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BSF Recruitment 2015 : 346 Constable (GD) Posts @ www.bsf.nic.in: Boarder Security Force (BSF) issued official notification for BSF 2015 Recruitment of 346 Constable (GD) on official website: http://www.bsf.nic.in (Recruitment of Sports Persons). There are total 346 vacancies of Constables (GD) (241 for male and 105 for female). We are putting further details as follows regarding the Recruitment.
Candidates must pass 10th class from any recognized board/university. Sportsperson must be represented a State or the Country in National or International level comptition in any sports event.
Candidates who are eligible for BSF Recruitment 2015 Constable they must take print of recruitment notification (as we giving below); fill the form. Attch the required documents and send to: Commandant, 25 Bn BSF, Chhawla Camp, Post Office-Najafgarh, New Delhi, Pin code-110071 within 30 days & for remote areas 45 days. | {
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Boeing and Continental Announce Deal for More 787s, Including the 787-9 Launch Order for the Americas
The Boeing Company [NYSE: BA] and Continental Airlines announced an order for five 787-9 Dreamliners.
SEATTLE — This order, combined with the Houston-based airline's earlier announced orders, brings Continental's total firm 787 order to 25 aircraft. The airline also has contracted to convert 12 previously ordered 787-8 jetliners to the larger 787-9s.
Continental was the first airline in the Americas to order the 787 Dreamliner, placing its initial order for 10 airplanes in 2004. With today's announcement, Continental also becomes the first customer in the Americas to order the 787-9. The 787-9 (250-290 passengers) is a slightly larger version of the 787-8 (210 to 250 passengers) with increased seating and range capability (approximately 8,500 nmi versus 8,200 nmi).
"We recognized the benefits of the 787 early, and today's order further demonstrates our confidence in Boeing and the 787 Dreamliner," said Continental's Chairman and Chief Executive Officer Larry Kellner. The 787-9 is a welcome addition to the 787 family. With its increased size and range we will have the ability to serve more cities with the lowest possible cost per seat."
"The 787 Dreamliner will enhance Continental's fleet with both its unmatched fuel efficiency and its flexibility for both regional and long-haul operations," said Ray Conner, Boeing Commercial Airplanes vice president of Sales, the Americas. "Passengers will choose to fly the 787 for its advanced interior environment including higher humidity, larger windows and lower cabin altitude."
[GADS_NEWS]Continental's 787s will be powered by all-new fuel efficient and environmentally progressive GEnx engines developed by General Electric. Continental is scheduled to receive its first 787 in 2009.
The Boeing 787 Dreamliner, scheduled for launch in 2008, provides passengers with a better flying experience and operators with a more efficient commercial jetliner. Thirty-seven airlines have logged 470 orders worth more than $70 billion at current list prices since the 787 launch in April 2004, making the Dreamliner the most successful commercial airplane launch in history.
Nicolaas Groeneveld-Meijer
International & Sales Communications
787 Program Communications
Allianzen Kooperationen Kartelle, Luftverkehr
Boeing and Continental Announce Deal for More 787s | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/1916457\/direct-product-commutes-with-tensor-product","text":"# direct product commutes with tensor product?\n\nLet $(A_i)_{i\\in I}$ be a family of right $R$-modules and $M$ be a left $R$-module, where $I$ is an index set. The natural homomorphism $$\\varphi:(\\prod_{i\\in I}A_i)\\otimes_RM\\to \\prod_{i\\in I}(A_i\\otimes_RM)$$ given by $(a_i)\\otimes m\\mapsto(a_i\\otimes m)$ is not always a bijection. It is easy to obtain $\\varphi$ is a surjection provided $M$ is finitely generated. If we assume that $M$ is finitely presented, can we prove $\\varphi$ is a bijection?\n\n$M$ being finitely presented means that there is an exact sequence $$R^m \\to R^n \\to M \\to 0.$$ This gives a commutative diagram $$\\require{AMScd}\\begin{CD} \\bigl(\\prod_i A_i\\bigr) \\otimes_R R^m @>>>\\bigl(\\prod_i A_i\\bigr) \\otimes_R R^n @>>> \\bigl(\\prod_i A_i\\bigr) \\otimes_R M @>>> 0\\\\ @VVV @VVV @VVV \\\\ \\prod_i (A_i \\otimes_R R^m) @>>> \\prod_i (A_i \\otimes_R R^n) @>>> \\prod_i (A_i \\otimes_R M) @>>> 0 \\end{CD}$$ with exact rows since tensoring is right exact. The first and second vertical arrows are easily seen to be isomorphisms. Then by the five lemma, the right vertical arrow is an isomorphism, too.","date":"2019-07-20 16:07:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.968463122844696, \"perplexity\": 64.02562667990983}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526536.46\/warc\/CC-MAIN-20190720153215-20190720175215-00081.warc.gz\"}"} | null | null |
Most of the Vegetable Oil Is Hot Pressed Oil In Our Daily Life. Namely, the oil raw materials were cleared and crushed, then heat them before pressing oil, it occurs a series of changes inside of oil raw materials: damage the oil materials cell, bring protein denaturation, reduce oil viscosity, etc, so that convenient to press oil and improve oil yield.
More importantly, it also provide palm oil mill plant turnkey construction. The need of palm oil mill equipment . Using traditional palm oil processing for the extraction of palm oil is the key factor for a low palm oil economy in Africa. By using the crude methods for extraction of pulp from the palm fruit leaves too much of residue on the fruit.
Dear, I hope this message finds you well. We are now prepared to proceed with the Crude Palm Oil Mill and Palm Kernel Oil Production. Timing is very critical for us as we are looking to complete the project in 2016 - initial scope of Mill production 2 tonnes Fresh Fruit Bunches Per hour and Palm Kernel Oil equivalent (from extraction of 2 FFB).
As well as from free samples, paid samples. There are 3,530 palm kernel prices suppliers, mainly located in Asia. The top supplying countries are China (Mainland), India, and United States, which supply 80%, 15%, and 1% of palm kernel prices respectively. Palm kernel prices products are most popular in Africa, Domestic Market, and South America.
More than 10 types of seeds can be pressed! (peanuts, rapeseeds, sesame, copra, cottonseeds, hazelnut, linseed, oil palm kernel, pumpkin seed, sun flower seeds, walnuts).
Regarding to the palm kernel: Of course the plant can deal with the palm kernel.The palm kernel oil contant is 45% more or less. So the oil output also is very high. I need one project for oil mill project of all.
Palm kernel oil can produce up to 15 to 18 percent MCT oil. To extract the MCTs, the oils must go through a process known as "fractionation," which includes distilling and isolating fatty acids.
The cold-pressed oil extracted from the Palm pulp is known as Crude Palm Oil (CPO). The Carotene content of crude Palm Oil contributes to its characteristic red color. When it is refined, bleached, and deodorized, its Carotenes are removed, thus the RBD oil becomes colorless as well as odorless.
Manufacturing Palm oil mill effluent treatment system station,In the process of palm oil production will produce a lot of waste water, the waste water will direct emissions pollution of land.Pool palm oil mill effluent treatment system station is the most used and effective treatment.
Expeller-pressed coconut oil can be produced more quickly than cold-pressed coconut oil as there is a wider range of acceptable temperatures during processing. Coconut Oil – Lauric Acid Source Coconut oil is known and revered by many as a great source of lauric acid.
Mianyang Guangxin Machinery Of Grain & Oil Processing Co., Ltd. Supplier from China. View Company.
2017 New Improved 1200 wet pan mill in Africa gold mine. 2017 New Improved New gold refinery mercury amalgamation machine for sale. New gold refinery Get Price Cold Pressed Coconut Oil Processing Machine In Nigeria - Buy Cold Get Price Palm Kernel Oil Processing Machine /sunflower Oil Production Line .
Pure and modified palm olein (palm olein + 5% palm oil) with an iodine value of 56.5 and 56.4 respectively were crystallized with a controlled cooling system and convenient stirring process.
Palm/ Palm Kernel Oil: These oils are the most popular plant-based alternative to tallow or lard when it comes to hardening a bar of soap. However, if you're shying away from tallow and lard for reasons of animal welfare or the environment, you should really think twice about using palm products.
Ndayishimiye and Tezerout used preheated palm oil and palm oil blended at 5, 10, 20 and 30% by wt. with diesel to investigate the performance and emission of a Direct Injection (DI) diesel engine. They found BTE of preheated palm oil blends were around 27% higher than OD.
Olive Oil: Most of the olive oils on the market are blended with (unlabeled) cheaper vegetable oils so be sure to buy only 100% extra virgin olive oil from a reliable source for making homemade salad dressings and low heat cooking.
It is used as cooking oil. about 50% and 80%.7% Polyunsaturated Linoleic C18:2 10.9%) Palmitic C16:0 44. monounsaturated oleic acid is also a constituent of palm oil while palm kernel oil contains mainly lauric acid. and amongst West African peoples. it may have now surpassed soybean oil as the most widely produced vegetable oil in the world.
Backed By Rich Industry Experiences, We Are Highly Engaged In Offering An Optimum Quality Grade Neem Oil Cake. Neem Cake Organic Manure Is The Bye Product Obtained In The Process Of Neem Oil Extraction From Neem Fruits/Seeds Through Oil Expellers Or Solvent Extraction Process.
Full text of "Oils Fats And Fuels"
All On Sale Page 6 - Goodness Me!
Chia Balance is a decadent, delicate treat. Indulge with these indulgent 62% Fair Trade dark chocolate bark bites containing perfectly caramelized coconut strips, nutrient-dense pumpkin and sunflower seeds, as well as probiotic chia in a feel-good snack that helps restore balance in your day.
The roasted plantain skin is mixed with palm oil, palm kernel oil to form the soap. The roasting of the plantains determines the brownish-black color of the soap. The longer the plantains are roasted, the darker the soap. Dr. Woods may be used by anyone who wishes to improve the quality of their skin.
Full text of "Oil; animal, vegetable, essential, and mineral"
What is the current price of iron per ton | ChaCha 20121113-What is the current price of iron per ton? ChaCha Answer: Prices for scrap iron have regressed to what they were in 1999.
The olive oil should be extra virgin, cold pressed (preferably the first pressing) in a tinted glass container (explained below). The use of butter, extra virgin coconut oil or palm fruit oil, is also an option.
Copra and palm kernels, however, must undergo a pre reduction by a hammer mill before being processed further by a cracking and a flaking mill. The flaking mill and its ability to produce consistently thin flakes are key to the efficient extraction of all oilseeds. | {
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Cet article relate les conséquences de la pandémie de Covid-19 sur la politique au Québec.
État d'urgence
Le , le gouvernement adopte un décret et déclare l'état d'urgence sanitaire, conformément à l'article 119 de la Loi sur la santé publique du Québec. L'état d'urgence permet au gouvernement d'adopter par décrets, sans débats, les mesures qu'il juge nécessaire pour gérer la pandémie. Ses mesures sont directement effectives, court-circuitant le pouvoir législatif de l'Assemblée nationale.
Le décret peut être renouvelé pour d'autres périodes maximales de dix jours ou, avec l'assentiment de l'Assemblée nationale, pour des périodes maximales de 30 jours. Entre et , le décret est renouvelé par le gouvernement près d'une centaine de fois pour des périodes de 10 jours au moins, ce qui fait que l'état d'urgence est maintenu tout au long de cette période.
Assemblée nationale
Aucune séance de l'Assemblée nationale du Québec n'a lieu avant le .
Le , de concert avec les chefs de groupes parlementaires, le président de l'Assemblée nationale, François Paradis, annonce que l'Assemblée nationale suspend l'accès du bâtiment de l'Assemblée nationale aux visiteurs en raison de la Covid-19. Les mesures entrent en vigueur dès le lendemain.
Le , les travaux parlementaires sont ajournés jusqu'au mardi .
Le , Simon Jolin-Barrette, leader parlementaire du gouvernement, annonce la prolongation de la suspension des travaux de l'Assemblée nationale jusqu'au . Il y aura cependant des commissions parlementaires virtuelles, une première historique, au cours desquelles les élus pourront poser des questions aux ministres du gouvernement. Ces commissions se tiendront du au .
Le a lieu la reprise des travaux à l'Assemblée nationale. Seuls 36 des peuvent siéger dans le Salon bleu, avec une distance minimale de entre chacun d'eux. Le drapeau du Québec est en berne de l'aube au crépuscule en mémoire des victimes de la pandémie.
Projet de loi sur la relance économique
Projet de loi 61
Le , le gouvernement Legault dépose le projet de loi 61, un projet de loi omnibus visant à faciliter la relance de l'économie québécoise. Portant surtout sur l'accélération de 202 travaux d'infrastructure, le projet inclut une mesure allégée d'expropriation et une accélération des consultations populaires du Bureau d'audiences publiques sur l'environnement. Le projet de loi permettrait également au gouvernement de prolonger l'état d'urgence sanitaire aussi longtemps qu'il le désire. Ce projet est salué par le Conseil du patronat, alors que les partis d'opposition craignent les risques de collusion.
Le projet de loi doit être adopté dans les suivant son dépôt, et doit obtenir l'unanimité du parlement en raison de son dépôt tardif. François Legault a indiqué que son gouvernement n'utiliserait pas le bâillon pour le faire adopter par le parlement, quitte à prolonger les travaux parlementaires au-delà du , date prévue de la fin de la session parlementaire.
Les courts délais pour se présenter en commission parlementaire ont été critiqués par les syndicats et ont empêché des joueurs importants, comme la vérificatrice générale et le Barreau du Québec, de participer aux audiences, bien qu'ils aient manifesté leur inquiétude dans les médias, notamment quant au renouvellement automatique de l'état d'urgence sanitaire.
Le , 300 personnes se réunissent à Val-David afin de manifester contre le projet de loi 61.
Le , le comité de suivi de la commission Charbonneau, ainsi que le Bureau de l'inspecteur général de Montréal soulignent que les modifications prévues dans le projet de loi 61 pourraient voir l'émergence d'un système parallèle, recréant un « environnement extrêmement favorable à la corruption, à la collusion et aux autres malversations ». Devant ces affirmations, François Legault a indiqué ne pas s'inquiéter de la corruption dans son parti, son équipe comptant sur l'ex-procureure en chef de la commission Charbonneau, la ministre de la Justice Sonia LeBel, de même que plusieurs comptables agréés, tel que le ministre du Trésor Christian Dubé. Le même jour, le gouvernement met en place un site web pour « séparer la fiction des faits ».
L'article 50 est l'objet de plus de critiques. Il permettrait au gouvernement de déroger à la Loi sur les contrats des organismes publics (LCOP), afin d'accorder des contrats de gré à gré, c'est-à-dire sans appel d'offres public. Les trois partis d'opposition et le député indépendant Guy Ouellette demandent l'abandon de ces pouvoirs extraordinaires.
À la suite des critiques, le président du Conseil du trésor Christian Dubé propose de faire des concessions sur plusieurs aspects du projet de loi 61, soient ceux dans l'article 50, (restreindre les pouvoirs discrétionnaires accordés au gouvernement, et revoir l'état d'urgence sanitaire à tous les ), ceux de l'article 51, permettant au gouvernement d'échapper à toute poursuite judiciaire, de même que d'autres amendements concernant la protection des milieux humides.
Malgré les concessions du gouvernement, les différents partis n'arrivent pas à une entente et la session parlementaire prend fin sans que le projet de loi ne soit adopté. Les mesures législatives contenues dans ce projet de loi ne pourront donc être adoptées avant la reprise des travaux parlementaires le .
Le , la présidente du Conseil du trésor, Sonia LeBel, annonce que le gouvernement abandonne le projet de loi 61, et déposera une nouvelle version de ce projet à la rentrée parlementaire de . Le nouveau projet de loi sera plus ciblé et inclura des contrepoids qui devraient répondre aux critiques qui avaient été faites à ce propos dans le projet de loi 61. Les partis d'opposition soulignent que cet abandon révèle l'improvisation du gouvernement Legault, et que ce projet de loi était abusif.
Projet de loi 66
Le , la présidente du Conseil du trésor, Sonia LeBel, présente le projet de loi 66, une version améliorée du projet 61 qui tient compte des critiques qui ont été faites. Le projet vise toujours à accélérer les projets d'infrastructures, ceux-ci passant de 202 à 181 projets. La liste est cependant fermée, contrairement à la précédente, et ne pourra pas faire l'objet de projets additionnels.
Les processus d'évaluations environnementales sont accélérés, ce qui suscite l'inquiétude des groupes environnementaux québécois, et il est possible de soustraire les projets de la Loi sur l'aménagement et l'urbanisme. Le projet de loi 66 accorde cependant un rôle à l'Autorité des marchés publics, ce que ne permettait pas le projet de loi 61 ce qui avait suscité de vives critiques. Ce nouveau rôle de l'Autorité des marchés publics (AMP) vise uniquement les 181 projets d'infrastructures et aucun autres chantiers du gouvernement du Québec. Le PDG de l'AMP a demandé à ce que les pouvoirs soient étendues à tous les chantiers et de façon permanente.
Les partis d'opposition, Équiterre et le Centre québécois du droit de l'environnement craignent que les allègements réglementaires en matière d'environnement ne servent de « Cheval de Troie » en vue d'affaiblir à long terme les protections environnementales. Alors que la Fédération des chambres de commerce du Québec appelait justement à en faire un projet pilote, Sonia LeBel, la présidente du Conseil du trésor et porteuse du dossier, se défend de vouloir chevaucher une telle monture.
Aide fédérale
Au mois de , Justin Trudeau indique que le gouvernement fédéral est prêt à dépenser une somme de de dollars pour relancer l'économie canadienne. Les champs de dépense pour ces sommes ne sont pas déterminés, mais le gouvernement du Canada indique que leur identification se fera en collaboration avec les provinces, et que sa distribution sera assortie de conditions, ce à quoi s'oppose les provinces canadiennes, dont le Québec.
En , devant le front commun des provinces, le gouvernement fédéral bonifie son programme, en le portant à , et assouplit ses demandes. Les provinces n'ont plus à remplir des conditions pour pouvoir profiter des sommes du gouvernement fédéral, mais devront investir dans des secteurs ciblés comme le dépistage de la Covid-19 ou encore l'augmentation du nombre des places en garderies.
Le détail du financement est donné le lendemain de l'annonce, et révèle que ce sera dans les faits qui seront accordés aux gouvernements provinciaux, une enveloppe de étant en fait réservé au gouvernement fédéral pour l'achat de matériel de protection individuel qui sera distribué au Canada. À ce montant s'ajoute d'aide aux municipalités et pour le transport en commun, pour un total de qui n'iront pas dans les coffres des gouvernements des provinces.
Éducation
Le , Justin Trudeau annonce que le gouvernement du Canada versera 2 milliards de dollars aux provinces pour la rentrée scolaire 2020, dont 432 millions pour le Québec. Cet argent devra servir à « adapter les milieux d'apprentissages ». Bien que l'éducation soit un champ de compétences du Québec, et que des provinces ont souligné qu'il s'agit là d'un empiétement du gouvernement fédéral, le gouvernement du Québec quant à lui s'est plutôt réjoui de cet ajout d'argent.
Références
Politique au Québec
2020 au Québec
Québec, Politique | {
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«Олимпик» () — украинский профессиональный футбольный клуб из Донецка. Основан 15 июня 2001 года.
История
15 июня 2001 года в Донецке был организован фонд развития футбола области «Олимпик». В этот же день, на его базе был создан одноименный футбольный клуб.
В течение 2003—2004 года «Олимпик» становился лучшим в Донецкой области, в сезоне 2004 в чемпионате области «Олимпик» занял второе место в клубном зачете, а команда футболистов 1985 года рождения представляла Донецкую область на вторых Всеукраинских летних спортивных играх и завоевала бронзовые медали. Этот же коллектив играл в чемпионате области среди взрослых и принимал участие в чемпионате Украины среди любителей.
В сезоне 2004/05 команда дебютировала во Второй лиге, в группе «В». В дебютном сезоне «Олимпик» занял 11-е место. В сезоне 2010/11 «Олимпик», лидируя на протяжении почти всего турнира, за тур до окончания чемпионата обеспечил себе первое место и выход в Первую лигу.
Сезон 2013/14 годов стал для «Олимпика» историческим — команда завоевала золотые медали и право выступать в Премьер-лиге. По ходу чемпионата «Олимпик» провел 20-матчевую беспроигрышную серию. Она длилась с 3 августа 2013 года по 13 апреля 2014 года.
Перед стартом в высшем дивизионе из-за вооружённого конфликта на востоке Украины «Олимпик» был вынужден переехать в Киев на стадион им. Виктора Банникова и распустить свою академию.
26 июля 2014 года «Олимпик» дебютировал в Премьер-лиге, где по итогам первого сезона занял 8-место в чемпионате и вышел в полуфинал Кубка, где по результатам двух матчей уступил киевскому «Динамо» с общим счётом 4:1.
Сезон 2016/2017 стал лучшим в истории «Олимпика» — по его итогам команда занял 4-е место и впервые получил право играть в еврокубках.
Свой первый еврокубковый матч «Олимпик» сыграл 27 июля 2017 года на стадионе «Динамо» имени Валерия Лобановского, в рамках 3-го квалификационного раунда Лиги Европы 2017/18 против греческого клуба ПАОК (1:1). Первый мяч донецкого клуба на международной арене забил Станислав Беленький. Второй матч на стадионе «Тумба» «Олимпик» проиграл 2:0 и выбыл из Лиги Европы.
В 2018 году «Олимпик» провёл ребрендинг и сменил эмблему впервые в своей истории. На принятом после открытого конкурса варианте красуются три белых розы — символы Донецка, — а также фирменные для клуба сине-голубые полосы. Автором новой эмблемы команды стал известный украинский дизайнер Дмитрий Штрайс.
28 июня 2019 года команду возглавил бразилец Жулио Сезар Сантос Корреа. До «Олимпика» тренеровал только любительский испанский «Кристо Атлетико». Во время игровой карьеры выступал «Реал Мадрид» (в составе которого стал победителем Лиги чемпионов), «Милан», «Бенфику», «Олимпиакос» и др. В августе того же года, после четырех поражений, отправлен в отставку. Команда под его руководством проиграла «Днепру-1» (0:2), «Колосу» (0:1), «Александрии» (1:2) и «Карпатам» (1:3).
8 июля 2021 года президент клуба Владислав Гельзин объявил о снятии «Олимпика» с чемпионата Украины и приостановке деятельности, однако позднее клуб был передан новому руководству и заявлен для участия в Первой лиге, где провёл ещё полгода, после чего был расформирован
Стадион
Домашней ареной команды является стадион спортивного комплекса «Олимпик» в Донецке.
Инфраструктура спортивного комплекса «Олимпик»:
Стадион с игровым футбольным полем с травяным покрытием размерами 105х68 метров и трибунами на 4500 индивидуальных посадочных мест;
Тренировочное футбольное поле с гаревым покрытием размерами 102x70 метров;
Тренировочное футбольное поле с травяным покрытием размерами 105x67 метров;
Тренировочное футбольное поле с синтетическим покрытием размерами 98x65 метров;
Мини футбольное поле с травяным покрытием размерами 50x30 метров;
Мини футбольное поле с синтетическим покрытием размерами 44x22 метра.
Предыдущие сезоны
Достижения клуба
Первая лига Украины
Победитель: 2013/14
Вторая лига Украины
Победитель: 2010/11
Чемпионат Донецкой области
Серебряный призёр: 2003
Мемориала Макарова
Победитель: 2016
Выступления в еврокубках
Примечания
Ссылки
ФК «Олимпик» Донецк
Футбольные клубы, основанные в 2001 году
Футбольные клубы Донецка | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,213 |
using NFluent;
using Xunit;
namespace LASI.Core.Tests
{
/// <summary>
/// This is a test class for IEntityTest and is intended to contain all IEntityTest Unit Tests
/// </summary>
public class IEntityTest
{
/// <summary>
///A test for EntityKind
/// </summary>
[Fact]
public void EntityKindTest()
{
IEntity target = new CommonSingularNoun("cat");
Check.That(target.EntityKind).IsEqualTo(EntityKind.Thing);
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,706 |
The university of the future: can the universities of today lead learning for tomorrow?
This report explores four divergent, yet plausible, views of Australia's higher education landscape in 2030. Each scenario introduces different opportunities and threats that challenge our thinking, question our assumptions, and help us think more broadly about the future. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,190 |
{"url":"http:\/\/technet.microsoft.com\/en-us\/library\/cc975532.aspx","text":"# Error Message:\n\nYou must type a valid Username for the user.\n\nExplanation:\n\nIn the New User dialog box, you have specified a user name that exceeds the limit of 20 characters, have left the option blank, or have included invalid characters. The following are invalid characters for a user name: ( ; : \" <> * + = \\\\ | ? , ).\n\nUser Action:\n\nSelect OK to return to the User Manager window, then specify a valid user name.","date":"2014-12-28 06:20:08","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9017215371131897, \"perplexity\": 3768.198115392082}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-52\/segments\/1419447555323.72\/warc\/CC-MAIN-20141224185915-00004-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
Dale is a HORNE Wealth Strategies manager at HORNE LLP. He specializes in individuals, corporations, S corporations and partnerships.
Dale joined HORNE in 2004 with more than 20 years of experience in public accounting.
Dale earned an Associate of Science in Accounting from Jackson State University, a Bachelor of Science in Business Administration with an emphasis in accounting and a Master of Accountancy from the University of Tennessee at Martin. He is also a Chartered Global Management Accountant. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,706 |
Common Risks for Common-Law Relationships
Marriage is on the decline in Canada and that cohabitation is on the rise. In 1961, less than 1 per cent of families lived in a cohabitation structure; now almost 20 per cent of families are spearheaded by common-law couples. Here are three things to consider before you shack up.
Ashley Redmond, Contributor
Writer and editor at Redmond Writes.
10/15/2014 05:19am EDT | Updated December 15, 2014
Ariel, 26, lived with her boyfriend in Oakville, Ont. for two years. The two planned on getting married, so they pooled their savings together. They accumulated about $26,800 in a tax-free savings account. When they realized that the relationship wasn't working, they split the money down the middle and each left with $13,400. For furniture and belongings, they each rotated on buying big-ticket items, so it was simple to divide assets.
"It was an amicable split, so it went pretty smoothly. But, I've heard horror stories of a partner leaving with all the joint savings, and it takes forever to get anything back," Ariel said.
From a legal standpoint, property rights are the main difference between being common-law and being married. When a marriage ends, the net family property is divided up based on a formula called equalization, says Christine Vanderschoot, a family law lawyer at the Toronto firm Jordan Battista LLP.
On the other hand, when common-law couples split up, the theory is that each person gets back what he or she brought into the relationship. This sounds simple enough. "But in reality it can be very hard to do that, especially when you start intermingling your funds, such as with a joint line of credit. Then it gets more complicated," Vanderschoot says.
She recommends that people obtain a cohabitation agreement, similar to a marriage contract (sometimes known as a pre-nuptial agreement), that outlines responsibilities and obligations early on.
This seems like a good idea, considering that marriage is on the decline in Canada and that cohabitation is on the rise. In 1961, less than 1 per cent of families lived in a cohabitation structure; now almost 20 per cent of families are spearheaded by common-law couples.
Here are three things to consider before you shack up:
Household and living expenses
There are obligations to meet such as rent and bills. Does it make sense to pool your resources to meet those expenses, or is it better to keep track of who pays for what? Robert Stammers, director of investor education for the CFA Institute, says that, unfortunately, there is no definite answer because there's nothing to indicate that one way is better than the other. A good idea may be taking a page out of Ariel's book and evenly distributing big-ticket purchases or keeping receipts. Vanderschoot says another way to divide living expenses is by room -- one person pays for the bedroom furniture and the other pays for the living room, for instance.
This topic has the ability to easily make or break any relationship. As with living expenses, you have to decide whether you wish to join your debt with that of your partner or have each person remain responsible for paying down his or her own debt. This can become especially dicey if one partner has bad credit because it can force the couple to take on new debt together by having the partner who has good credit co-sign for the one who doesn't. If the relationship goes sour, both people are then equally responsible for this debt.
Also, the reimbursement of debt during the relationship is treated differently depending on whether you are married or common law. "In a marriage it's assumed the debt was paid down using family money, so in the event of a divorce the amount of debt paid back will be added to your net worth on the date of separation and so be subject to the equalization process," Vanderschoot says. In essence, you will be giving your spouse credit for having paid half of your debt during the marriage. "Since there is no automatic joint property or equalization for common-law couples, each partner is responsible for the debt that is in their name." This means if you help pay down your boyfriend's student loan, for example, you can't expect him to pay you back if you split up, unless you had an explicit agreement to that effect.
We all need one. You need to sit down and figure out your short-term and long-term goals and whether you'll be saving together or independently. But that's not the only consideration. What are your respective investment philosophies? Are you going to use an advisor or stick to DIY investing? What's your risk profile? Stammers says risk profiles tend to change when people merge from singleton to relationship, as thinking switches to the long term.
After these discussions, if you are feeling somewhat uneasy, bring up a cohabitation agreement. Vanderschoot says this may be the best route to take in order to protect each person's assets, especially if one person is wealthier than the other.
Stammers emphasizes that overall the key to successful cohabitation, as in all relationships, is communication, because the more honest you are about your finances, the better off the relationship will be.
The legal definition of a common-law partner: A common-law partner is a person to whom you are not legally married, with whom you are living in a conjugal relationship, and to whom at least one of the following situations applies. He or she:
has been living with you in such a relationship for at least 36 continuous months;
is the parent of your child by birth or adoption;
or has custody and control of your child (or had custody and control immediately before the child turned 19 years of age) and your child is wholly dependent on that person for support.
This article originally appeared on www.morningstar.ca.
10 Items To Toss When Moving In With Someone
common lawcommon law marriagefinancelegalliving | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 169 |
<?php
// Copyright (c) 2016 Syndicate Plus B.V.
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
// The above copyright notice and this permission notice shall be included in all
// copies or substantial portions of the Software.
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
// SOFTWARE.
namespace Syndy\Api\Contracts\Retailer;
require_once dirname(__FILE__)."/../basecontract.class.php";
use Syndy\Api\Contracts;
class RetailerProductReference extends Contracts\BaseContract {
protected $id;
protected $dateLastUpdate;
protected $barcode;
protected $articleNumber;
public function __construct($rawData) {
$this->parse($rawData);
}
protected function parse($rawData) {
$rawData = parent::parse($rawData);
$this->id = $rawData->Id;
$this->dateLastUpdate = $rawData->DateLastUpdate;
$this->barcode = $rawData->Barcode;
$this->articleNumber = $rawData->ArticleNumber;
}
public function getId() {
return $this->id;
}
public function getDateLastUpdate() {
return $this->dateLastUpdate;
}
public function getBarcode() {
return $this->barcode->Value;
}
public function getArticleNumber() {
return $this->articleNumber;
}
public function __get($field) {
if ($field == "barcode") {
return $this->barcode->Value;
}
if (!isset($this->$field)) {
return null;
}
return $this->$field;
}
}
?> | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,011 |
Sabula is a city in Jackson County, Iowa, United States. The population was 506 at the 2020 census. Sabula is the site of Iowa's only island city. The island has a beach and a campground, as well as a harbor with boat docks and storage sheds to store boats during the winter. Because of its proximity to Chicago (three-hour drive), Sabula has become a popular vacation destination during the summer months. Sabula is the northern terminus of U.S. Route 67, a 1,560 mile (2,511 km) long north–south highway in the Central United States. The southern terminus of the route is at the United States–Mexico border in Presidio, Texas.
History
Sabula was established in 1835 when, according to legend, Isaac Dorman crossed the river from the Illinois side on a log and decided to settle on the present site of Sabula. Sabula is a name of French origin meaning "sand"; this refers to the sandy soil of the area.
In the late 19th century the principal industries in the community included a large "pearl button" factory—which produced buttons from clam shells harvested from large clam beds located in the river adjacent to the shoreline. (The factory is no longer there.) The thriving community also supported a large hog slaughtering industry.
The community did not actually become an island until the lock and dam system was constructed by the Army Corps of Engineers on the upper Mississippi in the 1930s. The construction of Lock and Dam No. 13 between Clinton, Iowa and Fulton, Illinois in 1939 left the lowlands west of the townsite permanently flooded, creating the "Island City," as the town is now known.
Geography
Sabula is located at (42.067866, −90.174270).
According to the United States Census Bureau, the city has a total area of , of which is land and is water.
Sabula is connected to Iowa via a roadway that runs between two lakes and with Savanna, Illinois, by another roadway that leads to a bridge that crosses the Mississippi River.
Demographics
2010 census
As of the census of 2010, there were 576 people, 270 households, and 157 families living in the city. The population density was . There were 321 housing units at an average density of . The racial makeup of the city was 99.1% White, 0.2% African American, 0.2% Native American, 0.2% from other races, and 0.3% from two or more races. Hispanic or Latino of any race were 0.7% of the population.
There were 270 households, of which 24.8% had children under the age of 18 living with them, 43.3% were married couples living together, 12.2% had a female householder with no husband present, 2.6% had a male householder with no wife present, and 41.9% were non-families. 37.0% of all households were made up of individuals, and 19.3% had someone living alone who was 65 years of age or older. The average household size was 2.13 and the average family size was 2.81.
The median age in the city was 45 years. 22.4% of residents were under the age of 18; 6% were between the ages of 18 and 24; 21.6% were from 25 to 44; 29.6% were from 45 to 64; and 20.3% were 65 years of age or older. The gender makeup of the city was 46.9% male and 53.1% female.
2000 census
As of the census of 2000, there were 670 people, 308 households, and 182 families living in the city. The population density was . There were 337 housing units at an average density of . The racial makeup of the city was 100.00% White. Hispanic or Latino of any race were 1.04% of the population.
There were 308 households, out of which 23.4% had children under the age of 18 living with them, 43.2% were married couples living together, 12.0% had a female householder with no husband present, and 40.6% were non-families. 36.0% of all households were made up of individuals, and 19.2% had someone living alone who was 65 years of age or older. The average household size was 2.18 and the average family size was 2.78.
In the city, the population was spread out, with 22.1% under the age of 18, 6.0% from 18 to 24, 23.7% from 25 to 44, 25.8% from 45 to 64, and 22.4% who were 65 years of age or older. The median age was 43 years. For every 100 females, there were 92.5 males. For every 100 females age 18 and over, there were 89.8 males.
The median income for a household in the city was $30,192, and the median income for a family was $39,688. Males had a median income of $29,000 versus $20,500 for females. The per capita income for the city was $16,901. About 11.5% of families and 14.4% of the population were below the poverty line, including 24.3% of those under age 18 and 4.7% of those age 65 or over.
Government
Sabula has a post office and a community center. Fire protection is provided by the Sabula Volunteer Fire Department. The Sabula Fire Department protects everything within city limits as well as accident response in the ambulance district and is also available to respond mutual aid to other cities in Iowa and Illinois. Most of the city firefighters are certified as Iowa Firefighter Ones and Hazmat Operations. Quartered with the Fire Department is an Ambulance service also made up of volunteers which provides ambulances to the city and outlying areas. The Ambulance service operates two Basic Life Support Ambulances. Prior to the Sabula Ambulance (Originally called the "Emergency Unit") the local funeral home provided a for profit ambulance. In 1974 the Fire Chief created the Emergency Unit and brought the first trained EMTs to the city drawn from firemen. Sabula has a public works department that plows the roads in winter and provides other services. Police protection is provided by the Sabula Police and Jackson County Sheriffs Office.
Sabula has a Mayor-Council city government. Meetings and elections are held in city hall.
Education
Sabula is a part of the Easton Valley Community School District, formed in 2013 by the merger of the East Central Community School District and the Preston Community School District. East Central formed in 1974 by the merger of the Sabula Community School District and the Miles Community School District. At one point the former East Central district entered into a grade-sharing relationship with the Northeast Community School District, in that East Central residents attended Northeast secondary schools.
Sabula previously had a school in the community, and it first met in a settler's home in 1838. New campuses were built in 1856 and then in 1883. By circa 2012 the leadership of what would be the Easton Valley district was seeking to close the Sabula school due to potential future upkeep costs and because of how old it was. The Sabula school remained vacant until 2015, when the Easton Valley board voted unanimously in favor of demolishing it. The school district and the city government both agreed to demolition after considering other options and uses.
Religion
Sabula is served by three churches: Sabula United Methodist church (dating to 1839), St. Peter's Catholic Church (dating to the 1840s), Calvary Lutheran Church (built in 1944).
References
Cities in Jackson County, Iowa
Cities in Iowa
Iowa populated places on the Mississippi River
1835 establishments in Michigan Territory | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,686 |
Q: Boost::Asio Transferred bytes while using an async operation I'm using Boost::Asio HTTP Server 3 example (http://www.boost.org/doc/libs/1_53_0/doc/html/boost_asio/examples.html)
but I need some "download information" such as number of transferred bytes in a certain amount of time in order to show a progress bar.
