteed in a particular state constitution, publicly funded education is far from being an alien or anomalous presence within the political culture of the United States. It is not viewed with suspicion and dread even though it requires the government to tax and spend. It is not seen as an insult to individual agency, or as part of a cult of "victimology." Publicly funded education is simply one method among others by which the country makes long-gestation investments in the human skills necessary to keep it afloat. In this sense, investment in education closely parallels investment in the enforcement of property rights and the protection of owners from arson and acquisitive crime.
If we want to know whether or not the United States can afford investments of this kind, we should not simply examine the contents of our collective bank account. We also have to calculate the expected returns to society over the long run of spending its money this way. Taxpayers invest more or less willingly in education, just as they invest in police protection, because both are thought to pay off in the long term. Both seem worthwhile investments, among other reasons, because they increase the self-discipline and cooperative behavior of citizens and, not incidentally, expand the tax base. Education may be an intrinsic good, but it is good for instrumental reasons as well.
This good, in a liberal society, is not distributed solely according to market principles. The nation's educational efforts are not concentrated exclusively on those who are "willing to pay." We train talented young people from all ranks of society to become heart surgeons and aeronautical engineers, rather than simply auctioning off such training to the children of parents who are in a position to make the highest bid. The nation enlists talent for collective purposes wherever such talent can be found.
How can the community help the poor without making them unduly reliant on community help and discouraging their own capacity for self-improvement? The most common and persuasive criticism of the regulatory-welfare state concerns incentives to antisocial behavior and other undesirable side effects. But "dependency" in and of itself should not be considered one of them. There are different kinds of dependency, and not all of them are bad. Although police and fire protection definitely make citizens dependent on "public assistance," such paternalistic support also increases the willingness of private individuals to embellish and add to their holdings. Publicly funded education, when operating well, has the same effect. It, too, is a form of state help designed to foster self-help. The question is not how to eliminate state intervention, but how to design welfare programs to enhance autonomy and initiative.
An early example of a successful American antipoverty program was the Homestead Act of 1862, which freely distributed Western land to all settlers willing to cultivate it. The act gave legal title to 160 acres of public land to homesteaders who lived on and worked it for five years. This give-away can only be described as an example of affirmative government action. But it was a relative success (eighty million acres had already been claimed by 1900) precisely because it was a selective investment of public resources designed to foster self-discipline, long-term planning, and economic growth. Most importantly, the Homestead Act viewed the poor as producers rather than consumers. It provided individuals and families with the means and opportunity of earning their own livelihood. In this sense, it was a transfer program modeled on publicly funded education.
"Compassion with a hard edge" (to borrow British Prime Minister Tony Blair's phrase) should be the broad principle underlying the ongoing reform of our welfare system. Rather than eliminating government assistance, we should channel public resources toward stimulating and underwriting private effort—for example, by providing business credits, financial incentives for those who hire and train low-income employees, and job training. Whenever possible, welfare recipients should be treated as potential producers, not as charity cases. The right to an education is a good model here: taxpayers provide schools, books, and teachers, but students do not simply receive benefits; on the contrary, they are required to study. That is the whole point of the idea of equality of opportunity (most reasonably understood as the provision of minimally decent opportunities for all), for the provision of opportunity is valuable only to those who seize and use it. Likewise, the government can create a right to freedom of speech, but this right is useless if people do not take the trouble to make their voices heard. The right to an education and the right to free speech (both of which require the rightsholder to act) are far better models for a reformed American welfare system than the rights of the sick, the handicapped, and the elderly, which tend to cast the rightsholder as a passive recipient of cash or services. That is to say, welfare rights should resemble the right to property or the right to sue for damages, rights that provide active individuals, at the public's expense, with some of the resources they need to pursue their ends.
Compared to simple cash grants, the EITC seems to be an unusually promising initiative for this very reason. It is an entitlement designed to reward self-discipline. It is less rigid, and less inefficient, than the minimum wage. Similar points can be made about child-care subsidies for working mothers and loan programs that seek to increase the incidence of home ownership among the borderline poor. While expensive, job training programs, meant to draw the unskilled into the workforce, are also promising. The point is not, however, to endorse particular reforms but to take the perspective that an understanding of the cost of rights implies: welfare rights, in effect, should be shaped on the model of classical rights—as public services, selective investments, incentives to self-discipline, and bargains meant to stimulate cooperation and stabilize productive interaction across ethnic lines.
RACE AND SOCIAL COOPERATION
In the United States, the questions raised in this book—"how much government?" "what kinds of rights?" "negative vs. positive rights?" "victimology vs. agency?" and "rights vs. responsibilities?"—are all thoroughly enmeshed with issues of race. Before the 1860s, the United States totally deprived a large swath of its population of both common-law rights and constitutional rights. Today, social programs that benefit white people, or that disproportionately benefit white people, rarely receive the level of social opprobrium reserved for programs that benefit black people or that disproportionately benefit black people. In many circles, rights are seen as having especially high costs, fiscal and otherwise, when they appear to be designed for, or mostly to be enjoyed by, African Americans.
To point this out is not to assert that programs that disproportionately benefit whites are working poorly or that those that disproportionately benefit African Americans are working well. Nor is it to imply that programs nominally designed to help African Americans actually help African Americans. The Supreme Court's attempt to compel local school districts to operate racially integrated schools, for instance, was not a resounding success. Many critics of the regulatory-welfare state are in perfectly good faith. But their claim that "positive rights" are somehow un-American and should be replaced by a policy of nonintervention is so implausible on its face that we may well wonder why it persists. What explains the survival of such a grievously inadequate way of thinking? There are many possible answers, but inherited biases—including racial prejudice, conscious and unconscious—probably play a role. Indeed, the claim that the only real liberties are the rights of property and contract can sometimes verge on a form of white separatism: prison-building should supplant Head Start. Withdrawal into gated communities should replace a politics of inclusion.
Upon careful inspection, the current American debate seems to be less about the choice between more or less government than about the old ideal (engraved on every coin) of e pluribus unum. At stake is our capacity and even our willingness to live together as a nation. To assert that society is a cooperative venture, and that rights can be understood as governmentally created agreements among heterogeneous individuals and groups, is simultaneously and for the same reason to cast doubt both on libertarian fairy tales (sometimes popular among the Right and astonishingly widespread in American culture) and on "identity politics" (sometimes popular on the Left and now enjoying a powerful resurgence). To focus on the cost of rights is to urge that the collectivity define rights, and spend resources on rights, in a way that is broadly defensible to a diverse public engaged in a common enterprise.
While the cooperation and coexistence of people with varying cultural backgrounds is fundamental to the American political experiment, multiculturalism becomes a problem when it degenerates into ethnic separatism. Rights may make the problem worse if they are enforced selectively. By expending resources on some rights, or some people's rights, while stinting on others, we may promote or discourage political divisions along ethnic lines. If the rights of all Americans are perceived to be splendidly beneficial to whites, for instance, but of scant use to African Americans, then the legitimacy of our rights-enforcing regime will suffer. If the right to be free from unreasonable searches and seizures is well enforced in some communities, but a meaningless paper guarantee in others, social cohesion and stable agreements will be extremely difficult. If rights are to be seen as social bargains, generating mutual benefits and providing the terms for social cooperation, these bargains must be the sort to which, in principle, all citizens can agree.
PERSONAL RIGHTS AS COMMUNITY ASSETS
The rights of stockholders are set down in a company's bylaws or certificate of incorporation. The rights of ocean fishermen are specified in international treaties. Such rights are not natural, but conventional. They are consciously designed, in the light of experience, to coordinate mutual expectations, maximize investment, promote fairness, and encourage competent management. This is not a bad model for understanding other rights as well, including constitutional rights.
The rights of Americans are artifices created and maintained by the community with the aim of improving the quality of collective and individual life. When a nation is divided along religious, economic, or racial lines, a strategic allocation of rights can alleviate social tensions and promote social cooperation. Religious liberty allows members of rival sects, in any multidenominational society such as the United States, to participate in shared processes of democratic decision-making. Properly conceived and implemented, freedom of religion strengthens society, guaranteeing that ultimate values of this kind will not be dragged through the mud of public contestation. (Think how different our political climate would be if debate such as that over the issue of abortion were the rule rather than the exception.) Underlying agreement on general principles of social ordering—many embodied in the Constitution—and on a range of particular practices makes a common life possible despite our "multiculturalism," that is, despite deep disagreements about personal and religious ideals. The privatization of religion in America allows a multidenominational society to resolve its other conflicts, those not involving ultimate values of religious conviction, by democratic compromise, fudging, and persuasion. Social coexistence and cooperation, including mutual respect, is enhanced by the protection of a private zone set aside for the exercise of freedom of religion. Taxpayers are willing to bear the costs of protecting religious liberty, not only because it helps ensure human dignity, but also because it helps keep a heterogeneous society in working order.
Other rights, too, are financed by the community at least in part because they solve difficult problems and provide widely shared benefits to the community. They are funded collectively because they are perceived to be collective goods. This is the principal reason why rights should not be opposed to duties; this is why individual liberty should not be casually associated with the corrosion of community. The contribution of rights to reconciling diverse social groups to each other, making them all feel a part of the nation and thereby encouraging public and private cooperation, is not limited to freedom of conscience. Just as important in this respect are all those rights designed to improve the conditions of relatively disadvantaged and vulnerable Americans.
When subsidizing legal services for the poor, the taxpaying public is accomplishing something concrete, but it is also making a highly visible gesture of inclusion. Welfare rights, broadly conceived, have the same purpose. This is hardly to deny that American welfare programs need to be rethought and revised. But the partisan attack on the very idea of the welfare state cannot be reasonably cast as a defense of rights in an authentic or genuine or original sense. As attention to the cost of rights makes clear, apparently nonwelfare rights are welfare rights too: public benefits designed to promote the voluntary participation of all rights wielders in society's common endeavors.
Conclusion
THE PUBLIC CHARACTER OF PRIVATE FREEDOMS
THE RIGHTS OF AMERICANS are neither divine gifts nor fruits of nature; they are not self-enforcing and cannot be reliably protected when government is insolvent or incapacitated; they need not be a recipe for irresponsible egoism; they do not imply that individuals can secure personal freedom without social cooperation; and they are not uncompromisable claims.