But how can I get this information if I'm using the boost::asio::async_write function? I know that I can access to boost::asio::placeholders::bytes_transferred in the handle_write completion handler but it is called just once asynchronous write operation has finished.
There is a way to do that?
A: You don't want to use the composed operation (boost::async_*).
Instead, roll your own composed operation that uses service_object.async_read_some repeatedly and report the progress where you want it.
*
*http://vinniefalco.github.io/beast/beast/core/tutorial.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,543 |
\section{Introduction}
In this paper, we study the incompressible steady flow in a planar bounded domain. The motion is governed by the following Euler equations
\begin{equation}\label{1-1}
\begin{cases}
(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla P,\\
\nabla\cdot\mathbf{v}=0,
\end{cases}
\end{equation}
where $\mathbf{v}=(v_1,v_2)$ is the velocity field and $P$ is the scalar pressure. The vorticity of the flow is defined by \[\omega=curl\mathbf{v}:=\partial_1v_2-\partial_2v_1.\]
If $\omega\equiv0$, then the flow is said to be irrotational.
Irrotational flows can be completely classified. Let $\mathbf{v}$ be a solution of \eqref{1-1}, then by the divergence-free condition $\nabla\cdot \mathbf{v}=0$ and Green's theorem, we have
\[\mathbf{v}=\nabla^\perp \psi:=(\partial_2\psi,-\partial_1\psi)\]
for some function $\psi$, which we call the stream function. It is easy to check that
\[-\Delta\psi=\omega.\]
So the flow is irrotational if and only if $\psi$ is harmonic. Conversely, it is easy to prove that for any harmonic function $\psi$, there exists an irrotational flow with $\psi$ as its stream function. In this sense, an irrotational flow is equivalent to a harmonic function.
In this paper, we are concerned with steady flows with nonvanishing vorticity. More precisely, we prove that for any given nontrivial irrotational flow(the velocity field is not zero), there exists a family of steady vortex patch solutions near this flow. Here by vortex patch we mean that the vorticity $\omega$ has the form $\omega=\kappa I_A$, where $\kappa$ is a real number representing the vorticity strength, $A$ is a measurable set, and $I_A$ denotes the characteristic function of $A$, namely, $I_A=1$ in $A$ and $I_A=0$ elsewhere.
Vortex patches are a special class of non-smooth solutions of the two-dimensional Euler equations appropriate for modeling an isolated region of constant vorticity. In the past few decades, the extensive study of vortex patches has led to many interesting and significant results. In this paper, we focus on the construction of steady vortex patches. There exist a great literatures dealing with this problem; see for example \cite{CPY,CW,CW2,HM,T,W} and the references listed therein. An efficient method to study the vortex patch problem is the vorticity method. It was first established by Arnold and Khesin \cite{A,AK} and later developed by many authors \cite{Ba,B,B2,EM,FB,T}. Roughly speaking, the vorticity method asserts that a steady flow is in fact a constrained critical point of the kinetic energy, and the stability of the flow is equivalent to the nondegeneracy of that critical point. By maximizing the kinetic energy in a weakly closed subset in $L^\infty$ that contains the class of isovortical patches and studying the limiting behavior, Turkington \cite{T} constructed a family of steady vortex patches concentrating at a global minimum point of the Robin function of the domain. Later Burton \cite{B,B2} considered the maximization of the kinetic energy on general rearrangement classes, and by which he found more dynamically possible equilibria of planar vortex flows. Another method to study the vortex patch problem is developed by Cao et al. \cite{CPY}. By using a reduction argument for the stream function, they obtained steady multiple vortex patches near any given nondegenerate critical point of the Kirchhoff-Routh function. They also proved the uniqueness in \cite{CGPY} under certain assumptions.
Most of the previous results in \cite{CPY,HM,T,W} were concerned with the desingularization of point vortices. According to the vortex model, the evolution of a finite number of concentrated vortices in two dimensions is described by a dynamical system involving the Kirchhoff-Routh function; See \cite{L} for a general discussion. A natural question is the connection between the vortex model and the Euler equations. More specifically, for any critical point of the Kirchhoff-Routh function, can we construct a family of steady solutions of the Euler equations concentrating near that point? The answer is yes in some situations, especially when that critical point is nondegenerate; See \cite{CPY}. When one deals with desingularization of point vortices, the vorticity converges to a Dirac measure, in which case the vorticity amount is usually fixed and the vorticity strength goes to infinity.
As a contrast, our results in this paper are essentially of perturbation type. By using an adaption of the method in \cite{T}, we consider a certain variational problem in which the vorticity strength is fixed and the vorticity amount goes to zero. In this situation, the limiting behavior is mostly determined by the background irrotational flow, rather than the Kirchhoff-Routh function. Finally by deriving asymptotic estimate for the Lagrange multiplier, we are able to show that the support of vorticity shrinks to the boundary of the domain.
It is also worth mentioning that in \cite{LYY} the authors obtained a similar existence result for vorticity without jump by considering a semilinear elliptic equation satisfied by the corresponding stream function. For the three dimensional case, steady flows with nonvanishing vorticity near an irrotational flow can also be constructed in some special cases; See \cite{Al,TX} for instance.
This paper is organized as follows. We first give a description of our problem and state the main results in Section 2. Then in Section 3 and 4 we solve a minimization problem and study the limiting behavior respectively to prove the main results. In Section 5 we briefly discuss the maximization case and obtain two similar results of existence.
\section{Main results}
Let $D\subset\mathbb{R}^2$ be a bounded and simply connected domain with a smooth boundary, $\partial D$.
Let $q$ be a harmonic function in $D$ corresponding to an irrotational flow $(\mathbf{v}_0,P_0)$ with $\mathbf{v}_0=\nabla^\perp q:=(\partial_2q,-\partial_1q)$. Then we have
\begin{equation}\label{1-2}
\begin{cases}
(\mathbf{v}_0\cdot\nabla)\mathbf{v}_0=-\nabla P_0 &\text{in }D,\\
\nabla\cdot\mathbf{v}_0=0 &\text{in }D,\\
\mathbf{v}_0\cdot\mathbf{n}= -\frac{\partial q}{\partial\mathbf{\nu}} &\text{on }\partial D,
\end{cases}
\end{equation}
where $\mathbf{n}=(n_1,n_2)$ is the exterior unit normal to the boundary $\partial D$, and $\mathbf{\nu}=\mathbf{n}^\perp:=(n_2,-n_1)$ denotes clockwise rotation through $\frac{\pi}{2}$ of $\mathbf{n}$.
To find a solution with nonvanishing vorticity near $\mathbf{v}_0$, we consider the following Euler equations with the same boundary condition as \eqref{1-2}
\begin{equation}\label{1-3}
\begin{cases}
(\mathbf{v}\cdot\nabla)\mathbf{v}=-\nabla P &\text{in }D,\\
\nabla\cdot\mathbf{v}=0 &\text{in }D,\\
\mathbf{v}\cdot\mathbf{n}= -\frac{\partial q}{\partial\mathbf{\nu}} &\text{on }\partial D.
\end{cases}
\end{equation}
Now we simplify \eqref{1-3} by using its vorticity formulation. Set $\omega=curl\mathbf{v}$. Taking the curl in the first equation in \eqref{1-3} we get
\begin{equation}\label{1-4}
\nabla\cdot(\omega\mathbf{v})=0.
\end{equation}
On the other hand, we can recover $\mathbf{v}$ from $\omega$ in the following way. Since
\begin{equation}\label{1-5}
\begin{cases}
\nabla\cdot(\mathbf{v}-\mathbf{v}_0)=0 &\text{in } D,\\
(\mathbf{v}-\mathbf{v}_0)\cdot\mathbf{n}=0 &\text{on } \partial D,
\end{cases}
\end{equation}
we obtain $\mathbf{v}-\mathbf{v}_0=\nabla^\perp\psi$ for some function $\psi$ with $\psi=$constant on $\partial D$. Since $D$ is simply connected, without loss of generality, we assume that $\psi=0$ on $\partial D$ by adding a suitable constant. Therefore $\psi$ can be uniquely determined by $\omega$
\begin{equation}\label{1-6}
\begin{cases}
-\Delta\psi=\omega &\text{in }D,\\
\psi=0 &\text{on } \partial D.
\end{cases}
\end{equation}
Set $G(\cdot,\cdot)$ to be the Green's function for $-\Delta$ in $D$ with zero Dirichlet data on $\partial D$. Then $\psi$ can be expressed in terms of the Green's operator as follows
\begin{equation}\label{1-8}
\psi(x)=G\omega(x):=\int_DG(x,y)\omega(y)dxdy.
\end{equation}
In other words, in order to solve \eqref{1-3}, it suffices to consider the following equation satisfied by $\omega$
\begin{equation}\label{1-7}
\nabla\cdot(\omega\nabla^\perp(G\omega+q))=0.
\end{equation}
Since we are going to deal vortex patches which are discontinuous, it is necessary to interpret $\eqref{1-7}$ in the weak sense, namely, we need to introduce the notation of weak solution to \eqref{1-7}.
\begin{definition}\label{1-9}
We call $\omega\in L^\infty(D)$ a weak solution of \eqref{1-7} if
\begin{equation}\label{1-10}
\int_D\omega\nabla^\perp(G\omega+q)\cdot\nabla\xi dx=0
\end{equation}
for any $\xi\in C^\infty_c(D)$.
\end{definition}
It should be noted that if $\omega\in L^\infty(D)$, then by $L^p$ estimate and Sobolev embedding $G\omega\in C^{1,\alpha}(D)$ for any $\alpha\in(0,1)$, therefore the integral in \eqref{1-10} makes sense.
Our first result is as follows.
\begin{theorem}\label{1-11}
Let $q\in C^2(D)\cap C^1(\overline{D})$ be a harmonic function and $\kappa$ be a positive real number. Set $\mathcal{S}:=\{x\in \overline{D}\mid q(x)=\min_{ \overline{D}}q\}$. Then
for any given positive number $\lambda$ with $\lambda<\kappa|D|$($|\cdot|$ denotes the two-dimensional Lebesgue measure), there exists a weak solution $\omega^\lambda$ of \eqref{1-7} having the form
\begin{equation}\label{1-12}
\omega^\lambda=\kappa I_{\Omega^\lambda}, \,\, \Omega^\lambda=\{x\in D\mid G\omega^\lambda(x)+q(x)<\mu^\lambda\},\,\,\kappa|\Omega^\lambda|=\lambda
\end{equation}
for some $\mu^\lambda\in\mathbb{R}$ depending on $\lambda$.
Furthermore, if $q$ is not a constant, then $\mathcal{S}\subset\partial D$ and $\Omega^\lambda$ approaches $\mathcal{S}$ as $\lambda\rightarrow0$, or equivalently, for any $\delta>0$, there exists $\lambda_0>0$, such that for any $\lambda<\lambda_0$, we have
\begin{equation}\label{1-14}
\Omega^\lambda\subset\mathcal{S}_\delta:=\{x\in D\mid dist(x,\mathcal{S})<\delta\}.
\end{equation}
\end{theorem}
\begin{remark}\label{2-101}
In Theorem \ref{1-11}, if $q$ is not a constant, then by the strong maximum principle
\[\{x\in \overline{D}\mid q(x)=\min_{ \overline{D}}q\}\subset \partial D,\]
so $\Omega^\lambda$ approaches the boundary of the domain as $\lambda\rightarrow0^+.$
\end{remark}
\begin{remark}\label{2-102}
In Theorem \ref{1-11} we show the existence of steady vortex patches with positive vorticity near the set of global minimum points of $q$. Since $(\omega,q)$ satisfies \eqref{1-7} if and only if $(-\omega,-q)$ satisfies \eqref{1-7}, so by reverting the signs of $\omega^\lambda$ and $q$ in Theorem \ref{1-11}, we can also prove the existence of steady vortex patches with negative vorticity near the set of global maximum points of $q$. In Section 5 we consider the other two cases: vortex patches with positive vorticity near the set of global maximum points of $q$ and vortex patches with negative vorticity near the set of global minimum points of $q$.
\end{remark}
Our next result shows that each finite collection of strict extreme points of $q$ corresponds to a family of steady multiple vortex patches shrinking to it.
\begin{theorem}\label{1-15}
Let $q\in C^2(D)\cap C^1(\overline{D})$ be a harmonic function, $k,l$ be two nonnegative integers and $\kappa_1,\cdot\cdot\cdot,\kappa_{k+l}$ be $k+l$ positive real numbers. Suppose that $\{x_1,x_2,\cdot\cdot\cdot,x_k\}\subset\partial D$ are $k$ different strict local minimum points of $q$ on $\overline{D}$, and $\{x_{k+1},x_{k+2},\cdot\cdot\cdot,x_{k+l}\}\subset\partial D$ are $l$ different strict local maximum points of $q$ on $\overline{D}$.
Then there exists a $\lambda_0>0$, such that for any $0<\lambda<\lambda_0$, there exists a weak solution $w^\lambda$ of \eqref{1-7} having the form
\begin{equation}\label{1-16}
w^\lambda=\sum_{i=1}^k\kappa_iI_{U_i^\lambda}-\sum_{j=k+1}^{k+l}\kappa_jI_{U_j^\lambda},
\end{equation}
where for each $1\leq i\leq k$
\begin{equation}\label{5-4-3}
U_i^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)<\nu_i^\lambda\}\cap{B_{\delta_0}(x_i)},\,\,|U_i^\lambda|=\lambda,
\end{equation}
and for each $k+1\leq j\leq k+l$
\begin{equation}\label{5-5-3}
U_j^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)>\nu_j^\lambda\}\cap{B_{\delta_0}(x_j)},\,\,|U_j^\lambda|=\lambda,
\end{equation}
where $\nu_p^\lambda\in\mathbb{R},$ for $1\leq p\leq k+l$, is the Lagrange multiplier depending on $\lambda$.
Here $\delta_0$ is chosen to be sufficiently small such that $x_i$ is the unique minimum point of $q$ on $\overline{B_{\delta_0}(x_i)\cap D}$ for $i=1,\cdot\cdot\cdot,k$, $x_i$ is the unique maximum point of $q$ on $\overline{B_{\delta_0}(x_j)\cap D}$ for $j=k+1,\cdot\cdot\cdot,k+l$, and $\overline{B_{\delta_0}(x_{p_1})\cap D}\cap\overline{B_{\delta_0}(x_{p_2})\cap D}=\varnothing$ for $1\leq p_1,p_2\leq k+l, p_1\neq p_2.$
Moreover, $U_p^\lambda$ shrinks to $x_p$ for each $1\leq p\leq k+l$ as $\lambda\rightarrow0^+$, or equivalently, for any $\delta>0$, there exists a $\lambda_0>0$, such that for any $0<\lambda<\lambda_0$, we have
\begin{equation}\label{1-17}
U^\lambda_p\subset B_{\delta}(x_p)\cap D.
\end{equation}
\end{theorem}
\section{Proof of Theorem \ref{1-11}}
In this section we prove Theorem \ref{1-11}. To begin with, we consider a minimization problem for the vorticity and study the limiting behavior of the minimizer.
\subsection{Minimization Problem}
Let $\kappa$ be a fixed positive number. For $0<\lambda<\kappa|D|$, we define the vorticity class $\mathcal{M}^\lambda$ as follows
\begin{equation}\label{3-1}
\mathcal{M}^\lambda:=\{\omega\in L^\infty(D)\mid 0\leq\omega\leq\kappa, \int_D\omega(x)dx=\lambda\}.
\end{equation}
Note that $\mathcal{M}^\lambda$ is not empty since $\lambda<\kappa|D|$. The variational problem is to minimize the functional
\begin{equation}\label{3-2}
E(\omega):=\frac{1}{2}\int_D\int_DG(x,y)\omega(x)\omega(y)dxdy+\int_Dq(x)\omega(x)dx
\end{equation}
in the class $\mathcal{M}^\lambda$, that is,
\begin{equation}
c_\lambda=\inf\{E(\omega)\mid\omega\in\mathcal{M}^\lambda\}.
\end{equation}
We call $\upsilon\in\mathcal{M}^\lambda$ an absolute minimizer if $E(\upsilon)\leq E(\omega)$ for all $\omega\in \mathcal{M}^\lambda$. In order to study the existence of an absolute minimizer of $E$, we need to establish two preliminary results.
\begin{lemma}
$\mathcal{M}^\lambda$ is a sequentially compact subset of $L^\infty(D)$ in the weak star topology, that is, for any sequence $\{\omega_n\}\subset L^\infty(D)$, $n=1,2,\cdot\cdot\cdot$, there exist a subsequence $\{\omega_{n_j}\}$ and $w_0\in\mathcal{M}^\lambda$ such that for any $\zeta\in L^1(D)$
\begin{equation}\label{3-344}
\lim_{j\rightarrow+\infty}\int_D\zeta(x)\omega_{n_j}(x)dx=\int_D\zeta(x)\omega_{0}(x)dx.
\end{equation}
\end{lemma}
\begin{proof}
As $\mathcal{M}^\lambda$ is clearly a bounded set in $L^\infty(D)$, there is, for any sequence $\{\omega_n\}\subset L^\infty(D)$, a subsequence $\{\omega_{n_j}\}$ such that $\omega_{n_j} \rightarrow \omega_0$ weakly star in $L^\infty$ as $j\rightarrow+\infty$ for some $\omega_0\in L^\infty(D)$. So it suffices to show $\omega_0\in\mathcal{M}^\lambda,$ namely, $\int_D\omega_0(x)dx=1$ and $0\leq\omega_0\leq \kappa$ a.e. in $D$.
Firstly, by choosing $\zeta\equiv1$ in \eqref{3-344} we have
\[\lim_{j\rightarrow +\infty}\int_D\omega_{n_j}(x)dx=\int_D\omega_0(x)dx=1.\]
Now we prove $ \omega_0\leq\kappa$ by contradiction. Suppose that $|\{x\in D\mid\omega_0(x)>\kappa\}|>0$, then there exists a $\varepsilon_0>0$ such that $|\{x\in D\mid\omega_0(x)\geq\kappa+\varepsilon_0\}|>0$. Denote $A=\{x\in D\mid\omega_0(x)\geq\kappa+\varepsilon_0\}$. Then by taking $\zeta=I_A$ in \eqref{3-344} we have
\[0=\lim_{j\rightarrow +\infty}\int_D(\omega_0(x)-\omega_{n_j}(x))\zeta(x)dx=\lim_{j\rightarrow +\infty}\int_{A}(\omega_0(x)-\omega_{n_j}(x))dx\geq\varepsilon_0|A|>0,\]
which is a contradiction. Similarly we can prove $\omega_0\geq 0$.
\end{proof}
\begin{lemma}\label{3-345}
$E$ is weakly star continuous in $L^\infty(D)$.
\end{lemma}
\begin{proof}
Let $\{f_n\}$ be a sequence in $L^\infty(D)$ such that for some $f_0\in L^\infty(D)$, $f_n\rightarrow f_0$ weakly star in $ L^\infty(D)$ as $n\rightarrow+\infty$. It suffices to prove that $\lim_{n\rightarrow+\infty}E(f_n)=E(f_0)$. First by $L^p$ estimate we have $Gf_n\rightarrow Gf_0$ weakly in $W^{2,p}$ for any $1<p<+\infty,$ then by Sobolev embedding
$Gf_n\rightarrow Gf_0$ in $C^1(\overline{D})$, as $n\rightarrow+\infty$. By the definition of weak star convergence we can easily deduce that $\lim_{n\rightarrow+\infty}E(f_n)=E(f_0)$.
\end{proof}
Now we are ready to show the existence of an absolute minimizer.
\begin{proposition}\label{3-3}
$c_\lambda$ is achieved by an absolute minimizer $\omega^\lambda\in\mathcal{M}^\lambda$ with the form
\begin{equation}\label{3-4}
\omega^\lambda=\kappa I_{\Omega^\lambda},\,\,\Omega^\lambda=\{x\in D\mid G\omega^\lambda(x)+q(x)<\mu^\lambda\}
\end{equation}
for some $\mu^\lambda$ depending on $\lambda$.
\end{proposition}
\begin{proof}
First we show that $E$ attains its minimum on $\mathcal{M}^\lambda$. By integration by parts, we have for any $\omega\in \mathcal{M}^\lambda$
\begin{equation}\label{3-5}
E(\omega)=\frac{1}{2}\int_D|\nabla G\omega(x)|^2dx+\int_Dq(x)\omega(x)dx\geq \lambda\min_{\overline{D}}q,
\end{equation}
which means that $E$ is bounded from below on $\mathcal{M}^\lambda$. Now we choose a sequence $\{\omega_n\}\subset\mathcal{M}^\lambda$ such that
\[\lim_{n\rightarrow+\infty}E(\omega_n)=\inf_{\omega\in\mathcal{M}^\lambda}E(\omega).\]
Since $\mathcal{M}^\lambda$ is a sequentially compact subset of $L^\infty(D)$ in the weak star topology, there exists a subsequence $\{\omega_{n_j}\}$ and a $\omega^\lambda\in \mathcal{M}^\lambda$ such that $\omega_{n_j}\rightarrow \omega^\lambda$ weakly star in $L^\infty(D)$ as $j\rightarrow+\infty$. Then by the weak star continuity of $E$ in $L^\infty(D)$ we obtain
\begin{equation}\label{3-6}
E(\omega^\lambda)=\lim_{j\rightarrow+\infty}E(\omega_{n_j})=\inf_{\omega\in\mathcal{M}^\lambda}E(\omega),
\end{equation}
which means that $\omega^\lambda$ is an absolute minimizer.
Now we prove that $\omega^\lambda$ has the following form
\begin{equation}\label{3-7}
\omega^\lambda=\kappa I_{\Omega^\lambda},\,\,\Omega^\lambda=\{x\in D\mid G\omega^\lambda(x)+q(x)<\mu^\lambda\}
\end{equation}
for some $\mu^\lambda$ depending on $\lambda$. To show this, we choose a family of test functions $\omega_s=\omega^\lambda+s(z_0-z_1)$, $s>0$, where $z_0$ and $z_1$ satisfy
\begin{equation}
\begin{cases}
z_0,z_1\in L^\infty(D),\,z_0,z_1\geq 0,\text{ a.e. in } D,
\\ \int_Dz_0(x)dx=\int_D z_1(x)dx,
\\z_0=0 \quad\text{in } D\setminus\{x\in D\mid\omega^\lambda(x)\leq\kappa-\delta\},
\\z_1=0 \quad\text{in } D\setminus\{x\in D\mid\omega^\lambda(x)\geq\delta\}.
\end{cases}
\end{equation}
Here $\delta$ is a positive number. It is not hard to check that for fixed $z_0,z_1$ and $\delta$, $\omega_s\in \mathcal{M}^\lambda$ provided that $s$ is sufficiently small. Since $\omega^\lambda$ is an absolute minimizer, we have
\[0\leq\frac{dE(\omega_s)}{ds}|_{s=0^+}=\int_Dz_0(x)(G\omega^\lambda(x)+q(x))dx-\int_Dz_1(x)(G\omega^\lambda(x)+q(x))dx.\]
By the choice of $z_0$ and $z_1$ we obtain
\begin{equation}\label{1-103}
\sup_{\{x\in D\mid\omega^\lambda(x)>0\}}(G\omega^\lambda+q)\leq\inf_{\{x\in D\mid\omega^\lambda(x)<\kappa\}}(G\omega^\lambda+q).
\end{equation}
Since $D$ is simply-connected and $G\omega^\lambda+q$ is continuous, \eqref{1-103} is in fact an equality, that is,
\begin{equation}
\sup_{\{x\in D\mid\omega^\lambda(x)>0\}}(G\omega^\lambda+q)=\inf_{\{x\in D\mid\omega^\lambda(x)<\kappa\}}(G\omega^\lambda+q).
\end{equation}
Now we can define
\[\mu^\lambda:=\sup_{\{x\in D\mid\omega^\lambda(x)>0\}}(G\omega^\lambda+q)=\inf_{\{x\in D\mid\omega^\lambda(x)<\kappa\}}(G\omega^\lambda+q).\]
It is clear that
\begin{equation}
\begin{cases}
\omega^\lambda=0\text{\,\,\,\,\,\,a.e.\,} \text{in }\{x\in D\mid G\omega^\lambda(x)+q(x)>\mu^\lambda\},
\\ \omega^\lambda=\kappa\text{\,\,\,\,\,\,a.e.\,} \text{in }\{x\in D\mid G\omega^\lambda(x)+q(x)<\mu^\lambda\}.
\end{cases}
\end{equation}
By the property of Sobolev functions, on the level set $\{x\in D\mid G\omega^\lambda(x)+q(x)=\mu^\lambda\}$, we have $\nabla (G\omega^\lambda+q)=0\text{\,\,a.e.}$, therefore $\omega^\lambda=-\Delta( G\omega^\lambda+q)=0\text{\,\,a.e.}$.
Altogether, we obtain
\[\omega^\lambda=\kappa I_{\{x\in D\mid G\omega^\lambda(x)+q(x)<\mu^\lambda\}},\]
which is the desired result.
\end{proof}
Turkington \cite{T} considered the maximization of $E$ on $\mathcal{M}^\lambda$. In that case, the maximizer may not be unique, especially for domains with certain symmetries. But here for the minimization problem we have
\begin{lemma}\label{3-1001}
There is a unique absolute minimizer of $E$ on $\mathcal{M}^\lambda$.
\end{lemma}
\begin{proof}
Suppose there are two minimizers $\omega^\lambda_1,\omega^\lambda_2$ of $E$ on $\mathcal{M}^\lambda$. By Proposition \ref{3-3}, $\omega^\lambda_i=\kappa I_{\Omega^\lambda_i}$ for $i=1,2$. Now we define a new function $\omega^\lambda_3=\frac{1}{2}(\omega^\lambda_1+\omega^\lambda_2),$ or equivalently,
\begin{equation}\label{3-8}
\omega^\lambda_3(x)=
\begin{cases}
\frac{1}{2}\kappa &x\in \Omega^\lambda_1\bigtriangleup\Omega^\lambda_2,\\
\kappa &x\in\Omega^\lambda_1\cap\Omega^\lambda_2,\\
0 &x\in(\Omega^\lambda_1\cup\Omega^\lambda_2)^c.
\end{cases}
\end{equation}
It is easy to see that $\omega^\lambda_3\in\mathcal{M}^\lambda$, and $\omega^\lambda_1=\omega^\lambda_2$ if and only if $\omega^\lambda_3$ has the form $\omega^\lambda_3=\kappa I_{\Omega^\lambda_3}$ for some $\Omega^\lambda_3\subset D$. Now we calculate $E(\omega^\lambda_3),$
\begin{equation}\label{3-9}
\begin{split}
E(\omega^\lambda_3)&=\frac{1}{8}\int_D\int_DG(x,y)(\omega^\lambda_1+\omega^\lambda_2)(x)(\omega^\lambda_1+\omega^\lambda_2)(y)dxdy +\frac{1}{2}\int_Dq(x)(\omega^\lambda_1+\omega^\lambda_2)(x)dx\\
&=\frac{1}{2}E(\omega^\lambda_1)+\frac{1}{4}\int_D\int_DG(x,y)\omega^\lambda_1(x)\omega^\lambda_2(y)dxdy
+\frac{1}{4}\int_Dq(x)(\omega^\lambda_1+\omega^\lambda_2)(x)dx\\
&\leq \frac{1}{2}E(\omega^\lambda_1)+\frac{1}{8}\int_D\int_DG(x,y)\omega^\lambda_1(x)\omega^\lambda_1(y)dxdy
+\frac{1}{8}\int_D\int_DG(x,y)\omega^\lambda_2(x)\omega^\lambda_2(y)dxdy\\
&+\frac{1}{4}\int_Dq(x)(\omega^\lambda_1+\omega^\lambda_2)(x)dx\\
&=E(\omega^\lambda_1).
\end{split}
\end{equation}
Here we use the fact that $\int_D\int_DG(x,y)(\omega^\lambda(x)-\omega^\lambda_2(x))(\omega^\lambda_1(y)-\omega^\lambda_2(y))dxdy\geq0$. By \eqref{3-9} we know that
$\omega^\lambda_3$ is also a minimizer of $E$, then again by Proposition \ref{3-3} $\omega^\lambda_3=\kappa I_{\Omega^\lambda_3}$ for some $\Omega^\lambda_3\subset D$, which implies $\omega^\lambda_1=\omega^\lambda_2.$
\end{proof}
\begin{remark}
$\mathcal{M}^\lambda$ is in fact a convex subset of $L^\infty(D)$ and $E$ is a strictly convex functional on $\mathcal{M}^\lambda$.
\end{remark}
\subsection{Limiting behavior of the minimizer}
Now we analyze the limiting behavior of the minimizer $\omega^\lambda$ obtained in Proposition \ref{3-3} as $\lambda\rightarrow 0^+$, which will also be used in the next subsection. For simplicity, we will use $C$ to denote various positive numbers not depending on $\lambda$.
\begin{lemma}\label{3-10}
We have the following upper bound for $E(\omega^\lambda)$
\begin{equation}
E(\omega^\lambda)\leq \lambda\min_{\overline{D}}q+C\lambda^{\frac{3}{2}}.
\end{equation}
\end{lemma}
\begin{proof}
The basic idea is to choose a suitable test function. Let $x_0\in\partial D$ be a minimum point of $q$ on $\overline{D}$. Since $\partial D$ is smooth, $D$ satisfies the interior sphere condition at $x_0\in \partial D$. Therefore for $\lambda$ sufficiently small we can choose a ball $B_\varepsilon(x^\lambda)\subset D$ with $|x^\lambda-x_0|=\varepsilon$, where $\varepsilon$ satisfies $\kappa\pi\varepsilon^2=\lambda$. Now we define the test function to be $\upsilon^\lambda=\kappa I_{B_\varepsilon(x^\lambda)}.$ It is obvious that $\upsilon^\lambda\in \mathcal{M}^\lambda$, and
\begin{equation}\label{3-11}
\begin{split}
E(\omega^\lambda)&\leq E(\upsilon^\lambda)=\frac{1}{2}\kappa\int_{B_\varepsilon(x^\lambda)}G\upsilon(x)dx+\kappa\int_{B_\varepsilon(x^\lambda)}q(x)dx.
\end{split}
\end{equation}
To estimate the first integral in \eqref{3-11}, we use $L^p$ estimate and Sobolev embedding to obtain
\begin{equation}\label{3-12}
|G\upsilon|_{L^\infty(D)}\leq C|G\upsilon|_{W^{2,2}(D)}\leq C|\upsilon|_{L^2(D)}=C\lambda^\frac{1}{2}.
\end{equation}
To estimate the second integral in \eqref{3-11}, we observe that for any $x\in B_\varepsilon(x^\lambda)$
\begin{equation}\label{3-13}
|q(x)-q(x_0)|\leq |\nabla q|_{L^\infty(D)}|x-x_0|\leq C\varepsilon.
\end{equation}
By combining \eqref{3-11}, \eqref{3-12} and \eqref{3-13} together we get
\begin{equation}\label{3-14}
E(\omega^\lambda)\leq \lambda\min_{\overline{D}}q+C\lambda^{\frac{3}{2}}.
\end{equation}
\end{proof}
\begin{lemma}\label{3-15}
The following estimate holds
\begin{equation}\label{3-16}
\int_D(G\omega^\lambda(x)+q(x)-\mu^\lambda)\omega^\lambda(x)dx\geq -C\lambda^{\frac{3}{2}}.
\end{equation}
\end{lemma}
\begin{proof}
For convenience we denote $\varphi^\lambda=G\omega^\lambda+q-\mu^\lambda$ and $\varphi^\lambda_-=\min{\{\varphi,0\}}$. First by H\"older's inequality
\begin{equation}\label{3-17}
\int_D(G\omega^\lambda(x)+q(x)-\mu^\lambda)\omega^\lambda(x)dx=\kappa\int_{\Omega^\lambda}\varphi(x)dx\geq
-\kappa|\Omega^\lambda|^{\frac{1}{2}}(\int_{\Omega^\lambda}|\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}.
\end{equation}
On the other hand, by the Sobolev embedding $W^{1,1}(D)\hookrightarrow L^2(D)$ and H\"older's inequality
\begin{equation}\label{3-18}
\begin{split}
(\int_{\Omega^\lambda}|\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}&= (\int_{D}|\varphi_-^\lambda(x)|^2dx)^{\frac{1}{2}}\\
&\leq C(\int_D|\varphi^\lambda_-(x)|dx+\int_D|\nabla\varphi^\lambda_-(x)|dx)\\
&=C(\int_{\Omega^\lambda}|\varphi^\lambda(x)|dx+\int_{\Omega^\lambda}|\nabla\varphi^\lambda(x)|dx)\\
&\leq C|\Omega^\lambda|^{\frac{1}{2}}(\int_{\Omega^\lambda}|\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}
+C|\Omega^\lambda|^{\frac{1}{2}}(\int_{\Omega^\lambda}|\nabla\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}.
\end{split}
\end{equation}
Since $|\Omega^\lambda|=\lambda/\kappa\rightarrow0$ as $\lambda\rightarrow0^+$, we get from \eqref{3-18}
\begin{equation}\label{3-19}
(\int_{\Omega^\lambda}|\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}\leq C|\Omega^\lambda|^{\frac{1}{2}}(\int_{\Omega^\lambda}|\nabla\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}.
\end{equation}
Combining \eqref{3-17} and \eqref{3-19} we obtain
\begin{equation}\label{3-20}
\int_D(G\omega^\lambda(x)+q(x)-\mu^\lambda)\omega^\lambda(x)dx\geq -C|\Omega^\lambda|(\int_{\Omega^\lambda}|\nabla\varphi^\lambda(x)|^2dx)^{\frac{1}{2}}.
\end{equation}
Notice that by $L^p$ estimate
\begin{equation}\label{3-21}
\begin{split}
\int_{\Omega^\lambda}|\nabla\varphi^\lambda(x)|^2dx&\leq2(\int_{\Omega^\lambda}|\nabla G\omega^\lambda(x)|^2dx+\int_{\Omega^\lambda}|\nabla q(x)|^2dx)\\
&\leq 2(|\nabla G\omega^\lambda|^2_{L^\infty(D)}|\Omega^\lambda|+|\nabla q|_{L^\infty(D)}^2|\Omega^\lambda|)\\
&\leq C\lambda|\nabla G\omega^\lambda|^2_{W^{1,3}(D)}+2\lambda|\nabla q|_{L^\infty(D)}^2|\\
&\leq C\lambda(1+|\omega^\lambda|^2_{L^3(D)}|)\\
&\leq C\lambda(1+\lambda^\frac{2}{3})\\
&\leq C\lambda.
\end{split}
\end{equation}
\eqref{3-20} and \eqref{3-21} together give the desired result.
\end{proof}
Now we are able to conclude the following crucial estimate for the Lagrange multiplier $\mu^\lambda$.
\begin{lemma}\label{3-100}
\begin{equation}
\min_{\overline{D}}q<\mu^\lambda\leq\min_{\overline{D}}q+C\lambda^{\frac{1}{2}}.
\end{equation}
\end{lemma}
\begin{proof}
It is easy to see that the following identity holds
\begin{equation}\label{3-22}
E(\omega^\lambda)=-\frac{1}{2}\int_DG\omega^\lambda(x)\omega^\lambda(x)dx+\int_D(G\omega^\lambda(x)+q(x)-\mu^\lambda)\omega^\lambda(x)dx+\lambda\mu^\lambda.
\end{equation}
Since $|G\omega^\lambda|_{L^\infty(D)}\leq C\lambda^\frac{1}{2}$, combining Lemma \ref{3-10} and Lemma \ref{3-15} we obtain
\begin{equation}\label{3-23}
\mu^\lambda\leq\min_{\overline{D}}q+C\lambda^{\frac{1}{2}}.
\end{equation}
On the other hand, since $\Omega^\lambda$ is not empty, we can choose $x\in\Omega^\lambda$, then
\begin{equation}\label{3-24}
\mu^\lambda>G\omega^\lambda(x)+q(x)\geq q(x)\geq \min_{\overline{D}}q.
\end{equation}
Here we use the fact $G\omega^\lambda\geq0$ in $D$ by the maximum principle.
\end{proof}
Now we are ready to give the limiting behavior of the minimizer $\omega^\lambda$, which is equivalent to the limiting behavior of $\Omega^\lambda$ as $\lambda\rightarrow0^+.$
\begin{lemma}\label{3-200}
\[\lim_{\lambda\rightarrow0^+}\sup_{x\in \Omega^\lambda}|q(x)-\min_{\overline{D}}q|=0.\]
\end{lemma}
\begin{proof}
Notice that \eqref{3-24} holds for any $x\in \Omega^\lambda$. Combining Lemma \ref{3-100} we get the desired result.
\end{proof}
\subsection{Proof of Theorem \ref{1-11}}
Now we are ready to prove Theorem \ref{1-11}.
\begin{proof}[Proof of Theorem \ref{1-11}]
Let $\omega^\lambda$ be the unique minimizer obtained in Proposition \ref{3-3}. First we show that $\omega^\lambda$ is a weak solution of \eqref{1-7}. For any $\xi\in C^{\infty}_c(D)$ and $x\in D$, we consider the following ordinary differential equation
\begin{equation}\label{4-1}
\begin{cases}\frac{d\Phi_t(x)}{dt}=\nabla^\perp\xi(\Phi_t(x)) &t\in\mathbb R, \\
\Phi_0(x)=x.
\end{cases}
\end{equation}
Since $\nabla^\perp\xi$ is a smooth vector field with compact support, $\eqref{4-1}$ has a global solution. It is easy to check that $\nabla^\perp\xi$ is divergence-free, so $\Phi_t$ is an area-preserving transformation from $D$ to $D$, that is, for any measurable set $A\subset D$, we have $|\{\Phi_t(x)\mid x\in A\}|=|A|$. Let $\{\omega_t\}_{t\in\mathbb R}$ be a family of test functions defined by
\begin{equation}
\omega_t(x):=\omega^\lambda(\Phi_t(x)).
\end{equation}
It is obvious that $\omega_t\in \mathcal{M}^\lambda$, so $\frac{dE(\omega_t)}{dt}|_{t=0}=0$.
Expanding $E(\omega_t)$ at $t=0$ we obtain for $|t|$ small
\[\begin{split}
E(\omega_t)=&\frac{1}{2}\int_D\int_DG(x,y)\omega^\lambda(\Phi_t(x))\omega^\lambda(\Phi_t(y))dxdy+\int_Dq(x)\omega^\lambda(\Phi_t(x))dx\\
=&\frac{1}{2}\int_D\int_DG(\Phi_{-t}(x),\Phi_{-t}(y))\omega^\lambda(x)\omega^\lambda(y)dxdy\int_Dq(\Phi_{-t}(x))\omega^\lambda(x)dx\\
=&E(\omega^\lambda)+t\int_D\omega^\lambda\nabla^\perp(G\omega^\lambda+q)\cdot\nabla\xi dx+o(t).
\end{split}\]
Therefore we get
\[\int_D\omega^\lambda\nabla^\perp(G\omega^\lambda+q)\cdot\nabla\xi dx=0.\]
Note that \eqref{1-12} has been verified in the construction of $\omega^\lambda$ in Section 3.
Now we prove \eqref{1-14} by contradiction.
Suppose that there exist $\lambda_j>0, x_j\in\Omega^{\lambda_j}$ for $j=1,2,\cdot\cdot\cdot,$ such that $\lambda\rightarrow0^+$ as $j\rightarrow+\infty$ and $x_j\notin \mathcal{S}_{\delta_0}$ for some $\delta_0>0$. By the continuity of $q$, we have
\begin{equation}\label{4-2}
\inf_j{q(x_j)}>\min_{\overline{D}}q.
\end{equation}
On the other hand, by Lemma \ref{3-200} we have
\begin{equation}\label{4-3}
\lim_{j\rightarrow+\infty}q(x_j)=\min_{\overline{D}}q,
\end{equation}
which is a contradiction.
\end{proof}
\section{Proof of Theorem \ref{1-15}}
To prove Theorem \ref{1-15}, we consider a similar variational problem.
Let $\delta_0$ be chosen in Theorem \ref{1-15}. For $\lambda>0$ sufficiently small, define
\begin{equation}\label{5-1}
\begin{split}
\mathcal{N}^\lambda:=&\{\omega\in L^\infty(D)\mid \omega=\sum_{p=1}^{k+1}\omega_p, supp(\omega_p)\subset B_{\delta_0}(x_p) \text{ for } p=1,\cdot\cdot\cdot k+l, \\
&\int_D\omega_i(x)dx=\lambda \text{ and } 0\leq\omega_i\leq\kappa_i \text{ for } i=1,\cdot\cdot\cdot,k,\\
&\int_D\omega_j(x)dx=-\lambda \text{ and } -\kappa_j\leq\omega_j\leq0 \text{ for } j=k+1,\cdot\cdot\cdot,k+l\}.
\end{split}
\end{equation}
The energy functional on $\mathcal{N}^\lambda$ is still defined by
\begin{equation}\label{5-2-1}
E(\omega)=\frac{1}{2}\int_D\int_DG(x,y)\omega(x)\omega(y)dxdy+\int_Dq(x)\omega(x)dx.
\end{equation}
We consider the minimization of $E$ on $\mathcal{N}^\lambda$, that is,
\begin{equation}
c^*_\lambda=\inf\{E(\omega)\mid\omega\in\mathcal{N}^\lambda\}.
\end{equation}
\subsection{Existence of a minimizer}
As in Section 3, we first establish the following result.
\begin{lemma}
$\mathcal{N}^\lambda$ is a sequentially compact set in $L^\infty(D)$.
\end{lemma}
\begin{proof}
Let $\{\omega_n\}$ be a sequence in $L^\infty(D)$ and $\omega_n\rightarrow\omega_0\in L^\infty(D)$ weakly star as $n\rightarrow+\infty$. It suffices to show $\omega_0\in\mathcal{N}^\lambda$. By the definition of weak star convergence it is easy to check that for each $p$, $1\leq p\leq k+l,$ $\omega_n I_{B_{\delta_0}(x_p)}\rightarrow\omega_0I_{B_{\delta_0}(x_p)}$ weakly star in $L^\infty(D)$. Then we repeat the argument in Lemma \ref{3-344} to obtain
\[0\leq\omega_0I_{B_{\delta_0}(x_i)}\leq\kappa_i, \int_D\omega_0I_{B_{\delta_0}(x_i)}(x)dx=\lambda, \text{ for }i=1,\cdot\cdot\cdot,k,\]
\[-\kappa_j\leq\omega_0I_{B_{\delta_0}(x_j)}\leq0, \int_D\omega_0I_{B_{\delta_0}(x_j)}(x)dx=-\lambda, \text{ for }j=1,\cdot\cdot\cdot,k.\]
Therefore $\omega_0=\sum_{p=1}^{k+l}\omega_n I_{B_{\delta_0}(x_p)}\in\mathcal{N}^\lambda.$
\end{proof}
\begin{proposition}\label{5-2}
$c^*_\lambda$ can be achieved by an absolute minimizer $w^\lambda\in \mathcal{N}^\lambda$ with the following the form
\begin{equation}\label{5-3}
w^\lambda=\sum_{i=1}^k\kappa_iI_{U_i^\lambda}-\sum_{j=k+1}^{k+l}\kappa_jI_{U_j^\lambda},
\end{equation}
where for each $1\leq i\leq k$
\begin{equation}\label{5-4}
U_i^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)<\nu_i^\lambda\}\cap{B_{\delta_0}(x_i)},\,\,|U_i^\lambda|=\lambda,
\end{equation}
and for each $k+1\leq j\leq k+l$
\begin{equation}\label{5-5}
U_j^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)>\nu_j^\lambda\}\cap{B_{\delta_0}(x_j)},\,\,|U_j^\lambda|=\lambda.
\end{equation}
Here $\nu_p^\lambda\in\mathbb{R}, 1\leq p\leq k+l$, is the Lagrange multiplier depending on $\lambda$.
\end{proposition}
\begin{proof}
First by $L^p$ estimate and Sobolev embedding
\begin{equation}\label{5-6}
|G\omega|_{L^\infty(D)}\leq C|G\omega|_{W^{2,2}(D)}\leq C|\omega|_{L^2(D)}\leq C\lambda^{\frac{1}{2}},\,\,\forall \omega\in\mathcal{N}^\lambda.
\end{equation}
Here $C$ still denotes various positive numbers not depending on $\lambda$. Therefore we obtain
\begin{equation}\label{5-7}
|\int_D\int_DG(x,y)\omega(x)\omega(y)dxdy|= |\int_DG\omega(x)\omega(x)dx|\leq C\lambda^{\frac{3}{2}}.
\end{equation}
On the other hand, for any $\omega\in\mathcal{N}^\lambda$ with $\omega=\sum_{p=1}^{k+l}\omega_p$, since $\omega_i\geq0$ for $1\leq i\leq k$ and $\omega_j\leq0$ for $k+1\leq j\leq k+l$, we have
\begin{equation}\label{5-8}
\begin{split}
\int_Dq(x)\omega(x)dx= \sum_{p=1}^{k+l} \int_Dq(x)\omega_p(x)dx
\geq \sum_{p=1}^{k+l}\int_Dq(x_p)\omega_p(x)dx=\lambda(\sum_{i=1}^kq(x_i)-\sum_{j=k+1}^{k+l}q(x_j)).
\end{split}
\end{equation}
From \eqref{5-7}\eqref{5-8} we can easily get
\begin{equation}\label{5-9}
\inf_{\mathcal{N}^\lambda}E\geq\lambda(\sum_{i=1}^kq(x_i)-\sum_{j=k+1}^{k+l}q(x_j))-C\lambda^\frac{3}{2},
\end{equation}
which implies that $E$ is bounded from below on $\mathcal{N}^\lambda$. Now we choose a minimizing sequence $\{\omega_n\}\subset\mathcal{N}^\lambda$ such that as $n\rightarrow+\infty$
\[E(\omega_n)\rightarrow c^*_\lambda.\]
Since $\mathcal{N}^\lambda$ is compact in the weak star topology of $L^\infty(D)$ and $E$ is weakly star continuous in $L^\infty(D)$, we deduce that there exists $w^\lambda\in\mathcal{N}^\lambda$ such that
\begin{equation}\label{5-10}
E(\omega_n)\rightarrow E(w^\lambda)=c^*_\lambda.
\end{equation}
Since $w^\lambda\in\mathcal{N}^\lambda$, we can write $w^\lambda=\sum_{p=1}^{k+l}w^\lambda_p$. Now we show that $w^\lambda$ satisfies \eqref{5-3}. We need to consider the following two different cases.
Case 1: For $1\leq p\leq k$,
\begin{equation}\label{5-11}
w_p^\lambda=\kappa_p I_{\{x\in D\mid Gw^\lambda(x)+q(x)<\nu^\lambda_p\}\cap B_{\delta_0}(x_p)}
\end{equation}
for some $\nu^\lambda_p\in\mathbb R$. To show this, we define a family of test functions $w_s^\lambda=w^\lambda+s(z_0-z_1), s>0$, where $z_0$ and $z_1$ satisfy
\begin{equation}
\begin{cases}
z_0,z_1\in L^\infty(D),\,\, z_0,z_1\geq 0\text{ a.e. in } D,
\\ \int_Dz_0(x)dx=\int_D z_1(x)dx,
\\ supp(z_0),supp(z_1)\subset B_{\delta_0}(x_p),
\\z_0=0 \quad\text{in } D\setminus\{x\in D\mid w^\lambda(x)\leq\kappa_p-\delta\},
\\z_1=0 \quad\text{in } D\setminus\{x\in D\mid w^\lambda(x)\geq\delta\}.
\end{cases}
\end{equation}
Here $\delta>0$ is small. It is not hard to check that for fixed $z_0,z_1$ and $\delta$, $\omega_s\in \mathcal{N}^\lambda$ for sufficiently small $s$. Since $w^\lambda$ is a minimizer, we get
\[0\leq\frac{dE(w_s)}{ds}|_{s=0^+}=\int_Dz_0(x)(Gw^\lambda(x)+q(x))dx-\int_Dz_1(x)(Gw^\lambda(x)+q(x))dx.\]
By the choice of $z_0$ and $z_1$ we obtain
\begin{equation}\label{5-12}
\sup_{\{x\in D\mid w^\lambda(x)>0\}\cap B_{\delta_0}(x_p)}(G w^\lambda+q)\leq\inf_{\{x\in D\mid w^\lambda(x)<\kappa_p\}\cap B_{\delta_0}(x_p)}(G w^\lambda+q).
\end{equation}
Since $D$ is simply-connected and $G w^\lambda+q$ is continuous in $\{x\in D\mid w^\lambda(x)>0\}\cap B_{\delta_0}(x_p)$, \eqref{5-12} is in fact an equality, i.e.,
\begin{equation}
\sup_{\{x\in D\mid w^\lambda(x)>0\}\cap B_{\delta_0}(x_p)}(G w^\lambda+q)=\inf_{\{x\in D\mid w^\lambda(x)<\kappa_p\}\cap B_{\delta_0}(x_p)}(G w^\lambda+q):=\nu^\lambda_p.
\end{equation}
Then it is easy to check that
\begin{equation}
\begin{cases}
w^\lambda=0 &\text{ a.e. } \text{in }\{x\in D\mid Gw^\lambda(x)+q(x)\geq\nu_p^\lambda\}\cap B_{\delta_0}(x_p),
\\ w^\lambda=\kappa_p &\text{ a.e. } \text{in }\{x\in D\mid Gw^\lambda(x)+q(x)<\nu_p^\lambda\}\cap B_{\delta_0}(x_p).
\end{cases}
\end{equation}
So we obtain
\[w_p^\lambda=\kappa_pI_{\{x\in D\mid G\omega^\lambda(x)+q(x)<\nu_p^\lambda\}\cap B_{\delta_0}(x_p)}.\]
Case 2: For $k+1\leq p\leq k+l$,
\begin{equation}\label{5-13}
w_p^\lambda=-\kappa_p I_{\{x\in D\mid Gw^\lambda(x)+q(x)>\nu^\lambda_p\}\cap B_{\delta_0}(x_p)}
\end{equation}
for some $\nu^\lambda_p\in\mathbb R$. In this case, we choose $w_s^\lambda=w^\lambda+s(z_0-z_1)$ as test function, where $s>0$ and $z_0, z_1$ satisfy
\begin{equation}
\begin{cases}
z_0,z_1\in L^\infty(D),\,\, z_0,z_1\leq 0\text{ a.e. in } D,
\\ \int_Dz_0(x)dx=\int_D z_1(x)dx,
\\ supp(z_0),supp(z_1)\subset B_{\delta_0}(x_p),
\\z_0=0 \quad\text{in } D\setminus\{x\in D\mid w^\lambda(x)\geq \delta-\kappa_p\},
\\z_1=0 \quad\text{in } D\setminus\{x\in D\mid w^\lambda(x)\leq-\delta\},
\end{cases}
\end{equation}
for $\delta>0$ small. The rest of the proof is similar to Case 1, therefore we omit it.
\end{proof}
\begin{remark}
Following the argument in Lemma \ref{3-1001}, we can also prove that $E$ has only one minimizer.
\end{remark}
\subsection{Limiting behavior of the minimizer as $\lambda\rightarrow0^+$}
Let $w^\lambda$ be the minimizer obtain in the last subsection. The following are several lemmas concerning the limiting behavior as $\lambda\rightarrow0^+$.
\begin{lemma}\label{5-101}
\begin{equation}\label{5-102}
E(w^\lambda)\leq\lambda(\sum_{i=1}^{k}q(x_i)-\sum_{j=k+1}^{k+l}q(x_j))+C\lambda^\frac{3}{2}.
\end{equation}
\end{lemma}
\begin{proof}
For $\lambda$ sufficiently small, we define a test function
\[v^\lambda=\sum_{i=1}^k\kappa_i I_{B_{\varepsilon_i}(x_i^\lambda)}-\sum_{j=k+1}^{k+l}\kappa_j I_{B_{\varepsilon_j}(x_j^\lambda)},\]
where $\varepsilon_p$ satisfies $\kappa_p\pi\varepsilon_p^2=\lambda$, $B_{\varepsilon_p}(x_p^\lambda)\subset B_{\delta_0}(x_p)$ and $|x_p-x^\lambda_p|=\varepsilon_p$ for each $1\leq p\leq k+l$. Note that such test function exists since $D$ satisfies the interior sphere condition. It is obvious that $v^\lambda\in\mathcal{N}^\lambda$. So we have
\begin{equation}\label{5-103}
\begin{split}
E(w^\lambda) \leq E(v^\lambda)
=\frac{1}{2}\int_D\int_DG(x,y)v^\lambda(x)v^\lambda(y)dxdy+\int_Dq(x)v^\lambda(x)dx.
\end{split}
\end{equation}
By \eqref{5-7},
\begin{equation}\label{5-104}
|\frac{1}{2}\int_D\int_DG(x,y)v^\lambda(x)v^\lambda(y)dxdy|\leq C\lambda^\frac{3}{2}.
\end{equation}
On the other hand,
\begin{equation}\label{5-105}
\begin{split}
\int_Dq(x)v^\lambda(x)dx &= \sum_{i=1}^k\kappa_i \int_{B_{\varepsilon_i}(x_i^\lambda)}q(x)dx-\sum_{j=k+1}^{k+l}\kappa_j \int_{B_{\varepsilon_j}(x_j^\lambda)}q(x)dx\\
&=\sum_{i=1}^k\kappa_i \int_{B_{\varepsilon_i}(x_i^\lambda)}(q(x)-q(x_i))dx-\sum_{j=k+1}^{k+l}\kappa_j \int_{B_{\varepsilon_j}(x_j^\lambda)}(q(x)-q(x_j))dx\\
&+\sum_{i=1}^k\kappa_i \int_{B_{\varepsilon_i}(x_i^\lambda)}q(x_i)dx-\sum_{j=k+1}^{k+l}\kappa_j \int_{B_{\varepsilon_j}(x_j^\lambda)}q(x_j)dx\\
&\leq\sum_{i=1}^k\kappa_i \int_{B_{\varepsilon_i}(x_i^\lambda)}|\nabla q|_{L^\infty(D)}|x-x_i|dx+\sum_{j=k+1}^{k+l}\kappa_j \int_{B_{\varepsilon_j}(x_j^\lambda)}|\nabla q|_{L^\infty(D)}|x-x_j|dx\\
&+\lambda(\sum_{i=1}^k q(x_i)-\sum_{j=k+1}^{k+l} q(x_j))\\
&\leq\sum_{p=1}^{k+l}\kappa_p \int_{B_{\varepsilon_p}(x_p^\lambda)}C\varepsilon_pdx
+\lambda(\sum_{i=1}^kq(x_i)-\sum_{j=k+1}^{k+l} q(x_j))\\
&\leq C\lambda^\frac{3}{2}
+\lambda(\sum_{i=1}^k q(x_i)-\sum_{j=k+1}^{k+l} q(x_j)).
\end{split}
\end{equation}
Combining \eqref{5-103},\eqref{5-104} and \eqref{5-105} we get the desired result.
\end{proof}
\begin{lemma}\label{5-106}
For each $p$, $1\leq p\leq k+l, $ we have
\begin{equation}\label{5-107}
\int_D(Gw^\lambda(x)+q(x)-\nu_p^\lambda)w_p^\lambda(x)dx\geq -C\lambda^\frac{3}{2}.
\end{equation}
\end{lemma}
\begin{proof}
We only prove the case $1\leq p\leq k$, for the other part the proof is similar. For simplicity we denote $\varphi^\lambda=Gw^\lambda+q-\nu^\lambda_p, \varphi^\lambda_-=\min{\{\varphi^\lambda,0\}}$ Then by H\"older's inequality
\begin{equation}\label{5-108}
\begin{split}
\int_D\varphi^\lambda(x)w^\lambda_p(x)dx
=\kappa_p\int_{U^\lambda_p}\varphi^\lambda(x)dx
\geq-\kappa_p|U^\lambda_p|^\frac{1}{2}(\int_{U^\lambda_p}|\varphi^\lambda(x)|^2dx)^\frac{1}{2}.
\end{split}
\end{equation}
On the other hand, by Sobolev embedding $W^{1,1}(B_{\delta_0}(x_p))\hookrightarrow L^2(B_{\delta_0}(x_p))$ and H\"older's inequality,
\begin{equation}\label{5-109}
\begin{split}
(\int_{U^\lambda_p}|\varphi^\lambda(x)|^2dx)^\frac{1}{2}=&(\int_{B_{\delta_0}(x_p)}|\varphi^\lambda_-(x)|^2dx)^\frac{1}{2}\\
\leq& C(\int_{B_{\delta_0}(x_p)}|\varphi^\lambda_-(x)|dx+\int_{B_{\delta_0}(x_p)}|\nabla\varphi^\lambda_-(x)|dx)\\
\leq&C|U^\lambda_p|^\frac{1}{2}(\int_{U^\lambda_p}|\varphi^\lambda(x)|^2dx)^\frac{1}{2}
+C|U^\lambda_p|^\frac{1}{2}(\int_{U^\lambda_p}|\nabla\varphi^\lambda(x)|^2dx)^\frac{1}{2}.
\end{split}
\end{equation}
Since $|U^\lambda_p|\rightarrow0$ as $\lambda\rightarrow0^+$, we get from \eqref{5-109}
\begin{equation}\label{5-110}
(\int_{U^\lambda_p}|\varphi^\lambda(x)|^2dx)^\frac{1}{2}\leq C|U^\lambda_p|^\frac{1}{2}(\int_{U^\lambda_p}|\nabla\varphi^\lambda(x)|^2dx)^\frac{1}{2}.
\end{equation}
Taking into account \eqref{5-108} and \eqref{5-110} and using $L^p$ estimate, we obtain
\begin{equation}\label{5-111}
\begin{split}
\int_D\varphi^\lambda(x)w^\lambda_p(x)dx&\geq -C\lambda (\int_{U^\lambda_p}|\nabla\varphi^\lambda(x)|^2dx)^\frac{1}{2}\\
&\geq-C\lambda (\int_{U^\lambda_p}|\nabla Gw^\lambda(x)|^2+|\nabla q(x)|^2dx)^\frac{1}{2}\\
&\geq -C\lambda^\frac{3}{2}(|\nabla Gw^\lambda|_{L^\infty(D)}+|\nabla q|_{L^\infty(D})\\
&\geq -C\lambda^\frac{3}{2},
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\begin{lemma}\label{5-200}
(i)For $1\leq i\leq k$,
\begin{equation}\label{5-301}
\nu^\lambda_i>q(x_i)-C\lambda^\frac{1}{2};
\end{equation}
(ii) For $k+1\leq j\leq k+l$,
\begin{equation}\label{5-302}
\nu^\lambda_j< q(x_j)+C\lambda^\frac{1}{2}.
\end{equation}
\end{lemma}
\begin{proof}
First recall that by $L^p$ estimate $|Gw^\lambda|_{L^\infty(D)}\leq C\lambda^\frac{1}{2}$.
Since $U^\lambda_p$ is not empty, we can choose $y_p\in U^\lambda_p$, then for $1\leq p\leq k$
\begin{equation}\label{5-115-2}
Gw^\lambda(y_p)+q(y_p)<\nu^\lambda_p,
\end{equation}
and for $k+1\leq p\leq k+l$
\begin{equation}\label{5-116}
Gw^\lambda(y_p)+q(y_p)>\nu^\lambda_p.
\end{equation}
From \eqref{5-115-2} we get for $1\leq p\leq k$
\begin{equation}\label{5-117}
q(x_p)-C\lambda^\frac{1}{2}\leq Gw^\lambda(y_p)+q(y_p)<\nu^\lambda_p.
\end{equation}
From \eqref{5-116} we get for $1\leq p\leq k$
\begin{equation}\label{5-118}
q(x_p)+C\lambda^\frac{1}{2}\geq Gw^\lambda(y_p)+q(y_p)>\nu^\lambda_p.
\end{equation}
\end{proof}
\begin{lemma}\label{5-112}
(i)For $1\leq i\leq k$,
\begin{equation}\label{5-303}
\nu^\lambda_i\leq q(x_i)+C\lambda^\frac{1}{2};
\end{equation}
(ii) For $k+1\leq j\leq k+l$,
\begin{equation}\label{5-304}
\nu^\lambda_j\geq q(x_j)-C\lambda^\frac{1}{2}.
\end{equation}
\end{lemma}
\begin{proof}
Notice that $E(w^\lambda)$ can be written as
\begin{equation}\label{5-113}
\begin{split}
E(w^\lambda)&=-\frac{1}{2}\int_D\int_DG(x,y)w^\lambda(x)w^\lambda(y)dxdy+\sum_{p=1}^{k+1}\int_D(Gw^\lambda(x)+q(x)-\nu^\lambda_p)w^\lambda_p(x)dx\\
&+\lambda(\sum_{i=1}^{k}\nu^\lambda_i-\sum_{j=k+1}^{k+l}\nu^\lambda_j).\\
\end{split}
\end{equation}
Taking into account Lemma \ref{5-101} and Lemma \ref{5-106}, we deduce from \eqref{5-113} that
\begin{equation}\label{5-114}
\sum_{i=1}^{k}\nu^\lambda_i-\sum_{j=k+1}^{k+l}\nu^\lambda_j\leq \sum_{i=1}^{k}q(x_i)-\sum_{j=k+1}^{k+l}q(x_j)+C\lambda^\frac{1}{2}.
\end{equation}
Now \eqref{5-301}, \eqref{5-302} and \eqref{5-114} together give the desired result.
\end{proof}
\begin{lemma}\label{3-500}
For each $p$, $1\leq p\leq k+l,$ we have
\begin{equation}\label{5-115}
\lim_{\lambda\rightarrow0^+}\sup_{x\in U^\lambda_p}|q(x)-q(x_p)|=0.
\end{equation}
\end{lemma}
\begin{proof}
First by Lemma \ref{5-112} and the definition of $U^\lambda_p$, for each $x\in U^\lambda_p$, we have
\[q(x_p)-C\lambda^\frac{1}{2}\leq Gw^\lambda(x)+q(x)<\nu^\lambda_p\leq q(x_p)+C\lambda^\frac{1}{2},\,\,1\leq p\leq k,\]
\[q(x_p)-C\lambda^\frac{1}{2}\leq\nu^\lambda< Gw^\lambda(x)+q(x)\leq q(x_p)+C\lambda^\frac{1}{2},\,\,1\leq p\leq k,\]
from which it is easy to see that for each $p$, $1\leq p\leq k+l$ and $x\in U^\lambda_p$,
\[|q(x)-q(x_p)|\leq |Gw^\lambda(x)+q(x)-q(x_p)|+|Gw^\lambda(x)|\leq C\lambda^\frac{1}{2},\]
which implies \eqref{5-115}.
\end{proof}
\subsection{Proof of Theorem \ref{1-15}}
Now we are ready to prove Theorem \ref{1-15}.
\begin{proof}[Proof of Theorem \ref{1-15}]
First we show that for each $p, 1\leq p\leq k+l$, $U^\lambda_p$ shrinks to $x_p$ as $\lambda\rightarrow0^+.$ Suppose that there exist $\delta_1>0, \lambda_n\rightarrow0^+, y_n\in U^\lambda_p$ such that $|y_n-x_p|\geq \delta_0.$ By the continuity of $q$ we deduce that $\inf_{n}|q(y_n)-q(x_p)|>0$, which is a contradiction to Lemma \ref{3-500}.
Now we show that $w^\lambda$ is a weak solution of \eqref{1-7} for $\lambda$ sufficiently small. For any $\xi\in C^{\infty}_c(D)$, let $\Phi_t(x)$ be defined by \eqref{4-1}. Then since $U^\lambda_p$ shrinks to $x_p$ for each $p$, we deduce that for $|t|<<1$, $w_t:=w^\lambda(\Phi_t(\cdot))\in \mathcal{N}^\lambda$, so we still have $\frac{dE(w_t)}{dt}|_{t=0}=0$, which gives
\[\int_Dw^\lambda\nabla^\perp(Gw^\lambda+q)\cdot\nabla\xi dx=0.\]
\end{proof}
\section{Further Discussion}
When we consider vortex patch solutions near a harmonic function, there are four possibilities:
\begin{enumerate}
\item a positive vortex patch near a minimum point of the harmonic function;
\item a negative vortex patch near a minimum point of the harmonic function;
\item a negative vortex patch near a maximum point of the harmonic function;
\item a positive vortex patch near a maximum point of the harmonic function.
\end{enumerate}
As has been mentioned in Remark \ref{2-102}, $(1)$ is equivalent to $(3)$ and $(2)$ is equivalent to $(4)$. Now we give an example in one dimension to illustrate $(1)$ and $(2)$.
Let $u$ be harmonic in $(0,1)\subset\mathbb R$ with boundary condition $u(0)=0, u(1)=1.$ We perturb $u$ near $x=0$ by constructing two kinds of vortex patch solutions with the same boundary condition, that is, we consider the following two problems:
\begin{equation}\label{6-1}
\mathcal{P}_1:
\begin{cases}
-\frac{d^2u^\lambda}{dx^2}=I_{\{u^\lambda<\mu^\lambda\}} &x\in(0,1),\\
|\{u^\lambda<\mu^\lambda\}|=\lambda,\\
u^\lambda(0)=u &\text{ on } \partial(0,1),
\end{cases}
\end{equation}
\begin{equation}\label{6-2}
\mathcal{P}_2:
\begin{cases}
-\frac{d^2v^\lambda}{dx^2}=-I_{\{v^\lambda>\nu^\lambda\}} &x\in(0,1),\\
|\{v^\lambda>\nu^\lambda\}|=\lambda,\\
v^\lambda(0)=u &\text{ on } \partial(0,1).
\end{cases}
\end{equation}
Explicit solutions to $\mathcal{P}_1$ and $\mathcal{P}_2$ are
\begin{equation}\label{6-3}
u^\lambda(x)=
\begin{cases}
-x^2+(2\lambda+1-\lambda^2)x &0\leq x\leq\lambda,\\
(1-\lambda^2)x+\lambda^2 &\lambda\leq x\leq 1,
\end{cases}
\end{equation}
\begin{equation}\label{6-4}
v^\lambda(x)=
\begin{cases}
x^2+(\lambda-1)^2x &0\leq x\leq\lambda,\\
(\lambda^2+1)x-\lambda^2 &\lambda\leq x\leq 1.
\end{cases}
\end{equation}
In Theorem \ref{1-11} and \ref{1-15}, we have solved $\mathcal{P}_1$ in two dimensions. In this section we consider $\mathcal{P}_2$.
More precisely, we construct steady vortex patches near maximum points of $q$ with positive vorticity and near minimum points of $q$ with negative vorticity.
The results are as follows.
\begin{theorem}\label{1-11-2}
Let $q\in C^2(D)\cap C^1(\overline{D})$ be a harmonic function and $\kappa$ be a positive real number. Set $\mathcal{G}:=\{x\in \overline{D}\mid q(x)=\max_{ \overline{D}}q\}$. Then
for any given positive number $\lambda$ with $\lambda<\kappa|D|$, there exists a weak solution $\omega^\lambda$ of \eqref{1-7} having the form
\begin{equation}\label{1-12-2}
\omega^\lambda=\kappa I_{\Omega^\lambda}, \,\, \Omega^\lambda=\{x\in D\mid G\omega^\lambda(x)+q(x)>\mu^\lambda\},\,\,\kappa|\Omega^\lambda|=\lambda
\end{equation}
for some $\mu^\lambda\in\mathbb{R}$ depending on $\lambda$.
Furthermore, if $q$ is not a constant, then $\mathcal{G}\subset\partial D$ and $\Omega^\lambda$ approaches $\mathcal{G}$, or equivalently, for any $\delta>0$, there exists $\lambda_0>0$, such that for any $\lambda<\lambda_0$, we have
\begin{equation}\label{1-14-2}
\omega^\lambda\subset\mathcal{G}_\delta:=\{x\in D\mid dist(x,\mathcal{G})<\delta\}.
\end{equation}
\end{theorem}
\begin{theorem}\label{1-15-2}
Let $q\in C^2(D)\cap C^1(\overline{D})$ be a harmonic function, $k,l$ be two nonnegative integers and $\kappa_1,\cdot\cdot\cdot,\kappa_{k+l}$ be $k+l$ positive real numbers. Suppose that $\{x_1,x_2,\cdot\cdot\cdot,x_k\}\subset\partial D$ are $k$ different strict local minimum points of $q$ on $\overline{D}$, and $\{x_{k+1},x_{k+2},\cdot\cdot\cdot,x_{k+l}\}\subset\partial D$ are $l$ different strict local maximum points of $q$ on $\overline{D}$.
Then there exists a $\lambda_0>0$, such that for any $0<\lambda<\lambda_0$, there exists a weak solution of \eqref{1-7} $w^\lambda$ having the form
\begin{equation}\label{1-16-2}
w^\lambda=-\sum_{i=1}^k\kappa_iI_{U_i^\lambda}+\sum_{j=k+1}^{k+l}\kappa_jI_{U_j^\lambda},
\end{equation}
where for each $1\leq i\leq k$
\begin{equation}\label{5-4-2}
U_i^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)<\nu_i^\lambda\}\cap{B_{\delta_0}(x_i)},\,\,|U_i^\lambda|=\lambda,
\end{equation}
and for each $k+1\leq j\leq k+l$
\begin{equation}\label{5-5-2}
U_j^\lambda =\{x\in D\mid Gw^\lambda(x)+q(x)>\nu_j^\lambda\}\cap{B_{\delta_0}(x_j)},\,\,|U_j^\lambda|=\lambda,
\end{equation}
where $\nu_p^\lambda\in\mathbb{R},$ for $1\leq p\leq k+l$, is the Lagrange multiplier depending on $\lambda$.
Here $\delta_0$ is chosen to be sufficiently small such that $x_i$ is the unique minimum point of $q$ on $\overline{B_{\delta_0}(x_i)\cap D}$ for $i=1,\cdot\cdot\cdot,k$, $x_i$ is the unique maximum point of $q$ on $\overline{B_{\delta_0}(x_j)\cap D}$ for $j=k+1,\cdot\cdot\cdot,k+l$, and $\overline{B_{\delta_0}(x_{p_1})\cap D}\cap\overline{B_{\delta_0}(x_{p_2})\cap D}=\varnothing$ for $1\leq p_1,p_2\leq k+l, p_1\neq p_2.$
Moreover, $U_p^\lambda$ shrinks to $x_p$ for each $1\leq p\leq k+l$ as $\lambda\rightarrow0^+$, or equivalently, for any $\delta>0$, there exists a $\lambda_0>0$, such that for any $\lambda<\lambda_0$, we have
\begin{equation}\label{1-17-2}
U^\lambda_p\subset B_{\delta}(x_p)\cap D.
\end{equation}
\end{theorem}
Proofs for Theorem \ref{1-11-2} and Theorem \ref{1-15-2} are similar to those for Theorem \ref{1-11} and Theorem \ref{1-15}. More specifically, we consider the maximization of $E$ on $\mathcal{M}^\lambda$ and $\mathcal{N}^\lambda$ respectively and then analyze the limiting behavior of the maximizers as $\lambda\rightarrow0^+$. Since the details are almost the same, we omit it here.
~\\
\noindent{\bf Acknowledgments:}
Daomin Cao was partially supported by Hua Luo Geng Center of Mathematics,
AMSS, CAS, he was also partially supported by NNSF of China grant No.11331010 and No.11771469.
Guodong Wang was supported by NNSF of China grant No.11771469.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,579 |
Q: Why we used "this" keyword in this snippet? var someObject = {
someArray : new Array(),
someInt : 0,
Total: function(){
this.someInt = 0;//we used "this" keyword here, why?Cant we just say "someInt = 0"?
for(var i=0;i<this.someArray.length;i++){//and here..
var c = this.someArray[i];//again we use "this"
this.someInt += c.value;//also here
}
so why did we use "this" keyword? cant we just type the name of the variable?
A: The this keyword refers to the object on whose behalf the call is made later on, i.e. if you call the function like this:
someObject.Total()
then this will refer to someObject inside the function. Thanks to this keyword the function can modify someInt and read from someArray which are members of someObject. If you dropped this from the function body, all those references would be to global variables or variables local to the function body.
A: No, the statement someInt = 0 would not modify the someInt property of someObject. Instead, it would modify a property named someInt on the global/default object (window in a browser), which is obviously not want you want.
Note that (depending on how you intend to invoke the Total function) you could also write this as someObject.someInt. However, when calling the function like this:
someObject.Total()
...the value of this in the function is equal to someObject.
A: No, because the variable is not fully created. By using the 'this' keyword you can access a variable from itself.
A: Looks to me like it's just for clarity. Always using this. is probably a good practice when you don't use special naming conventions for instance variables vs. local variables (in other languages as well, not just javascript).
Even though it may not be required in this case, if you had a larger function, with lots of local and instance variables, it makes things much clearer when you distinguish.
A: In fact, Douglas Crockford in his Javascript The Good Parts, suggests not using this keywords in the code, for the functions from the object may be applied to other objects and this may cause errors. so it's better sometimes to use just variable names.