A more adequate approach to rights has a disarmingly simple premise: private liberties have public costs. This is true not only of rights to Social Security, Medicare, and food stamps, but also of rights to private property, freedom of speech, immunity from police abuse, contractual liberty, free exercise of religion, and indeed of the full panoply of rights characteristic of the American tradition. From the perspective of public finance, all rights are licenses for individuals to pursue their joint and separate purposes by taking advantage of collective assets, which include a share of those private assets accumulated under the community's protection.
Taking seriously the budgetary costs of all rights means loosening a number of settled convictions about the nature of American liberalism. That tax dollars must be collected before rights can be reliably enforced implies above all that individual liberty, in the United States, is more dependent upon the joint efforts of the community than is commonly acknowledged. That all rights require political officials to tax and spend suggests the speciousness of the overused distinction between positive and negative rights. That the legal rights of Americans draw on a limited pool of public resources makes clear why they can never be treated as trumps or uncompromisable claims. And finally, that rights enforcement requires public expenditures raises urgent but neglected questions of democratic accountability and distributive justice: according to what principles are tax dollars allocated for the enforcement of legal rights? And who decides how many resources are spent to subsidize which specific rights for which specific groups of individuals?
Conceived as a matter of public finance, legal rights emerge as politically created and collectively funded instruments designed to promote human welfare. Because returns from equal rights protection—such as the benefit of living in a relatively just society where, for the most part, groups with different ethnic backgrounds can peaceably coexist and cooperate—are diffuse and hard to capture, initial investments in such protection must be made by the public power.
Rights in contract law, which transform promises into binding obligations, are a model in this regard. The basic right of all Americans to enter into legally binding contracts supports habits of promise-keeping on which economic prosperity, beneficial to society as a whole, depends. Similarly, the rights to be notified, to submit evidence, to confront adverse witnesses, and so forth, are crafted to increase the accuracy of civil and criminal procedures and to decrease the risk of factual errors and mistaken decisions. Efficiency in the economy and truth in the administration of justice are public, not merely private, goods. They are secured to a substantial extent by the artful design, thoughtful allocation, reliable enforcement, and public funding of individual rights.
Like law in general, rights are institutional inventions by which liberal societies attempt to create and maintain the preconditions for individual self-development and to solve common problems, including settling conflicts and facilitating intelligently coordinated responses to shared challenges, disasters, and crises. As a means of collective self-organization and a precondition for personal self-development, rights are naturally costly to enforce and protect. As government-provided services aimed at enhancing individual and collective welfare, all legal rights, including constitutional ones, presuppose political decisions (which could have been different) about how to channel scarce resources most effectively given the shifting problems and opportunities at hand.
All of our legal rights—in constitutional law as well as private law—originally arose as practical responses to concrete problems. This is one reason why they vary over time and across jurisdictions. As instruments forged to serve evolving human interests and moral views, they are repeatedly recast, or respecified, by new legislation and adjudication. Rights also mutate because obstacles to human welfare—the problems that rights are designed to mitigate or overcome—change, along with technology, the economy, demography, occupational roles, styles of life, and many other factors.
When the need arises, state and federal lawmakers (which includes judges as well as legislators proper) have been known to recast or even abolish some traditional rights. American legislators did this, for instance, when they concluded that the best way to improve the welfare of employees and their dependents was to provide them with fixed awards in the case of on-the-job injuries. Workers' compensation statutes bar common-law remedies, that is to say, they legally extinguished the right that workers previously enjoyed to sue their employers for employment-related accidents. So rights are routinely unmade as well as made. Changing impediments to human well-being and shifting legislative strategies effect a reconfiguration of liberty because all legal rights are, or aspire to be, welfare rights—politically and judicially designed attempts to achieve human well-being in changing social contexts. When these attempts fail, as they sometimes do, rights will be, and should be, created and suspended, redesigned and reassigned.
Constitutional rights provisions, especially, contain broad and ambiguous generalities that must be interpreted and specified by ever-new judicial personnel possessing moral sensibilities and commitments that vary over time. The concrete meaning of freedom of speech is not unambiguously fixed in the original text of the First Amendment, for instance, but has evolved significantly, along with both the Court and the country, during a long historical process. But rights cannot be enforced in an unchanging manner for a more mundane reason as well: enforcement is subject to budgetary constraints which differ from year to year. Indeed, the enforcement of rights is largely a matter of public outlays for infrastructure and skills of a legal kind. It involves, for instance, public investment in judicial salaries and real estate and auxiliary staff and in police and prison-guard training and monitoring. To take the cost of rights into account is therefore to think something like a government procurement officer, asking how to allocate limited resources intelligently while keeping a wide array of public goods in mind. Legal rights have "opportunity costs"; when rights are enforced, other valuable goods, including rights themselves, have to be forgone (because the resources consumed in enforcing rights are scarce). The question is always, might not public resources be deployed more sensibly in some other way?
This question may at first sound pettily economistic. Does not inquiry into costs tarnish the lofty majesty of the law? Should we consign our most precious liberties to bookkeepers or introduce mean considerations of cost-effectiveness where ultimate vulnerabilities are involved? Should courts or other government agencies sacrifice rights simply because they are expensive? Such apprehensions are well directed against some forms of cost-benefit analysis, but they are misplaced when aimed against the approach and arguments urged here.
Far from being crudely economistic, a study of the fiscal conditions of rights enforcement is fundamentally political. Attending to costs forces us to take a broad rather than a narrow view of the public weal. It prevents us from tackling problems sequentially, as they happen to catch our attention, and forces us to propose "package" solutions to a wide array of social problems. Above all, it lays bare the indispensability of public investments, made and evaluated collectively. Rather than reflecting a blind worship of market outcomes, that is to say, the study of the cost of rights is meant to encourage thoughtful public policy. It is also a kind of communitarian or collectivist theme, though with deep roots in the liberal political tradition.
The difficulties it raises are myriad, however. For one thing, cost-consciousness in the field of rights enforcement presents a serious challenge to the judiciary, precisely because it demands attention to a broad range of competing demands upon the public budget, while judges are necessarily riveted to a particular controversy, narrowly defined. Without paying serious attention to possible alternative uses of scarce taxpayer dollars, for instance, American judges regularly compel big-city governments to dole out millions of taxpayers' dollars in tort remedies. Is this a democratic and morally responsible way to expend scarce public resources? Why should this money not be spent on public education or public health?
We cannot even ask such questions, it should be noted, until we candidly acknowledge the cost of rights. The fact that American courts—the principal guardians of our most precious liberties—are poorly positioned to make intelligent allocative decisions is a reason to worry about the implications of judicial decision-making for a responsible system of public finance. But since judges are entrusted by law with the task of protecting costly rights, students of adjudication cannot reasonably ignore the cost of rights.
For in a democracy, collective expenditures should be collectively overseen. Since the enforcement of basic rights presupposes the outlay of scarce public moneys, the public is entitled to know if the game is worth the candle, if the benefits received are roughly equivalent to the expenses incurred. To the extent that it is funded by the community, a particular pattern of rights enforcement must be justified to the community, with appropriate safeguards for members of minority groups. The benefit-cost ratio must not only be positive, it must also be perceived to be positive. So should not rights enforcers—or those who hire, pay, and supervise them—be seen as financial fiduciaries? Should they not account publicly for their necessarily controversial decisions about how scarce public moneys are put to use? Should they not make clear the principles they use when allocating benefits and burdens? And should they not explain why a chosen distribution is preferable to its feasible alternatives?
The cost of rights raises not only questions of democratic accountability and transparency in the process of allocating resources; it also brings us unexpectedly into the heart of moral theory, to problems of distributional equity and distributive justice. To describe rights as public investments is to encourage rights theorists to pay attention to the question of whether rights enforcement is not merely valuable and prudent, but also fairly allocated. The question here is whether, as currently designed and implemented, disbursements for the protection of rights benefit society as a whole, or at least most of its members, or only those groups with special political influence. Do our national priorities, in the area of rights enforcement, merely reflect the influence of powerful groups, or do they promote the general welfare? To study costs is not to shortchange politics and morality, but rather to compel consideration of such questions. The subject is so important precisely because it draws attention to the relation between rights on the one hand and democracy, equality, and distributive justice on the other.
Rights elicit public support because—and to the extent that—they permit a large collectivity of differently situated individuals to reap the substantial rewards, personal and social, of nonpredatory coexistence and mutual cooperation. To interpret rights as welfare-enhancing investments, extracted by society for society's purposes, should improve our understanding not only of the rationale for rights, but also of their inevitably redistributive character. Such a conceptualization may conceivably stimulate a richer public debate about various neglected questions, such as whether private resources (presumably extracted, in a democracy, only for public purposes publicly explained) are invested in a way that produces adequate public gains and whether these benefits and burdens are fairly shared.
PUBLIC WILLINGNESS TO PAY
To classify rights as costly public goods is not to encourage heartless policy analysts—leagued with cadres of accountants—to settle unilaterally the question of what rights citizens should or should not enjoy. On the contrary, the inevitability of trade-offs reminds us of the need for democratic control and even "civic virtue," that is, for careful taxpayer scrutiny of budgetary allocations in the area of rights protection and enforcement. Needless to say, it is much easier to call for democratic accountability in such matters than to achieve it.
Well-trained and competent specialists have a role to play, here as elsewhere. They are indispensable for uncovering, interpreting, and translating into easily intelligible speech the often-complex information required for meaningful public consultations and decision-making about rights. But experts should be on tap, not on top. Where disputable judgments of value are involved, decision-making should occur in an open and democratic fashion. Because rights result from strategic choices about how best to deploy public resources, there are good democratic reasons why decisions about which rights to protect, and to what degree, should be made in as open a manner as possible by a citizenry as informed as possible, to whom political officials, including judges, must address their reasonings and justifications.
Judgments about which rights in which forms should be granted protection and about how much social wealth should be invested in protecting these rights ought to be subject to ongoing public criticism and debate in processes of democratic deliberation. Such decisions should be guided by the basic principles of the American legal system, including, of course, those set forth in the Constitution. How judges can retain their independence while becoming more fiscally accountable presents serious and important challenges to institutional reform. But it cannot be denied that, in the United States today, important allocative decisions concerning basic rights are often made in secretive ways, with little public consultation and control. At the very least, such judgments should become publicly scrutinizable as judgments that could have been made differently and that require justification in processes of public deliberation, subject to constitutional constraints that must themselves be justified.