A: someInt is defined as a property of someObject. It would need to be defined as a variable in order to access it that way.
http://jsfiddle.net/Z7mSK/
A: Understand this keyword in detail HERE
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,556 |
\section{Introduction}
\label{sec:intro}
The success of deep neural network (DNN) in both machine translation (MT) \cite{bahdanau_15_attention, sutskever_14_seq2seq, luong_15_attention, chen_18_best2worlds, bahar_2018_2d_mt} and automatic speech recognition (ASR) \cite{bahdanau_2016_asr, chan_2016_las, chorowski_2015_attention_asr, zeyer_2018_att_asr, bahar_2019_2d_asr} has inspired the work of end-to-end speech to text translation (ST) systems.
The traditional ST methods are based on a consecutive cascaded pipeline of ASR and MT systems.
In contrast,
the end-to-end stand-alone model \cite{berard_2016_proof, goldwater_2017_noasr, weiss_2017_directly, gangi-etal-2019-enhancing} translates speech in source language directly into target language text.
The end-to-end model has advantages over the cascaded pipeline,
however, its training requires a moderate amount of paired speech-to-text data which is not easy to acquire.
Therefore, recently some techniques such as multi-task learning \cite{weiss_2017_directly, anastasopoulos_2018_tied_multitask, Sperber_19_attention_passing, st_kd_interspeech2019},
pre-training different components of the model \cite{berard_2018_librispeech, bansal_2019_pretraining_asr, bansal_2018_low_resource} and generating synthetic data \cite{jia_2019_synthatic} have been proposed to mitigate the lack of ST parallel training data. These methods aim to use weakly supervised data, i.e. speech-to-transcription or text-to-translation pairs in addition to fully supervised data, i.e. speech-to-translation pairs. As reported in the literature, all of these methods give a boost up to some degree, however, each has its own problems.
Multi-task learning has become a significant method that aims at improving
the generalization performance of a task using other related tasks \cite{luong_2015_multitask}. For this technique, we need to compromise between multiple tasks and the parameters are updated independently for each task, which might lead to a sub-optimal solution for the entire multi-task optimization problem.
The pre-training methods and synthesis systems rely on given previously trained models. The component of the ST model can be trained using only an ASR model \cite{bansal_2019_pretraining_asr} or both an ASR and an MT model \cite{bansal_2018_low_resource}. Similarly, the synthetic techniques depend on a pre-trained MT or a text-to-speech (TTS) synthesis model. Both cases require more effort to build the models and thus more computational power. Furthermore, both scenarios rely on paired speech-to-transcription and/or text-to-translation data. In contrast, an unsupervised ST system uses only independent monolingual corpora of speech and text, though, its performance is behind the aforementioned approaches \cite{chung_2019_unsupervised_st}.
There are many aspects to be considered for training end-to-end models which we are exploring in this work.
Our contribution is as follows:
We review and compare different end-to-end methods for building an ST system and we analyze their effectiveness.
We confirm that multi-task learning is still beneficial up to some extent.
We also adopt the idea of connectionist temporal classification (CTC) auxiliary loss from ASR. Although the first intuition is that the CTC module can not cooperate with the ST task since the translation does not necessarily require the monotonic alignment of the speech and target sequences, surprisingly we show that including this loss helps to gain better performance.
To facilitate the model with weakly supervised data, we study an extensive quantitative comparison of various possible pre-training schemes which to our best of knowledge, no such comparison for ST exists yet.
We discover that directly coupling the pre-trained encoder and decoder hurts the performance and present a solution to incorporate them in a better fashion.
We demonstrate our empirical results on two ST tasks, IWSLT TED-talks En$\to$De, and LibriSpeech Audio-books En$\to$Fr.
\begin{figure*}[h]
\centering
\begin{subfigure}[b]{0.25\textwidth}
\centerline{\includegraphics[width=3.5cm,height=5cm,keepaspectratio]{figures/st-architect-direct-pretrain}}
\caption{direct}
\label{fig:direct}
\end{subfigure}
~\hfill
\begin{subfigure}[b]{0.35\textwidth}
\centerline{\includegraphics[width=6.5cm,height=6.5cm,keepaspectratio]{figures/st-architect-multitask-one2many-pretrain.pdf}}
\caption{multitask-one2many}
\label{fig:multitask-one2many}
\end{subfigure}
~\hfill
\begin{subfigure}[b]{0.35\textwidth}
\centerline{\includegraphics[width=6.5cm,height=6.5cm,keepaspectratio]{figures/st-architect-multitask-many2one-pretrain.pdf}}
\caption{multitask-many2one}
\label{fig: multitask-many2one}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centerline{\includegraphics[width=6.8cm,height=6.8cm,keepaspectratio]{figures/st-architect-tied-triangle-pretrain.pdf}}
\caption{multitask-tied cascade}
\label{fig:multitask- tied cascaded}
\end{subfigure}
~\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centerline{\includegraphics[width=6.8cm,height=6.8cm,keepaspectratio]{figures/st-architect-tied-cascade-pretrain.pdf}}
\caption{multitask-tied triangle}
\label{fig:multitask- tied triangle}
\end{subfigure}
\caption{Overview of end-to-end speech translation model architectures. Blue
blocks correspond to pre-trained components, and orange blocks are fine-tuned on the ST task. Green depicts the layers of architecture.}\label{fig:arch}
\end{figure*}
\section{Network Architectures}
\label{sec:network architecture}
We assume a speech input observation of variable length $T$, $x_1^{T}$, a sequence of tokens of unknown length $J$ in the source language, $f_1^{J}$, and a sequence of target tokens of unknown length $I$, $e_1^{I}$.
We define translation posterior probability as
$p(e_1^{I} | x_1^{T}) = \prod_{i=1}^{I} p(e_i| e_1^{i-1}, x_1^{T})$
where $T> I$, $J$. This can be modeled by either explicit use of the source sequence ($f_1^{J}$) as a pivot step or implicitly learn the representations.
The former can be a cascade of an ASR followed by an MT system and the latter is the usage of an end-to-end model. Here, we aim to address various kinds of end-to-end approaches.
\subsection{Direct Model}
\label{sec:direct}
The direct model is the vanilla end-to-end network based on long short-term memories (LSTMs) \cite{hochreiter_97_LSTM} attention encoder-decoder architecture \cite{bahdanau_15_attention}.
Here, we focus on LSTM-based models rather than the transformer \cite{vaswani_17_transformer, gangi-etal-2019-enhancing,st_kd_interspeech2019}.
The model is summarized in Equation \ref{eq:att}.
A bidirectional LSTM (BLSTM) converts the input sequence into a sequence of encoder representations $h_1^{T}= h_1,\ldots, h_{T}$.
For speech encoders, to reduce the audio sequence length, we apply 3 max-pooling layers with a pool-size of 2 in the time-dimension at multiple steps between BLSTM layers.
For the input sequence $x_1^T$, we condense to $h_1^{T'}$, where $T' = T / 8$.
In the decoder, an LSTM generates an output sequence using an attention function outputting attention weights $\alpha_{i,t}$.
The context vector $c_i$ is obtained as a weighted sum of encoder representations using these weights. $e_{i}$ is predicted by a linear transformation followed by a softmax layer, and the decoder state is updated to $s_i$ by a stack of LSTM layers. $L$ is the number of layers.
Figure \ref{fig:direct} illustrates an abstract overview of the direct model.
\begin{align}
&h_1^{T'} = (\operatorname{BLSTM}_{\rm{L}}
\circ \cdots
\circ \operatorname{max-pool}_1
\circ \operatorname{BLSTM}_1) (x_1^T) \nonumber \\
&\alpha_{i,t} = \operatorname{softmax} \big(\tanh (s_{i-1},h_{t}) \big) \text{,\hspace{0.5cm} } c_i = \sum_{t=1}^{T'} \alpha_{i,t} h_t \nonumber \\
&p(e_{i}|e_1^{i-1},x_1^T) = \operatorname{softmax} \big( \operatorname{linear}(e_{i-1}, s_{i-1}, c_{i}) \big) \nonumber\\
&s_i = \operatorname{LSTM}_{L}
\circ \cdots
\circ \operatorname{LSTM}_1 (e_{i}, s_{i-1}, c_{i}) \label{eq:att}
\end{align}
\subsection{Mutli-Task Model}
\label{sec:multitask}
As suggested in \cite{weiss_2017_directly}, in multi-task learning an auxiliary model is co-trained with the speech translation model by sharing some parameters. This auxiliary model can be an ASR or an MT model. We compare different multi-task training strategies as follows:
\\
\textbf{One-to-many}
As shown in Figure \ref{fig:multitask-one2many}, in the one-to-many method, an ASR model is used as the auxiliary co-trainer in which a speech encoder is shared between both tasks, while two independent decoders correspond to transcription and translation texts.
The error is back-propagated via two decoders into the input and thus the final loss is computed as the weighted sum of the two losses.
Here, we choose $\lambda=0.5$.
\begin{align}
L = \lambda \log p_{s2s}(e_1^{I} | x_1^{T}) + (1-\lambda) \log p_{s2s}(f_1^{J} | x_1^{T})
\label{eq:multi:one2many}
\end{align}
\textbf{Many-to-one}
In many-to-one architecture, a text decoder is shared to generate a target translation by attending on two independent encoders, a speech encoder, and a text encoder.
Then the text decoder interchangeably attends on one of them.
Here, an MT model is co-trained along with the ST model (see Figure \ref{fig: multitask-many2one}).
\\
\textbf{Tied cascade}
Unlike the multitask network discussed above, where information shared either in the encoder or in the decoder, in the tied multi-task architectures \cite{anastasopoulos_2018_tied_multitask}, a higher-level intermediate
representation is provided.
In the tied cascade model, the decoder of the ST task is stacked on top of an ASR decoder and the ASR decoder is stacked on top of a speech encoder respectively (see Fig. \ref{fig:multitask- tied cascaded}).
Therefore, the second (ST) decoder attends only to the output states of the first (ASR) decoder.
\\
\textbf{Tied triangle}
In the tied triangle model, the second decoder attends both on the encoder states of a speech encoder and on the states of the first text decoders as shown in Fig. \ref{fig:multitask- tied triangle}.
In this architecture, two context vectors form the final representation.
We use the greedy search to generate the first decoder's output and combine losses like Eq. \ref{eq:multi:one2many} in both tied cascade and triangle models.
\section{CTC Loss}
\label{sec:ctc}
CTC has been introduced in \cite{graves_2006_ctc} to solve the problem of unknown segmentation of the input sequence. It introduces a special blank state, which can appear at any time and represents that currently no token is recognized. Note that the blank state is not the same as whitespace, and is modeled as an own token.
A simple intuition is that the CTC module can not cooperate with the ST task since the translation task does not necessarily require the monotonic alignment of the speech and target sequences, unlike ASR.
But, based on the fact that an auxiliary CTC loss function has shown to help convergence in the training of ASR models \cite{zeyer_2018_att_asr}, we also apply it on top of the speech encoder only during training for the ST models.
During training, given an input sequence $x_1^{T}$, the decoder predicts the frame-wise posterior distribution of $e_1^{I}$, whilst the CTC module predicts posterior distribution of $f_1^{J}$, referred to $p_{s2s}$ and $p_{ctc}$ respectively. We simply use the sum of their log-likelihood values as:
\begin{align}
L = \log p_{s2s}(e_1^{I} | x_1^{T}) + \log p_{ctc}(f_1^{J} | x_1^{T})
\end{align}
We note that we also use CTC loss in the multi-task learning setups, however, one of the decoders predicts the frame-wise posterior distribution of the transcriptions. We believe CTC loss can help the ASR sub-task. Thus, Eq. \ref{eq:multi:one2many} is written as:
\begin{align}
L &= \lambda \log p_{s2s}(e_1^{I} | x_1^{T}) \nonumber \\
&+ (1-\lambda) \Big(\log p_{s2s}(f_1^{J} | x_1^{T}) + \log p_{ctc}(f_1^{J} | x_1^{T}) \Big)
\end{align}
\section{Pre-training}
\label{sec:pretraining}
Pre-training greatly improves the performance of end-to-end ST networks.
In this context, pre-training refers to pre-train the model on a high-resource task, and then fine-tune its parameters on the ST data.
In Figure \ref{fig:arch}, the blue blocks refer to the pre-trained components and the orange blocks mean fine-tuning on the ST data.
Pre-training can be done in different ways as proposed in the literature. The common way is to use an ASR encoder and an MT decoder to initialize the parameters of the ST network correspondingly \cite{bansal_2018_low_resource}. Surprisingly, using an ASR model to pre-train both the encoder and the decoder of the ST model works well \cite{bansal_2019_pretraining_asr}.
\begin{table}[h]
\begin{center}
\caption{Training data statistics.}
\scalebox{0.9}{%
\label{tab:stat}
\begin{tabular}{lll|ll}
\hline
\multirow{2}{*}{\bfseries Task} & \multicolumn{2}{c}{\bfseries IWSLT En$\to$De} & \multicolumn{2}{c}{\bfseries LibriSpeech En$\to$Fr} \\ \cline{2-5}
& \# of seq. & hours & \# of seq. & hours \\ \hline
ASR & 92.9k & 207h & 61.3k & 130h \\
ST & 171.1k & 272h & 94.5k & 200h \\
MT & 32M & - & 94.5k & - \\ \hline
\end{tabular}
}
\end{center}
\end{table}
\section{Experiments}
\label{sec:expriments}
\subsection{Datasets and Metrics}
\label{sec:dataset}
We have done our experiments on two ST tasks: the IWSLT TED En$\to$De \cite{Cho_2014_ted,Niehues_2018_iwslt2018}\footnote{https://sites.google.com/site/iwsltevaluation2018/Lectures-task} and the LibriSpeech En$\to$Fr \cite{Kocabiyikoglu_2018_librispeech, berard_2018_librispeech}\footnote{https://persyval-platform.univ-grenoble-alpes.fr/DS91/detaildataset}. Table \ref{tab:stat} shows the training data statistics.
\\
\textbf{IWSLT En$\to$De:}
We use the TED-LIUM corpus (excluding the black-listed talks) including 207h and the IWSLT speech translation TED corpus with 272h of speech data, in total 390h.
80-dimensional Mel-frequency cepstral coefficients (MFCC) features are extracted.
Similar to \cite{apptek_2018_st}, we automatically recompute the provided audio-to-source-sentence alignments to reduce the problem of speech segments without a translation.
We randomly select a part of our segments as our cross-validation set and choose dev2010 and test2015 as our development and test sets with 888 and 1080 segments respectively. We select our checkpoints based on the dev set.
For the MT training, we use the TED, OpenSubtitles2018, Europarl, ParaCrawl, CommonCrawl, News Commentary, and Rapid corpora resulting in 32M sentence pairs after filtering noisy samples.
\\
\textbf{LibriSpeech En$\to$Fr:}
Similar to \cite{berard_2018_librispeech}, to increase the training data size, we add the original translation and the Google Translate reference provided in the dataset package. It results in 200h of speech corresponding to 94.5k segments for the ST task.
We extract 40-dimensional Gammatone features \cite{schluter_2007_gammatone} using
the RASR \cite{wiesler_2014_rasr}.
For MT training, we utilize no extra data.
The dev and test sets contain 2h and 4h of speech, 1071 and 2048 segments respectively. The dev set is used as our cross-validation set and checkpoint selection.
For the IWSLT, in addition to the ST data, we benefit from weakly supervised data (paired ASR and MT data) both in multi-task learning and in pre-training. Whereas for the LibriSpeech, we only use the ST data in the multi-task scenario to see whether any gain comes from the data or the model itself.
For both tasks, we remove the punctuation only from the transcriptions (i.e. the English text) and keep the punctuation on the target side. We note that by doing so we tend to explore whether the MT models can implicitly capture the punctuation information without any additional component. Furthermore, we do not need to automatically enrich the ASR's output with punctuation marks for the cascade pipeline. \texttt{Moses} toolkit \cite{koehn_07_moses}\footnote{http://www.statmt.org/moses/?n=Moses.SupportTools} is used for tokenization. We employ frequent casing for the IWSLT tasks while lowercase for the LibriSpeech. There, the evaluation of the IWSLT En$\to$De is case-sensitive, while that of the LibriSpeech is case-insensitive\footnote{We do the case-insensitive evaluation to be comparable with the other works, however, it is not clear which \BLEU script they used \cite{chung_2019_unsupervised_st, berard_2018_librispeech}.}.
The translation models are evaluated using the official scripts of WMT campaign, i.e. \BLEU~\cite{papineni_02_bleu} computed by \texttt{mteval-v13a}\footnote{ftp://jaguar.ncsl.nist.gov/mt/resources/mteval-v13a.pl} and \TER~\cite{snover_06_ter} computed by \texttt{tercom}\footnote{http://www.cs.umd.edu/~snover/tercom/}. \WER~is computed by \texttt{sclite}\footnote{http://www1.icsi.berkeley.edu/Speech/docs/sctk-1.2/sclite.htm}.
\begin{table}
\begin{center}
\caption{ASR results measured in \WER~[\%].}
\label{tab:asr_results}
\begin{tabular}{lrr}
\hline
\multirow{2}{*}{\bfseries Task} & \multicolumn{2}{c}{\bfseries \WER[$\downarrow$]} \\
& \bfseries dev & \bfseries test \\ \hline
IWSLT En$\to$De & 12.36 & 13.80 \\
LibriSpeech En$\to$Fr & \phantom{0} 6.47 & \phantom{0} 6.47 \\ \hline
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Models}
\label{sec:models}
In our experiments, we build an ASR, an MT, and different ST models. The ASR and MT models are used for building the cascade pipeline as well as pre-training the ST networks. All models are based on the attention model described in Section \ref{sec:network architecture} with one or two encoder(s) or decoder(s) according to the architecture.
For both tasks, byte pair encoding (BPE) \cite{sennrich_16_bpe} with $20$k merge operations on the MT data (both source and target side), and $10$k symbols on the ASR transcriptions is used.
We map all tokens into embedding vectors of size 620. Both speech and text encoders are built with 6 stacked BLSTM layers equipped with 1024 hidden size.
Similar to \cite{zeyer_2018_att_asr}, we apply layer-wise pre-training, where we start with two encoder layers.
As explained before, for speech encoder, we utilize max-pooling between BLSTM layers to reduce the audio sequence length.
The decoders are a 1-layer unidirectional LSTM of size 1024.
Our attention component is composed of a single head additive attention with alignment feedback \cite{tu2016ACL,bahar_2017_rwth}.
To enable pre-training, for ST models, we use the same architecture to the ASR encoder and the same architecture to the MT decoder.
We train models using Adam update rule \cite{kingma_14_adam} with a learning rate between 0.0008 to 0.0001. We apply dropout of 0.3 \cite{srivastavad_14_dropout}, and label smoothing \cite{pereyra_2017_label_smoothing} with a ratio of 0.1.
We lower the learning rate with a decay factor of 0.9 and wait for 6 consecutive checkpoints. Maximum sequence length is set to 75 sub-words. All batch sizes are specified to be as big as possible to fit in a single GPU memory.
A beam size of 12 is used during the search.
The models are built using our in-house implementation of end-to-end models in
\texttt{RETURNN} \cite{zeyer_18_returnn}
that relies on \texttt{TensorFlow} \cite{tensorflow}.
The code and the configurations of the setups are available online~\footnote{https://github.com/rwth-i6/returnn}~\footnote{https://github.com/rwth-i6/returnn-experiments}.
\begin{table}
\begin{center}
\caption{MT results measured in \BLEU~[\%] and \TER~[\%] using ground truth source text.}
\label{tab:mt_results}
\begin{tabular}{lllll}
\hline
\multirow{2}{*}{\bfseries Task} & \multicolumn{2}{c}{\bfseries \BLEU[$\uparrow$]} & \multicolumn{2}{c}{\bfseries \TER[$\downarrow$]} \\
& \bfseries dev & \bfseries test & \bfseries dev & \bfseries test\\ \hline
IWSLT En$\to$De & 30.50 & 31.50 & 50.57 & - \\
LibriSpeech En$\to$Fr & 20.11 & 18.22 & 65.27 & 67.71 \\ \hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{ST results using cascaded pipeline of end-to-end ASR and MT measured in \BLEU~[\%] and \TER~[\%].}
\label{tab:st_results}
\begin{tabular}{lllll}
\hline
\multirow{2}{*}{\bfseries Task} & \multicolumn{2}{c}{\bfseries \BLEU[$\uparrow$]} & \multicolumn{2}{c}{\bfseries \TER[$\downarrow$]} \\
& \bfseries dev & \bfseries test & \bfseries dev & \bfseries test\\ \hline
IWSLT En$\to$De & 24.67 & 24.43 & 58.86 & 62.52 \\
LibriSpeech En$\to$Fr & 17.31 & 15.74 & 69.08 & 70.59 \\ \hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table*}
\begin{center}
\caption{ST results for different architectures measured in \BLEU~[\%] and \TER~[\%].}
\scalebox{0.9}{%
\label{tab:archs}
\begin{tabular}{lllllllll}
\toprule
\multirow{2}{*}{\bfseries Method} & \multicolumn{4}{c}{\bfseries En$\to$De} & \multicolumn{4}{c}{\bfseries En$\to$Fr} \\
& \multicolumn{2}{c}{\bfseries dev} & \multicolumn{2}{c}{\bfseries test} & \multicolumn{2}{c}{\bfseries dev} & \multicolumn{2}{c}{\bfseries test}\\
& \bfseries \BLEU & \bfseries \TER & \bfseries \BLEU & \bfseries \TER & \bfseries \BLEU & \bfseries \TER & \bfseries \BLEU & \bfseries \TER \\
\midrule
direct & 14.80 & 69.81 & 14.86 & 72.49 & 15.71 & 75.86 & 14.69 & 76.53 \\
\quad + CTC & 17.86 & 66.32 & 16.50 & 70.40 & \textbf{16.41} & \textbf{74.17} & \textbf{15.11} & \textbf{75.76} \\
\midrule
one-to-many & 17.25 & 66.98 & 16.29 & 71.05 & 15.31 & 74.79 & 14.28 & 76.84 \\
\quad + CTC & 17.77 & \textbf{66.15} & 16.94 & 70.04 & 16.01 & 73.93 & 14.46 & 76.25\\
\midrule
many-to-one & \textbf{18.17} & 67.23 & \textbf{18.06} & \textbf{69.83} & 11.88 & 79.69 & 11.45 & 81.69 \\
\quad + CTC & 17.54 & 65.47 & 16.96 & 71.20 & 11.76 & 79.51 & 11.52 & 80.95 \\ \midrule
tied cascaded & 15.36 & 69.91 & 14.85 & 73.21 & 13.28 & 77.08 & 12.46 & 79.14 \\
\quad + CTC & 16.60 & 68.73 & 15.68 & 72.44 & 13.75 & 76.88 & 12.83 & 79.15 \\ \midrule
tied triangle & 14.24 & 70.83 & 13.83 & 74.30 & 12.76 & 78.97 & 12.09 & 79.43 \\
\quad + CTC & 15.94 & 68.50 & 14.96 & 71.83 & 13.39 & 77.24 & 12.68 & 78.87 \\
\bottomrule
\end{tabular}
}
\end{center}
\end{table*}
\section{Results}
Table \ref{tab:asr_results} and \ref{tab:mt_results} show the ASR and MT results on IWSLT and LibriSpeech tasks respectively. On the test sets, we gain 13.80\% and 6.47\% \WER.
We obtain 31.50\% \BLEU on the IWSLT and respectively 18.22\% \BLEU on the LibriSpeech by pure MT.
Table \ref{tab:st_results} also shows our baseline that is the traditional cascade pipeline where the output of our ASR model, a sequence of tokens, is fed as the input to our MT system. We achieve 24.43\% \BLEU and 62.52\% \TER on the IWSLT test set and 15.74\% \BLEU and 70.59\% \TER on the LibriSpeech. As expected, the ST systems are behind the pure MT models trained using ground truth source text (cf. \ref{tab:mt_results} and \ref{tab:st_results}).
\subsection{Network Architectures}
The results of five different end-to-end ST architectures are listed in Table \ref{tab:archs}.
On the IWSLT task, for one-to-many and many-to-one approaches we utilize additional ASR and MT data for the respective auxiliary tasks. Therefore, we observe improvements in the performance of both compared to the direct model. The one-to-many method improves by an average of 1.98\% \BLEU and 2.13\% \TER and the many-to-one approach by an average of 3.29\% \BLEU and 2.61\% \TER. \\
On LibriSpeech, we only work with the ST data and notice no real improvement over the direct model. It supports the intuition that the gain in the multi-task training relies on more data rather than better learning, however, it does not hurt. \\
Both the tied cascade and triangle models perform worse than the direct models. We also notice that the cascade models generally seem to perform slightly better than the triangle models (see Table \ref{tab:archs}).
Our results show that the auxiliary data in different multi-task learning is beneficial only up to some degree and that the multi-task models are significantly dependent on the end-to-end data. Our justification is that the multi-task training that utilizes auxiliary models on additional data, discards many of the additionally learned parameters, hence it does not lead to better results than the direct model at the end.
We also think a stack of 2 decoders in the tied methods makes the optimization harder. That's why their performance is behind the other architectures.
\subsection{CTC Loss}
Similar to \cite{zeyer_2018_att_asr}, we use CTC as an additional loss mainly to help the initial convergence, also as a mean for regularization.
The results are listed in Table \ref{tab:archs} for each of the discussed architectures. Interestingly, due to better optimization, in almost all cases, the CTC loss helps the performance in terms of both \BLEU and \TER. As shown, the largest improvement is on the direct model with 1.64\% in \BLEU and 2.09\% in \TER on the test set and 3.06\% in \BLEU and 3.50\% in \TER on the dev set for the IWSLT task. The same trend can be seen on the LibriSpeech with a smaller impact. We believe that, the easier the optimization is, the more beneficial the CTC loss is. That's why it helps the direct model more than the other architectures. \\
It is important to highlight that in principle, the CTC loss is similar to adding a second decoder to predict the transcription equipped with an empty token. On the many-to-one model, CTC loss leads to degradation in performance. Our interpretation is that since we have both speech and text encoders and we only equip the speech encoder with the CTC, it might hurt the learning phase of the text encoder. Because the parameters are updated independently for each encoder.
\begin{table}[t]
\begin{center}
\caption{Different pre-training scheme for En$\to$De corresponding to the blue blocks of Fig. \ref{fig:arch}. \lq\lq-\rq\rq: failed results. $^1$: similar to \cite{bansal_2019_pretraining_asr}, we pre-train German ST/MT decoder using a pre-trained English ASR decoder.}
\scalebox{0.72}{%
\label{tab:pre-training}
\begin{tabular}{lllll}
\toprule
\multirow{2}{*}{\bfseries Method} & \multicolumn{2}{c}{\bfseries dev} & \multicolumn{2}{c}{\bfseries test} \\
& \bfseries \BLEU & \bfseries \TER & \bfseries \BLEU & \bfseries \TER \\ \hline
\bfseries direct \\
ASR enc. & 20.15 & 64.29 & 19.41 & 67.52 \\
MT dec. & - & - & - & - \\
ASR enc.+MT dec. (w ASR dec)$^1$ & 19.92 & 62.48 & 19.56 & 66.68 \\
ASR enc.+MT dec. & -& - & - & - \\
\quad+ adopter & \textbf{21.07} & 62.14 & \textbf{20.74} & \textbf{65.45} \\
\midrule
\bfseries one-to-many \\
MT dec. & - & - & - & - \\
ASR enc.+ASR dec & 18.89 & 64.39 & 17.68 & 68.73 \\
ASR enc.+ASR dec.+MT dec. (w ASR dec)$^1$ & 18.69 & 64.31 & 17.27 & 68.75 \\
ASR enc.+ASR dec.+MT dec. & 19.39 & 63.92 & 18.07 & 68.35 \\
\quad + adopter & \textbf{21.07} & \textbf{61.80} & 19.17 & 67.86 \\
\midrule
\bfseries many-to-one \\
ASR enc. & 17.84 & 65.95 & 16.75 & 70.65 \\
MT dec. & - & - & - & - \\
ASR enc.+MT enc.+MT dec. (w ASR dec)$^1$ & 18.95 & 64.78 & 18.23 & 68.93 \\
ASR enc.+MT enc.+MT dec. & 18.13 & 65.78 & 17.10 & 70.33 \\
\quad+ adopter & \textbf{21.17} & 62.12 & 19.77 & 66.79 \\
\midrule
\bfseries tied cascade \\
MT dec. & - & - & - & - \\
ASR enc.+ASR dec. & 16.44 & 73.13 & 15.45 & 78.03 \\
ASR enc.+ASR dec.+MT dec. & 14.98 & 75.55 & 14.56 & 79.73 \\
\quad+ adopter & 15.91 & 73.50 & 14.19 & 83.62 \\
\midrule
\bfseries tied triangle \\
MT dec. & - & - & - & - \\
ASR enc.+ASR dec. & 16.67 & 67.96 & 15.59 & 71.88 \\
ASR enc.+ASR dec.+MT dec. & 16.49 & 67.14 & 15.37 & 71.65 \\
\quad+ adopter & 18.52 & 64.38 & 18.20 & 67.98 \\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
\subsection{Pre-training}
We have also done several experiments to figure out the optimal way of utilizing pre-training for the end-to-end ST models. In theory, pre-training should not hurt but in practice, it often does because the pre-trained components are not incorporated properly together. In this case, it might lead to an optimization problem in which the training is not robust to new training data variations. We carry out different schemes in various architectures. The results are shown in Table \ref{tab:pre-training}. The first column of the table corresponds to blue blocks in Figure \ref{fig:arch}. For instance, \texttt{ASR enc+MT dec} refers to the case in which we initialize the network's components using a pre-trained ASR encoder and MT decoder.
In the first attempt,
we have tried to only initialize the speech encoder using a pre-trained ASR encoder. As it can be seen in the table, it gives a boost in terms of both \BLEU and \TER for the direct but not for the many-to-one model where it leads to worse performance.
Our explanation is that, for multi-task models, we have an extra component that has to be trained jointly and it makes the optimization task harder, even with better initialization using pre-trained models. We should note that we have not tried using \texttt{ASR enc} pre-training for the other models since we can initialize both the encoder and the first decoder using an entire ASR model, \texttt{ASR enc+ASR dec} (discussed later).
Then, we take an MT decoder to initialize the text decoder.
We observe that pre-training the text decoder seems to harm the models in all cases such that no reasonable performance is obtained (shown by \lq\lq-\rq\rq). Our observations indicate that the pre-trained MT decoder expects to be jointly trained with a text encoder, not a speech encoder, as it has been trained with a text encoder beforehand.
Therefore, one should integrate these two components in a better way.
This huge degradation has led to further investigations where we study why pre-training of text decoder using an MT model hurts.
To explain this justification, we first try pre-training both the encoder and the decoder using our ASR model as suggested in \cite{bansal_2019_pretraining_asr}. Since the ASR decoder is already familiar with the ASR encoder, this problem should be disappeared. As shown for the direct model, it performs comparable with the case where we only used ASR encoder. We also apply the method for the other multi-task models with ASR decoders. As listed in Table \ref{tab:pre-training}, in almost all cases, it greatly outperforms the non-pre-trained baseline model.
In the next step, we pre-train both the speech encoder and the text decoder(s) at the same time. Again as it has been shown, for the direct model, coupling the pre-trained ASR encoder and MT decoder fails training, however, in the other architectures, this setup outperforms the baseline. We have also observed that if we pre-train all layers except the attention component, \texttt{ASR enc+ASR MT} works well for the direct model.
Based on these observations we add an additional layer (one BLSTM) as an \texttt{adapter} component to familiarize
the input of the pre-trained decoder with the output of
the pre-trained encoder.
We insert it on the top of the component where we want to be coupled smoothly. It means, for direct, one-to-many and many-to-one methods, we add it on the top of the pre-trained speech encoder. For the tied cascade and triangle, we add it on the top of the ASR pre-trained decoder.
We train the adaptor component jointly without freezing the parameters.
This helps the fine-tuning stage on the ST data by 6.08\% in \BLEU and 7.36\% in \TER on average for the direct model at most. In general, we observe better results for all architectures except for the tied cascade system.
Interestingly the adopter approach works better for the triangle architecture compared to the tied cascade. Possibly here the decoder can utilize the adopter layers better to focus more and connect to the output of the ASR encoder rather than the decoder, mimics similar to the direct approach and provides better results.
\begin{table}
\begin{center}
\caption{Comparison of test sets with the literature. $^1$: the evaluation is without punctuation. To be comparable with other works, we note that the LibriSpeech results in this table are case-insensitive \BLEU computed using \texttt{multi-bleu.pl} script \cite{koehn_07_moses}.}
\scalebox{0.90}{%
\label{tab:others}
\begin{tabular}{lcc}
\toprule
\multirow{2}{*}{\bfseries Method} & \multicolumn{1}{c}{\bfseries En$\to$De} & \multicolumn{1}{c}{\bfseries En$\to$Fr} \\
&\bfseries test & \bfseries test \\
\midrule
\bfseries other works \\
\quad direct \cite{liu_2018_iwslt} & 20.07 & - \\
\quad cascade pipeline \cite{liu_2018_iwslt} & 25.99 & - \\
\quad direct \cite{berard_2018_librispeech} & - & 13.30\\
\quad multi-task \cite{berard_2018_librispeech} & - & 13.40\\
\quad unsupervised$^1$ \cite{chung_2019_unsupervised_st} & - &12.20 \\
\quad transformer \cite{gangi-etal-2019-enhancing} & - & 13.80 \\
\quad transformer+pretraining \cite{st_kd_interspeech2019} & - & 14.30 \\
\quad\quad + knowledge distillation \cite{st_kd_interspeech2019} & - & 17.02 \\
\toprule
\bfseries this work \\
\quad direct+pretraining+adaptor & 20.74& 16.80 \\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
Finally, we also compare our models with the literature in Table \ref{tab:others}. We use our best setup which is a direct model using an adapter on top of pre-training for the comparison in this table. On the IWSLT test set, our model outperforms the winner of the IWSLT2018 evaluation campaign by 0.67\% in \BLEU. On the LibriSpeech test set, our model outperforms both the LSTM-based and the transformer models and slightly behind the knowledge distillation method.
\section{Conclusion}
\label{sec:conclusion}
In this work, we have studied the performance of various architectures for end-to-end speech translation, where a moderate amount of speech translation data, as well as weakly supervised data, i.e. ASR or MT pairs, are available.
We have demonstrated the effect of pre-training of the network's components and explored how to efficiently couple the encoder and the decoder by adding an additional layer in between. This extra layer allows for better joint learning and gives performance boosts.
Moreover, CTC loss can be an important factor for the end-to-end ST training since it leads to better performance as well as faster convergence.
\section{Acknowledgements}
\label{sec:acknowledgements}
\begin{wrapfigure}{l}{0.15\textwidth}
\vspace{-4mm}
\begin{center}
\includegraphics[width=0.17\textwidth]{figures/euerc.jpg} \\
\vspace{1mm}
\includegraphics[width=0.17\textwidth]{figures/dfg_logo_blau.jpg}
\end{center}
\vspace{-4mm}
\end{wrapfigure}
This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 694537, project "SEQCLAS"), the Deutsche Forschungsgemeinschaft (DFG; grant agreement NE 572/8-1, project "CoreTec") and from a Google Focused Award. The work reflects only the authors' views and none of the funding parties is responsible for any use that may be made of the information it contains.
\bibliographystyle{IEEEbib}
\let\OLDthebibliography\thebibliography
\renewcommand\thebibliography[1]{
\OLDthebibliography{#1}
\setlength{\parskip}{-2pt}
\setlength{\itemsep}{0pt plus 0.07ex}
}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,369 |
package cn.limw.summer.time;
import java.text.DateFormat;
import java.text.ParsePosition;
import java.util.Date;
import cn.limw.summer.java.text.exception.ParseRuntimeException;
/**
* @author li
*/
public class DateFormatUtil {
/**
* @see java.text.DateFormat#parse(String)
*/
public static Date parse(DateFormat dateFormat, String source) {
ParsePosition pos = new ParsePosition(0);
Date result = dateFormat.parse(source, pos);
if (pos.getIndex() == 0) {
throw new ParseRuntimeException("Unparseable date: \"" + source + "\"", pos.getErrorIndex());
}
return result;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,453 |
Wigmore Mondays – Florian Boesch & Malcolm Martineau in Schumann & Wolf
Florian Boesch (baritone), Malcolm Martineau (piano)
Wigmore Hall, London, 4 July 2016
http://www.bbc.co.uk/programmes/b07j3vm5
Available until 3 August
Schumann Die beiden Grenadiere Op.49/1; Abends am Strand Op.45/3; Die feindlichen Brüder Op.49/2
Märzveilchen Op.40/1; Muttertraum Op.40/2; Der Soldat Op.40/3; Der Spielmann Op.40/4 (all 1840) (20 minutes)
Wolf Goethe Lieder: Der Schäfer; Phänomen; Wandrers Nachtlied; Anakreons Grab; Harfenspieler I – III (18 minutes)
Schumann Belsatzar Op.57 (1840) (5 minutes)
Florian Boesch has recorded a disc of Schumann but only one of the songs in this concert (Belsatzar). Here is a playlist containing all of the songs, using recordings made by the great baritone Dietrich Fischer-Dieskau:
1840 was an extraordinary year for Robert Schumann's musical productivity. His so-called 'year of song', it saw him write 138 songs in total – including the eight in this recital program. Among the choice are four settings of Heine, which certainly preyed on the composer's dark side.
In a similar vein, the year 1888 was a hugely productive one for the song composer Hugo Wolf. The tenor Ian Bostridge wrote this very fine introduction to the songs of Wolf for the Guardian in 2006. He wrote a whole songbook setting some of Goethe's poetry, collected in 22 songs through 1888 and 1889, in the composer's late twenties. The seven we hear are illustrations of the composer's ability to combine melodic originality and a piano part that helps set the words in context, including the three songs of the downtrodden harpist.
Florian Boesch's baritone is an extraordinary instrument, and it is perfectly suited to the darker recesses of these Schumann settings, especially the Heine songs. Here is some of the composer's most descriptive vocal music, and it is incredibly effective in this performance, not just for Boesch's insights but for Malcolm Martineau's ever colourful piano pictures. Here the colours are predominantly grey and black, but the steely edge to his lower register tone is crucial to the impact of the text and makes the moments of lighter relief – for there are a few! – ever more telling.
Similar forces are at work in the music of Wolf, which Boesch brings to thoughtful life. He is particularly effective in the slower songs such as Wandrers Nachtlied, where he and Martineau exhibit wonderful control of the drawn out phrases.
1:48 Die beiden Grenadiere (The two grenadiers) text
The piano's terse introduction is quickly picked up by the baritone, who sings of the battle in dark tones. At 4:18 the song breaks into the melody of La Marseillaise, as the French grenadier expresses his wish to be buried on home soil should he die.
5:13 Abends am Strand text
A chilling song.
8:47 Die feindlichen Brüder (The hostile brothers) text
The singer and piano are closely aligned here. Initially the mood is a brooding one in preparation for the brothers' fight, but then hostilities break out and the tempo quickens considerably, the piano stooping ever lower, well below the range of the singer.
11:20 Märzveilchen Op.40/1 (March violets) text
The mood lightens a little for Schumann's celebration of the flowers, described by the poet as 'a pair of laughing blue eyes'.
12:54 Muttertraum Op.40/2 (A Mother's Dream) (Adelbert von Chamisso) text
The piano part is characteristically intimate for this soft reverie – but the peace does not last long, for there is a dark side in the form of a raven outside the window (from 14:10) at which point the singer's tone gets progressively darker, to the depths of the end.
15:26 Der Soldat Op.40/3 (The Soldier) (Adelbert von Chamisso) text
There is a military air from the start of the piano introduction, with fanfares and ceremony, but again the mood is steely dark, right through to the drama of the bullets fired in the last verse, where the poet 'shot him through the heart'.
18:20 Der Spielmann Op.40/4 (The Fiddler) (Adelbert von Chamisso) text
There are bright festivities at the start of this song, but again it is not long until darker thoughts emerge, the baritone sinking lower in his range as he sings of the bride of the story, who 'looks like whitewashed death'.
23:29 Der Schäfer (The Shephard) text
A darkly humourous song about a lazy shepherd, set by Wolf with some far-reaching harmonies and lazily decorated piano lines.
24:57 Phänomen (Phenomenon) text
A slow song, offering consolation at its end.
26:52 Wandrers Nachtlied (Wanderer's Night Song) text
A slow and deeply sorrowful song, with long, drawn-out phrases – completed by Martineau's soft postlude, lost in thought.
30:06 Anakreons Grab (Anakreon's Grave) text
The contemplation at Anacreon's Grave is not as sorrowful as one might think, 'beautifully graced with verdant life' in Goethe's words. The song speaks of rest rather than torment.
32:45 Harfenspieler I text
Not surprisingly the piano imitates the harp beautifully at the start, though the vocal line that follows is quite stern, the singer imploring 'leave me to my torment'!
36:25 Harfenspieler II text
Another predominantly slow setting, portraying a wretched man with dark tone in the singer's voice and a reserved piano part.
38:49 Harfenspieler III text
The most dramatic of the three Harfenspieler settings, a tormented singer, in ringing tones, lamenting how the heavenly powers 'let the wretched man feel guilt'.
43:26 Belsatzar (Belshazzar) text
This extraordinary song runs through a whole gamut of moods and emotions. It begins with the Babylonian king singing with great bravado, his boasting and the piano's tumbling figures adding to the sense of giddiness. At 45:24 he proclaims, 'I am the king of Babylon!' After this the song turns, the king fearful, until the famous writing on the wall passage, which sends a chill through the spine from 46:17. There is no coming back from here for the king, murdered by the end.
49:45 Described as 'Twitter of the nineteenth century' by Florian Boesch, this is Schumann's Verratene Liebe Op 40/5 – another von Chamisso text – and it's over in 45 seconds!
Florian Boesch is a remarkable talent – and has forged a formidable partnership with Malcolm Martineau. Here they are in a complete album of Schumann, including the first of the composer's Liederkreis cycles:
This entry was posted in In concert, Small scale and tagged BBC Radio 3, Florian Boesch, Hugo Wolf, Malcolm Martineau, Robert Schumann, Wigmore Hall by arcanafm. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,902 |
\section{Introduction}
Federated learning (FL)~\cite{mcmahan2017communication,li2020federated, 8889996} has grown rapidly in recent years thanks to the popularity of hand-held devices such as mobile phones and tablets~\cite{mothukuri2021survey,rieke2020future,sheller2020federated}. Since the training data are usually distributed on clients and are mostly private, FL needs to establish a network of clients connected to a server and train a global model in a privacy-preserving manner. One of the main challenge of FL is the statistical diversity, i.e., the data distributions among clients are distinct (i.e., non-i.i.d.). Recently, personalized FL~\cite{fallah2020personalized,t2020personalized} is proposed to address this problem, which motivates us to develop the approach to achieve better personalization. Another challenge is that the server and clients need multiple rounds of communication to guarantee the convergence performance. However, the communication latency from clients to the cloud server is high and the communication bandwidth is limited. Hence, a communication-efficient approach has to be proposed~\cite{8889996, 8945292, 9305988}.
Many federated learning algorithms~\cite{mcmahan2017communication,t2020personalized,li2018federated} need to equalize the dimensions of the client models and the global model. However, the global model generally requires more parameters than the client model to cover the main features of all clients, which means there might be too many parameters in client models. As a result, it is possible to prune many parameters from client networks without affecting performance. In this way, parameter traffic between the client models and the global model can be directly reduced, and the required storage and computing amount are reduced~\cite{he2019bag}, which is crucial for applications running on low-capacity hand-held devices. Unfortunately, traditional sparse optimizers~\cite{srinivas2017training,han2015learning,gale2019state,li2021structured} cannot obtain personalized models in client networks, which results in poor performance and convergence rate on statistical diversity data~\cite{t2020personalized}.
Inspired by broad applications of personalized models in healthcare, finance and AI services~\cite{deng2020adaptive}, this paper proposes a sparse personalized federated learning scheme via maximizing correlation ({\texttt{FedMac}}). By incorporating an approximated $\ell_1$-norm and the correlation (inner product) of global model and local models into the loss function, the proposed method achieves better personalization ability and higher communication efficiency. This reason is as follows: the introduction of correlation makes each client leverage global model to optimize its personalized model w.r.t. its own data and so improves the performance on statistical diversity data; the incorporation of approximated $\ell_1$-norm makes the models become sparse and so the amount of communication is greatly reduced.
\subsection{Main Contributions}
In this paper, we propose a novel sparse personalized FL method based on maximizing correlation, and further formulate an optimization problem designed for {\texttt{FedMac}} by using $\ell_1$-norms and the correlation between the global model and client models in a regularized loss function. The $\ell_1$-norm constraints can generate sparse models, and hence the communication loads between the server and the clients are greatly reduced, i.e., there are many zero weights in models and only the non-zero weights need to be uploaded and downloaded. Maximizing the correlation between the global model and client models encourages clients to pursue their own models with different directions, but not to stay far away from the global model.
In addition, we propose an approximate algorithm to solve {\texttt{FedMac}}, and present convergence analysis for the proposed algorithm, which indicates that the sparse constraints do not affect the convergence rate of the global model. Furthermore, we theoretically prove that the performance of {\texttt{FedMac}} under sparse conditions is better than that of the personalization methods based on $\ell_2$-norm~\cite{t2020personalized,li2018federated,mcmahan2017communication}. The training data required for the proposed {\texttt{FedMac}} is significantly less than that required for the methods based on $\ell_2$-norm.
Finally, we evaluate the performance of {\texttt{FedMac}} using real datasets that capture the statistical diversity of clients' data. Experimental results show that {\texttt{FedMac}} is superior to various advanced personalization algorithms, and our algorithm can make the network sparse to reduce the amount of communication. Experimental results based on different deep neural networks show that our {\texttt{FedMac}} achieves the highest accuracy on both the personalized model (98.95\%, 99.37\%, 90.90\% and 89.06\%) and the global model (96.90\%, 85.80\%, 86.57\% and 84.89\%) on the MNIST, FMNIST, CIFAR-100 and Synthetic datasets, respectively.
\subsection{Related Works}
\subsubsection{Personalized FL}
FedAvg~\cite{mcmahan2017communication} is known as the first FL algorithm to build a global model from a subset of clients with decentralized data, where the client models are updated by the local SGD method. However, it cannot handle the problem of non-i.i.d. data distribution well~\cite{zhu2021federated}. To address this problem, various personalized FL methods have been proposed, which can be divided into different categories mainly including local fine-tuning based methods~\cite{fallah2020personalized, huang2021personalized, li2018federated, t2020personalized}, personalization layer based methods~\cite{arivazhagan2019federated}, multi-task learning based methods~\cite{smith2017federated}, knowledge distillation based methods~\cite{hitaj2017deep} and lifelong learning based methods~\cite{liu2019lifelong}. The methods based on the personalization layer require the client to permanently store its personalization layer without releasing it. And the methods based on multi-task learning, knowledge distillation, and lifelong learning introduce other complex mechanisms to achieve personalization in FL, which may encounter difficulties in computation, deployment, and generalization. Therefore, in this paper, we pay more attention to personalized methods based on local fine-tuning.
\subsubsection{Local fine-tuning based personalized FL}
Local fine-tuning is the most classic and powerful personalization method~\cite{zhu2021federated}. It aims to fine-tune the local model through meta-learning, regularization, interpolation and other techniques to combine local and global information. Per-FedAvg~\cite{fallah2020personalized} sets up an initial meta-model that can be updated effectively after one more gradient descent step.
By facilitating pairwise collaborations between clients with similar data, FedAMP~\cite{huang2021personalized} uses federated attentive message passing to promote similar clients to collaborate more.
$\ell_2$-norm regularization based methods such as Fedprox~\cite{li2018federated} and pFedMe~\cite{t2020personalized} also achieve good personalization. However, none of these works are specifically designed for the communication-efficient and computation-friendly FL framework, which prompts us to propose \texttt{FedMac}, a sparse personalized FL method.
\begin{figure*}[htbp]
\centering
\subfloat{
\includegraphics[height=1.3in]{figure1-1.pdf}
}
\subfloat{
\includegraphics[height=1.3in]{figure1-2.pdf}
}
\subfloat{
\includegraphics[height=1.3in]{figure1-3.pdf}
}
\caption{The changes of normal cone and tangent cone after different shifts. \textbf{Left:} Maximizing correlation constraint $\bar f( \bm \theta, {\bm w}) = - \langle \bm \theta , {\bm w} \rangle$; \textbf{Middle:} $\ell_2$-norm constraint $\bar f( \bm \theta, {\bm w}) = \frac{1}{2} \| \bm \theta - {\bm w} \|_2^2$; \textbf{Right:} The tangency occurs iff ${\mathcal{E}}_{\bm \theta}$ is disjoint from the spherical part of the tangent cone at point $\bm \theta$. We draw the standard tangent cone and normal cone in blue, while draw the shifted tangent cones and normal cones in red and green for $\bar f( \bm \theta, {\bm w}) = - \langle \bm \theta , {\bm w} \rangle$ and $\bar f( \bm \theta, {\bm w}) = \frac{1}{2} \| \bm \theta - {\bm w} \|_2^2$, respectively. Exact sparse recovery happens when the random subspace ${\mathcal{E}}_{\bm \theta}$ is tangent to the tangent cone. Good personalization constraints narrow the tangent cone, thereby reducing the amount of training data required.}
\label{fig_superisorty}
\end{figure*}
\section{Towards Sparse Personalized FL}
\subsection{Conventional FL Methods}
Suppose there are $N$ clients communicating with a server, a federated learning system aims to find a {\em global model} $\bm w \in \mathbb{R}^{d}$ by minimizing
\begin{align}
\label{conv-FL}
\min _{\bm w \in \mathbb{R}^{d}}\left\{\ell(\bm w)\triangleq \frac{1}{N} \sum_{i=1}^{N} \ell_{i}(\bm w)\right\},
\end{align}
where $\ell_{i}(\cdot): \mathbb{R}^{d} \rightarrow \mathbb{R},~ i=1, \ldots, N$ is the expected loss over the data distribution of the $i$-th client. Let $Z_{i}=\left\{(X_i,Y_i)\right\}$ denote the training data randomly drawn from the distribution of client $i$, where $X_i$ is the input variable while $Y_i$ is the response variable. Note that clients might have non-i.i.d. data distributions, i.e., the distributions of $Z_{i}$ and $Z_j (i \neq j)$ might be distinct, since their data may come from different environments, contexts and applications~\cite{t2020personalized}. Let $h(x;\bm w)$ be the output of an $L$-layer feedforward overparameterized deep neural network, and $\mathcal{L}(h;y)$ be the loss function, then we have
\begin{align}
\ell_{i}(\bm w)\triangleq \mathbb{E}_{{\mathcal Z}_i}\left[ \mathcal{L}_i(h(X_i;\bm w);Y_i) \right],
\end{align}
where $\mathcal{Z}_i$ is the distributions of $Z_{i}$.
\subsection{Towards Better Sparse Personalization}
On the client side, we only consider one client at a time and omit the subscript $i$ for simplicity. To solve the statistical diversity problem and reduce the communication burden, we aim to find the {\em sparse personalized model} $\bm \theta$ by minimizing
\begin{align}
\label{optimal-SP}
\min _{\bm \theta \in \mathbb{R}^{d}} \left\{\ell \left(\bm \theta \right) + \gamma \| \bm \theta \|_1 + \lambda \bar f( \bm \theta, {\bm w}) \right\},
\end{align}
where $\gamma$ and $\lambda$ are weighting factors that respectively control the sparsity level and the degree of personalization, and $\bar f( \bm \theta, {\bm w})$ is a personalization constraint, allowing clients to benefit from the abundant data aggregation in the global model while maintaining a certain degree of personalization. In this subsection, our goal is to find a better personalization constraints under sparse conditions. We think that a good personalization constraint can enable clients to achieve better personalization capabilities, and can also improve the convergence speed of personalized model during the training process. In other words, even when the amount of client data is small, a high-precision personalized model can be trained with reference to the global model.
To do this, we first simplify the problem to explore the potential better personalization constraint. Consider a linear neural network system, e.g., a single-layer network. Suppose that $\bm X \in \mathbb{R}^{N_D \times N_I}$ is the training input with $N_D$ being the number of training data and $N_I = d$ being the length of data, $\bm y \in \mathbb{R}^{N_D \times 1}$ is the training output (label) and $\bm \theta \in \mathbb{R}^{d \times 1} $ is the parameter. Assume that the mean squared error is used as the training loss, then solving {\eqref{optimal-SP}} is equivalent to solving
\begin{align}
\label{TPS-0}
\min _{\bm \theta \in \mathbb{R}^{d} } \norm{\bm y - \bm X \bm \theta}_2^2 + \gamma f(\bm \theta),
\end{align}
where $f(\bm \theta):= \| \bm \theta \|_1 + \zeta \bar f( \bm \theta, {\bm w})$ with $\zeta = {\lambda}/{ \gamma }$. It is known that when $\zeta = 0$, \eqref{TPS-0} is reduced to the standard unconstrained compressed sensing problem under noisy conditions~\cite{donoho2006compressed,eldar2012compressed}, and the size of the tangent cone can be used to analyze the probability of obtaining the optimal sparse solution~\cite{zhang2018recovery,8006522}. We then analyze the change in the size of the tangent cone under different personalization constraints. Note that the normal cone at the optimal solution $\bm \theta^{\star}$ is the polar of its tangent cone, which can be generated by using the sub-differential of the objective function at $\bm \theta^{\star}$. We hence hope to find a personalized constraint that allows the sub-differential to move to a position close to zero. At this time, the normal cone is the largest, while the tangent cone is the smallest, and the probability of being tangent to the solution space is the highest. This motivates us to maximize the correlation between the client model and the global model to obtain personalization capabilities, i.e., $f(\bm \theta) = f_1(\bm \theta) := \| \bm \theta \|_1 - {\zeta_1} \langle \bm \theta , {\bm w} \rangle $, then we have the corresponding normal cone is
\begin{align}
\mathcal{N}_{f_1} = \text{cone}\{ {\partial \| \bm \theta^{\star} \|_1 - \zeta_1 \bm w } \},
\nonumber
\end{align}
which is a shifted version of the standard normal cone (when $\zeta=0$) $\mathcal{N}_{0} = \text{cone}\{ {\partial \| \bm \theta^{\star}\|_1 } \}$, and is close to zero, so it can increase the probability of obtaining the optimal sparse solution (see Figure~\ref{fig_superisorty}).
\begin{remark}
$\mathcal{N}_{f_1}$ is close to zero since it is known that the angle between a network $\bm w$ and its sign ${\rm sign}(\bm w)$ is usually very small after sufficient training~\cite{chao2020directional}, and the converged client model is not too far from the global model, we have ${\partial \| \bm \theta^{\star} \|_1 - \zeta \bm w } \approx 0$ with a suitable $\zeta$. The detailed theoretical analysis will be given in the next section.
\end{remark}
In contrast, when the $\ell_2$-norm distance is used as the personalization constraint~\cite{mcmahan2017communication,t2020personalized,li2018federated}, i.e., $f(\bm \theta) = f_2(\bm \theta) := \| \bm \theta \|_1 + \frac{\zeta_2}{2} \| \bm \theta - {\bm w} \|_2^2$, the corresponding normal cone is given by
\begin{align}
\mathcal{N}_{f_2} &= \text{cone}\{ {\partial (\| \bm \theta^{\star}\|_1 + \frac{\zeta_2}{2} \|\bm \theta^{\star} - \bm w\|_2^2 ) } \} \nonumber \\
&= \text{cone}\{ {\partial \| \bm \theta^{\star} \|_1 + \zeta_2 (\bm \theta^{\star} - \bm w) } \}.
\nonumber
\end{align}
Since the converged client model is not too far from the global model, i.e., $(\bm \theta^{\star} - \bm w)$ is small, the normal cone $\mathcal{N}_{f_2}$ does not move much (see Figure~\ref{fig_superisorty}(b)) and hence the probability of obtaining the optimal sparse solution of \eqref{TPS-0} does not change much. That is, although the $\ell_2$-norm based personalization method can obtain personalization capabilities, it cannot help the client to train a better sparse model with a small amount of data.
\section{FedMac}
\subsection{{\texttt{FedMac}}: Problem Formulation}
Instead of solving the problem in \eqref{conv-FL}, we aim to find a {\em sparse global model} $\bm w$ based on {\em sparse personalized models} $\bm \theta_{i},~i=1,...,N$ via maximizing correlation between $\bm w$ and $\bm \theta_{i}$ by minimizing
\begin{align}
\label{FedMac-1n}
{{\texttt{FedMac:}}} \min _{{\bm w} \in \mathbb{R}^{d}}\left\{F(\bm w)\triangleq \frac{1}{N} \sum_{i=1}^{N} F_{i}(\bm w)\right\},
\end{align}
where
\begin{align}
F_{i}(\bm w) &\triangleq \min _{\bm \theta_{i} \in \mathbb{R}^{d}} \left\{ H_i(\bm\theta_i;\bm w) + \frac{\lambda}{2} \| \bm w \|_2^2 + \gamma_w \| \bm w\|_1 \right\} \nonumber \\
H_i(\bm\theta_i;\bm w) &\triangleq \ell_{i} \left(\bm \theta_{i} \right) + \gamma \| \bm \theta_{i} \|_1 - \lambda\left \langle \bm \theta_{i}, {\bm w}\right \rangle .\nonumber
\end{align}
On the server side, since the client data only serve for the update of $\bm \theta_{i}$ and there is no data on the server to restrict $\bm w$, the global model ${\bm w}$ obtained by maximizing the correlation $\left \langle \bm \theta_{i}, \bm w \right \rangle$ may gradually become larger. We hence add a penalty term $\frac{\lambda}{2} \| {\bm w} \|_2^2 $ to counteract the divergence of the global model.
\subsection{\texttt{FedMac}: Algorithm}
Next we propose the algorithm {\texttt{FedMac}}. To start with, we use a twice continuously differentiable approximation to replace the $\ell_{1}$-norm used in {\texttt{FedMac}}, i.e., replacing $\|\bm x\|_1$ by~\cite{7755794}
\begin{align}
\phi_{\rho}(\boldsymbol{\bm x})
&=\rho \sum_{n=1}^{d} \log \cosh \left(\frac{{x}_{n}}{\rho}\right), \notag
\end{align}
where $x_n$ is the $n$-th element in $\bm x$ and $\rho$ is a weight parameter, which controls the smoothing level. And we have the $n$-th element of $\nabla\phi_{\rho}(\bm x)$ is $ \left[\nabla\phi_{\rho}(\bm x)\right]_n=\tanh \left(x_{n} / \rho\right)$.
Note that we use $\phi_{\rho}({\cdot})$ instead of $\|\cdot\|_1$ to exploit the sparsity in the proposed algorithm because it makes the loss function has continuous differentiable property, which enables us to analyze the convergence of the proposed algorithm and facilitates gradient calculation and back propagation in the network.
Next, we present the following useful assumptions, which are widely used in FL gradient calculation and convergence analysis~\cite{karimireddy2020scaffold, fallah2020personalized, li2019communication, yu2019linear}.
\begin{assumption}\label{assumption_cs} (Strong convexity)
Assume that $\ell_i(\cdot):\mathbb{R}^{d} \rightarrow \mathbb{R} $ is $\mu$-strongly convex on $\mathbb{R}^{d}$, where $\mathbb{R}^{d}$ is endowed with the $\ell_2$-norm.
\end{assumption}
\begin{assumption}\label{assumption_bv} (Bounded variance)
The variance of stochastic gradients (sampling noise) in each client is bounded by
\begin{align}
\mathbb{E}_{Z_i}\left[\left\|\nabla \tilde{\ell}_{i}\left(\bm w ; Z_{i}\right)-\nabla \ell_{i}(\bm w)\right\|^{2}_2\right] \leq \gamma_{\ell}^{2},\notag
\end{align}
where $Z_{i}$ is a training data randomly drawn from the distribution of client $i$.
\end{assumption}
On this basis, we propose the smoothness and strong convexity proportion of $F_{i}(\bm w)$ in Proposition~\ref{proposition_convex_analyse}, which is proved in the appendix.
\begin{proposition}
\label{proposition_convex_analyse}
If $\ell_{i}$ is $\mu$-strongly convex on $\mathbb{R}^{d}$ w.r.t. $\|\cdot\|_2$, then $F_i$ with
\begin{align}
\nabla_{\bm w} F_i = \lambda (\bm w - \hat{\bm \theta}(\bm w)) + \gamma_w \nabla \phi_{\rho}(\bm w)
\end{align}
is $\mu_F$-strongly convex with $\mu_F=\frac{\lambda(\mu-\lambda)}{\mu}$ (with the condition that $\lambda < \mu$) and $L_F$-smooth with $L_F=\lambda + \frac{\lambda^2}{\mu} + \frac{\gamma_w}{\rho}$.
\end{proposition}
Our algorithm updates $\bm \theta_i$ and $\bm w$ by alternately minimizing the two subproblems of {\texttt{FedMac}}. In particular, we first update $\bm \theta_i$ by solving
\begin{align}
\label{A-theta}
\bm{\hat\theta}_{i} = \arg\min _{\bm \theta_{i} \in \mathbb{R}^{d}}\left\{\ell_{i}\left(\bm \theta_{i}\right)
+\gamma\phi_{\rho}(\bm \theta_{i})
-{\lambda}\left\langle \bm \theta_{i}, \bm w \right\rangle \right\} ,
\end{align}
then update the corresponding $\bm w_i$ (named {\em local global model}) by solving
\begin{align}
\label{A-w}
\bm{\hat w}_{i}= \min_{\bm w_i \in \mathbb{R}^{d}}\left\{ - \lambda \langle \bm{\hat \theta}_i , \bm w_i \rangle + \frac{\lambda}{2} \| {\bm w_i} \|_2^2 +\gamma_w\phi_{\rho}(\bm w_{i}) \right\}.
\end{align}
After that, we upload $\bm{\hat w}_i,~i = 1,...,N$ to the server and aggregate them to get the updated $\bm w$.
\begin{algorithm}[!ht]
\label{Algorithm_FedMac-S}
\begin{minipage}[!ht]{\linewidth}
\caption{\small{\texttt{FedMac}}: Sparse Personalized Federated Learning via Maximizing Correlation Algorithm}
\begin{tabular}{l}
{\bf{Server executes:}} \\
\hspace{0.5cm}{Input} $T, R, S, \lambda, \eta, \gamma,\eta_p, \gamma_w, \beta, \rho, \bm w^{0}$ \\
\hspace{0.5cm}\bf{for} $t=0,1,...,T-1$ \bf{do} \\
\hspace{1.0cm}\bf{for} $i=1,2,...,N$ \bf{in parallel do} \\
\hspace{1.5cm}$\bm w_i^{t+1} \leftarrow \text{ClientUpdate}(i,\bm w^t)$ \\
\hspace{1.0cm}$\mathcal{S}^{t} \leftarrow \text{(random set of $S$ clients)}$ \\
\hspace{1.0cm}$\bm w^{t+1}=(1-\beta) \bm w^{t}+\frac{\beta}{S} \sum_{i \in S^t} {\bm w_i^{t+1}}$ \\
\\
$\textbf{ClientUpdate}(i,\bm w^{t})\textbf{:}$ \\
\hspace{0.5cm}$\bm \theta_{i}^{t,0} = \bm w_{i}^{t,0} = \bm w^{t}$ \\
\hspace{0.5cm}{\bf{for}} $r=0,1,...,R-1$ \bf{do} \\
\hspace{1.0cm}$\mathcal{D}_{i} \leftarrow \text{(sample a mini-batch with size $|\mathcal{D}|$ )}$ \\
\hspace{1.0cm}Update $\bm{\tilde\theta}_{i}^{t,r}$ according to \eqref{A-H-min}\\
\hspace{1.0cm}$\bm w_{i}^{t, r+1} = \bm w_{i}^{t,r}-\eta \nabla_{\bm w} F_{i}\left(\bm w_{i}^{t,r}\right)$\\
\hspace{0.5cm}Return $\bm{\bm w}_{i}^{t,R}$ to the server
\end{tabular}
\end{minipage}
\end{algorithm}
Note that \eqref{A-theta} can be easily solved by many first order approaches, for example the stochastic gradient descent~\cite{zinkevich2010parallelized}, based on the gradient
\begin{align}
\nabla_{\bm \theta} F_{i} = \nabla \ell_{i}(\bm \theta_{i}) +\gamma \nabla \phi_\rho(\bm \theta_{i}) - \lambda \bm w \notag
\end{align}
with a learning rate $\eta_p$.
However, calculate the exactly $\nabla \ell_{i}(\bm \theta_{i})$ requires the distribution of $Z_{i}$, we hence use $\nabla \tilde \ell_{i}(\bm\theta_{i},\mathcal{D}_{i}) = \frac{1}{|\mathcal{D}_{i}|}\sum_{Z_{i} \in \mathcal{D}_{i}} \nabla \ell_{i}(\bm\theta_{i},Z_{i})$ instead, i.e., we sample a mini-batch of data $\mathcal{D}_{i}$ to obtain the unbiased estimate of $\nabla \ell_{i}(\bm \theta_{i})$,
such that $\mathbb{E}[ \nabla \tilde \ell_{i}(\bm\theta_{i},\mathcal{D}_{i}) ] = \nabla \ell_{i}(\bm \theta_{i})$. Therefore, we solve the minimization problem
\begin{align}
\label{A-H-min}
\bm{\tilde \theta}_{i}^{t,r}(\bm{ w}_{i}^{t,r}) = \arg\min _{\boldsymbol{\theta}_{i} \in \mathbb{R}^{d}} \tilde H(\bm \theta_{i};\bm{ w}_{i}^{t,r},\mathcal{D}_{i})
\end{align}
instead of solving \eqref{A-theta} to obtain an approximated personalized client model, where $\bm{ w}_{i}^{t,r}$ is the current local global model w.r.t. the $i$-th client, $t$-th global round, and $r$-th client round, $\bm{\tilde \theta}_{i}^{t,r}$ is the corresponding estimated client model and
\begin{align}
\tilde H(\bm \theta_{i};\bm{w}_{i}^{t,r},\mathcal{D}_{i}) = \tilde \ell_{i}\left(\boldsymbol{\theta}_{i},\mathcal{D}_{i} \right) +\gamma \phi_{\rho}(\bm \theta_{i}) - \lambda \left\langle \bm \theta_{i}, \bm{w}_{i}^{t,r} \right\rangle . \notag
\end{align}
Similarly, \eqref{A-H-min} can be solved by the stochastic gradient descent. We let the iteration go until the condition $\| \nabla \tilde H(\bm \theta_{i};\bm{w}_{i}^{t,r},\mathcal{D}_{i}) \|^2_2 \leq \nu$ is reached, where $\nu$ is an accuracy level.
Once the client model is updated, the corresponding local global model is updated by stochastic gradient descent as follows
\begin{align}
\bm w_{i}^{t, r+1} = \bm w_{i}^{t,r}-\eta \nabla_{\bm w} F_{i}\left(\bm w_{i}^{t,r}\right) \notag,
\end{align}
where $\eta$ is a learning rate. Finally, we summarize our algorithm in Algorithm~1. Similar to~\cite{t2020personalized,karimireddy2020scaffold}, an additional parameter $\beta$ is used for global model update to improve convergence performance, and we average the global model over a subset of clients $\mathcal{S}^t$ with size $S$ to reduce the occupation of bandwidth.
\section{Theoretical Analysis}
\subsection{Convergence Analysis}
For unique solution $\bm w^{\star}$ to {\texttt{FedMac}}, which always exists for strongly convex $F_i$, we have the following important Lemmas and Theorem~\ref{theorem_convergence_FedMac}, which are proved in Appendix.
\begin{lemma}[Bounded diversity of $\bm \theta_i$ w.r.t. mini-batch sampling]
\label{lemma_theta-theta}
Let $\tilde{\bm \theta}_i (\bm w_i^{t,r})$ be a solution to $\| \nabla \tilde{H}_i ( \tilde{\bm \theta}_i; \bm w_i^{t,r}, \mathcal{D}_i ) \|_2^2 \leq \nu $, if Assumptions~\ref{assumption_cs} and \ref{assumption_bv} hold, we have
$$ \mathbb{E}\left[ \| \tilde{\bm \theta}_i(\bm w_i^{t,r}) - \hat{\bm \theta}_i(\bm w_i^{t,r}) \|_2^2 \right] \leq \delta^2 = \frac{2}{\mu^2} \left( \frac{\gamma^2_\ell}{|D|} + \nu \right).$$
\end{lemma}
\begin{lemma}[Bounded client drift error]
\label{lemma_g-F}
If $\eta \leq \frac{1}{2L_F\sqrt{R(1+R)}} \Leftrightarrow \tilde\eta \leq \frac{\beta \sqrt{R}}{2L_F\sqrt{1+R}} $ and Assumptions~\ref{assumption_cs} and \ref{assumption_bv} hold, we have
\begin{align}
&~\frac{1}{N R} \sum_{i, r=1}^{N, R} \mathbb{E}\left[\left\|g_{i}^{t,r} - \nabla F_{i}\left(\bm w^{t}\right)\right\|_2^{2}\right]
\leq 64\tilde\eta L_F^2 \mathbb{E} \left[ F\left(\bm w^{t}\right) - F\left(\bm w^{\star}\right) \right] + 8\sigma_{F}^{2} + 10 \lambda^{2} \delta^{2}, \notag
\end{align}
where $g_{i}^{t,r} = \lambda\left(\bm w_{i}^{t,r}-\tilde{\bm \theta}_{i}(\bm w_{i}^{t,r})\right)+\gamma_w\nabla\phi_\rho(\bm w_{i}^{t,r})$ and $\sigma_{F}^{2} \triangleq \frac{1}{N} \sum_{i}^{N}\left\|\nabla F_{i}\left(\bm w^{\star} \right) \right\|_2^{2}$.
\end{lemma}
\begin{lemma}[Bounded diversity of $F_i$ w.r.t. client sampling] If Assumption~\ref{assumption_cs} holds, we have
\label{lemma_Fi-F}
\begin{align}
&~ \mathbb{E}_{S_t}\left\|\frac{1}{S} \sum_{i \in \mathcal{S}^{t}} \nabla F_{i}\left(\bm w^{t}\right)-\nabla F\left(\bm w^{t}\right)\right\|_2^{2}
\leq \frac{N/S-1}{N(N-1)}\sum_{i=1}^{N}\left( \left\|\nabla F_{i}\left(\bm w^{t}\right)-\nabla F\left(\bm w^{t}\right)\right\|_2^{2}\right). \notag
\end{align}
\end{lemma}
\begin{lemma}[Bounded diversity of $F_i$ w.r.t. distributed training] If Assumption~\ref{assumption_cs} holds, we have
\label{lemma_Fi-F_s}
$$ \frac{1}{N} \!\sum_{i=1}^{N}\left\|\nabla F_{i}(\bm w) \!- \nabla F(\bm w)\right\|_2^{2} \leq 4 L_{F}\left(F(\bm w) \!- F\left(\bm w^{\star}\right)\right) \!+ 2\sigma_{F}^{2}, $$
where $\sigma_{F}^{2} \triangleq \frac{1}{N} \sum_{i}^{N}\left\|\nabla F_{i}\left(\bm w^{\star}\right)\right\|_2^{2}$.
\end{lemma}
\begin{lemma}[One-step global update]
\label{lemma_one_step}
If Assumption~\ref{assumption_cs} holds, we have
\begin{align}
\mathbb{E}\left[\left\|\bm w^{t+1}-\bm w^{\star}\right\|_2^{2}\right]
\leq&~ \mathbb{E}\left[\left\|\bm w^{t}-\bm w^{\star}\right\|_2^{2}\right]-\tilde{\eta}\left(2-6 L_{F} \tilde{\eta}\right)\mathbb{E}\left[F\left(\bm w^{t}\right)-F\left(\bm w^{\star}\right)\right]\notag\\
&~ +\frac{\tilde{\eta}\left(3 \tilde{\eta}+1 / \mu_{F}\right)}{N R} \sum_{i, r}^{N, R} \mathbb{E}\left[\left\|g_{i}^{t,r}-\nabla F_{i}\left(\bm w^{t}\right)\right\|^{2}_2\right]\notag\\
&~ + 3 \tilde{\eta}^{2}\mathbb{E}\left[\left\|\frac{1}{S} \sum_{i \in \mathcal{S}^{t}}\nabla F_{i}\left(\bm w^{t}\right)-\nabla F\left(\bm w^{t}\right)\right\|_2^{2}\right]. \notag
\end{align}
\end{lemma}
\begin{remark}
Lemma~\ref{lemma_theta-theta} shows the diversity of $\bm \theta_i$ w.r.t. mini-bath sampling is bounded. Lemma~\ref{lemma_g-F} shows the client drift error caused by mini-batch training and local update is bounded. Lemma~\ref{lemma_Fi-F} and \ref{lemma_Fi-F_s} show the diversities of $F_i$ w.r.t. client sampling and distributed training are bounded. Using all the above lemmas, we can get the error bound
of the one-step update of the global model in Lemma~\ref{lemma_one_step}.