Justice need have no special quarrel with cost-effectiveness. No one can object to innovative methods that allow us to deliver the same level of Social Security benefits or food stamps at half the cost. No one suggests that such efficiencies undermine the moral purpose of the programs of the welfare state. The same should be said of all rights, for cost-effectiveness can be improved everywhere, including, say, in the delivery of rights protection to suspects during interrogation or to pretrial detainees. But we can begin to consider a more efficient delivery of rights protection only after we have recognized that rights have costs.
Public deliberation should therefore be focused on the following issues. (1) How much do we want to spend on each right? (2) What is the optimal package of rights, given that the resources that go to protect one right will no longer be available to protect another right? (3) What are the best formats for delivering maximum rights protection at the lowest cost? (4) Do rights, as currently defined and enforced, redistribute wealth in a publicly justifiable way? These questions have important empirical dimensions, and it is important to bring them to the fore. But their resolution depends on judgments of value as well. The empirical dimensions should be identified as such; the judgments of value should be made openly and be subjected to criticism, review, and public debate.
REDISTRIBUTION
Having barely touched upon the uses of governmental power to help the disadvantaged, this book obviously cannot conclude with a blueprint for redesigning the American welfare state. Particular judgments depend on particular facts. Like other policy initiatives, efforts to help the disadvantaged sometimes backfire. But blanket attacks on redistribution as such make little sense. Redistribution is omnipresent. It does not occur only when the government takes money from taxpayers and hands it over to the needy. Redistribution also occurs, for example, when the public force is made available, at the expense of taxpayers generally, to protect wealthy individuals from private violence and threats of violence. Even the so-called minimal state requires the extraction of private revenue for public purposes. The most dramatic example of such regressive taxation occurs when the poor are drafted into military service in wartime to defend, among other things, the property of the rich from foreign predators. Even the most minimal state redistributes resources from those "able to pay" to protect vulnerable people. In some cases, those who are protected (like the Westhampton homeowners threatened by fire) are wealthier than those who shoulder most of the burden of the protection.
Strength and weakness are not physical conditions or brute facts. The relative strength of social actors depends less on sheer muscle or raw brains than on legal institutions and entitlements and the sheer capacity for social organization and coordination. Property holders, in the late twentieth century, are comparatively strong only as a result of government support, that is, because of deftly crafted laws, enforced at public expense, that enable them to acquire and to hold onto what is "theirs." It is impossible to define who is strong and who is weak socially, therefore, without knowing on whose side political authority will stand—that is, without reference to prior decisions about the political allocation of scarce social public resources. The rich are strong because they are protected by judicially managed systems of enforceable property rights and criminal justice.
So all the troubling questions remain: are current public investments in rights enforcement wise or foolish? Are they biased or fair? In a democracy, presumably, public investments are made by the taxpaying citizenry with the anticipation of good social returns, very broadly understood. So are the returns on our investments truly good or even acceptable? Are property rights, for instance, worth what we, as a nation, spend on protecting them?
Such questions cannot be answered in the abstract, without knowing, for instance, how scarce public resources might otherwise be productively employed. But one thing is certain. The dependency of clearly defined and robustly enforced property rights on law, government, and public resources does not lessen their value. The right to private property fuels economic growth. It also lengthens the time horizons and enhances the psychological security of individual citizens, assuring them, for example, that their expressions of political dissent will not place their holdings at risk. Even though the right to private property is costly up front, it is a shrewd and even self-funding investment. (Of course, systems of private property differ among themselves, and reasonable people can disagree about the advantages and disadvantages of each. But some form of private property is an indispensable part of any well-functioning modern society.)
The right to public education can be justified in similar terms; good education is a precondition for many other things, and it has both intrinsic and instrumental value. For children, especially, rights to health care make a good deal of sense; health is valuable in itself and makes other good things possible. Thus substantial public expenditures in both areas are justified in exactly the same way as expenditures that go to the protection of private property. All such rights establish and stabilize the conditions of individual self-development and collective coexistence and cooperation.
To say that rights enforcement presupposes the strategic allocation of public resources is above all to recall how the parts fit into the whole, how liberal individualism—as opposed to the unbridled anarchism of the state of nature—presupposes a politically well organized community. Individual freedom is both constituted and bolstered by collective contributions. The cost of rights is merely the easiest to document of such contributions. Focusing on the issue of cost forces us therefore to rethink and modify the familiar but exaggerated opposition between individual and society.
American citizens can successfully protect themselves against the unwanted intrusion of society in their private affairs, but only with society's consistent support. This is true for the most self-confidently individualistic of rightsholders. For the liberty of individuals cannot be protected unless the community pools its resources and applies them in a shrewd fashion to deter and remedy violations of individual rights. Rights presuppose effective government because only through government can a complex modern society achieve the degree of social cooperation necessary to transform paper declarations into claimable liberties. Indeed, rights can be depicted as antigovernmental, as walls constructed against the state, only if the public authority's indispensable contribution to wall construction and maintenance is unjustifiably overlooked. For government is still the most effective instrument available by which a politically organized society can pursue its common objectives, including the shared aim of securing the protection of legal rights for all.
Appendix
SOME NUMBERS ON
RIGHTS AND THEIR COSTS
Although we have occasionally referred to some numbers, it has not been our purpose in this book to offer a quantitative assessment of the cost of rights. The task of producing a quantitative assessment requires acceptance of our conceptual claims, and then some further judgments, themselves complex both empirically and conceptually, about how to disaggregate various expenditures so as to come up with dollar expenditures per right. For reasons discussed in the text, it is possible to make some progress toward that task, but precise figures may be impossible to produce.
In this appendix we offer a simple table, from the 1996 Budget of the United States, in order to present some information on the amount spent on various activities and institutions. This information should be taken with some grains of salt, for it does not allow specification of the cost of particular rights. It does, however, give a sense of how much taxpayers spend, in federal dollars, to carry out various programs and activities and to protect various rights. The vast amount spent by the states on various aspects of rights protection are of course not included.
ACTIVITY OR INSTITUTION | DOLLARS
(IN MILLIONS)
---|---
1. Operating the system of justice
United States Courts of Appeals | 303
---|---
United States Tax Court | 33
District Courts | 1,183
United States Sentencing Commission | 9
United States Supreme Court | 26
Legal Activities of Department of Justice | 537
Legal Services Corporation | 278
Violent crime reduction program | 30
Expenses relating to U.S. prisoners | 351
Court of Veterans Appeals | 9
Federal prison system | 2,465
2. Monitoring government
Office of Government Ethics | 8
---|---
General Accounting Office | 362
Federal Election Commission | 26
3. Facilitating market arrangements
Securities and Exchange Commission | 103
---|---
Federal Trade Commission | 35
Animal and plant inspection | 516
Food safety and inspection | 545
Consumer Product Safety Commission | 41
4. Protecting property rights
Patent and trademark protection | 82
---|---
Disaster relief and insurance | 1,160
Federal emergency management | 3,614
Community disaster loans | 112
Management and protection of forests | 1,283
Real property activities | 68
Fund for rural America (agricultural support) | 100
Records management connected with property | 203
5. National defense
Pay and allowances of officers in military | 5,808
---|---
Pay and allowances of enlisted personnel | 12,457
Pay and allowances of cadets | 35
Veterans' benefits and services | 3,830
Subsistence of enlisted personnel | 769
Total obligations of defense
department, military | 20,497
6. Education
Educational expenditures, e.g., state
and local education | 530
---|---
Elementary, secondary, and vocational
education | 1,369
Equal Employment Opportunity
Commission | 233
7. Income distribution
Administration of food stamp program | 108
---|---
Food and nutrition assistance | 4,200
Social Security Administration | 6,148
8. Environmental protection
Environmental Protection Agency | 41
---|---
Clean Air Act | 217
Hazardous waste | 159
Pesticides | 64
Natural resources conservation | 644
Water quality | 244
9. Others
Printing government publications | 84
---|---
Postal service | 85
National Archives and Records
Administration | 224
National Labor Relations Board | 170
Occupational Safety and Health
Review Commission | 8
Bureau of the Census | 144
NOTES
INTRODUCTION
Charles Murray, What It Means to Be a Libertarian: A Personal Interpretation (New York: Broadway Books, 1997), p. 5; David Boaz, Libertarianism: A Primer (New York: Free Press, 1997), p. 12.
Budget of the United States Government, Fiscal Year 1998 (Washington, D.C.: U.S. Government Printing Office, 1997), p. 231. The 1996 budget of the Federal Emergency Management Agency ran to something over $3.6 billion (p. 1047).
Enforced by informal opprobrium, rather than by legal sanction, social norms also play a role in inducing private individuals to respect each other's rights and public officials to respect the rights of private actors. Such norms do not operate independently, however; they are always enmeshed, in complex ways, with government efforts to redesign and enforce criminal law, tort law, contract law, labor law, and so forth.
Not discussed here are certain hard questions about the extent to which moral considerations do or must enter into the interpretation of legal terms. See Ronald Dworkin, Law's Empire (Cambridge, Mass.: Harvard University Press, 1985); Frederick Schauer, Playing by the Rules (Oxford: Oxford University Press, 1992).
Article 13 of the European Convention on Human Rights (signed in 1950 and ostensibly entered into force in 1953). The rights announced in the convention and its protocols are reliably enforced when the subscribing states treat them as part of domestic law.
H. L. A. Hart, Essays on Bentham (Oxford: Clarendon, 1982), p. 171.
Patent rights are among the few exceptions to this rule. Since the Omnibus Budget Reconciliation Act of 1990, the Patent and Trademark Office (PTO) has been required fully to fund itself through user fees. Congress allocates money to the PTO every year, but a complicated mix of paybacks (from fees collected) into the agency budget and the general budget is supposed to compensate for this up-front allocation.
Under 26 U.S.C. 501(c)(3), organizations that provide legal services to promote human and civil rights are, under specified conditions, exempt from the federal income tax.
U.S. Department of Justice, Bureau of Justice Statistics, Justice Expenditure and Employment Extracts, 1992: Data from the Annual General Finance and Employment Surveys (Washington, D.C.: U.S. Government Printing Office, 1997), Table E.
Billy L. Wayson and Gail S. Funke, What Price Justice? A Handbook for the Analysis of Criminal Justice Costs (Washington, D.C.: Department of Justice, U.S. National Institute of Justice, August 1989).
Paul A. Crotty, "Containing the Tort Explosion: The City's Case," CityLaw, vol. 2, no. 6 (December 1996).