\end{remark}
\begin{theorem}
\label{theorem_convergence_FedMac}
Let Assumptions~\ref{assumption_cs} and \ref{assumption_bv} hold. If $\eta \leq \frac{\hat\eta}{\beta R}$, where $\hat\eta = \frac{1}{(18+256\kappa_F)L_F}$ and $\beta \geq 1$, then we have
\begin{align}
(a)~&~\frac{1}{T} \sum_{t=0}^{T-1} \mathbb{E} \left [F(\bm w^t) -F(\bm w^{\star}) \right] \leq \mathcal{O}(\Delta)
\triangleq~ \mathcal{O} \left\{ \frac{\Delta_0}{\hat\eta T} + \frac{ \sigma_F^2 }{\mu_F} + \frac{ \lambda^2\delta^2}{\mu_F} + \frac{ (\sigma_F^2 \Delta_0(N/S-1))^{1/2}}{\sqrt{TN}} \right\}, \notag\\
(b)~&~\frac{1}{N} \sum_{i=1}^N \mathbb{E} \left[ \| \tilde{\bm \theta}^T_i(\bm w^T) - \bm w^{\star} \|^2_2 \right]
\leq ~ \frac{L_F^2+\lambda^2}{\lambda^2\mu_F} \mathcal{O} ( \Delta ) + \mathcal{O} \left( \delta^2 + \frac{\sigma_F^2 + \gamma_w^2 d_s^2}{\lambda^2} \right) , \notag
\end{align}
where $\sigma_{F}^{2} \triangleq \frac{1}{N} \sum_{i=1}^{N}\left\|\nabla F_{i}\left(\bm w^{\star}\right)\right\|_2^{2}$, $\delta^2 = \frac{2}{\mu^2} \left( \frac{\gamma^2_\ell}{|D|} + \nu \right)$, $\Delta_0 \triangleq \mathbb{E} \left[ \| \bm w^0 - \bm w^{\star} \|^2_2 \right]$ and $d_s$ denotes the number of non-zero elements in $\bm w$.
\end{theorem}
\begin{remark}
Theorem~\ref{theorem_convergence_FedMac}(a) shows the convergence of the global model. The first term is caused by the initial error $\Delta_0$, which decreases linearly with the increase of training iterations. The second term is caused by client drift with multiple local updates. The third term shows that {\texttt{FedMac}} converges towards a $\frac{ \lambda^2\delta^2}{\mu_F}$-neighbourhood of $\bm w^{\star}$. The last term is due to the client sampling, which is 0 when $S=N$. We can see that the sparse constraints in {\texttt{FedMac}} do not affect the convergence rate of the global model. Theorem~\ref{theorem_convergence_FedMac}(b) shows the convergence of personalized models in average to a ball of center $\bm w^{\star}$ and radius $\mathcal{O} \left\{ \delta^2 + \frac{\sigma_F^2 + \gamma_w^2 d_s^2}{\lambda^2} + \frac{ \lambda^2\delta^2}{\mu_F}\right\} $, which means that a less sparse model requires a larger $\lambda$ to strengthen the connection between the client models and the global model.
\end{remark}
\subsection{Theoretical Performance Superiority}
In this subsection, we first present some useful definitions, then present the performance superiority of {\texttt{FedMac}} based on Assumptions~\ref{assumption_cs}-\ref{assumption_l1}. Suppose that the mean squared error is used as the training loss, i.e., $\ell(\bm \theta) = \norm{\bm y - \bm X \bm \theta}_2^2$.
For fixed $\bm w$, we have the following Theorem~\ref{theorem_Base}, which is proved in Appendix.
\begin{definition}
A random variable $\theta$ is called \emph{sub-Gaussian} if it has finite Orlicz norm
$$
\norm{\theta}_{\psi_2}=\inf \{t>0: \E \exp(\theta^2/t^2)\le 2\},
$$
where $\norm{\theta}_{\psi_2}$ denotes the sub-Gaussian norm of $\theta$. In particular, Gaussian, Bernoulli and all bounded random variables are sub-Gaussian.
\end{definition}
\begin{definition}
A random vector $\bm \theta \in \mathbb{R}^{N_I}$ is called \emph{sub-Gaussian} if $\left\langle \bm \theta,\bm{w} \right\rangle$ is sub-Gaussian for any $\bm{w} \in \mathbb{R}^{N_I}$, and its sub-Gaussian norm is defined as
$$
\norm{\bm \theta}_{\psi_2} = \sup \limits_{\bm{w} \in \mathbb{S}^{N_I-1}} \norm{\left\langle \bm \theta,\bm{w} \right\rangle}_{\psi_2}.
$$
\end{definition}
\begin{definition}
A random vector $\bm \theta \in \mathbb{R}^{N_I}$ is \emph{isotropic} if it satisfies $\E \bm \theta \bm \theta^T = \bm{I}_{N_I}$, where $\bm{I}_{N_I} \in \mathbb{R}^{N_I \times N_I}$ is the identity matrix.
\end{definition}
\begin{definition}
The \emph{subdifferential} of a convex function $f: \mathbb{R}^{N_I} \to \mathbb{R}$ at $\bm \theta^\star$ for all $\bm{d} \in \mathbb{R}^{N_I}$ is defined as the set of vectors
$$
\partial f(\bm \theta^\star) = \{\bm{u} \in \mathbb{R}^{N_I}: f(\bm \theta^\star + \bm{d}) \geq f(\bm \theta^\star) + \langle \bm{u}, \bm{d} \rangle.
$$
\end{definition}
\begin{definition}
The \emph{Gaussian width} of a subset $\mathcal{E} \subset \mathbb{R}^{N_I}$ is defined as
$$
w(\mathcal{E})= \E \sup \limits_{\bm \theta \in \mathcal{E}} \ip{\bm{g}}{\bm \theta},~\bm{g} \sim N(0,\bm{I}_{N_I})
$$
and the \emph{Gaussian complexity} of a subset $\mathcal{E} \subset \mathbb{R}^{N_I}$ is defined as
$$
\xi (\mathcal{E})= \E \sup \limits_{\bm \theta \in \mathcal{E}} |\ip{\bm{g}}{\bm \theta}|,~\bm{g} \sim N(0,\bm{I}_{N_I}).
$$
These two geometric quantities have the following relationship \cite{Chen2019}
\begin{equation}\label{Relation}
\xi(\mathcal{E}) \le 2 w(\mathcal{E})+\norm{\bm{y}}_2~~~\textrm{for every}~\bm{y} \in \mathcal{E}.
\end{equation}
\end{definition}
\begin{definition}
The \emph{Gaussian squared distance} $\eta(\mathcal{E})$ is defined as
$$
\eta^2 (\mathcal{E})= \E \inf \limits_{\bm \theta \in \mathcal{E}} \norm{\bm{g}-\bm \theta}_2^2,~\bm{g} \sim N(0,\bm{I}_{N_I}).
$$
\end{definition}
\begin{definition}
Define the error set
\begin{equation} \label{def: error_set}
\mathcal{E}_f=\{\bm{d} \in \mathbb{R}^{N_I}: f(\bm \theta^\star+ \bm{d}) \le f(\bm \theta^\star)\},
\end{equation}
then it belongs to the following convex set
\begin{equation} \label{def: convex_cone}
\mathcal{C}_f =\{\bm{d} \in \mathbb{R}^{N_I}: \ip{\bm{d}}{\bm{u}} \le 0~~\textrm{for any}~\bm{u} \in \partial f(\bm \theta^\star)\}.
\end{equation}
\end{definition}
\begin{assumption}\label{assumption_net}
Assuming that the network is a single-layer network, such that $\bm y = \bm X \bm \theta^\star$ with solution $\bm \theta^{\star} \in \mathbb{R}^{d \times 1} $.
\end{assumption}
\begin{assumption}\label{assumption_data}
Assuming that $\bm{X} \in \mathbb{R}^{N_D \times N_I}$ is a random matrix whose rows $\{\bm{X}_{i}\}_{i=1}^{N_D}$ are independent, centered, isotropic and sub-Gaussian random vectors.
\end{assumption}
\begin{assumption}\label{assumption_sign}
After sufficient training iterations $T$, there exists a constant $\zeta>0$ such that $\zeta\bm w - {\rm sign}(\bm w) \approx 0$ and $\zeta\bm \theta - {\rm sign}(\bm \theta) \approx 0$.
\end{assumption}
\begin{assumption}\label{assumption_l1}
Define $I=\{n: \theta_n^\star \neq 0\}$ and $J=\{n: \theta_n^\star \neq w_n\}$, assume that $N_I \gg q$ with $q=|I \cup J|$.
\end{assumption}
\begin{remark}
{\rm{\textbf{[Condition (A3)]:}}}
Although Assumption~\ref{assumption_net} is a strong assumption, we only use it for quantitative analysis of superiority. Extensive experiments in Section 4 show that {\texttt{FedMac}} has superior performance on various networks.
{\rm{\textbf{[Condition (A4)]:}}}
Assumption~\ref{assumption_data} is standard for compressed sensing analysis and training data usually meets this property.
{\rm{\textbf{[Condition (A5)]:}}}
Empirically, Assumption~\ref{assumption_sign} holds as the Figure~2 in~\cite{chao2020directional} shows that the angle between a network parameter $\bm{w}$ and its sign ${\rm{sign}}(\bm{w})$ is usually very small after sufficient training.
{\rm{\textbf{[Condition (A6)]:}}}
Assumption~\ref{assumption_l1} can be understood as we assume that the network is sparse, for which it is not difficult to obtain $N_I \gg q$.
\end{remark}
\begin{theorem}\label{theorem_Base} Let Assumptions~\ref{assumption_net} and \ref{assumption_data} hold. Let $\hat{\bm \theta}$ be the solution of the following problem
\begin{align}
\label{TPS-1T}
\min _{\bm \theta \in \mathbb{R}^{d} } \norm{\bm y - \bm X \bm \theta}_2^2 + \gamma f(\bm \theta),
\end{align}
and $f(\cdot)$ satisfies $|f(\bm \theta^\star)-f(\hat{\bm \theta})|\le \alpha_f ||\bm \theta^\star-\hat{\bm \theta}||_2$ for some $\alpha_f>0$. Let $\mathcal{C}_f$ denote the convex set $\mathcal{C}_f =\{\bm{d} \in \mathbb{R}^{N_I}: \ip{\bm{d}}{\bm{u}} \le 0~~\textrm{for any}~\bm{u} \in \partial f(\bm \theta^\star)\}$. If $\gamma \leq 1 / \alpha_f$ and the number of training data satisfies
\begin{equation}\label{NumberofMeasurements}
\sqrt{N_D} \ge CK^2 \xi(\mathcal{C}_f \cap \mathbb{S}^{N_I-1})+ \epsilon,
\end{equation}
then with probability at least $1- 2\exp(-\xi^2(\mathcal{C}_f \cap \mathbb{S}^{N_I-1}))$, the solution $\hat{\bm \theta}$ satisfies
\begin{equation}
\| \bm \theta^\star-\hat{\bm \theta}\|_2 \le \frac{1}{\epsilon^2},
\end{equation}
where $\epsilon,C$ are absolute constants, $\xi (\mathcal{E})= \E \sup_{\bm \theta \in \mathcal{E}} |\ip{\bm{g}}{\bm \theta}|$ denotes the Gaussian complexity with $\bm{g} \sim N(0,\bm{I}_{N_I})$ and $K=\max_i \norm{\bm{X}_{i}}_{\psi_2}$ denotes the maximum sub-Gaussian norm of $\{\bm{X}_{i}\}_{i=1}^{N_D}$.
\end{theorem}
\begin{remark}
Theorem~\ref{theorem_Base} indicates that once Assumptions~\ref{assumption_net} and \ref{assumption_data} hold, by using a small $\gamma$ such that $\gamma \leq 1 / \alpha_f$ and enough training data satisfying \eqref{NumberofMeasurements}, we can obtain the robust estimation of the sparse parameters of the network with high probability by minimizing the loss function \eqref{TPS-1T}.
\end{remark}
Next, we apply Theorem \ref{theorem_Base} to two sparse cases $f_1(\bm \theta) = \| \bm \theta \|_1 - \zeta_1 \left \langle \bm \theta, {\bm w}\right \rangle $ and $f_2(\bm \theta) = \| \bm \theta \|_1 + \zeta_2/2 \norm{\bm \theta - {\bm w}}_2^2 $ to compare their performance by calculating the upper bounds of $\xi(\mathcal{C}_{f_1} \cap \mathbb{S}^{N_I-1})$ and $\xi(\mathcal{C}_{f_2} \cap \mathbb{S}^{N_I-1})$, respectively. The results are presented in Theorem~\ref{UpperBound}, which is proved in the appendix.
\begin{theorem} \label{UpperBound} Let Assumptions~\ref{assumption_cs}-\ref{assumption_l1} hold, then we have
\begin{align}
\xi(\mathcal{C}_{f_1} \cap \mathbb{S}^{N_I-1}) &\le \mathcal{O} ( \sqrt{v_{\zeta_1} \log ({N_I}) } ), \notag \\
\xi(\mathcal{C}_{f_2} \cap \mathbb{S}^{N_I-1}) &\le \mathcal{O} ( \sqrt{v_{\zeta_2} \log ({N_I}) } ), \notag
\end{align}
where
$v_{{\zeta_1} } = \sum_{i \in I}({\rm{sign}}(\bm \theta_i^\star)-{\zeta_1} \bm{w}_i)^2+\sum_{i \in K_{\zeta_1}^{\neq}} ({\zeta_1} |\bm{w}_i|-1)^2$,
$v_{\zeta_2}= \sum_{i \in I}({\rm{sign}}(\bm \theta_i^\star)+\zeta_2 (\bm \theta_i^\star-\bm{w}_i))^2+\sum_{i \in K_{\zeta_2}^{\neq}} (\zeta_2 |\bm{w}_i|-1)^2,
$
$I=\{i: \bm \theta_i^\star \neq 0\}$, $J=\{i: \bm \theta_i^\star \neq \bm{w}_i\}, $
and $ K_\zeta^{\neq}=\{ i \in I^c\cap J: |\bm{w}_i|>1/{\zeta}\}$.
Moreover, if $\eta \leq \frac{\hat\eta}{\beta R}$, $\bm w = \bm w^{\star}$ and $\gamma \leq 1 / \alpha_f$ we further have
\begin{align}
v_{{\zeta_1} } \leq \Delta^v(\zeta_1), \quad
v_{\zeta_2} \leq 2 |I| + \Delta^v(\zeta_2) , \notag
\end{align}
where $\Delta^v(\zeta) = {2 {\zeta}^2} \left( {2\gamma_w^2 d_s^2}/{\lambda^2} + {1}/{\epsilon^4} \right) $.
\end{theorem}
\begin{remark}
Theorem~\ref{UpperBound} indicates that the difference between the upper bounds of $\xi(\mathcal{C}_{f_1} \cap \mathbb{S}^{N_I-1})$ and $\xi(\mathcal{C}_{f_2} \cap \mathbb{S}^{N_I-1})$ is determined by the difference between $v_{{\zeta_1} }$ and $v_{{\zeta_2} }$. Note that from the components in $\Delta^v(\zeta)$, it can be seen that it mainly represents the error after the algorithm converges, so it is a relatively small value, especially when $\zeta$ is small. Therefore, the training data required for \eqref{TPS-1T} using $f_1(\bm \theta)$ is much less than that required for \eqref{TPS-1T} using $f_2(\bm \theta)$.
\end{remark}
\section{Experimental Results}
\begin{table*}[htbp]
\caption{Fine-tune results on MNIST, FMNIST, CIFAR-100 and Synthetic datasets. Best results are bolded.}
\label{fine-tune-table}
\centering
\scalebox{1.0}{
\begin{tabular}{cccccccccccc}
\toprule
\multirow{2}{*}{Dataset} & FedAvg & Fedprox & HeurFedAMP &Per-FedAvg &\multicolumn{2}{c}{pFedMe} & \multicolumn{2}{c}{FedMac} \\
\cmidrule(r){2-2}
\cmidrule(r){3-3}
\cmidrule(r){4-4}
\cmidrule(r){5-5}
\cmidrule(r){6-7}
\cmidrule(r){8-9}
& GM & GM & PM & PM & GM & PM & GM & PM \\
\midrule
{MNIST} & 96.8603 & 96.8564 & 98.4425 & 98.1474 & 96.8698 & 98.9223 & \textbf{96.8964} & \textbf{98.9547} \\
{FMNIST} & 85.6949 & 85.6569 & 99.1756 & 99.3051 & 84.9582 & 99.2327 & \textbf{85.7958} & \textbf{99.3698} \\
{CIFAR-100} & 85.7712 & 86.0150 & 90.4033 & 89.7828 & 84.9512 & 90.1374 & \textbf{86.5691} & \textbf{90.8987} \\
{Synthetic} & 84.6333 & 84.5475 & 84.2741 & 88.7006 & 83.5622 & 87.4969 & \textbf{84.8889} & \textbf{89.0583} \\
\bottomrule
\end{tabular}}
\end{table*}
\subsection{Experimental Setting}\label{sec_set}
The proposed $\texttt{FedMac}$ is a personalized FL based on local fine-tuning, so we compare the performance of {\texttt{FedMac}} with FedAvg~\cite{mcmahan2017communication} and local fine-tuning personalized FL methods, including
Fedprox~\cite{li2018federated},
Per-FedAvg~\cite{NEURIPS2020_24389bfe},
HeurFedAMP~\cite{huang2021personalized} and pFedMe~\cite{t2020personalized}, on non-i.i.d. datasets. In communication cost simulations, we only compare \texttt{FedMac} with methods based on sparse constraints. Since other methods (e.g. ternary compression~\cite{8889996}, asynchronous learning~\cite{8945292} and quantization~\cite{9305988}) that can reduce the amount of communication can be combined with methods based on sparse constraints
We generate the non-i.i.d. datasets based on four public benchmark datasets, MNIST~\cite{lecun2010mnist,lecun1998gradient}, FMNIST (Fashion-MNIST)~\cite{xiao2017fashion}, CIFAR-100~\cite{krizhevsky2009learning} and Synthetic datasets \cite{li2018federated}. For MNIST, FMNIST and CIFAR-100 datasets, we follow the non-i.i.d. setting strategy in ~\cite{t2020personalized}. Each client occupies a unique local data with different data sizes and only has 2 of the 10 labels.
The number of clients for MNIST/FMNIST is $N=20$ and the number of clients for CIFAR-100 is $N=10$. For the Synthetic dataset, we follow the non-i.i.d. setting strategy in \cite{li2018federated} for $N=100$ clients by setting $\bar \alpha=\bar\beta=0.5$ to control the differences in the local model and dataset of each client.
We set $S=10$ for experiments on MNIST and FMNIST datasets, while set $S=2$ and $S=20$ for experiments on CIFAR-100 and Synthetic datasets, respectively. A two-layer deep neural network with a hidden layer size of 100 is used for experiments on MNIST and FMNIST datasets, and a hidden layer of size 20 is used on the Synthetic dataset. A VGG-Net~\cite{simonyan2014very} is used for experiments on CIFAR-100 dataset. For all algorithms, we follow the testing strategy in \cite{huang2021personalized} and evaluate the performance through the highest mean testing accuracy in all communication rounds of training.
\begin{figure}[!t]
\centering
\subfloat{
\includegraphics[width=2.6in]{Mnist_FedMac_R_test.pdf}
}
\subfloat{
\includegraphics[width=2.6in]{Mnist_FedMac_R_train.pdf}
}
\subfloat{
\includegraphics[width=2.6in]{Mnist_FedMac_beta_test.pdf}
}
\subfloat{
\includegraphics[width=2.6in]{Mnist_FedMac_beta_train.pdf}
}
\caption{Effect of hyperparameters on the convergence rate of \texttt{FedMac} algorithm on MNIST dataset. \textbf{Top:} Test accuracy and training loss with different $R$. \textbf{Bottom:} Test accuracy and training loss with different $\beta$.}
\label{fig_change_hyperparameters}
\end{figure}
We did all experiments in this paper using servers with a GPU (NVIDIA Quadro RTX 6000 with 24GB memory), two CPUs (each with 12 cores, Inter Xeon Gold 6136), and 192 GB memory. The base DNN and VGG models and federated learning environment are implemented according to the settings in \cite{t2020personalized}. In particular, the DNN model uses one hidden layer, ReLU activation, and a softmax layer at the end. For the MNIST dataset, the size of the hidden layer is 100, while that is 20 for the Synthetic dataset. The VGG model is implemented for CIFAR-100 dataset with ``[16, `M', 32, `M', 64, `M', 128, `M', 128, `M']" cfg setting. We use PyTorch~\cite{NEURIPS2019_bdbca288} for all experiments.
For a fair comparison, we allow the stochastic gradient descent (SGD) algorithm used to solve \eqref{A-H-min} in Algorithm~1, i.e.,
$$
\bm{\tilde \theta}_{i}^{t,r}(\bm{ w}_{i}^{t,r}) = \arg\min _{\boldsymbol{\theta}_{i} \in \mathbb{R}^{d}} \tilde H(\bm \theta;\bm{ w}_{i}^{t,r},\mathcal{D}_{i})
$$
to be iterated only once to make it consistent with the settings in FedAvg, Fedprox and Per-FedAvg. And we also modify HeurFedAMP and pFedMe to make the corresponding algorithms iterate only once.
The code based on PyTorch 1.8~\cite{NEURIPS2019_bdbca288} will be available online.
\subsection{Effect of Hyperparameters}\label{sec_result_hyper}
We first empirically study the effect of different hyperparameters in {\texttt{FedMac}} on MNIST dataset. More results are provided in the appendix.
\textbf{Effects of $R$:} In {\texttt{FedMac}} algorithm, $R$ denotes the local epochs. An appropriately large $R$ allows the algorithm to converge faster but requires more computations at local clients, while a small $R$ requires more communication rounds between the server and clients. According to the top part of Figure~\ref{fig_change_hyperparameters}, where we fix $\eta=3000$, $\beta=1$, $S=10$, $|\mathcal{D}|=20$, $\lambda=0.0001$, $\gamma=\gamma_w=0$, we set $R=20$ for the remaining experiments to balance this trade-off.
\begin{figure}[!t]
\centering
\subfloat{
\includegraphics[height=2.6in]{Mnist_test_results.pdf}
}
\subfloat{
\includegraphics[height=2.6in]{Mnist_train_results.pdf}
}
\caption{Comparison results of the convergence rate of different algorithms on the MNIST dataset. \textbf{Left:} Test accuracy. \textbf{Right:} Training loss.}
\label{fig_mnist_results}
\end{figure}
\textbf{Effects of $\beta$:} The bottom part of Figure~\ref{fig_change_hyperparameters} shows the test accuracy and training loss of {\texttt{FedMac}} algorithm with different values of $\beta$, where we set $\eta=3000$, $R=20$, $S=10$, $|\mathcal{D}|=20$, $\lambda=0.0001$, $\gamma=\gamma_w=0$. We can see that increasing $\beta$ can improve the test accuracy of the global model and weaken the performance of the personalized model. For balance, we set $\beta=1$ for the remaining experiments.
\subsection{Performance Comparison Results}
We first compare the performance of our {\texttt{FedMac}} with other methods under non-sparse conditions to show its advantages, i.e., we set $\gamma=\gamma_w=0$ for {\texttt{FedMac}}. Table~\ref{fine-tune-table} shows the fine-tune performances on different datasets and models. We test these algorithms through the same settings with $|\mathcal{D}|=20$ and $T=800$, and appropriately fine-tune other hyperparameters to obtain the highest mean testing accuracy in all communication rounds of training. We run each experiment at least 3 times to obtain statistical reports. More detailed results on hyperparameters are listed in the appendix. We can see that the personalized model (PM) of {\texttt{FedMac}} outperforms other models in all settings, while the global model (GM) of {\texttt{FedMac}} outperforms all other global models.
Figure~\ref{fig_mnist_results} shows the convergence rate of our {\texttt{FedMac}} algorithm and other algorithms on the MNIST dataset. The accuracy of the personalized model of {\texttt{FedMac}} is 1.79\%, 0.55\%, 2.29\%, 0.75\%, 0.13\%, 1.96\% and 1.92\% higher than that of the global model of {\texttt{FedMac}}, the personalized model of pFedMe, the global model of pFedMe, Per-FedAvg, HeurFedAMP, Fedprox and FedAvg, respectively. In these experiments, we set $R=20$, $S=10$, $|\mathcal{D}|=20$, $\beta=1$ for all algorithms. For {\texttt{FedMac}}, we use small $\lambda = 0.0001$ because Theorem~\ref{UpperBound} shows that a small $\zeta = \lambda/\gamma$ can reduce the upper bound of the required training data. For pFedMe, we set $\lambda = 15$, which is a recommended value in~\cite{t2020personalized}. To balance the effects of different $\lambda$ on different convergence speeds, we set $\eta=0.02$ for pFedMe and $\eta=3000$ for {\texttt{FedMac}} to make $\eta \times \lambda$ equal. For other hyperparameters in other algorithms, we try to set the same value for fair comparison. See the appendix for detailed settings.
\begin{figure}[!t]
\centering
\subfloat{
\includegraphics[height=2.6in]{Mnist_test.pdf}
}
\subfloat{
\includegraphics[height=2.6in]{Mnist_sparse.pdf}
}
\caption{Comparison results of {\texttt{FedMac}} and modified pFedMe under sparse conditions. \textbf{Left:} Test accuracy. \textbf{Right:} Model sparsity, i.e., non-zero element ratio. We set $\eta_p=0.08$, $\beta=1$, $S=10$, $|\mathcal{D}|=20$ and $\gamma_w=0$.}
\label{fig_sparsity_results}
\end{figure}
Finally, we show the performance advantages of our {\texttt{FedMac}} over the $\ell_2$-norm based personalization methods under sparse conditions. To control the variables, we set $\gamma_w=0$ in this experiments, that is, we only analyze the difference in sparsity caused by different optimization functions in \eqref{TPS-0}. For comparison, we obtain a modified pFedMe algorithm by adding approximate $\ell_1$-norm constraints to the loss function. Figure~\ref{fig_sparsity_results} shows that our {\texttt{FedMac}} algorithm has significantly better performance than the modified pFedMe algorithm under sparse conditions. {\texttt{FedMac}} can obtain a sparser model faster, and the accuracy of {\texttt{FedMac}} with 50\% sparsity is still about 2\% higher than that of modified pFedMe with 86\% sparsity. Such obvious performance difference also verifies our previous theoretical analysis.
\begin{table}[!t]
\caption{The detailed hyperparameters of the fine-tune results in Table~\ref{fine-tune-table}}
\label{fine_parameters_table}
\centering
\scalebox{0.84}{
\begin{tabular}{llc}
\toprule
Algorithm & Basic Settings & Fine-tune Hyperparameters\\
\midrule
{FedAvg} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=0.02$} & $R,~\eta$\\
{Fedprox} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=0.02,~ \mu=0.0001$} & $R,~\eta, \mu$\\
{HeurFedAMP} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=0.02,~ \eta_p=0.05,~\alpha=0.5,~\lambda=5,~\sigma=50$} & $R,~\eta,~\eta_p,~\alpha,~\lambda,~\sigma$\\
{Per-FedAvg} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=0.02,~ \eta_p=0.05$} & $R,~\eta,~\eta_p$\\
{pFedMe} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=0.02,~ \eta_p=0.05,~\lambda=15$} & $R,~\eta,~ \eta_p,~\lambda$\\
{FedMac} & {$|\mathcal{D}|=20,~ R=20,~T=800,~\beta=1,~\eta=3000,~ \eta_p=0.05,~\lambda=0.0001,~\gamma=\gamma_w=0$} & $R~,\eta,~ \eta_p,~\lambda,~\gamma$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{table}[!t]
\caption{The detailed hyperparameters of results in Figure~\ref{fig_mnist_results}}
\label{parameters-table}
\centering
\scalebox{1.0}{
\begin{tabular}{llc}
\toprule
Algorithm & Basic Settings & Specific Settings \\
\midrule
{FedAvg} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=0.02$\\
{Fedprox} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=0.02,~ \mu=0.0001$\\
{FedAMP} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=0.02,~ \eta_p=0.03,~\alpha=0.5,~\lambda=1,~\sigma=50$\\
{Per-FedAvg} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=0.02,~ \eta_p=0.03$\\
{pFedMe} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=0.02,~ \eta_p=0.03,~\lambda=15$\\
{FedMac} & {$|\mathcal{D}|=20,~ R=20,~S=10,~T=800,~\beta=1$} & $\eta=3000,~ \eta_p=0.03,~\lambda=0.0001,~\gamma=\gamma_w=0$\\
\bottomrule
\end{tabular}}
\end{table}
\subsection{Experimental Setup Details}
For experiments in Table~\ref{fine-tune-table}, we set $|\mathcal{D}| = 20, T = 800$, and $\beta = 1$ for all algorithms and fine-tuned other hyperparameters respectively in Table~\ref{fine_parameters_table}. We fine-tune one hyperparameter while maintaining the basic settings of other hyperparameters. Under most hyperparameter settings, the FedMac algorithm achieves better performance than other algorithms. Each experiment is run at least 5 times to obtain statistical reports.
For experiments in Figure~\ref{fig_mnist_results}, and we set $|\mathcal{D}|=20,~ R=20,~S=10,~T=800$, and $\beta=1$ for all algorithms. To balance the effects of different $\lambda$ on different convergence speeds, we set $\{\eta=0.02,~\lambda=15\}$ for pFedMe and $\{\eta=3000,~\lambda=0.0001\}$ for {\texttt{FedMac}} to make $\eta \times \lambda$ equal. For FedAMP, $\lambda$ is different from that in pFedMe and \texttt{FedMac}, we set $\lambda=1$ to obtain a better result according to our fine-tune results, which is also the same as that used in~\cite{huang2021personalized}. For other hyperparameters, we set reasonable values based on experimental results to promote fairer comparisons, which are summarized in Table~\ref{parameters-table}.
\begin{figure}[!t]
\centering
\subfloat{
\includegraphics[height=2.6in]{Com_cost_sparse2.pdf}
}
\quad
\subfloat{
\includegraphics[height=2.6in]{Com_cost_bit2.pdf}
}
\caption{The model sparsity and communication cost for experiments in Figure~\ref{fig_sparsity_results}, where 50\% sparsity (non-zero parameters ratio) is the lower bound of sparsity that we set to preserve performance. With this setting, the communication cost is reduced by almost half after 200 global round. \textbf{Left:} Model sparsity. \textbf{right:} Accumulated communication cost.}
\label{fig_sparse_communication}
\end{figure}
\subsection{Additional Experimental Results under Sparse Conditions}
The model sparsity and the accumulative communication cost against the global rounds for experiments in Figure~\ref{fig_sparsity_results} are presented in Figure~\ref{fig_sparse_communication}. Sparsity is defined as the proportion of the number of non-zeros in the model, where we set 50\% sparsity as a lower bound to preserve performance. The accumulated communication cost is calculated based on a model with 79510 parameters. Non-zero parameters are quantized by 64 bits, while zero parameters are quantized by 1 bit.
And the {\texttt{FedMac}} algorithm needs to communicate an additional index matrix to represent the location of the zero value, which requires $79510\times1$ bits of communication cost.
Therefore, for each round of iterative training, the communication cost required by the server and each client for non-sparse model is $79510\times2\times64$ bits for upload and broadcast, while the communication cost required by the server and each client for sparse model is $\text{sparsity}\times 79510\times2\times64 + (1-\text{sparsity})\times 79510\times2\times1 + 79510\times1 $ bits for upload and broadcast. With 50\% sparsity setting, the communication cost is reduced by almost half.
Figure~\ref{fig_gw_effect} shows the convergence rate of \texttt{FedMac} with different value of $\gamma_w$, while fixing $\eta=2000,~\eta_p=0.08,\gamma=0.001,~\lambda=0.0001,\beta=1,S=10,|\mathcal{D}|=20$. We can see that an appropriate increase of $\gamma_w$ can effectively improve the speed of model sparseness without affecting the final accuracy of \texttt{FedMac}.
\begin{figure}[!t]
\centering
\subfloat{
\includegraphics[height=2.6in]{Mnist_gw_test.pdf}
}
\quad
\subfloat{
\includegraphics[height=2.6in]{Mnist_gw_sparse.pdf}
}
\caption{Effect of hyperparameter $\gamma_w$ on the convergence rate of \texttt{FedMac} algorithm on the MNIST dataset. We set $\eta=2000,~\eta_p=0.08,\gamma=0.001,~\lambda=0.0001,\beta=1,S=10,|\mathcal{D}|=20$. \textbf{Left:} Test accuracy. \textbf{Right:} Model sparsity.}
\label{fig_gw_effect}
\end{figure}
\section{Conclusions}
In this paper, we propose {\texttt{FedMac}} as a sparse personalized FL algorithm to solve the statistical diversity issue, which has better performance than the $\ell_2$-norm based personalization methods under sparse conditions. Our approach makes use of an approximated $\ell_1$-norm and the correlation between the global model and client models in the loss function. Maximizing correlation decouples the personalized model optimization from the global model learning, which allows {\texttt{FedMac}} to optimize personalized models in parallel. Convergence analysis indicates that the sparse constraints in {\texttt{FedMac}} do not affect the convergence rate of the global model. Moreover, theoretical results show that {\texttt{FedMac}} performs better than the $\ell_2$-norm based personalization methods and the training data required is significantly reduced. Experimental results demonstrate that {\texttt{FedMac}} outperforms many advanced personalization methods under both sparse and non-sparse conditions.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 921 |
\section{Introduction}
\label{sec:intro}
In~\cite{rieckyamashita} we defined an invariant of closed orientable 3-manifolds that measures how
efficiently a given manifold $M$ can be represented as a cover of $S^{3}$ where the
branch set is a hyperbolic link (such a cover can be constructed using Hilden~\cite{hilden}
or Montesinos~\cite{montesinos}).
We use the notation $M \stackrel{p}\to (S^{3},L)$ to denote a $p$-fold cover $M \to S^{3}$ branched
over $L$. To the cover $M \stackrel{p}\to (S^{3},L)$
we associate the complexity $p \vol[S^{3} \setminus L]$.
The {\it link volume} of $M$ is defined to be
the infimum of the complexities of all possible covers, that is
$$\lv[M] = \inf\{\ p \vol[S^{3} \setminus L]\ |\ M \stackrel{p}\to (S^{3},L)\}.$$
We observed that for any hyperbolic manifold $M$, $\lv[M] > \vol[M]$, and conjectured that the link volume
cannot be bounded in terms of the hyperbolic volume. In this paper we prove this conjecture.
Our main result is (see the next two sections for definitions; by $M(\alpha_{i})$ we mean the manifold obtained
by filling $M$ along slope $\alpha_{i}$ and $\ensuremath{d_{\mbox{Farey}}}$
denotes the distance in the Farey graph):
\begin{thm}
\label{thm:main}
Let $M$ be a hyperbolic manifold with one cusp. Let $\{\alpha_{i}\}$
be a set of slopes of $\partial M$.
Then there exists $L>0$ so that $\lv[M(\alpha_{i})] < L$ for all $i$ if and only if
there exists $d>0$ so that $\ensuremath{d_{\mbox{Farey}}}(\alpha_{i},\alpha_{i'})<d$ for all $i,i'$.
\end{thm}
\begin{rmk} One direction is known:
if there exists $d>0$ so that $\ensuremath{d_{\mbox{Farey}}}(\alpha_{i},\alpha_{i'})<d$ for all $i,i'$,
then~\cite{rieckyamashita} implies that there exists $L>0$ so that $\lv[M(\alpha_{i})] < L$ for all $i$.
\end{rmk}
Let $M$ denote the figure eight knot exterior; then $\vol[M] = 2.02988\dots$.
By Cao and Meyerhoff~\cite{caomeyerhoff}, $M$ has the smallest volume
among all cusped hyperbolic 3-manifolds.
Applying Theorem~\ref{thm:main} to manifolds obtained by filling $M$ we get
the following corollary:
\begin{cor}
For every $V>0$, there exists a hyperbolic manifold $M_{V}$, so that
$$\vol[M_{V}] < 2.02988\dots \hspace{10pt} \mbox{and} \hspace{10pt} \lv[M_{V}] > V.$$
\end{cor}
This corollary can be interpreted (negatively) as saying that representing
manifolds as branched covers of $S^{3}$ is inefficient. On the positive side,
it shows that the link volume is a truly new invariant.
Not much is known about the link volume of specific manifolds; Rieck and Remigio--Ju\'arez~\cite{remigiorieck}
calculated the link volume of certain prism manifolds (prism manifolds are small Seifert fibered spaces),
but the link volume is not known for {\it any} hyperbolic manifold. We refer the reader to~\cite{rieckyamashita}
for basic facts and open questions about the link volume. In particular, the upper bound obtained in
\cite{rieckyamashita} is explicit, and linear in terms of distance in the Farey tessellation. It would be
nice if a similar lower bound could be proved.