See Robert L. Spangenberg and Tessa J. Schwartz, "The Indigent Defense Crisis Is Chronic," Criminal Justice, Summer 1994; John B. Arango, "Defense Services for the Poor: Tennessee Indigent Defense System in Crisis," Criminal Justice (Spring 1992), p. 42 (which reports that the public defender's budget was cut by 5.3 percent); Rorie Sherman, "N.J. Shuts Down Its Advocate; Was Unique in Nation," National Law Journal, July 20, 1992, p. 3 (which notes that New Jersey eliminated $6 million designated to provide appointed counsel); Richard Klein and Robert Spangenberg, "The Indigent Defense Crisis" (report prepared for the American Bar Association Section of Criminal Justice, Ad Hoc Committee on the Indigent Defense Crisis, 1993), which reports that of the $74 billion spent on criminal justice by federal, state, and local governments in 1990, only 2.3 percent was spent on public defense nationally (pp. 1–3).
Matthews v. Eldridge, 424 U.S. 319 (1976). Although it admitted that American taxpayers have a financial interest in getting people off the welfare rolls as soon as they are determined to be ineligible for public aid, the Court had earlier awarded those about to lose their welfare benefits a full administrative hearing before termination. Goldberg v. Kelly, 397 U.S. 254 (1970). But the Court has never dismissed the importance of cost out of hand. In Goss v. Lopez, 419 U.S. 565 (1975), for instance, the Court ruled that public schools could provide an extremely modest hearing to a student threatened with expulsion because "even truncated trial-type procedures might well overwhelm administrative facilities in many places and by diverting resources, cost more than it would save in educational effectiveness."
This book does not address what philosophers call the "deontological" thesis that moral reasoning should be concerned with principles rather than consequences. But the cost of rights does bear on the misuse of deontological arguments to foster illusions about the purely apolitical nature of adjudication and the nondependence of the judiciary on the extraction and channeling of public resources. Even rights that are of intrinsic rather than instrumental worth are costly and must therefore be subject to trade-offs of some kind.
CHAPTER ONE
410 U.S. 113 (1973).
432 U.S. 464 (1977).
The distinction between negative rights and positive rights should not be confused with the similar-sounding distinction between negative and positive liberty, popularized by Isaiah Berlin in Four Essays on Liberty (Oxford: Oxford University Press, 1969), pp. 118–172. True, negative rights and negative liberty have roughly the same meaning (freedom from interference), but positive liberty, as Berlin used the term, refers either to democratic self-government (ibid., pp. 160–63) or to human self-realization, particularly the mastery of passion by reason. "Positive rights," which Berlin did not discuss at all and which the Court, by contrast, had in mind in these decisions, refer to individual claims upon resources provided by the taxpayer and managed by the government.
Among academic writers, the master taxonomist of legal rights remains Wesley Hohfeld. In his Fundamental Legal Conceptions (New Haven: Yale University Press, 1923), Hohfeld distinguished among permissions, claims, powers, and immunities. While interesting, this four-part schema is unsatisfactory for several reasons, including the notable fact that powers, immunities, and even permissions all involve implicit claims on governmental performance and public resources in a sense to be discussed later in this book.
Liberals often cite, in this regard, T. H. Marshall's Class, Citizenship and Social Development (Chicago: University of Chicago Press, 1964), which identifies three phases in the evolution of British and European liberalism. Roughly speaking, civil rights developed in the eighteenth century, political rights in the nineteenth, and social rights in the twentieth. Marshall's three-part taxonomy, in other words, makes it misleadingly easy to use the negative-positive polarity to interpret Western historical development.
According to Hans Kelsen, "it is always a potential plaintiff who is the subject of a right." General Theory of Law and State (New York: Russell & Russell, 1973), p. 83.
Budget of the United States Government, Fiscal Year 1998 (Washington, D.C.: U.S. Government Printing Office, 1997), Appendix, p. 1019.
Budget of the United States Government, Fiscal Year 1998, Appendix, pp. 662, 1084, 1095, 1029.
CHAPTER TWO
Bentham referred to such legal powers not as negative rights but as rights to "negative services." See H. L. A. Hart, "Bentham and Legal Rights," in Oxford Essays in Jurisprudence, Second Series, ed. A. W. B. Simpson (Oxford: Oxford University Press, 1973), pp. 171–201.
Tinker v. Des Moines School Dist., 393 U.S. 503 (1969).
Baron de la Bréde et de Montesquieu, The Spirit of the Laws, trans. Thomas Nugent (New York: Hafner, 1949), Vol. I (Book XI, chap. 4), p. 150.
CHAPTER THREE
Jeremy Bentham, The Theory of Legislation, trans. C.K.Ogden and Richard Hildreth (Oxford, Eng.: Oxford University Press, 1931), p.113.
William Blackstone, Commentaries on the Laws of England (Chicago: University of Chicago Press, 1979), Vol. II, p. 11.
3 Budget of the United States Government, Fiscal Year 1998, pp. 137 and 142–43.
U.S. Department of Justice, Bureau of Justice Statistics, Justice Expenditure and Employment Extracts, 1992, Table E.
Budget of the United States Government, Fiscal Year 1998, p. 246.
Ibid., pp. 225–26.
Ibid., Appendix, p. 395.
Ibid., Appendix, p. 764.
9 Ibid., Appendix, p. 28.
Robert Ellickson, Order without Law (Cambridge, Mass.: Harvard University Press, 1993) shows that some forms of social ordering can occur, and do quite well, without much understanding of law and on the basis of social norms generated and followed by the relevant community. But even such a system of ordering—which is likely to be local, not national—probably depends on background norms accompanied by legal guarantees.
Friedrich A. von Hayek, The Road to Serfdom (Chicago: University of Chicago Press, 1944), pp. 80–81.
Budget of the United States Government, Fiscal Year 1998, Appendix, p. 1062.
Ibid., Appendix, p. 1112.
Ibid., Appendix, pp. 1011, 1112–13.
CHAPTER FOUR
U.S. Department of Justice, Bureau of Justice Statistics, Justice Expenditure and Employment Extracts, 1992, Table E.
Wilson v. Seiter et al., 501 U.S. 294, 298 (1990).
Budget of the United States Government, Fiscal Year 1998, Appendix, p. 689.
Bounds v. Smith 430 U.S. 817 (1977).
The Supreme Court has mentioned the possibility that "fiscal constraints" beyond the control of prison officials might "prevent the elimination of inhumane conditions," but has not explicitly ruled on "the validity of a 'cost' defense" in suits for damages under the Eighth Amendment. Wilson v. Seiter supr., 301–302.
Budget of the United States, Fiscal Year 1998, p. 670.
"There is lacking that equality demanded by the Fourteenth Amendment where the rich man, who appeals as of right, enjoys the benefit of counsel's examination into the record, research of the law, and marshaling of arguments on his behalf, while the indigent . . . is forced to shift for himself." Douglas v. People of State of California, 372 U.S. 353 (1963).
CHAPTER FIVE
DeShaney v. Winnebago County Department of Social Services, 489 U.S. 189 (1989).
At first glance, there is much to be said for this way of thinking. Many commentators argue that the Constitution is directed "against" government intrusion, and that it does not compel government intrusion, even if the intrusion can be called protection. Of course, the First Amendment prevents state and federal governments from interfering with freedom of speech. Only by subconstitutional statutes are private organizations inhibited from doing the same. This is also true of the ban on racial discrimination. The Fourteenth Amendment's requirement of equal protection of the laws does not apply to private organizations, even to large businesses, which are required to act in a nondiscriminatory fashion by statutes, not by the Constitution.
Some of the issues discussed here are dealt with in more detail in Cass R. Sunstein, The Partial Constitution (Cambridge, Mass.: Harvard University Press, 1993).
Shelley v. Kraemer, 334 U.S. 1 (1948).
Edmonson v. Leesville Concrete Co., 500 U.S. 614 (1991).
6 Smith v. Allwright, 321 U.S. 649 (1944); Terry v. Adams, 345 U.S. 461 (1953).
Lebron v. National Railroad Passenger Corp., 115 S.Ct. 961 (1995).
Burton v. Wilmington Parking Authority, 365 U.S. 715 (1961).
Dee Farmer v. Brennan, 511 U.S. 825 (1994).
Budget of the United States Government, Fiscal Year 1998, Appendix, p. 670.
This point is made in Richard A. Posner, Overcoming Law (Cambridge, Mass.: Harvard University Press, 1996), though not in his opinion for the lower court in DeShaney, an opinion that relied on the negative-rights/positive-rights distinction.
CHAPTER SIX
"Individual rights are political trumps held by individuals." Ronald Dworkin, Taking Rights Seriously (Cambridge, Mass.: Harvard University Press, 1977), p. xi.
Dworkin, Taking Rights Seriously, p. 193.
Korematsu v. United States, 323 U.S. 214 (1944).
See the discussion of exclusionary reasons in Joseph Raz, Practical Reason and Norms, 2d ed. (Princeton: Princeton University Press, 1993).
See Jean Dreze and Amartya Sen, India (Oxford: Oxford University Press, 1996).
So argues Leonard Levy in Original Intent and the Framers' Constitution (New York: Macmillan, 1988), pp. 174–220.
See Fordyce County, Georgia v. The Nationalist Movement, 505 U.S. 123 (1992).
CHAPTER SEVEN
This rough estimate is based on the debatable assumption that the cost per voter of an election is between $2 and $5. Interview with Melissa Warren, Elections Division, Secretary of State of California, July 25, 1997.
Hans Kelsen, General Theory of Law and the State (Cambridge, Mass.: Harvard University Press, 1945), p. 88.
O'Brien v. Skinner, 414 U.S. 524 (1974).
Interview with John Cloonan, Elections Division, Secretary of the Commonwealth of Massachusetts, July 22, 1997. The city of Boston spends roughly $300,000 for each election.
Interview with Melissa Warren, Elections Division, Secretary of State of California, July 25, 1997.
Joseph Raz, Ethics in the Public Domain: Essays in the Morality of Law and Politics (Oxford: Clarendon, 1995), p. 39.
CHAPTER EIGHT
Franklin D. Roosevelt, "Message to the Congress on the State of the Union" (January 11, 1944), in The Public Papers and Addresses of Franklin D. Roosevelt Vol. 13 (New York: Random House, 1969), p.41.
Deference to the phraseology of international human rights instruments helps account for the inclusion of many prohibitively expensive rights in postcommunist constitutions. The 1966 International Covenant on Economic, Social, and Cultural Rights (now signed by 61 nations, and putatively "entered into force" in 1976) includes the right to work (Art. 6), the right of everyone to the enjoyment of just and favorable conditions of work (Art. 7), special protections for mothers and children (Art. 10), the right of everyone to an adequate standard of living and the fundamental right of everyone to be free from hunger (Art. 11), and the right of everyone to the enjoyment of the highest attainable standard of physical and mental health (Art. 12). Twenty-Five Human Rights Documents (New York: Columbia University Center for Human Rights, 1994), pp. 10–16.
John Rawls, Political Liberalism (New York: Columbia University Press, 1996). Rawls refers to basic health care, employment, and a decent distribution of income, and appears to imply a right to a minimum income of some kind.
B. Guy Peters, The Politics of Taxation (Cambridge, Mass.: Blackwell, 1991), p. 3.
See John Graham and Jonathan Weiner, Risk vs. Risk (Cambridge, Mass.: Harvard University Press, 1996).
See Stephen Breyer, Breaking the Vicious Circle (Cambridge, Mass.: Harvard University Press, 1993).
CHAPTER NINE
Zablocki v. Redhail, 434 U.S. 374 (1978).
See Mary Ann Glendon, Rights Talk (New York: Free Press, 1993).
A qualification of these claims about rights and duties, not crucial to our discussion here, emerges from Joseph Raz, The Morality of Freedom (Oxford: Oxford University Press, 1986), pp. 183–86.
For details see Richard Posner and Kate Silbaugh, Sex Laws in America (Chicago: University of Chicago Press, 1996).
CHAPTER TEN
Thomas Hobbes, Leviathan (Harmondsworth, Eng.: Penguin, 1968), p. 85.
The rights of union members, downstream landowners, and noncustodial parents—to be genuine rights and not merely empty phrases—presuppose well-designed systems of positive and negative incentives. The rights in question are false promises in the absence of such spurs and reins. Liberal rights, in this sense, rest not upon laissez-faire but upon a deliberate governmental sculpting of personal behavior for social ends. To claim legal rights is to act within established institutions according to well-defined rules.
Alasdair MacIntyre, After Virtue (Notre Dame: University of Notre Dame Press, 1981).
Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations (New York: Modern Library, 1937), p. 98.
See William Galston, "Causes of Declining Well-Being among U.S. Children," in Sex, Preference, and Family, ed. David Estlund and Martha Nussbaum (Oxford: Oxford University Press, 1996); Derek Bok, The State of the Nation (Cambridge, Mass.: Harvard University Press, 1996); David Ellwood, Poor Support: Poverty in the American Family (New York: Basic Books, 1988).
Glendon, Rights Talk, p. x.
Blackstone, Commentaries on the Laws of England, Vol. 3, p. 4. In the common-law tradition, that is to say, "rights talk" was invented to replace bully-boy talk.
CHAPTER ELEVEN
The same might be said of the authors of the American Bill of Rights.
U.S. Department of Health and Human Services, Office of Child Support Enforcement, Twentieth Annual Report to Congress for the 1995 Fiscal Year. The American taxpayer paid one dollar for every four dollars of child support collected. (Appendix B, Table 1, p. 78).
Christopher Jencks, "The Hidden Paradox of Welfare Reform," American Prospect, no. 32 (May–June 1997), p. 36.
CHAPTER TWELVE
This exchange of rights protection for political legitimacy is one of the central themes of Jürgen Habermas, Between Facts and Norms (Cambridge, Mass.: MIT Press, 1996).
This empirical thesis should not be confused with David Gauthier's argument that principles of justice can be defended by showing that they would have emerged from bargains based on mutual advantage, given existing distributions of talents, entitlements, and so forth. See David Gauthier, Morals by Agreement (Oxford: Clarendon, 1986). Philosophical arguments that invoke mutual advantage must justify, or assume the justice of, the starting point from which the bargaining parties set out. The difficulty of justifying any particular starting point is an enduring problem for those who want to use social contract theory to demonstrate the correctness of their moral conclusions. See John Rawls, A Theory of Justice (Cambridge, Mass.: Harvard University Press 1971); Brian Barry, Theories of Justice (Berkeley: University of California Press, 1989). The present chapter aims not to justify rights philosophically, but only to defend the claim that rights emerge from cooperation and enable cooperation.
"This model of rights as the product of interest-driven bargains looks at least as plausible as the common notions that rights derive from mentalities, Zeitgeisten, general theories, or the sheer logic of social life." Charles Tilly, "Where Do Rights Come From?" in Contributions to the Comparative Study of Development, Vol. 2 ed. Lars Mjoset (Oslo: Institute for Social Research, 1992), pp. 27–28.
In Employment Division, Department of Human Resources v. Smith, 494 U.S. 872 (1990), the Supreme Court upheld a general criminal prohibition on the religiously inspired use of peyote. The most controversial aspect of the holding was the conclusion that a neutral law, applying to all, is constitutionally unproblematic even if it has an adverse effect on religion. The Court thus read narrowly (without overruling) its prior holding in Sherbert v. Verner, 374 U.S. 398 (1963), which had held that unemployment compensation cannot be denied to a religious believer under a general law requiring people to seek work on Saturday. But the Court has not completely reached closure on the issue of religiously compelled exemptions from general laws. Congress attempted to overrule the Smith decision by statute; the Court said that Congress lacked the power to do so, but in the process revealed an internal division on the whole question of whether (costly) exemptions may sometimes be compelled by the free exercise clause.
Employment Division, Department of Human Resources v. Smith. See discussion in preceding note.
The definition of coercion is no simple matter, admittedly, and religious groups are often legally permitted to be "coercive" in some possible sense of that elastic concept.
CHAPTER THIRTEEN
Richard Posner, Economic Analysis of Law, 4th ed. (Boston: Little, Brown, 1992), pp. 463–64.
See the striking empirical finding of people's judgments to this effect in Norman Frohlich and Joe Oppenheimer, Choosing Justice (Berkeley: University of California Press, 1993).
John Stuart Mill, "Principles of Political Economy," in Collected Works, ed. J. M. Robson, vol. 3 (Toronto: University of Toronto Press, 1965), p. 962.
Theda Skocpol, Protecting Soldiers and Mothers: The Political Origins of Social Policy in the United States (Cambridge, Mass.: Harvard University Press, 1992).
CHAPTER FOURTEEN
See the data in Nada Eissa and Jeffrey B. Liebman, "Labor Supply Responses to the Earned Income Tax Credit," Quarterly Journal of Economics 111 (1996) p. 605, which shows significant increases in the workforce participation after the introduction of EITC.
Margaret Levy, Of Rule and Revenue (Berkeley: University of California Press, 1988).
Entitlements for those unable to help themselves will always be necessary, of course, especially in emergencies.
A good discussion, showing vices as well as virtues in the EITC, is Daniel Shaviro, "The Minimum Wage, The Earned Income Tax Credit, and Optimal Subsidy Policy," University of Chicago Law Review 64, (1997), p. 405.
Related themes are developed in Stephen Holmes, Passions and Constraint (Chicago: University of Chicago Press, 1995); and Cass R. Sunstein, Legal Reasoning and Political Conflict (New York: Oxford University Press, 1996).
INDEX
abortion rights, 163
public funding issue, 35–36
religious liberty and, 187–88
as social answer to failure of social responsibility, 166–68
"absoluteness" of rights:
fallacy of absoluteness, 93–94, 97–98, 102–3
inadmissible arguments and, 105–6, 107
trade-offs in rights enforcement and, 128–32
accountability in rights-related decision-making, 227–29
accused's rights, 21, 23, 79–83, 241n7
see also prisoners' rights
addictive substances, 142
affirmative action programs, 166
affirmative government, see government performance
African Americans, 141, 166, 169–70, 203
government performance in enforcement of rights and, 56–57
racial dimension of rights, 215–17
smoking rate, 143
agricultural subsidies, 65–66
Aid to Families with Dependent Children (AFDC), 157, 170
American Civil Liberties Union (ACLU), 21
antifraud law, 73–74
antimajoritarian dimension of rights, 115–16
Aristotle, 149
Articles of Confederation, 58
aspirational (open-ended) nature of rights, 119–20
bail and release on recognizance systems, 80
bargains, 245n2, 3
property rights as, 195–96, 208–9
rights in general as, 177–80, 188, 191–92, 199–200
welfare rights as, 197, 208–9
Bentham, Jeremy, 16, 59, 61, 240n1
bequeathing one's private property, 47
Berlin, Isaiah, 239n3
bigotry, 144
Bill of Rights, 57
black Americans, see African Americans
Blackstone, William, 61, 161
Blair, Tony, 213–14
Boaz, David, 13
Bounds v. Smith, 241n4
Brazil, 123
Breyer, Stephen, 125
Burton v. Wilmington Parking Authority, 242n8
California, 114
child abuse, 83, 87–89, 94
child support, 47–48, 170, 171
civil rights movement, 203
Class, Citizenship and Social Development (Marshall), 239n5
Clinton, Bill, 137
collective ownership, 149
communal decency, rights' promotion of, 159–61, 178–79, 217–19
compliance with law, 148–49, 155, 175–77
conscription, 62
conservatives:
"corrosive" effect of rights and, 137–38, 147
cost of rights, reluctance to acknowledge, 25, 28–29
negative/positive rights dichotomy and, 40–41, 42
rights favored and disfavored by, 191
constitutional rights, 223
changing interpretations of, 122–23
environmental rights as, 126–28
as negative rights, 51–54, 89
privately inflicted harms, protection from, 88, 89–92, 93, 96–97, 241n2
welfare rights as, 120–21
see also specific constitutional amendments
Consumer Credit Protection Act, 74
Consumer Product Safety Commission, 46
consumer protection bureaus, 46
"contingency fee" system, 201
contractual rights, 15, 49–50, 221–22
credit and, 70
privately inflicted harms and, 91
responsible behavior and, 140
"victimology" and, 165
copyrights, 48, 66
cost of rights, 15, 220–21, 231
amounts spent on various activities and institutions (1996), 233–36
antifraud law, 73–74
calculation of, 23–24
cost-benefit analysis of rights,
concerns regarding, 224
courts, financing of, 45–46
courts' attitude toward, 29–30
courts' cost-conscious approach to dealing with rights, 26–27, 224–25
criminal justice system, 77–80, 83
definition of, 15–16
descriptive nature, 18
enforcement costs, 154, 223–24
environmental rights, 124
freedom of speech, 111–12
"hidden costs" fallacy, 25
indirect costs, 22–23
moral dimension, 18–19, 226
opportunity costs, 223–24
philosophical implications, 131
as political issue, 224–27
private costs, 21
property rights, 61–66, 73–74
public nature, 20–21
realistic acknowledgment of costs, benefits of, 97–98
redistributive approach to covering costs, 113–17, 229–30
reducing costs, 25–28
religious liberty, 180–83, 187
responsible behavior and, 146–47
retaliation, protecting rightsholders from, 92–93, 171
"scarce resources" issue, 88–89, 94–98, 101–2, 107, 131–32
self-financing rights, 22, 237n7
separation of powers and, 122
social costs, 21–22
as taboo subject, 24–25, 28–31
voting rights, 113–15
welfare rights, 209, 210
worth of a right and, 21, 28
courts, 19, 122
accountability in rights-related decision-making, 228
as branch of government, 44–45
cost-conscious approach to dealing with rights, 26–27, 224–25
"cost of rights" issue, attitude toward, 29–30
curtailment of rights, regulation of, 106
enforcement of rights and, 44–46
environmental rights and, 126–27
financing of courts as cost of rights, 45–46
property rights and, 67–68
"scarce resources" problem, 88–89, 94–97, 98
scope of rights, influence on, 80–83
see also Supreme Court, U.S.