The following question is obtained by simply reversing the inequality
in the upper bound of \cite{rieckyamashita} (by $M(\alpha_{1},\dots,\alpha_{n})$ we
mean the manifold obtained by filling $M$ along slopes $\alpha_{1},\dots,\alpha_{n}$):
\begin{que}
Let $M$ be a compact orientable connected 3-manifold with toral boundary.
We will denote the components of $\partial M$ as $T_{1},\dots,T_{n}$. For each $i$,
fix a slope $\beta_{i}$ of $T_{i}$.
Then do there exist $A,\ B>0$,
so that for any choice of slope $\alpha_{i}$ of $T_{i}$,
$$\lv[M(\alpha_{1},\dots,\alpha_{n})] > A + B(\Sigma_{i=1}^{n} \ensuremath{d_{\mbox{Farey}}}(\alpha_{i},\beta_{i}))?$$
\end{que}
The work in this paper is an application of the Structure Theorem of~\cite{rieckyamashita}.
The Structure Theorem states that for any $V>0$, there is a finite set of ``parent systems''
$\{\phi_{i}:X_{i} \to E_{i}\}_{i=1}^{n}$, where $X_{i}$ and $E_{i}$ are hyperbolic manifolds and
$\phi_{i}:X_{i} \to E_{i}$ is an unbranched cover, so that for any manifold $M$
with $\lv[M] < V$, there is some $i$ so that the following diagram commutes:
\medskip
\begin{center}
\begin{picture}(200,60)(0,0)
\put( 0, 0){\makebox(0,0){$E_{i}$}}
\put( 0,50){\makebox(0,0){$X_{i}$}}
\put( 0, 40){\vector(0,-1){30}}
\put( 10, 0){\vector(1,0){70}}
\put(100, 0){\makebox(0,0){$(S^3,L)$}}
\put(100, 40){\vector(0,-1){30}}
\put(10,28){\makebox(0,0){$/\phi_{i}$}}
\put(108,28){\makebox(0,0){$/\phi$}}
\put(108,28){\makebox(0,0)}
\put(100,50){\makebox(0,0){$M$}}
\put( 10,50){\vector(1,0){80}}
\end{picture}
\end{center}
\medskip
where here the horizontal arrows denote inclusions induced by fillings (that is, attaching solid tori) and
$\phi:M \to S^{3}$ is a cover that realizes the link volume, that is, a cover for which
$\deg(\phi) \vol[S^{3} \setminus L] = \lv[M]$.
The reader may observe the similarity to the celebrated result of J\o rgensen and Thurston~\cite{thurston}
that states that for every $V>0$ there are finitely many ``parent manifolds'' $\{E_{i}\}_{i=1}^{n}$ so that
every hyperbolic manifold of volume less than $V$ can be obtained by filling $E_{i}$
for some $i$.
(For a detailed exposition see, for example,~\cite{kobayashirieck}.) This is no coincidence;
the Structure Theorem is a consequence of J\o rgensen--Thurston and the manifolds $E_{i}$
appearing in it are the parent manifolds of
J\o rgensen andThurston.
\bigskip\bigskip\bigskip\noindent
In order to obtain Theorem~\ref{thm:main} from the Structure Theorem above,
we are forced to study several questions
about fillings. Specifically, we study several questions about {\it cosmetic surgery},
that is, surgery on a link in a manifold $M$ that results in a manifold diffeomorphic to $M$.
Before describing our methods and the results obtained, we
introduce the basic setup; detailed
description is given in Subsection~\ref{subsec:notaion}.
After two preliminary sections (\ref{sec:background} and~\ref{sec:BoundedSets}),
in Sections~\ref{sec:mnh} and~\ref{sec:T(X)} we construct our main tool, a rooted tree denoted
$T(X)$ which we associate with a manifold $X$. The vertices of $T(X)$ are labeled by
labels that correspond to manifolds; $X$ itself corresponds to the root of $T(X)$.
If $X$ is hyperbolic, its immediate descendants are certain non-hyperbolic
manifolds obtained by filling $X$; if $X$ is not prime, its immediate descendants are the factors of its
prime decomposition; if $X$ is JSJ (that is, if $X$ is prime and the collection of tori in the
JSJ decomposition of $X$ is not empty),
its immediate descendants are the components of its torus decomposition. If $X$ is Seifert
fibered or solv it has no descendants. We prove
(Proposition~\ref{prop:T(X)isFinite}) that $T(X)$ is finite. All the results described below
are proved by induction on $|T(X)|$, the number of vertices in $T(X)$.
The various applications of $T(X)$ are somewhat independent, and we made an effort
to make the following sections (especially
Sections~\ref{sec:CosmeticSurgeryOnT2XI}--\ref{section:CosmeticSurgeryOnS3}
and~\ref{sec:FillingsThatDontFactorThroughM}) independently readable.
Throughout this paper a set of slopes of a torus is called {\it bounded}
if it is bounded in the Farey graph.
In Section~\ref{sec:CosmeticSurgeryOnT2XI} we study cosmetic surgery on a link $L \subset T^{2} \times [0,1]$.
Let $B$ be a bounded set of slopes of $T^2 \times \{1\}$.
By a {\it multislope} $\alpha$ of $L$ we mean a vector whose components are slopes on the components
of $L$ or $\ensuremath\infty$ (see Subsection~\ref{subsec:notaion} for a precise definition).
We will denote the manifold obtained by surgery on $L$ with multislope $\alpha$ as $L(\alpha)$.
Let $\mathcal{A}$ be the multislopes of $L$ that yield cosmetic surgery, that is,
$\mathcal{A} = \{\alpha\ |\ L(\alpha) \cong T^{2} \times [0,1]\}$.
Given $\alpha \in \mathcal{A}$, we may use the product
structure of $L(\alpha)$ to project the set $B$ and obtain a set of slopes of $T^{2} \times \{0\}$.
Since this set depends on $\alpha$ we will denote it as $B_{\alpha}$.
We prove the $T^{2} \times [0,1]$ Cosmetic Surgery Theorem (\ref{thm:CosmeticSurgeryOnT2XI})
that says that
$$\bigcup_{\alpha \in \mathcal{A}} B_{\alpha}$$
is a bounded set of slopes of $T^{2} \times [0,1]$.
In Section~\ref{sec:CosmeticSurgeryOnSolidTorus} we study cosmetic surgery on a link
$L \subset D^{2} \times S^{1}$.
We prove the Solid Torus Cosmetic Surgery Theorem (\ref{thm:SlopesOnSolidTorus}), that
says that the set of slopes of $\partial D^{2} \times S^{1}$
that bound a disk after cosmetic surgery on $L$ is bounded
(unless some component of $L$ is a core of the solid torus after surgery, in which case the
claim is obviously false). For use in later sections we also prove
Proposition~\ref{pro:SolidTorusSurgery2}, which gives certain constraints
on multislopes of $L$ that yield a cosmetic surgery.
Sections~\ref{section:HyperSurgSlopes} and~\ref{section:HyperSurgRadInj} are
devoted to cosmetic surgery on hyperbolic manifolds. Let $M$ be a hyperbolic
manifold, $L \subset M$ a link, $T$ a component of $\partial M$,
$B$ a bounded set of slopes of $T$, and
$\mathcal{X} = \{(\alpha,f_{\alpha})\}$ so that for every $(\alpha,f_{\alpha})$,
$\alpha$ is a multislope of $L$ and $f_{\alpha}$
is a diffeomorphism $f_{\alpha}:L(\alpha) \cong M$ that maps $T$ to itself.
Then for every $(\alpha,f_{\alpha}) \in \mathcal{X}$ the image of $B$ under
$f_{\alpha}$ is a set of slopes of $T$ that we will denote as $B_{\alpha,f_{\alpha}}$.
In Section~\ref{section:HyperSurgSlopes} we prove
Theorem~\ref{thm:CosmeticSurgeryOnM} that says that
$$\bigcup_{(\alpha,f_{\alpha}) \in \mathcal{X}} B_{\alpha,f_{\alpha}}$$
is a bounded set of slopes of $T$.
In Section~\ref{section:HyperSurgRadInj} we consider multislopes $\alpha$ of a manifold $X$
that yield a hyperbolic manifold $X(\alpha)$, so that every geodesic in $X(\alpha)$ is
longer than $\epsilon$ (for some fixed $\epsilon>0$; here $\alpha$ is a multislope on
$\partial X$ and $X(\alpha)$ represents filling rather than surgery). We prove two things: first, there
are only finitely many such manifolds $X(\alpha)$ (although there certainly may be infinitely
many such multislopes $\alpha$). Second, we prove that all but finitely many of these multislopes
factor through a non-hyperbolic filling. In other words, there is a subset of the boundary components
so that the manifold obtained by filling only these components is not hyperbolic
(we call this a non hyperbolic {\it partial filling}; this and
other useful terminology is introduced in Subsection~\ref{subsec:notaion}).
Moreover, one of the non hyperbolic partial fillings corresponds to an edge out
of the root of $T(X)$; this is the key that allows us to use induction.
In Section~\ref{section:CosmeticSurgeryOnS3} we prove the $S^{3}$ Cosmetic Surgery
Theorem~(\ref{thm:cosmeticSurgeryOnS3}): let $L \subset S^{3}$ be a link and $K$ a component
of $L$. Let $\mathcal{A} = \{\alpha \ | \ L(\alpha) = S^{3}\}$, and $\mathcal{A}' \subset \mathcal{A}$
be the multislopes for which the core of the solid torus attached to $\partial N(K)$ does
not form a Hopf link with the core of any other attached solid torus. Given
a multislope $\alpha$, we denote its value on $K$ as $\alpha|_{K}$. We prove that
$$\{\alpha|_{K} \ | \ \alpha \in \mathcal{A}'\}$$
is bounded.
Finally, in Sections~\ref{sec:SetUpOfProof}--\ref{sec:BothFillingsFactor} we apply these
results to prove Theorem~\ref{thm:main}. In particular, in Section~\ref{sec:FillingsThatDontFactorThroughM}
we prove Theorem~\ref{thm:fillingsThatDontFactor} which is of independent interest.
In it we consider manifolds $X$ and $M$, where $M$ is a one cusped hyperbolic manifold.
We consider the set of multislopes $\mathcal{A}$ of $\partial X$ so that any $\alpha \in \mathcal{A}$
fulfills the following condition:
the manifold obtained by filling all but one boundary component of $X$
is not $M$. We describe this by saying that $\alpha$ does not admit a partial filling $\alpha'$
for which $X(\alpha') \cong M$.
In Theorem~\ref{thm:fillingsThatDontFactor}
we show that the
set of slopes $\beta$ so that $M(\beta) \cong X(\alpha)$ for some $\alpha \in \mathcal{A}$
is bounded.
\bigskip
\noindent{\bf Acknowledgements.} We thank Matt Day, Chaim Goodman--Strauss,
Mike Hilden, Kazuhiro Ichihara, Tsuyoshi Kobayashi,
Marc Lackenby, Kimihiko Motegi, John Retcliffe, and Masakazu Teragaito for helpful conversations and
correspondence. A very special thanks to Michael Yoshizawa whose patient
reading of an early version of this paper helped us greatly.
YR: this research took place during three visits to Nara Women's University. I
would like to thank the Math Department and the Department of Information and Computer Science
for their hospitality during my long stays.
\section{Background}
\label{sec:background}
Throughout this paper, by {\it manifold} we mean a compact orientable connected 3-dimensional smooth
manifold. We only consider manifold with toral boundary, that is, manifolds
whose boundary consists of a (possibly empty) collection of tori. A manifold is called {\it hyperbolic} if its
interior admits a complete finite volume Riemannian metric locally isometric to hyperbolic space $\mathbb H^{3}$;
we sometimes refer to the boundary components of a hyperbolic manifold as {\it cusps}.
We denote closed normal neighborhood, closure, and interior by $N(\cdot)$,
$\mbox{cl}$, and $\mbox{int}$, respectively. The geometric intersection number
between curves on a torus is denoted $\Delta(\cdot,\cdot)$. Given a knot or a link $L \subset M$,
we call $M \setminus \mbox{int}N(L)$ the {\it exterior} of $L$ and denote it as $E(L)$.
We {\it always} assume transversality.
\subsection{Notation}
\label{subsec:notaion}
The following notation will be used extensively throughout the paper.
Let $X$ be a manifold, fix $n$ components of $\partial X$ denoted as $T_{1},\dots,T_{n}$,
and denote their union as
$\mathcal{T} = \cup_{i=1}^{n} T_{i}$
(note that $\mathcal{T} \subset \partial X$, but possibly $\mathcal{T} \neq \partial X$).
\begin{enumerate}
\item By a {\it multislope} of $\mathcal{T}$, say $\alpha$, we mean a vector $\alpha = (\alpha_{1},\dots,\alpha_{n})$
so that for each $i$, $\alpha_{i}$ is either the homology class of a connected simple closed curve on $T_{i}$ or
$\alpha_{i} = \infty$. By a multislope of a link $L$ we mean a multislope of $\partial N(L) \subset \partial E(L)$.
\item By {\it filling} $X$ along $\alpha$ we mean the manifold $X(\alpha)$ that is obtained by the following
operation:
\begin{enumerate}
\item To components $T_{i} \subset \mathcal{T}$ for which $\alpha_{i} \neq \infty$ we attach
a solid torus $V_{i}$ so that the meridian of $V_{i}$ is identified with a connected simple closed
curve representing $\alpha_{i}$.
\item Nothing is done to components $T_{i} \subset \mathcal{T}$ for which $\alpha_{i} = \ensuremath\infty$ and
components of $\partial X \setminus \mathcal{T}$.
\end{enumerate}
\item If $\alpha = (\alpha_{1},\dots,\alpha_{n})$ and $\alpha'= (\alpha'_{1},\dots,\alpha'_{n})$ are multislopes,
we say that $\alpha'$ is a {\it partial filling} of $\alpha$, which we will
denote as $\alpha' \pf \alpha$, if for each $i$, $\alpha'_{i} \in \{\alpha_{i},\infty\}$. If $\alpha' \pf \alpha$ and
$\alpha' \neq \alpha$ we say that $\alpha'$ is a {\it strict} partial filling of $\alpha$.
We will also use the notation $(\alpha_{1},\dots,\hat{\alpha}_{i},\dots,\alpha_{n})$ for the multislope
obtained from $(\alpha_{1},\dots,{\alpha}_{i},\dots,\alpha_{n})$ by replacing $\alpha_{i}$ with
$\ensuremath\infty$ (intuitively, toosing $\alpha_{i}$ out).
\item Assume that $X$ is hyperbolic. A multislope $\alpha$ is called {\it hyperbolic} if $X(\alpha)$ is hyperbolic;
$\alpha$ is called {\it totally hyperbolic} if every partial filling of $\alpha$ is hyperbolic.
\item Assume that $X$ is hyperbolic. A multislope $\alpha$ is called {\it non-hyperbolic}
if $X(\alpha)$ is not hyperbolic. If $\alpha$ is non-hyperbolic, and every strict partial filling of $\alpha$ is hyperbolic, then $\alpha$ is called {\it minimally non hyperbolic}. Minimally non hyperbolic fillings are
studies extensively in Section~\ref{sec:mnh}.
\item Let $\mathcal{T}' \subset \partial X$ be a union of components of $\partial X$ and $\alpha$
a multislope of $\mathcal{T}$. Then $\alpha$ defines a multislope on $\mathcal{T}'$
by removing the components of $\alpha$ that correspond to components of
$\mathcal{T} \setminus \mathcal{T}'$ and assigning the value $\ensuremath\infty$ to every
component of $\mathcal{T}' \setminus \mathcal{T}$.
This multislope is called the {\it restriction} of $\alpha$ to $\mathcal{T}'$ and
we will denote it as $\alpha|_{\mathcal{T}'}$.
In particular, we will denote the value of $\alpha$ on $T_{i}$ as $\alpha|_{T_{i}}$.
$\alpha|_{\mathcal{T}'}$
is also called the multislopes {\it induced} by $\alpha$ on $\mathcal{T}'$.
\item Induced multislopes also appear in a more general setting: let $F$ be a collection of tori in $\mbox{int}(X)$.
Suppose that $X$ cut open along $F$ consists of $N_{1}$ and $N_{2}$, that is: $X = N_{1} \cup_{F} N_{2}$.
Here we are {\it not} assuming that $N_{1}$ or $N_{2}$ is connected. Let $\alpha$ be a multislope of
components of $\partial X$ (denoted $\mathcal T$) so that, for each component $Y$ of $N_{2}$,
$Y(\alpha|_{\partial Y}) \cong D^{2} \times S^{1}$.
In other words, after filling, $N_{2}(\alpha|_{\partial N_{2}})$ consists of solid tori.
We will denote $(\mathcal{T} \cap \partial N_{1}) \cup F$ as $\mathcal{T}_{1} \subset \partial N_{1}$.
Then the multislope of $\mathcal{T}_{1}$ {\it induced} by $\alpha$ is the multislope defined by
$\alpha|_{\mathcal{T} \cap \partial N_{1}}$
(on the components of $\mathcal{T} \cap \partial N_{1}$) and the homology classes of the meridians of
$N_{2}(\alpha|_{\partial N_{2}})$ (on the components of $F$).
\item If $L \subset M$ is a link then a {\it multislope} of $L$ is a multislope of
$\partial N(L) \subset \partial E(L)$. By {\it surgery} on $L$ with multislope $\alpha$,
which we will denote as $L(\alpha)$, we mean $E(L)(\alpha)$.
\end{enumerate}
\subsection{JSJ-manifolds}
In this subsection we summarize the information we need about manifolds with non-trivial JSJ
decomposition. JSJ decompositions were studied by Jaco and Shalen~\cite{jacoshalen} and,
independently, by Johannson~\cite{Johannson}. We assume familiarity with this subject;
further details can be found in~\cite{JacoBook}. We summarize what we need in the definition below;
note that since we restrict our attention to manifolds with boundary tori, we may assume that the JSJ
decomposition is along tori (and no annuli).
\begin{dfn}[JSJ-manifold]
\label{dfn:jsj}
Let $X$ be a compact 3-manifold so that $\partial X$ consists of a (possibly empty) collection of tori. We say that
$X$ is a {\it JSJ-manifold} if $X$ is irreducible and the JSJ-decomposition of $X$ consists
of a non-empty collection of tori which we will denote as $\mathcal{T}$. In that case we also say that $X$
has a {\it non-trivial} JSJ decomposition. The manifolds obtained by cutting $X$ along $\mathcal{T}$
are called the {\it components of the torus decomposition of $X$}. The graph dual to the JSJ decomposition
of $X$ has one vertex for every component of the torus decomposition of $X$, and an edge for every
torus in $\mathcal{T}$; the edge corresponding to $T \in \mathcal{T}$ connects the (not necessarily distinct)
vertices that correspond to the components of the torus decomposition of $X$ that have $T$ in their boundary.
\end{dfn}
\subsection{Topological preliminaries}
We will need the following simple lemma which says that knot exteriors can't be
``linked'' with each other:
\begin{lem}
\label{lem:KnotExteriorsAreDisjoint}
Let $Y$ be a manifold and for $i=1,\dots,p$, let $E_{i}$ be a non-trivial knot exterior embedded in a ball
$D_{i} \subset Y$. Suppose that for $i \neq i'$, $E_{i} \cap E_{i'} = \emptyset$.
Then we may choose the balls $D_{i}$ so that for $i \neq i'$, $D_{i} \cap D_{i'} = \emptyset$.
\end{lem}
\begin{proof}
We assume that $E_{i} \cap E_{i'} = \emptyset$ during all the isotopies considered in this proof (for $i \neq i'$).
Assume, for a contradiction, that the lemma is false and let $Y$ and
$E_{i}$ form a counterexample that minimizes $p$; note that $p \geq 2$. Then
there exist disjoint balls $D_{1},\dots,D_{p-1}$ so that
$E_{i} \subset D_{i}$.
Assume first that there is no isotopy of $E_{p}$ so that
$E_{p} \cap (\cup_{i=1}^{p-1} D_{i}) = \emptyset$. Let $m_{i} \subset D_{i}$ be a meridian disk for $E_{i}$,
$i=1,\dots,p-1$; note that $D_{i}$ is isotopic to $E_{i} \cup N(m_{i})$.
Denote $M = \cup_{i=1}^{p-1} m_{i}$. Minimize $|M \cap E_{p}|$;
since $E_{p}$ cannot be isotoped to be disjoint from $\cup_{i=1}^{p-1} D_{i}$,
$|M \cap E_{p}| > 0$. Let $\delta \subset M$ be an innermost disk of $M \cap \partial E_{p}$.
If $\partial \delta$ is inessential in $\partial E_{p}$, we use an innermost disk from
$\partial E_{p}$ to surger $M$ and reduce $|M \cap E_{p}|$.
This gives new (and not necessarily isotopic)
meridian disks for $E_{1},\dots,E_{p-1}$; we still denote these disks by $m_{1},\dots,m_{p-1}$
and $\cup_{i=1}^{p-1}m_{i}$ by $M$. We replace the balls $D_{i}$ with $E_{i} \cup N(m_{i})$,
which we will continue to denote by $D_{i}$. We repeat this
process until one of the following holds:
\begin{enumerate}
\item $|M \cap E_{p}| > 0$ and $\partial\delta$ is essential in $\partial E_{p}$:
in that case, $\delta$ is a meridian disk for $E_{p}$.
Let $\hat D_{p} = E_{p} \cup N(\delta)$. Then $\hat D_{p} \cap (\cup_{i=1}^{p-1} E_{i}) = \emptyset$.
By isotopy, we move $M$ out of $\hat D_{p}$; thus we see that
$\{D_{i},\dots,D_{p-1},\hat D_{p}\}$ is a collection of disjointly embedded balls contradicting our assumption.
\item $|M \cap E_{p}| = 0$: in this case $E_{p} \cap (\cup_{i=1}^{p-1} D_{i}) = \emptyset$. Let $m_{p}$
be a meridian disk for $E_{p}$; by isotopy we move
$m_{p}$ out of $\cup_{i=1}^{p-1} D_{i}$. Then
$\{D_{i},\dots,D_{p-1},E_{p} \cup N(m_{p})\}$ is a collection of
disjointly embedded balls contradicting our assumption.
\end{enumerate}
\end{proof}
\begin{dfn}
\label{dfn:unlink}
Let $Y$ be a connected manifold. Let $U \subset Y$ be a link.
We say the $L$ is an {\it unlink} if the components of $L$
bound disjointly embedded disks.
\end{dfn}
In the following lemma we consider two links in a manifold $Y$, which we will denote as
$\mathcal{L}$ and $\mathcal{U}$. $\mathcal{U}$ is an unlink; hence we can perform
$1/n$ surgery about any component $K$ of $\mathcal{U}$ without changing
the ambient manifold; here the framing is chosen so that the boundary of the disk bound by
$K$ corresponds to $0/1$. After this surgery say that the components of $\mathcal{L}$
are {\it twisted about $K$ $n$ times}.
This process may be repeated on the other component of $\mathcal{U}$.
We assume that $Y \setminus \mbox{int}N(\mathcal{L} \cup \mathcal{U})$ is irreducible,
but since $\mathcal{L}$ and $\mathcal{U}$ play different roles we phrase this condition differently,
see Condition~(2) below.
We are interested in the effect on $\mathcal{L}$ when $n$ is large:
\begin{lem}
\label{lem:TwistingToGetIrreducible}
Let $Y$ be a manifold and $\mathcal{L}$, $\mathcal{U}$ links in $Y$. Assume that the following conditions hold:
\begin{enumerate}
\item $\mathcal{U}$ is an unlink.
\item $\mathcal{L}$ is irreducible in the complement of $\mathcal{U}$.
\end{enumerate}
Then the link $\mathcal{L}'$ obtained from $\mathcal{L}$ by twisting its components about each component of
$\mathcal{U}$ sufficiently many times has an irreducible exterior.
\end{lem}
\begin{proof}
We induct on $|\mathcal{U}|$, the number of components in $\mathcal{U}$. If $|\mathcal{U}| = 0$
there is nothing to prove. Otherwise, let $K$ be a component of $\mathcal{U}$.
We will denote $Y \setminus \mbox{int}N(\mathcal{L} \cup \mathcal{U})$ as $X$. Let $\alpha$
be any slope of $\partial N(K)$ so that the manifold obtained by filling $\partial N(K)$ with slope
$\alpha$ is reducible. Denote a reducing sphere that minimizes the intersection with the attached solid torus
by $S$. Then it is easy to see that $S_{E} = S \cap Y \setminus \mbox{int}N(\mathcal{L} \cup \mathcal{U})$
is an essential surface with $\partial S_{E} \subset \partial N(K)$ a non-empty collection of essential curves,
all defining the slope $\alpha$. By Hatcher~\cite{hatcher} only finitely many slopes of $\partial N(K)$
bounds such a surface. Twisting $n$ times about $K$ is equivalent to filling $\partial N(K)$ with slope
$\alpha = 1/n$ (for any choice of basis for $H_{1}(\partial N(K);\mathbb Z)$ for which the
boundary of the disk bound by $K$ corresponds to $0/1$). Thus for all but
finitely many values of $n$ the manifold obtained is irreducible. We pick one such $n$, and after twisting
$\mathcal{L}$ about $K$ $n$ times, we remove $K$ from $\mathcal{U}$. Induction completes the proof.
\end{proof}
The following lemma was proved in~\cite{BHW} by Bleiler, Hodgson and Weeks.
It says that the action induced by the mapping class group of $M$ on the
slopes on $\partial M$ (which is assumed to be a single torus) is trivial.
We bring a new argument here. Our argument holds for
many non-hyperbolic manifolds as well: all we require is that $M$
does not admit infinitely many fillings that result in diffeomorphic manifolds;
this is well known to hold for hyperbolic manifolds as well as all Seifert
fibered spaces except $D^{2} \times S^{1}$. In Conclusion~(2) below
we use a basis for $H_1(\partial M)$ to identify the slopes of $\partial M$
with $\bar{\mathbb{Q}} = \mathbb{Q} \cup \{1/0\}$.
\begin{lem}
\label{lem:slopes}
Let $M$ be a hyperbolic manifold with $\partial M$ a single torus. Let $\phi:M \to M$
be a orientation preserving diffeomorphism.
Then for any simple closed curve $\gamma \subset \partial M$,
$\gamma$ and $\phi\circ \gamma$ represent the same slope.
Moreover, if $M$ admits orientation reversing diffeomorphisms then one of the following holds:
\begin{enumerate}
\item For any orientation reversing diffeomorphism $\gamma$ and
$\phi\circ \gamma$ represent the same slope.
\item There is a basis for $H_{1}(\partial M)$ so that
for any orientation reversing diffeomorphism $\phi$,
if a curve $\gamma$ represents the slope $p/q$ then
$\phi\circ \gamma$ represent the slope $- p/q$.
\end{enumerate}
\end{lem}
\begin{proof}
We use the notation introduced in the statement of the lemma.
By a well known homological argument $\ker i_{*} \cong \mathbb Z$, where here
$i_{*}:H_{1}(\partial M;\mathbb Z) \to H_{1}(M;\mathbb Z)$ is the homomorphism
induce by the inclusion; moreover, any generator of $\ker i_{*}$ is primitive in
$H_{1}(\partial M;\mathbb Z) $. Let $l$ be a simple curve so that $[l]$ is a generator of
$\ker i_{*}$ (we denote homology classes with $[ \ ]$). Then $[\phi \circ l] = \pm [l]$.
Let $m$ be a simple curve so that $[l]$ and $[m]$ generate $H_{1}(\partial M;\mathbb Z)$.
Since $\phi$ induces an isomorphism on $H_{1}(\partial M;\mathbb Z)$,
$[\phi \circ m]$ and $[\phi \circ l]$ generate $H_{1}(\partial M;\mathbb Z)$.
Thus $[\phi \circ m] = \pm [m] + n [l]$, for some $n \in \mathbb Z$.
If $n \neq \pm 1$, the orbit of $[m]$ is infinite. But then the slope
represented by $m$ has an infinite orbit, and hence $M$ admits infinitely many fillings
(namely, fillings along the slopes represented by $\phi^{k}\circ m$ for $k \in \mathbb Z$)
that produce diffeomorphic manifolds. As $M$ is hyperbolic, this is impossible.
Thus $[\phi\circ m] = \pm m$.
We will use $[m]$ and $[l]$ as a basis for $H_{1}(\partial M)$.
We conclude that $\phi_*$ is one of the following maps:
\begin{enumerate}
\item $\phi_{*}((p,q)) = (p,q)$.
\item $\phi_{*}((p,q)) = (-p,-q)$.
\item $\phi_{*}((p,q)) = (-p,q)$.
\item $\phi_{*}((p,q)) = (p,-q)$.
\end{enumerate}
Note that cases~(3) and ~(4) imply that $\phi$ is orientation reversing;
hence if $\phi$ is orientation preserving then
$\phi_{*}:H_{1}(\partial M;\mathbb Z) \to H_{1}(\partial M;\mathbb Z)$
is either the identity or the antipodal map; since a slope is defined as the homology class
of an {\it unoriented} curve, both maps fix all slopes.
All that remain is to show that if $\phi_{1}$ and $\phi_{2}$ are orientation reversing
diffeomorphisms of $M$, then $\phi_{1 *}$ and $\phi_{2*}$ are both
as in Cases~(1) or~(2), or both as in Cases~(3) or~(4). If one is
as in Cases~(1) or~(2) and the other as in Cases~(3) or~(4), then
$\phi_{2}^{-1} \circ \phi_{1}$ is an orientation preserving diffeomorphism
and $(\phi_{2}^{-1} \circ \phi_{1})_{*}$ is as in Case~(3) or~(4), which is impossible.
\end{proof}
\subsection{Cores of solid tori}
In this subsection we prove the following necessary condition for a curve in $T^{2} \times [0,1]$
to become a core of the solid torus obtained by filling:
\begin{lem}
\label{lem:CoresOfSolidTori}
Let $K \subset T^{2} \times [0,1]$ and
$c \subset T^{2} \times \{0\}$ be curves, and assume that $c$ is simple and
essential.
Let $V$ be the solid torus obtained by filling $T^{2} \times \{0\}$ along $c$.
Suppose that $[c]$ and $[K]$ do not generate $H_{1}(T^{2} \times [0,1];\mathbb Z)$.
Then $K$ is not isotopic to a core of $V$.
\end{lem}
\begin{proof}
We identify $T^{2} \times \{1\}$ with $T^{2}$. We will denote the projections
of $c$ and $K$ to $T^{2}$ as $c'$ and $K'$ respectively.
We obtain two classes in $H_{1}(T^{2};\mathbb Z)$, defined up-to sign,
which we will denote as $\pm [c']$ and $\pm [K']$.
Suppose that $K$ is isotopic to a core of $V$.
Then the signed intersection of $K$ and the meridian disk of $V$ is $\pm 1$.
Up-to isotopy, $c'$ is the boundary of the meridian disk of $V$.
Therefore, the signed intersection of $K'$ with $c'$
as curves in $T^{2}$ is $\pm 1$ (the sign may not agree with that of the intersection of $K$ and the meridian disk of $V$).
Any class of $H_{1}(T^{2};\mathbb{Z})$ is represented by $n$ parallel copies of a simple closed curve (possibly $n=0$).
Let $\gamma$ be a simple closed curve so that $\pm [n\gamma] = \pm [c']$. Since $c'$ intersects
$K'$ algebraically once, $n = \pm 1$ and we may assume that $\gamma$ and $K'$ intersect once.
Thus $[\gamma]$ and $[K']$ generate $H_{1}(T^{2};\mathbb Z)$.
Since $[\gamma] = [c'] = [c]$ and $[K'] = [K]$,
$[c]$ and $[K]$ generate $H_{1}(T^{2} \times [0,1];\mathbb Z)$.
\end{proof}
\subsection{Hyperbolic Dehn Surgery}
Let $M$ be a one cusped hyperbolic manifold. It is well known that for all but finitely many slopes
on $\partial M$, $M(\beta)$ is hyperbolic and the core of the attached solid torus, which we will denote as
$\gamma$, is a geodesic. As we vary $\beta$, the length of the geodesics $\gamma$ obtained fulfill
$$\lim_{l(\beta) \to \infty} l(\gamma) = 0$$
where here $l(\beta)$ is measured in the Euclidean metric induced on $\partial M$ after some truncation
of the cusp. Moreover, after an appropriate choice of basepoints,
as $l(\beta) \to \infty$, the manifolds $M(\beta)$ converge in the sense of
Gromov--Hausdorff to $M$. With this we obtain the following lemma, which is well known to many experts,
but since we could not find a reference we sketch its proof here.
\begin{lem}
\label{lem:HyperbolicDehnSurgery}
With the notation as above, there exists $\epsilon>0$ and a finite set of slopes of $\partial M$,
which we will denote as $B_{f}$, so that for any slopes $\beta$ of $\partial M$, if $\beta \not \in B_{f}$
then the following three conditions hold:
\begin{enumerate}
\item $M(\beta)$ is hyperbolic.
\item $l(\gamma) < \epsilon$.
\item If $\delta \subset M(\beta)$ is a geodesic and $l(\delta) < \epsilon$, then $\delta = \gamma^{n}$ for some $n$.
\end{enumerate}
\end{lem}
\begin{proof}[Sketch of proof]
Fix $\mu>0$ a Margulis constant that is shorter than half the length of the shortest geodesic in $M$.
The {\it thick part} of $M$, which we will denote $M_{\geq \mu}$, consists of all the points
of $M$ that have radius of injectivity at least $\mu$; note that $M_{\geq \mu}$ is $M$ with its
cusp truncated. The thick part of $M(\beta_{i})$, which we will denote as
$M(\beta_{i})_{\geq \mu}$, is defined similarly.
By the discussion above,~(1) and ~(2) are established in the literature. Assume, for a contradiction, that
there does not exist a finite set $B_{f}$ for which~(3)
holds. Then there is a sequence $\beta_{i}$ of slopes of $\partial M$ with $l(\beta_{i}) \to \infty$,
and geodesics $\delta_{i} \subset M(\beta_{i})$ so that $l(\delta_{i}) < 1/i$ and $\delta_{i}$ is
not a power of $\gamma_{i}$, the core of the attached solid torus.
Let $V_{i}$ be the component of $M(\beta_{i}) \setminus M(\beta_{i})_{\geq \mu}$ that contains
$\gamma_{i}$. We will denote $N_{\frac{1}{2i}}(M(\beta_{i}) \setminus V_{i})$ as $N_{i}$, where
here $N_{\frac{1}{2i}}$ denotes the $\frac{1}{2i}$ neighborhood. By construction, $\delta_{i} \subset N_{i}$.
For an appropriate choice of basepoints $x_{i} \in M(\beta_{i})$ and $x \in M$, $(M(\beta_{i}),x_{i})$
converges to $(M,x)$ is the Gromov--Hausdorff sense.
Then $f_{i} \circ \delta_{i}$
are closed curves in $M_{\geq \mu}$ with $l(f_{i} \circ \delta_{i}) \to 0$. Thus, for sufficiently large $i$,
$l(f_{i} \circ \delta_{i}) < \mu$, and hence $f_{i} \circ \delta_{i}$ is null homotopic
or is homotopic into the cusp. In the former case, let $D \subset M_{\geq \mu}$ be an immersed
disk whose boundary is $f_{i} \circ \delta_{i}$. By isotopy we may move $D$ out of the cusp.
The image of $D$ under $f_{i}^{-1}$
shows that $\delta_{i}$ bounds an immersed disk, which is a contradiction. In the latter case,
we may use an immersed annulus given by the trace of the homotopy of $f_{i} \circ \delta_{i}$
to the cusp to conclude that $\delta_{i}$ is homotopic into a neighborhood of the core
of the attached solid torus; this implies that the geodesic $\delta_{i}$ is itself a power of that core.
\end{proof}
\section{Bounded sets in the Farey graph}
\label{sec:BoundedSets}
\bigskip
\begin{figure}[h!]
\includegraphics[height=3.5in]{FareywithLabels}
\caption{The Farey tessellation}
\label{fig:Farey}
\end{figure}
\noindent
The Farey graph is a connected graph whose vertices encodes the slopes of the torus;
we begin this section with a brief description of this well known construction,
see Figure~\ref{fig:Farey}.