covenants between buyers and sellers, 91
creationism, teaching of, 185
credit, 70, 74
criminal justice system:
cost of, 77–80, 83
see also courts
curtailment of rights, 100–103, 105–7, 108–11
Declaration of Independence, 15
Dee Farmer v. Brennan, 242n9
defense spending, 62–63
democratic citizenship, 196
deontological thesis, 238n14
dependency of individuals and subgroups on the state, 204–5
desegregation, 56, 216
DeShaney case, 87–89, 93–94, 95, 96–97, 131
Dole, Robert, 137
domestic violence, 22
Douglas v. People of State of California, 241n7
due process rights, 25, 26–27, 53, 56, 160–61
duties, see responsible behavior
Dworkin, Ronald, 99, 100
earned income tax credit (EITC), 206, 214–15
economic regulation, 64–65, 68–76
Edmonson v. Leesville Concrete Co., 242n5
education:
desegregation of, 56, 216
public education, 205, 206, 210, 211–12, 213
religious liberty and, 181, 185–86
right to, 22, 211–12, 214, 231
Eighth Amendment, 28, 78
Eleventh Amendment, 27
Ellickson, Robert, 240n10
Employment Division, Department of Human Resources v. Smith, 245n4, 246n5
enforcement of rights, 19
compliance with law and, 175–77
as conflict management, 178
cost of, 154, 223–24
"positive" nature of all rights and, 43–48
privately inflicted harms and, 92–93
trade-offs in, 127–32
win-lose situations resulting from, 55
see also government performance
entitlement mentality, 138, 157–58
entitlements, see positive rights
environmental programs, 142–43
environmental rights, 123–28
Epstein, Richard, 63
Equal Employment Opportunity Commission (EEOC), 47
equality of opportunity, 214
equal rights, 199–203
Etzioni, Amitai, 137
European Convention on Human Rights, 237n5
exclusionary rule, 81
fairness, 206–7
Federal Aviation Administration (FAA), 95
Federal Tort Claims Act, 204
Federal Trade Commission (FTC), 73
fees, 20–21, 201
feminism, 164
Fifteenth Amendment, 170
Fifth Amendment, 79
First Amendment, 52, 53, 92, 184, 185, 223, 241n2
flag burning, 109
flood control, 66
food stamps, 157–58, 210
Fordyce County, Georgia v. The Nationalist Movement, 242n7
Fourteenth Amendment, 56, 92, 159, 241n2
Fourth Amendment, 79, 115, 128
fraud, 73–74
freedom of speech, 17, 52, 117, 159, 214
communal character, 159–60
costs of, 111–12
evolving meaning of, 110, 123, 223
framers' intentions regarding, 110
importance of, 107–8
regulation of, 108–11
Fundamental Legal Conceptions (Hohfeld), 239n4
Galston, William, 137
Gauthier, David, 245n2
Germany, 121, 123
gerrymandering, 114
Glendon, Mary Ann, 137, 158–59
Goldberg v. Kelly, 238n13
Goss v. Lopez, 238n13
government:
inevitability of, 31
liberty's dependence on, 204–5
market economy and, 64–65, 68–76
opposition to, 13–14
rights' dependence on, 14–15, 19, 30–31, 232
rights-enforcing functions, 44–48
government performance:
constitutional rights and, 51–54, 58
enfranchisement of citizens through, 53
private-law rights and, 49–51
protection "against" government "by" government, 55–58
Great Society, 41, 42
gypsies, 198
habeas corpus, 77
Hart, H. L. A., 16
Hayek, Friedrich, 69
health care rights, 196–97, 231
health-health trade-offs, 124–26
Himmelfarb, Gertrude, 137
Hobbes, Thomas, 152
Hohfeld, Wesley, 239n4
Holmes, Oliver Wendell, Jr., 16
homeless persons, 206
home ownership, 206
Homestead Act of 1862, 213
human rights, 19
Hume, David, 61
Hungary, 123
identity politics, 216
illegitimacy rates, 157–58
immigrants' rights, 46
immunities, see negative rights
inclusion, see politics of inclusion
incorporation doctrine, 56
individual rights, 154
interests:
rights, comparison with, 17, 99–100
rights as interests of a special kind, 104–6
International Covenant on Economic, Social, and Cultural Rights, 119, 243n2
involuntary servitude, prohibition of, 91
Japanese-Americans, internment of, 100
Johnson, Lyndon, 41
judiciary, see courts
jury trial, right to, see due process rights
Justice Department, U.S., 46, 93
Kelsen, Hans, 16, 113
King, Martin Luther, Jr., 166
Korematsu v. United States, 242n3
laissez-faire theory, 71
Lebron v. National Railroad Passenger Corp., 242n7
legal aid to the poor, 201
legal rights, 16–20, 152
responsible behavior and, 152–53, 244n2
liberals:
"corrosive" effect of rights and, 137–38, 147
cost of rights, reluctance to acknowledge, 25, 28
negative/positive rights dichotomy and, 41–42
rights favored and disfavored by, 191
libertarians, 13–14, 63, 71
liberty, 20
dependence on government, 204–5
privately inflicted harms and, 90
life, right to, 159
long-term stability, 70–71
Maher v. Roe, 35
marine safety, 66
market economy, government and, 64–65, 68–76
marriage rights, 47, 135–36
Marshall, T. H., 239n5
Marshall, Thurgood, 166
Marxist theory, 207
Massachusetts, 114
Matthews v. Eldridge, 238n13
Medicaid, 35, 210
Medicare, 210
Mexico, 123
Mill, John Stuart, 194
minimal state, 63, 229
mixed regime, 209
Montesquieu, 55
moral censure, 162–64
morality:
cost of rights and, 18–19, 226
law and, 147–49
see also responsible behavior
moral rights, 16, 17–20
Morals by Agreement (Gauthier), 245n2
motorcycle helmet laws, 23
Murray, Charles, 13, 63
National Labor Relations Board (NLRB), 46
negative liberty, 239n3
negative/positive rights dichotomy:
appeal of, 43
conservative perspective, 40–41, 42
futility of categorizing rights, 37–39, 43
liberal perspective, 41–42
political implications, 42
usefulness in analysis of rights, 50–51
widespread acceptance of, 39–40
negative rights:
constitutional rights as, 51–54, 89
as immunity from government meddling, 35–36
religious liberty as, 181
see also negative/positive rights dichotomy
New Deal, 41, 42
New York City, 25
no-fault auto insurance, 27
norms, see social norms
Nozick, Robert, 63
O'Brien v. Skinner, 243n3
Occupational Safety and Health Act, 143
Occupational Safety and Health Administration (OSHA), 46–47
opportunity costs, 223–24
Order without Law (Ellickson), 240n10
Palestinians, 63
patent rights, 237n7
peremptory challenges, 92
Peters, B. Guy, 121
petition for redress of grievances, right to, 53
Plato, 149
police abuse, 128–29
police training, 24, 79
Political Liberalism (Rawls), 243n3
political primaries, 92
politics of inclusion, 197–99, 209, 219
poor nations, 119–20
positive liberty, 239n3
positive rights:
all legally enforced rights as, 43–48
as entitlement to benefits, 35–36
see also negative/positive rights dichotomy
Posner, Richard, 190
poverty relief, see welfare rights
Powell, Colin, 137
power, rights and, 55–58, 203, 205–6
pre-legal rights, 152
priority-setting, 125
prisoners' rights, 22, 44, 77–79, 81–83, 92, 241n5
see also accused's rights
privacy rights, 163
private-law rights, 49–51
privately inflicted harms:
constitutional rights and, 88, 89–92, 93, 96–97, 241n2
ordinary rights enforcement and, 92–93
"scarce resources" issue and, 88–89, 94–97
private property, see property rights
procedural republic, 160–61
Prohibition, 170
property rights, 15, 22
as bargain, 195–96, 208–9
bequeathing one's private property, 47
cooperative nature of society and, 192–93
costs of, 61–66, 73–74
democratic citizenship and, 196
deterrence of crimes against property, 60–61, 64–65
economic regulation and, 64–65, 68–76
as entitlements, 68
foreign threats, 62–63
as government-delivered service, 29, 48, 59–61
origin as veterans benefits, 194–95
poverty relief as means of safeguarding property, 189–91, 193–94, 196–97
privately inflicted harms and, 90–91
redistributive nature, 229–30
responsible behavior and, 140, 146, 149–51
rules of ownership, creation and enforcement of, 66–68
selective enforcement of, 129–30
social purposes, 61–62, 116, 192, 198, 230–31
sovereignty and, 63
uncompensated takings, prohibition on, 52, 57
"victimology" and, 165
public education, 205, 206, 210, 211–12, 213
quasi-rights, 127
racial dimension of rights, 215–17
rape, 144
Rawls, John, 243n3
Raz, Joseph, 117
Reagan, Ronald, 13
Redhail, John, 135
redistribution of resources, 113–17, 229–30
Rehnquist Court, 82–83
religious liberty, 105, 179–80
costs of, 180–83, 187
free exercise/no-establishment conflict, 180
as negative right, 181
peaceable social coexistence and, 182, 186–88, 218
regulation of, 182–86, 245n4
as social pact among churches and sects, 186
threat from private sects, 184–86
remedies, see enforcement of rights
rescue, proposed duty to, 158–59
responsible behavior:
cost of rights and, 146–47
creation of rights by imposition of responsibilities, 158–59
definition of, 139
enforcement of moral responsibilities, dangers of, 171
by government agents, 145
increases in, 142–45
legal rights and, 152–53, 244n2
legal system and, 141–42, 148–49
misuse of rights and, 140, 153–54
moral censure and, 162–64
mutual dependence of rights and responsibilities, 140–41, 145–47, 151, 155, 171
rights' "corrosive" effect on, 135–39, 147, 151
rights' role in encouraging, 149–51, 153–55, 159–61, 170
social norms and, 142, 143, 144, 145, 156–57, 158, 168–70
society's irresponsibility, rights claims as responses to, 166–70
"victimology" and, 164–66
welfare rights and, 157–58
see also morality
retaliation, protection against, 92–93, 171
"right order," 154–55
rights:
accountability in rights-related decision-making, 227–29
antimajoritarian dimension, 115–16
aspirational (open-ended) nature, 119–20
balancing of competing rights, 100, 106–7, 207–8
as bargains, 177–80, 188, 191–92, 199–200
changes in interpretation and application, 80–83, 122–23, 222–23
collective interests and individual benefits, 115–17
communal decency, promotion of, 159–61, 178–79, 217–19
as compensatory responses to an original dereliction of social responsibility, 166–70
curtailment of, 100–103, 105–7, 108–11
definition of, 15–16, 123
equality issue, 199–203
government, dependence on, 14–15, 19, 30–31, 232
inadmissible arguments regarding, 105–7
interests, comparison with, 17, 99–100
as interests of a special kind, 104–6
intrinsic value of some rights, 103–4
misuse, potential for, 17, 103, 140, 153–54, 161
plethora of rights entrenched in American law, 37–39
political stability through precise balance of rights, 207–8
politics of inclusion and, 197–99
power and, 55–58, 203, 205–6
racial dimension, 215–17
regulatory nature, 106
as trumps, 99, 101
"victimology" and, 164–66
see also "absoluteness" of rights; cost of rights; enforcement of rights; responsible behavior; specific rights
"rights talk," 104–5, 137
Roe v. Wade, 35
Roosevelt, Franklin D., 41, 118
Russia, 123, 149
"scarce resources" issue, 88–89, 94–98, 101–2, 107, 131–32
school prayer, 181
searches and seizures, 80–81, 115, 128–29
Second Bill of Rights, proposed, 118
Securities and Exchange Commission (SEC), 73
selective attention, problem of, 125
self-defense, right to, 103
self-financing rights, 22, 237n7
separation of powers, 122
sex equality, 164, 166–68
sexual harassment, 110–11, 144
sexual mores, 156–57
shame, 137
Shelley v. Kraemer, 242n4
Sherbert v. Verner, 245n4
Sixth Amendment, 79, 83
Skocpol, Theda, 194
slavery, prohibition of, 91
Smith, Adam, 157
Smith v. Allwright, 242n6
smoking, 142, 143
social contract, 176, 177–79, 188, 198, 202, 207, 245n2
social norms, 81, 237n3, 240n10
responsible behavior and, 142, 143, 144, 145, 156–57, 158, 168–70
Social Security, 210
social services, 87–89, 94
South Africa, 22–23
sovereignty, 63
speech, freedom of, see freedom of speech
speech codes, 169
state of nature, 152, 161
state power, federal encroachment on, 57–58
sue, right to, 17, 25–28, 50, 103
Superfund, 124
Supreme Court, U.S., 55, 57, 100, 216
on abortion rights, 35–36
on accused's rights, 80, 82–83, 241n7
cost-consciousness of, 26–27, 238n13
on due process rights, 26–27
on freedom of speech, 111, 112
lavishing of rights, alleged, 136, 137
on marriage rights, 135
on negative and positive rights, 35–36
on prisoners' rights, 241n5
Rehnquist Court, 82–83
on religious liberty, 183, 245n4
on state protection of individuals, 88, 89, 93, 96–97
Warren Court, 25, 41, 82, 136
symbolic rights, 127
taxes, 20–21, 75–76
compliance with tax laws, 148–49, 155
Third Amendment, 23
third-generation rights, 127–28
Thirteenth Amendment, 52, 91
Thomas, Clarence, 137
Tilly, Charles, 245n3
Tinker v. Des Moines School Dist., 240n2
tobacco advertising, 142
tort law, 49–50
trade-offs:
in enforcement of rights, 127–32
in environmental protection, 124–26
trespass, law of, 52, 90–91
"trumps" perspective on rights, 99, 101
United Nations, 119
United Nations Declaration of Human Rights, 19
value of liberty, 20
veterans benefits, 194–95, 197
"victimology," 164–66
Victims' Rights Amendment, proposed, 168
violent competition, 71–72
voting rights, 53, 57
cost of, 113–15
Voting Rights Act of 1964, 57
Warren Court, 25, 41, 82, 136
welfare rights, 15, 118–19, 223
aspirational (open-ended) nature, 119
autonomy and initiative as goals of welfare programs, 205, 211–15
as bargain, 197, 208–9
constitutionalization issue, 120–21
cost of, 209, 210
fairness and, 206–7
intergenerational redistribution through welfare programs, 209–10
negative/positive rights dichotomy and, 40–42
origin as veterans benefits, 194, 197
politics of inclusion and, 198–99, 209, 219
property protection through poverty relief, 189–91, 193–94, 196–97
responsible behavior and, 157–58
Westhampton fire of 1995, 13, 14
"Where Do Rights Come From?" (Tilly), 245n3
Will, George, 137
William the Conqueror, 194–95
Wilson v. Seiter et al., 241n2
Wilson v. Seiter supr., 241n5
Winnebago County (Wisconsin) Department of Social Services (DSS), 87–89
witness protection programs, 93
workers' compensation, 222–23
workplace safety, 46–47, 143
Zablocki, Mary, 135
Copyright © 1999 by Stephen Holmes and Cass R. Sunstein
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Library of Congress Cataloging-in-Publication Data
Holmes, Stephen, 1948–
The cost of rights : why liberty depends on taxes / Stephen Holmes and
Cass R. Sunstein.
p. cm.
Includes bibliographical references and index.
ISBN 0-393-04670-2
1. Civil rights—United States—Costs. 2. Finance, Public—United States.
3. Government spending policy—United States. I. Sunstein, Cass R. II. Title.
JC599.U5 H55 1999
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Elizaveta Danilovna Drozd (Russisch: Елизавета Даниловна Дрозд; Krasnojarsk, 18 september 1999), beter bekend als Liza Drozd, is een Russisch zangeres.
Drozd vormt al geruime tijd een zangduo met Sasja Lazin. Samen namen zij al aan verschillende zangwedstrijden deel. In 2010 wonnen ze het festival Moscow - Kansk Transit en Zolotoy Petushok in Nizjni Tagil met het nummer Boy and girl. Met dit nummer mochten zij vervolgens ook deelnemen aan de Russische voorronde voor het Junior Eurovisiesongfestival. Wederom met succes, want ze wonnen opnieuw en mochten daarom Rusland vertegenwoordigen op het Junior Eurovisiesongfestival 2010, in de Wit-Russische hoofdstad Minsk. Dit deden ze ook weer met het nummer Boy and girl.
Drozd en Lazin kwamen in Minsk zeer dicht bij de winst. Ze eindigden op de tweede plaats met slechts één punt achter winnaar Armenië.
Drozd is naast zangeres ook presentatrice van een kinderprogramma (Odni doma) op de regionale zender TRK Afontovo.
Russisch zanger
Junior Eurovisiesongfestival 2010 | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: Sending messages and roadside to roadside (R2R) communication in Veins I'm new in Omnet Veins. I try to create my own application. So first of all, I have done this in the existing TraciDemo11p files (I have just kept the files name and modify the code).
In the first step, I want to make all nodes sending a HelloMsg (a new packet that I have created .msg .h and .cc).
To well understand how messages are exchanged between nodes, I launched the simulation and all is well, but I cannot realize if the messages are received by nodes or not.
This is a screenshot of what I have:
enter image description here
I followed the transmission of the message between the application, mac and phy layers. I can see that the message is successfully transmitted by node1 for example. But does the message on node[0] "packet was not detected by the card. power was under sensitivity threshold" mean that the packet was not received by node[0]?. If it is the case, how can I fix that? Also, I cannot find the source file of this message (apparently, in PhyLayer80211p.cc or BasehyLayer.cc but I cannot find it).
In the second step, I want to use two RSUs. Nodes broadcast a helloMessage and then each RSU will repeat the received signal. To clarify more, this exactly what I have:
First of all. I add another RSU to the veins example as follows:
##########################################################
# RSU SETTINGS #
# #
# #
##########################################################
*.rsu[0].mobility.x = 6490
*.rsu[0].mobility.y = 1000
*.rsu[0].mobility.z = 3
*.rsu[1].mobility.x = 7491
*.rsu[1].mobility.y = 1000
*.rsu[1].mobility.z = 3
*.rsu[*].applType = "TraCIDemoRSU11p"
*.rsu[*].appl.headerLength = 80 bit
*.rsu[*].appl.sendBeacons = false
*.rsu[*].appl.dataOnSch = false
*.rsu[*].appl.beaconInterval = 1s
*.rsu[*].appl.beaconUserPriority = 7
*.rsu[*].appl.dataUserPriority = 5
Also, I made two maxInterferenceDistance, one of the nodes and the other for the RSUs:
##########################################################
# 11p specific parameters #
# #
# NIC-Settings #
##########################################################
*.connectionManager.sendDirect = true
*.connectionManager.maxInterfDist = 1000m #2600m
*.connectionManager.drawMaxIntfDist = false #false
*.connectionManager.maxInterfDistNodes = 300m
*.connectionManager.drawMaxIntfDistNodes = false
*.**.nic.mac1609_4.useServiceChannel = false
*.**.nic.mac1609_4.txPower = 20mW
*.**.nic.mac1609_4.bitrate = 6Mbps
*.**.nic.phy80211p.sensitivity = -89dBm
*.**.nic.phy80211p.useThermalNoise = true
*.**.nic.phy80211p.thermalNoise = -110dBm
*.**.nic.phy80211p.decider = xmldoc("config.xml")
*.**.nic.phy80211p.analogueModels = xmldoc("config.xml")
*.**.nic.phy80211p.usePropagationDelay = true
*.**.nic.phy80211p.antenna = xmldoc("antenna.xml", "/root/Antenna[@id='monopole']")
To make the transmission range of RSU different on that of nodes, I made this change in the isInRange function of the baseConnectionMannager:
bool BaseConnectionManager::isInRange(BaseConnectionManager::NicEntries::mapped_type pFromNic, BaseConnectionManager::NicEntries::mapped_type pToNic)
{
double dDistance = 0.0;
if ((pFromNic->hostId == 7) || (pFromNic->hostId == 8)) {
EV<<"RSU In range from: "<<pFromNic->getName()<<" "<<pFromNic->hostId<<" to: "<<pToNic->getName()<<" "<<pToNic->hostId<<"\n";
if(useTorus) {
dDistance = sqrTorusDist(pFromNic->pos, pToNic->pos, *playgroundSize);
} else {
dDistance = pFromNic->pos.sqrdist(pToNic->pos);
}
return (dDistance <= maxDistSquared);
} else {
if(useTorus) {
dDistance = sqrTorusDist(pFromNic->pos, pToNic->pos, *playgroundSize);
} else {
dDistance = pFromNic->pos.sqrdist(pToNic->pos);
}
return (dDistance <= maxDistSquaredNodes);
}
}
Where node IDs 7 and 8 are the RSUs in the scenario I run.
In addition, I have the TraciDemo11p (for nodes) and TraciDemoRSU11p (for RSUs) modified as follow:
- In the TraciDemo11p, nodes when enter the network broadcast a Hello message to all their neighbors. The code is:
void TraCIDemo11p::initialize(int stage) {
BaseWaveApplLayer::initialize(stage);
if (stage == 0) {
HelloMsg *msg = createMsg();
SendHello(msg);
}
}
HelloMsg* TraCIDemo11p::createMsg() {
int source_id = myId;
double t0 = 0;
int port = 0;
char msgName[20];
sprintf(msgName, "send Hello from %d at %f from gate %d",source_id, t0, port);
HelloMsg* msg = new HelloMsg(msgName);
populateWSM(msg);
return msg;
}
void TraCIDemo11p::SendHello(HelloMsg* msg) {
findHost()->getDisplayString().updateWith("r=16,green");
msg->setSource_id(myId);
cMessage* mm = dynamic_cast<cMessage*>(msg);
scheduleAt(simTime() + 10 + uniform(0.01, 0.02), mm);
}
void TraCIDemo11p::handleSelfMsg(cMessage* msg) {
if (dynamic_cast<HelloMsg*>(msg)) {
HelloMsg* recv = dynamic_cast<HelloMsg*>(msg);
ASSERT(recv);
int sender = recv->getSource_id();
if (sender == myId) {
EV <<myId <<" broadcasting Hello Message \n";
recv->setT0(SIMTIME_DBL(simTime()));
sendDown(recv->dup());
}
}
else {
BaseWaveApplLayer::handleSelfMsg(msg);
}
}
void TraCIDemo11p::onHelloMsg(HelloMsg* hmsg) {
if ((hmsg->getSource_id() == 7) || (hmsg->getSource_id() == 8)) {
EV <<"Node: "<<myId<<" receiving HelloMsg from rsu: "<<hmsg->getSource_id()<<"\n";
} else {
EV <<"Node: "<<myId<<" receiving HelloMsg "<<hmsg->getKind()<<" from node: "<<hmsg->getSource_id()<<"\n";
NBneighbors++;
neighbors.push_back(hmsg->getSource_id());
EV <<"Node: "<<myId<<" neighbors list: ";
list<int>::iterator it = neighbors.begin();
while (it != neighbors.end()) {
EV <<*it<<" ";
it++;
}
}
}
void TraCIDemo11p::handlePositionUpdate(cObject* obj) {
BaseWaveApplLayer::handlePositionUpdate(obj);
}
On the other hand, RSUs just repeat the message they received from nodes. So, I have on the TraciDemoRSU11p:
void TraCIDemoRSU11p::onHelloMsg(HelloMsg* hmsg) {
if ((hmsg->getSource_id() != 7) && (hmsg->getSource_id() != 8))
{
EV <<"RSU: "<<myId<<" receiving HelloMsg "<<hmsg->getKind()<<" from node: "<<hmsg->getSource_id()<<" at: "<<SIMTIME_DBL(simTime())<<" \n";
//HelloMsg *msg = createMsg();
//SendHello(msg);
hmsg->setSenderAddress(myId);
hmsg->setSource_id(myId);
sendDelayedDown(hmsg->dup(), 2 + uniform(0.01,0.2));
}
else {
EV<<"Successful connection between RSUs \n";
EV <<"RSU: "<<myId<<" receiving HelloMsg "<<hmsg->getKind()<<" from node: "<<hmsg->getSource_id()<<"\n";
}
}
After the execution of this code, I can see:
*
*a few numbers of vehicles receiving the hello message from their neighbors.
*also, just a few messages were received by the two RSUs.
*Each RSUs repeats the signal it receives, but there is no communication between the two RSU, which are supposed in the transmission of one another.
*And always I have a lot of this message "packet was not detected by the card. power was under sensitivity threshold" printed on my screen.
Is there any problem in the transmission range or it is a question of interference? Also, I would like to mention that in the analysis there is no packet loss.
Thanks in advance.
Please help.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,640 |
Badara peut désigner :
Toponymes
Badara, la commune rurale située dans le département de Bama au Burkina Faso ;
Badara, la commune rurale située dans le département de Soubakaniédougou du Burkina Faso ;
Badara, la commune du Haut-Karabagh.
Patronymes
Alpha 5.20, de son vrai nom Ousmane Badara, rappeur français d'origine sénégalaise.
Homonymie de toponyme | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,662 |
Q: Getting "RangeError: Maximum call stack size exceeded" when doing form.reset() with Angular custom text field I created a custom text field with Angular Material. Everything seems to be working correctly except when I try to reset the FormGroup I get RangeError: Maximum call stack size exceeded.
I set up the custom text field using this StackOverflow answer: https://stackoverflow.com/a/60071669/2026659.
Here's my custom text field component typescript:
import { Component, Input, Optional, Self, AfterViewInit } from '@angular/core';
import { ControlValueAccessor, NgControl, FormControl } from '@angular/forms';
import { of } from 'rxjs';
import { skipWhile, take } from 'rxjs/operators';
export class RequiredFormControl extends FormControl {
required = true;
}
@Component({
selector: 'input-text-field',
templateUrl: './input-text-field.component.html',
styleUrls: ['./input-text-field.component.scss']
})
export class InputTextFieldComponent implements AfterViewInit, ControlValueAccessor {
public _formControl = new FormControl();
public onChange = (value: any) => {};
@Input() public fieldId: string;
@Input() public isRequired: boolean;
@Input() public fieldClass: string;
@Input() public labelText: string;
@Input() public tabNumber: number;
constructor(@Self() @Optional() public ngControl: NgControl) {
if(this.ngControl) {
this.ngControl.valueAccessor = this;
}
}
ngAfterViewInit(): void {
if (this.ngControl) {
of(this.ngControl.control)
.pipe(
skipWhile(fc => !fc),
take(1)
)
.subscribe(fc => {
this.formControl = fc as FormControl;
});
}
}
get formControl() :FormControl|RequiredFormControl {
return this._formControl;
}
set formControl(forControl:FormControl|RequiredFormControl) {
this._formControl = forControl;
}
registerOnChange(fn: (value: any) => void): void {
this.onChange = fn;
}
registerOnTouched(fn: (value: any) => void): void {}
writeValue(value: any): void {
if(this.formControl) this.formControl.setValue(value, { emitEvent: false });
}
}
Here's my custom text field html:
<mat-form-field>
<mat-label><span *ngIf="isRequired" class="input-required">*</span> {{labelText}}</mat-label>
<input type="text"
matInput
[formControl]="formControl"
[id]="fieldId"
[class]="fieldClass"
[tabIndex]="tabNumber"
>
</mat-form-field>
I create the associated FormGroup and FormControl in my parent component:
const userProfile = new FormGroup({
firstName: new FormControl(null, Validators.required),
lastName: new FormControl(null, Validators.required),
....
});
I have a button in the parent component that triggers a function that resets the form:
public resetFormEventClick() {
this.userProfile.reset();
}
I looked at the stack trace and I found the error occurs when the code gets to the writeValue function in my custom text field component.
I tried adding a form reset function to this Stackblitz and got the same RangeError. I was trying to see if the error was something peculiar to my code.
It seems the error is when the reset function tries to change the value of the custom text field. All the function is doing relating to the value is changing it back to null. I don't understand why it would throw an error relating to the call stack. Any help would be appreciated. I've been having a tough time trying to get this custom component to completely work.
UPDATE:
I tried using patchValue instead of setValue in the writeValue function like this: this.formControl.patchValue(value, {onlySelf: true, emitEvent: false}); but still got the same RangeError.
I added a console.log inside the writeValue function and when this.userProfile.reset() is triggered the console.log message keeps repeating then the RangeError occurs. There seems to be an infinite loop where the code keeps triggering the writeValue function.
A: Thanks to @Eliseo's suggestion I was able to solve the error.
I changed the writeValue function to this:
writeValue(value: any): void {
if(this.formControl && this.formControl.value!=value) this.formControl.setValue(value, { emitEvent: false });
}
Updating the if statement stopped the infinite loop.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,859 |
using NUnit.Framework;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Reflection;
using FluentAssertions;
namespace EasyReflectionTests
{
[TestFixture]
public class MemberInfoExtensionsTests
{
private Type type;
[SetUp]
public void SetUp()
{
this.type = typeof(TestClass);
}
[Test]
public void PropertyInfoGetNamesReturnsNames()
{
var names = type.GetGettersAndSetters().GetNames();
var expectedGettersAndSetters = new[] { "PublicProperty", "PrivateProperty", "ProtectedProperty", "InternalProperty" };
names.Should().Contain(expectedGettersAndSetters).And.HaveCount(expectedGettersAndSetters.Count());
}
[Test]
public void FieldInfoGetNamesReturnsNames()
{
var names = type.GetAllFields().GetNames();
var expectedFields = new[] { "publicField", "privateField", "protectedField", "internalField" };
names.Should().Contain(expectedFields).And.HaveCount(expectedFields.Count());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,640 |
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