By viewing each edge as a length one interval the Farey graph induces a metric on the slopes
which is instrumental for our study: throughout this paper we make extensive use of sets of
slopes that are bounded in this metric. In~\cite{rieckyamashita} we noted that
that any cover between tori induces a bijection on their slopes and argued that
branched covers between manifolds can be completed after filling if and only if the
slopes filled correspond under this bijection; in Subsection~\ref{subsection:FareyAndCovering}
we recall these facts and prove that this bijection is a bilipschitz map, and hence the image
of a bounded set is bounded (in either direction). In Subsection~\ref{subsection:boundedsets}
we prove that bounded sets are closed under certain operations, notably Dehn twists and
adding slopes of bounded intersection (see Proposition~\ref{prop:PropertiesOfBoundedSets}
for a precise statement). Moreover, we show that bounded sets form the smallest non empty
collection of sets that is closed under these operations;
hence, from our point of view, they are the smallest collection of sets we may use.
\bigskip
\noindent
We now describe the Farey graph; it is best seen as embedded in
$\mathbb H^{2} \cup S^{1}_{\infty}$ as the 1-skeleton of the Farey tessellation of
the hyperbolic plane, although
the metric we will use (as described above) is {\it not} induced by the hyperbolic metric.
For this construction see Figure~\ref{fig:Farey}. Pick an ideal triangle in $\mathbb H^{2}$
and label its vertices as $0/1$, $1/0$, and $1/1$. The Farey tessellation is constructed
recursively: after reflecting a triangle by an edge whose endpoints are labeled
$p/q$ and $r/s$, we obtain a new triangle and
label the new vertex by $(p \pm r)/(q \pm s)$; the sign
depends on the direction of the reflection. This addition rule
is induced by the addition in $\mathbb Z \times \mathbb Z$.
At the end of the day we obtain a tessellation of $\mathbb H^{2}$ by ideal triangles and
it is a well known consequence of Euclid's Algorithm that every element of
$\bar{\mathbb{Q}} = \mathbb{Q} \cup \{1/0\}$ appears as
the label for exactly one vertex. The {\it Farey graph} is the graph
given by the 1-skeleton of the Farey tessellation.
After choosing a basis for $H_{1}(T^{2};\mathbb Z)$ we
identify the slopes of $T^{2}$ with $\bar{\mathbb{Q}}$.
Thus we have a bijection between the slopes of $T^{2}$ and the vertices of the
Farey tessellation. It is easy to see that the claims below
do not depend on the choice of basis, as a change of basis induces an isomorphism
of the Farey graphs.
Let $x$ and $y$ be vertices that correspond to slopes $\alpha_{1}$
and $\alpha_{2}$; it is well known that $x$ and $y$ are connected by an edge in the Farey
tessellation if and only if $\alpha_{1}$ and $\alpha_{2}$ can be represented by
curves that intersect exactly once.
The {\it distance}
in the Farey tessellation is the minimal number of edges traversed to get
from one vertex to another.
We define the distance between slopes $\alpha_{1}$ and $\alpha_{2}$ to be the distance
between the corresponding vertices of the Farey tessellation and denote it as
$$\ensuremath{d_{\mbox{Farey}}}(\alpha_{1},\alpha_{2}).$$
Using this mertic we can define:
\begin{dfn}
\label{dfn:bounded}
A set of slopes is called {\it bounded} if it is has a bounded diameter using the
metric induced by $\ensuremath{d_{\mbox{Farey}}}$.
\end{dfn}
It is well known that the Farey graph has infinite diameter, hence:
\begin{lem}
If $B$ is a bounded set of slopes and then there are infinitely many slopes not in $B$.
\end{lem}
\subsection{Coverings and slopes}
\label{subsection:FareyAndCovering}
For this subsection fix tori $T$, $T'$ and $\phi:T \to T'$ a cover.
In~\cite{rieckyamashita} we showed that $\phi$ induces a bijection between the
slopes of $T$ and those of $T'$; we refer the reader to that paper for a detailed discussion.
The correspondence is defined as follows: let $\alpha$ be a slope of $T$ and $\gamma$ an essential connected
simple closed curve on $T$ representing $\alpha$. Then $\phi \circ \gamma$ is an essential connected
(not necessarily simple) closed curve on $T'$, and hence there exist a positive integer $m$ and
$\beta'$, a connected simple closed curve on $T'$,
so that $(\beta')^{m}$ is freely homotpic to $\phi \circ \gamma$. We define the
slope represented by $\beta'$ to be the slope that corresponds to $\alpha$. Conversely, let $\alpha'$
be a slope of $T'$ represented by an essential simple closed curve $\gamma'$. Then $\phi^{-1}(\gamma')$
is an essential (not necessarily connected) simple closed curve. We define
the slope represented by a component of $\phi^{-1}(\gamma')$ to be the slope that corresponds to $\alpha'$.
It is not hard to see that these correspondences are inverses of each other.
We refer to slopes that correspond under this bijection as {\it corresponding slopes}.
The branched covers between manifolds
that we consider in this paper induce covers on boundary components.
In~\cite{rieckyamashita} we showed that a branched cover $X \to E$ (where $X$ and $E$ are manifolds
with toral boundary and the branch set is closed, that is, disjoint from $\partial E$)
extends to a branched cover after filling if and only if the slopes filled are corresponding slopes.
{\it This simple fact will often be used without reference throughout this paper.}
\begin{rmk}
In the example below it will be convenient to take an alternate view of the correspondence between the slopes of
$T$ and those of $T'$, where $\phi: T \to T$ is a cover.
By lifting $\phi:T \to T'$ appropriately to the universal covers
we obtain an matrix in $SL(2,\mathbb Q)$. Any such matrix gives a correspondence between
lines of rational slope in the universal covers, which are naturally identified with slopes on the tori;
the reader can verify that this correspondence is equivalent to the correspomndence described above.
\end{rmk}
Let $T$ and $T'$ be tori and $\phi:T \to T'$ a cover as above.
We will often consider bounded sets of $T$ or $T'$; our goal in this subsection
is to understand their behavior under the correspondence induced by $\phi$.
We begin with a simple example: consider $T$ and $T'$ as
$\mathbb{R}^{2}/\sim$, where $\sim$ is given by $(x,y)\sim (x+n,y+m)$ for $n,m \in \mathbb Z$.
Let $\phi:T \to T'$ be the double cover induced by the map $\tilde\phi:\mathbb{R}^{2} \to \mathbb{R}^{2}$
given by $\tilde\phi((x,y)) = (x,2y)$. The four segments in $\mathbb{R}^{2}$ connecting
$(0,0)$ to $(1,0)$, $(2,1)$, $(1,1)$, and $(0,1)$ map to four curves on $T$, defining slopes
that can be naturally denoted as $0/1$, $1/2$, $1/1$, and $1/0$, see Figure~\ref{fig:Farey2}.
\begin{figure}
\includegraphics[height=2.5in]{Farey2}
\caption{The two quadrilaterals in the Farey tessellation}
\label{fig:Farey2}
\end{figure}
The images these segments
under $\tilde\phi$ are the segments connecting $(0,0)$ to $(1,0)$, $(2,2)$, $(1,2)$, and $(0,2)$,
and the corresponding slopes are $0/1$, $1/1$, $2/1$, and $1/0$.
Thus the correspondence maps a quadrilateral in the Farey graph of $T$
to a quadrilateral in the Farey graph of $T'$; each quadrilateral has exactly one diagonal edge.
In the first, the diagonal edge connects $0/1$ to $1/1$, and in the second it
connects $1/0$ to $1/1$.
These edges do {\it not} correspond, showing that the correspondence
between the slopes of $T$ and those of $T'$ (viewed as a bijection between the vertices of
the Farey graphs) does {\it not} induce an isomorphism of the Farey
graphs. Equivalently, the bijection between the slopes of $T$ and $T'$ (considered as metric
spaces with the metric $\ensuremath{d_{\mbox{Farey}}}$) is {\it not} an isometry. It is easy it see that in this example
some distances decrease while others increase.
Since we always consider Farey graphs as metric spaces with the metric $\ensuremath{d_{\mbox{Farey}}}$ we need
the following lemma:
\begin{lem}
\label{lem:CorrespondingSlopesFeray}
Let $T$, $T'$ be tori and $\phi:T \to T'$ be a cover.
Then the correspondence between slopes of $T$ and $T'$ is bilipschitz
and hence it sends bounded sets of $T$ to
bounded sets of $T'$ and vice versa.
\end{lem}
\begin{proof}
For convenience we endow $T'$ with a Euclidean metric and $T$ with the pullback metric.
Since we are only interested in the homology classes represented by curves, throughout the proof of
this lemma we assume as we may that the essential curves of $T$ and $T'$ considered are geodesic.
We first show that the map from the slopes of $T$ to those of $T'$ is Lipschitz.
Fix a positive integer $d$ and two slopes of $T$ of distance exactly $d$, which we will
denote as $\alpha_{0}$ and $\alpha_{d}$. By definition of $\ensuremath{d_{\mbox{Farey}}}$,
there exist slopes $\alpha_{1},\dots,\alpha_{d-1}$ so that
$\ensuremath{d_{\mbox{Farey}}}(\alpha_{i-1},\alpha_{i}) = 1$ (for $i=1,\dots,d$).
Let $\gamma_{0},\dots,\gamma_{d}$ be geodesics on $T$ representing
$\alpha_{0},\dots,\alpha_{d}$ (respectively). Since geodesics on a torus minimize intersection,
$\ensuremath{d_{\mbox{Farey}}}(\alpha_{i-1},\alpha_{i}) = 1$ is equivalent to
$|\gamma_{i-1}\cap \gamma_{i}|=1$.
Let $\gamma_{i}'$ be a connected simple closed geodesic on $T'$
and $m_{i}$ a positive integer so that $\phi(\gamma_{i})$ is obtained by
traversing $\gamma_{i}'$ exactly $m_{i}$ times.
By definition, $\gamma_{i}'$ is a geodesic representing $\alpha_{i}'$,
where here $\alpha_{i}'$ is the slope of $T'$ that corresponds to $\alpha_{i}$.
We will denote $|\gamma'_{i-1} \cap \gamma'_{i}|$ as $c_{i}$.
Let $\Gamma_{i}$ denote $\phi^{-1}(\phi(\gamma'_{i}))$ and denote the number of components of
$\Gamma_{i}$ as $n_{i}$.
Each component of $\Gamma_{i}$ is a geodesic parallel to $\gamma_{i}$ and is an $m_{i}$ fold
cover of $\beta_{i}$; it follows that $n_{i} m_{i} = \deg(\phi)$, and in particular
$$n_{i} \leq \deg(\phi).$$
Since $\Gamma_{i-1} \cap \Gamma_{i} = \phi^{-1}(\gamma_{i-1} \cap \gamma_{i})$,
$|\Gamma_{i-1} \cap \Gamma_{i}| = c_{i} \deg(\phi)$.
Since $|\gamma_{i-1} \cap \gamma_{i}|=1$, every component of $\Gamma_{i-1}$ intersects
every component of $\Gamma_{i}$ exactly once; if follows that
$|\Gamma_{i-1} \cap \Gamma_{i}| = n_{i-1} n_{i}$, the number of pairs of
curves from $\Gamma_{i-1}$ and $\Gamma_{i}$. Combining these facts we see
that
$$c_{i} \deg(\phi) = |\Gamma_{i-1} \cap \Gamma_{i}| = n_{i-1}n_{i} \leq \deg(\phi)^{2},$$
showing that $c_{i} \leq \deg(\phi)$.
Thus $\Delta(\gamma_{i-1}',\gamma_{i}') \leq |\gamma_{i-1}' \cap \gamma_{i}'| \leq \deg(\phi)$.
It then follows from Euclid's algorithm
that $\ensuremath{d_{\mbox{Farey}}}(\alpha_{i-1}',\alpha_{i}') \leq 2 \log_{2}{(\deg(\phi))}$.
(For the relation between Euclid's Algorithm and the Farey tessellation see, for example,
\cite{hardywright}.)
We get:
\begin{eqnarray*}
\ensuremath{d_{\mbox{Farey}}}(\alpha'_{0},\alpha'_{d}) &\leq& \sum_{i=1}^{d} \ensuremath{d_{\mbox{Farey}}}(\alpha'_{i-1},\alpha'_{i}) \\
&\leq& 2 \log_{2}(\deg(\phi)) d \\
&=& 2 \log_{2}(\deg(\phi)) \ensuremath{d_{\mbox{Farey}}}(\alpha_{0},\alpha_{d}).
\end{eqnarray*}
As $d$, $\alpha_{0},$
and $\alpha_{d}$ were arbitrary we conclude that:
$$\ensuremath{d_{\mbox{Farey}}}(\alpha_{0}',\alpha_{d}') \leq 2 \log_{2}{(\deg(\phi))} \ensuremath{d_{\mbox{Farey}}}(\alpha_{0},\alpha_{d}),$$
showing that the map from the slopes of $T$ to the slopes of $T'$ is Lipschitz with
constant $2\log_{2}(\deg(\phi))$.
\medskip
The converse is essentially identical and we only sketch it here.
Similar to the argument above, fix $d$
and let $\alpha_{0}',\alpha_{d}'$ be slopes of $T'$ so that $\ensuremath{d_{\mbox{Farey}}}(\alpha'_{0},\alpha'_{d})=d$.
Let $\alpha_{0}',\dots,\alpha_{d}'$
be a sequence of slopes that realizes the distance, that is, $\ensuremath{d_{\mbox{Farey}}}(\alpha_{i-1}',\alpha_{i}') = 1$.
Let $\gamma_{i}'$ be a geodesic on $T'$ that represents $\alpha_{i}'$;
then $|\gamma_{i-1}' \cap \gamma_{i}'|=1$.
Let $\gamma_{i}$ be a component of $\phi^{-1}(\gamma_{i})$. By definition, $\gamma_{i}$
represents $\alpha_{i}$, the slope of $T$ that corresponds to $\alpha_{i}'$.
Since $\gamma_{i-1} \cap \gamma_{i} \subset \phi^{-1}(\gamma'_{i-1} \cap \gamma_{i}')$ we have:
$$|\gamma_{i-1} \cap \gamma_{i}| \leq |\phi^{-1}(\gamma'_{i-1} \cap \gamma_{i}')|
= \deg(\phi) |\gamma'_{i-1} \cap \gamma_{i}'| = \deg(\phi).$$
As above this implies that $\ensuremath{d_{\mbox{Farey}}}(\alpha_{i-1},\alpha_{i}) \leq 2 \log_{2}{(\deg(\phi))}$,
and hence
$$\ensuremath{d_{\mbox{Farey}}}(\alpha_{0},\alpha_{d}) \leq 2 \log_{2}{(\deg(\phi))} \ensuremath{d_{\mbox{Farey}}}(\alpha_{0}',\alpha'_{d}),$$
showing that the map from the slopes of $T'$ to the slopes of $T$ is Lipschitz with
constant $2\log_{2}(\deg(\phi))$.
\end{proof}
\bigskip
\noindent
\subsection{On bounded sets}
\label{subsection:boundedsets}
In this subsection we study properties of bounded sets, that is, sets of slopes that are bounded
in the metric $\ensuremath{d_{\mbox{Farey}}}$ (as defined in~\ref{dfn:bounded}).
Our main goal is to show (Proposition~\ref{prop:PropertiesOfBoundedSets}) that bounded sets are closed
under certain operations, the most important of which is Dehn twists, which we explain below.
We then note (Lemma~\ref{lem:BoundedIsNecessary})
that any non-empty collection of sets of slopes
that is closed under the operations discussed in Proposition~\ref{prop:PropertiesOfBoundedSets}
contains all bounded sets; therefore boundedness
is the {\it weakest} condition that suffices for our work.
Let $\alpha$ be a slope on $T$ represented by a connected simple closed curve $\gamma$.
By definition $\alpha$ determines $\gamma$ up-to isotopy. Hence the Dehn twist about $\gamma$
is completely determined by $\alpha$. The Dehn twist about $\gamma$ induces
a map on the slopes of $T$, which we will denote as $D_{\alpha}$. Although the proper way
to refer to $D_{\alpha}$ is ``the map induced on the slopes of $T$ by Dehn twisting
about a simple connected curve representing $\alpha$'' we will, for simplicity's sake,
refer to it as {\it the Dehn twist about $\alpha$}.
Since any Dehn twist is a homeomorphism of $T$, it sends
any pair of curves that intersect once to a pair of curves that intersect once;
hence, for any slope $\alpha$, $D_{\alpha}$ induces a graph isomorphism
of the Farey graph, and in particular
$D_{\alpha}$ induces an isometry on the slopes
of $T$ with the metric given by $\ensuremath{d_{\mbox{Farey}}}$.
We are now ready to state:
\begin{prop}
\label{prop:PropertiesOfBoundedSets}
Let $T^{2}$ be a torus. The following conditions hold for sets of slopes on $T^{2}$:
\begin{enumerate}
\item A subset of a bounded set is bounded.
\item Finite unions of bounded sets are bounded.
\item For any slope $\alpha$ of $T^{2}$ and any bounded set of slopes $B$, the set
$$\cup_{n \in \mathbb Z}\{D_{\alpha}^{n}(\beta) \ | \ \beta \in B \}$$
is bounded (where $D_{\alpha}$ is as above).
\item For any integer $c \geq 0$ and any bounded set $B$, the set of slopes
$$\{\alpha \ | \ \exists \beta \in B, \Delta(\alpha,\beta) \leq c \}$$
is bounded (where here we use $\Delta(\alpha,\beta)$ to represent the geometric intersection
between simple closed curves that represent $\alpha$ and $\beta$).
\end{enumerate}
\end{prop}
\begin{proof}
\begin{enumerate}
\item Obvious from the definition of a metric space.
\item Obvious from the definition of a metric space.
\item Since $D_{\alpha}$ is induced by a homeomorphism
of $T^{2}$, it is clearly an isometry of the Farey graph;
moreover, $\alpha$ is fixed under $D_{\alpha}$.
Let $d>0$ be the diameter of $B$; fix $\beta \in B$.
By the triangle inequality, for any $n_{1},n_{2}\in\mathbb Z$
and any $\beta_{1},\beta_{2} \in B$ we have:
\begin{eqnarray*}
\ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta_{1}),D_{\alpha}^{n_{2}}(\beta_{2}))
&\leq& \ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta_{1}),D_{\alpha}^{n_{1}}(\beta)) \\
&& +\ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta),\alpha) \\
& & + \ensuremath{d_{\mbox{Farey}}}(\alpha,D_{\alpha}^{n_{2}}(\beta)) \\
&& + \ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{2}}(\beta),D_{\alpha}^{n_{2}}(\beta_{2})).
\end{eqnarray*}
Since $\alpha$ is fixed under the isometry $D^{n_{1}}_{\alpha}$, for the second term
we have
$$\ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta),\alpha) =
\ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta),D^{n_{1}}_{\alpha}(\alpha)) =
\ensuremath{d_{\mbox{Farey}}}(\beta,\alpha).$$
Similarly, the third term is bounded above by $\ensuremath{d_{\mbox{Farey}}}(\beta,\alpha)$.
For the first term we have
$$ \ensuremath{d_{\mbox{Farey}}}(D_{\alpha}^{n_{1}}(\beta_{1}),D_{\alpha}^{n_{1}}(\beta)) \leq
\ensuremath{d_{\mbox{Farey}}}(\beta_{1},\beta) \leq d.$$
Similarly, the fourth term is bounded by $d$.
Combining these bounds we see that set
$\cup_{n \in \mathbb Z}\{D_{\alpha}^{n}(\beta) \ | \ \beta \in B \}$
is bounded with diameter at most
$$2d+2\ensuremath{d_{\mbox{Farey}}}(\alpha,\beta).$$
\item We first prove the claim when $B$ has only one element which we will
denote as $\beta$
By an appropriate choice of basis for $H_{1}(T^{2};\mathbb Z)$ we may identify the slopes of $T$
with $\mathbb Q \cup \{1/0\}$ so that $\beta$ corresponds to $1/0$. It is well know that for any slope
$p/q$, $\ensuremath{d_{\mbox{Farey}}}(1/0,p/q)$ is exactly the length of the shortest continued fraction expansion of $p/q$
(see, for example, \cite{series}).
On the other hand, $\Delta(1/0,p/q) = |\det (\begin{smallmatrix} 1&p\\ 0&q \end{smallmatrix})| = |q|$.
Thus the slopes under consideration correspond to $p/q$ with $|q| \leq c$.
By a well known application of Euclid's Algorithm, such a number has a continued
fraction expansion of length at most $2 \log_{2}(|q|)$. Thus every slope in $B$
has distance at most $2 \log_{2}(|q|)$ from $\beta$, showing that $B$ is bounded with diameter
at most $4 \log_{2}(c)$.
For the general case, let $B$ be a bounded set of slopes and $c \geq 0$ an integer.
We will denote the diameter
of $B$ as $d$. Let $\alpha_{1}$ and $\alpha_{2}$ be slopes in
$\{\alpha \ | \ \exists \beta \in B, \Delta(\alpha,\beta) \leq c \}$. Let $\beta_{1}$
and $\beta_{2}$ be slopes of $B$ so that $\Delta(\alpha_{1},\beta_{1}) \leq c$ and
$\Delta(\alpha_{2},\beta_{2}) \leq c$. By the argument above,
$\ensuremath{d_{\mbox{Farey}}}(\alpha_{1},\beta_{1}), \ensuremath{d_{\mbox{Farey}}}(\alpha_{2},\beta_{2}) \leq 4 \log_{2}(c)$.
Since $\beta_{1},\beta_{2} \in B$,
$\ensuremath{d_{\mbox{Farey}}}(\beta_{1},\beta_{2}) \leq d$. By the triangle inequality
$\{\alpha \ | \ \exists \beta \in B, \Delta(\alpha,\beta) \leq c \}$
in bounded with diameter at most $4 \log_{2}(c) + d$.
\end{enumerate}
\end{proof}
\bigskip\noindent
The reader may wonder if ``bounded sets'' is the right concept to use. First we give an example that will explain
why using unbounded sets may be tricky. Using the upper half plane model of $\mathbb H^{2}$, we construct the
Farey tessellation starting with the triangle 0, 1, and $1/0$ (the point at infinity).
Then the slopes are naturally identified with $\mathbb Q \cup 1/0$; this identification
is used throughout this example.
Let $q_{i}'$ be an enumeration of $\mathbb{Q} \cap [0,1)$, $i=1,2,\dots$. Let $D_{1/0}$ denote
the map on the slopes induced by Dehn twist about $1/0$; it is not hard to see that for any slope $\alpha \neq 1/0$,
$D_{1/0}(\alpha) = \alpha+1$. Let $q_{i} = D^{i}(q_{i}')$. Finally, let $S = \{q_{i}\}$.
Now $S$ is a fairly ``thin'' set of slopes; it has only one member in every interval $[n,n+1), n=1,2,\dots$, and so
its only accumulation point is $1/0$. However, if we allow twisting about $1/0$, we see the following: for any slope
$\alpha \neq 1/0$,
there is a unique $j \in \mathbb Z$, so that $\alpha + j \in [0,1)$. Hence,
there is a unique $i$ so that $\alpha+j = q_{i}'$. Thus
$\alpha = D_{1/0}^{-j}(q_{i}') = D_{1/0}^{-(i+j)}(q_{i})$. In other words,
$\{D_{1/0}^{n}(q_{i}) | q_{i} \in S, n \in \mathbb Z\}$ is the set of {\it all} slopes but $1/0$.
As we shall see below repeatedly, when considering cosmetic surgery
one must allow for Dehn twists; thus, the analogue
of Proposition~\ref{prop:PropertiesOfBoundedSets} for the set $S$ is
very much false, and such a set
cannot be used in our work.
On the other hand we have the following lemma, that tells us that all
bounded sets {\it must} be permitted; in that sense,
our results cannot be improved:
\begin{lem}
\label{lem:BoundedIsNecessary}
Let $S$ denote the set of all slopes and suppose $Z \subset \mathcal{P}(S)$ fulfills
the following condition:
\begin{enumerate}
\item For some slopes $\alpha$, $\{\alpha\} \in Z$.
\item $Z$ is closed under conditions~(1)--(4) of Proposition~\ref{prop:PropertiesOfBoundedSets}.
\end{enumerate}
Then $Z$ contains all the bounded sets.
\end{lem}
\begin{proof}
By assumption there exists a slope $\alpha$ so that $\{\alpha\} \in Z$. Let $\alpha'$ be any other slope.
By condition~(4), $\{\alpha'' | \Delta(\alpha,\alpha'') \leq \Delta(\alpha,\alpha')\} \in Z$.
By~(1), $\{\alpha'\} \in Z$. Hence $Z$ contains all the singletons.
For a slope $\alpha_{0}$ and a positive integer $d$, we will denote the set
of all slopes of distance at most $d$ from $\alpha_{0}$ as
$D_{d}(\alpha_{0})$. We claim that for any $\alpha_{0}$ and any $d$,
$D_{d}(\alpha_{0}) \in Z$. The proof is inductive. For $d=0$,
$D_{0}(\alpha_{0}) = \{\alpha_{0}\}$ and hence $D_{0}(\alpha_{0}) \in Z$.
Assume that $d>0$ and that $D_{d-1}(\alpha_{0}) \in Z$. Then
for any $\alpha_{d} \in D_{d}(\alpha_{0}) \setminus D_{d-1}(\alpha_{0})$, there is
$\alpha_{d-1}(\alpha_{0}) \in D_{d-1}(\alpha_{0})$, so that $\ensuremath{d_{\mbox{Farey}}}(\alpha_{d},\alpha_{d-1}) = 1$.
Hence $\Delta(\alpha_{d},\alpha_{d-1}) = 1$. Thus
$$D_{d}(\alpha_{0}) \subset \{\alpha | (\exists \beta \in D_{d-1}) \Delta(\alpha,\beta) \leq 1 \}.$$
By (4), the latter is in $Z$. By~(1), $D_{d} \in Z$.
Let $B$ be a non empty bounded set. We will denote the diameter of $B$ by $d$.
Then for any slope $\beta \in B$,
$$B \subset D_{d}(\beta).$$
Thus by~(1), $B \in Z$.
\end{proof}
We conclude this section by revisiting Lemma~\ref{lem:slopes}:
\begin{lem}
\label{lem:MCGisometryOfSlopes}
Let $M$ be a hyperbolic manifold, $\partial M$ a single torus.
There exists a (possibly trivial) involution $i$ on the set of slopes of $\partial M$,
that induces an isometry on the Farey graph,
so that if $\phi:M \to M$ is a diffeomorphism and $\alpha$ a slopes of $\partial M$,
the image of $\alpha$ under $\phi$ is either $\alpha$ or $i(\alpha)$.
\end{lem}
\begin{proof}
This follows immediately from Lemma~\ref{lem:slopes} and the fact that
$(p,q) \mapsto (-p,q)$ induces an isometry on the Farey graph (this isometry is the
reflection by the edge connecting $1/0$ to $0/1$, and the endpoints of this edge are its
fixed slopes).
\end{proof}
\section{Minimally non hyperbolic fillings}
\label{sec:mnh}
\noindent
From this section on, the notation introduced in Subsection~\ref{subsec:notaion} will be used regularly;
we assume the reader is comfortable with it.
In this section we study {\it minimally non hyperbolic\ fillings}, a concept which is designed for studying exceptional
surgeries on links with many boundary components, or equivalently, exceptional fillings
on manifold with many boundary components.
There are two reasons for studying minimally non hyperbolic\ fillings: first, every non hyperbolic filling admits a minimally non hyperbolic\ partial filling;
second, in Proposition~\ref{pro:mnhIsFinite} we
show that any manifold admits only {\it finitely many} minimally non hyperbolic\ fillings.
This is essential for finiteness of $T(X)$, the tree that we will construct in the next section.
\bigskip\bigskip\noindent
Let us begin with a simple example. Let $X$ be the exterior of the Whitehead link endowed with the
natural meridian and longitude on each boundary component.
For $j=1,2\dots$, let $(\alpha_{1}^{j},\alpha_{2}^{j}) = (1/0,p^{j}/q^{j})$
for some $p_{j},q_{j}$. Then for
each $j$, $X(\alpha_{1}^{j},\alpha_{2}^{j})$ is a lens space and hence $(\alpha_{1}^{j},\alpha_{2}^{j})$
is a non hyperbolic multislope. There is no mystery here: all the multislopes $(\alpha_{1}^{j},\alpha_{2}^{j})$
have a common partial filling, namely $(1/0,\ensuremath\infty)$,
and $X(1/0, \ensuremath\infty) \cong D^{2} \times S^{1}$ is non-hyperbolic.
In this situation, it is better to consider the single multislope $(1/0,\infty)$ and not the
infinite set of multislopes $\{(\alpha_{1}^{j},\alpha_{2}^{j}) \}_{j=1}^{\infty}$.
This leads us to the following definition
which is central to our work:
\begin{dfn}[minimally non hyperbolic\ filling]
Let $X$ be a hyperbolic manifold, $\mathcal{T} =T_1 \cup \cdots \cup T_{n}$ components of $\partial X$,
and $\alpha$ a multislope of $\mathcal{T}$.
We say that $\alpha$ is {\it minimally non hyperbolic} if
$X(\alpha)$ is non-hyperbolic and any strict partial
filling $\alpha' \pf \alpha$ is hyperbolic.
\end{dfn}
So in the example of the Whitehead link exterior discussed above, the slopes $(1/0,p^{j}/q^{j})$ are not minimally non hyperbolic\
and $(1/0, \ensuremath\infty )$ is.
\bigskip\noindent
We prove:
\begin{prop}
\label{pro:mnhIsFinite}
Let $X$ be a hyperbolic manifold and $\mathcal{T} = T_{1} \cup \cdots \cup T_{n}$
components of $\partial X$. Then there are only finitely
many minimally non hyperbolic\ fillings on $\mathcal{T}$.
\end{prop}
\begin{proof}
We assume as we may that there are infinitely many multislopes on
$\mathcal{T}$ yielding non hyperbolic manifolds.
Let $\alpha^{j} = (\alpha_{1}^{j},\dots,\alpha_{n}^{j})$ be an infinite set of distinct
non hyperbolic fillings ($j=1,2,\dots$). We will prove the theorem by showing that
one of these is not minimally non hyperbolic. After subsequencing $n$ times we may assume that
for each $1 \leq i \leq n$, $\alpha_{i}^{j}$ (the restrictions of
$\alpha^{j}$ to $T_{i}$) fulfill one of the following conditions:
\begin{enumerate}
\item For $j \neq j'$, $\alpha^{j}_{i} \neq \alpha^{j'}_{i}$; we assume in addition
that for all $j$, $\alpha_{{i}}^{j} \neq \ensuremath\infty$.
\item For any $j,j'$, $\alpha^{j}_{{i}} = \alpha^{j'}_{i}$ (possibly $\alpha^{j}_{i} = \ensuremath\infty$).
\end{enumerate}
To avoid overly complicated notation we do not rename $\alpha^{j} = (\alpha_{1}^{j},\dots,\alpha_{n}^{j})$.
After reordering the boundary components if necessary, we may assume that
$\alpha_{i}^{j}$ are distinct for $1 \leq i \leq k$, and $\alpha_{i}^{j}$ is constant for $k+1 \leq i \leq n$
(for some $0 \leq k \leq n+1$). Note that $k=0$ is impossible since $\{\alpha^{j}\}$
is infinite. We drop the superscript from $\alpha_{i}$ for $i>k$.
Let $\widehat X = X(\ensuremath\infty ,\dots,\ensuremath\infty,\alpha_{k+1},\dots,\alpha_{n})$. Then the manifolds
$\widehat X(\alpha_{1}^{j},\dots,\alpha_{k}^{j}) = X(\alpha_{1}^{j},\dots,\alpha_{n}^{j})$
are all non hyperbolic by assumption. We claim that $\widehat X$ is not hyperbolic.
Assume for a contradiction it is. Since $\alpha_{i}^{j} \to \ensuremath\infty$ for all $i$,
by Thurston's Dehn Surgery Theorem, for large enough $j$,
$\widehat X(\alpha_1^{j},\dots,\alpha_{k}^{j})$ is hyperbolic as well.
This contradicting our assumptions and shows that
$\widehat X$ is not hyperbolic. Recall that $\alpha_i^{j} \neq \ensuremath\infty$ for $1 \leq i \leq k$ and $k\geq 1$;
hence $(\ensuremath\infty,\dots,\ensuremath\infty,\alpha^{j}_{k+1},\dots,\alpha^{j}_{n})$
is a strict partial filling of $(\alpha^{j}_{1},\dots,\alpha^{j}_{n})$, showing that the latter is not minimally non hyperbolic.
The proposition follows.
\end{proof}
\section{$T(X)$}
\label{sec:T(X)}
\bigskip
\noindent
In this section we construct the tree $T(X)$, which is the main tool for our work on cosmetic
surgery. The construction relies heavily on the concept of {\it minimally non hyperbolic\ fillings} from the previous section.
After constructing $T(X)$ we prove (Proposition~\ref{prop:T(X)isFinite}) that it is
finite. We then explain (Proposition~\ref{prop:ObtainedByFilling})
why $T(X)$ can be used to study fillings. Since $T(X)$ was designed
for studying exceptional surgeries on hyperbolic manifolds, it is perhaps a little
surprising that it can also be used to study hyperbolic fillings; this
is explained and proved in Proposition~\ref{prop:UsingT(X)forHyperbolicFilling}.
\bigskip\bigskip\noindent
Let $X$ be a compact orientable manifold whose boundary consists of tori
and $\mathcal{T} = T_{1} \cup \dots \cup T_{n}$
components of $\partial X$.
We wish to associate to $(X,\mathcal{T})$ a finite rooted tree,
denoted $T(X,\mathcal{T})$ (or $T(X)$ when no confusion can arise), that
encodes exceptional fillings on $X$.
Before constructing $T(X)$ we comment about its structure. The vertices of $T(X)$
correspond to manifolds with $X$ as the root. We direct every edge away from the root.
Branches, which are always assumed to follow this direction, encode the filling process, and
therefore we may have distinct vertices that correspond to diffeomorphic manifolds.
It follows from the construction below that the vertices are arranged along levels. The levels are grouped
into block of the form $3m$, $3m+1$, and $3m+2$ and obey the following rules:
\begin{enumerate}
\item Geometric manifolds (that is, hyperbolic manifolds, Seifert manifolds, and sol manifolds)
are arranged on levels $3m$ (for $m\in \mathbb Z_{\geq 0}$).
\item Reducible manifolds are arranged on levels $3m+1$ (for $m\in \mathbb Z_{\geq 0}$).
\item JSJ manifolds (recall Definition~\ref{dfn:jsj}) are arranged on levels $3m+2$ (for $m\in \mathbb Z_{\geq 0}$).
\item Every edge in $T(X)$ is directed from the initial vertex to the terminal vertex (say from $u$
to $v$) so that if $u$ is at level $3m$, $3m+1$, or $3m+2$, then $v$ is at level $3m+1$, $3m+2$,
or $3m+3$; moreover, the level of $v$ is strictly greater than that of $u$. We call $u$ the {\it predecessor} of
$v$ and $v$ is the {\it direct descendant} of $u$.
\end{enumerate}
\bigskip\noindent
We are now ready to construct $T(X)$. The root of $T(X)$ is a vertex labeled $X$.
Assume first that $X$ is geometric. Then $X$ is placed in
level $0$. If $X$ is Seifert fibered or a sol manifold,
then the corresponding vertex is a leaf: there
are no edges out of $X$. If $X$ is hyperbolic, we place one edge $e$ out of $X$ for each
minimally non hyperbolic\ filling on $X$ (say $\alpha$), and the terminal vertex of $e$ is labeled $X(\alpha)$.
Since $\alpha$ is a non-hyperbolic filling, $X(\alpha)$ is one of the following:
\begin{enumerate}
\item Reducible: then $X(\alpha)$ is placed at level $1$.
\item JSJ: then $X(\alpha)$ is placed at level $2$.
\item Seifert fibered or sol manifold: then $X(\alpha)$ is placed at level $3$.
\end{enumerate}
Next suppose that $X$ is not prime. Then the corresponding vertex is placed at
level $1$. Let $X_{1},\dots,X_{k}$ be the factors of the prime decomposition of $X$.
We place $k$ edges out of
$X$ with terminal vertices labeled $X_{1},\dots,X_{k}$. Each $X_{i}$ is either
JSJ, hyperbolic, Seifert fibered, or sol.
Accordingly, the corresponding vertex is placed at level $2$ (if it is JSJ) or $3$ (in all other cases).
Note that if $i \neq i'$ then $X_{i}$ and $X_{i'}$ correspond to distinct vertices
even if $X_{i} \cong X_{i'}$.
Finally, let $X$ be a JSJ manifold. The corresponding vertex is placed at level
$2$. Let $X_{1},\dots,X_{k}$
be the components of the torus decomposition of $X$. We place $k$ edges out of
$X$, with terminal vertices labeled $X_{1},\dots,X_{k}$. Each $X_{i}$ is
hyperbolic or Seifert fibered. Accordingly, it is placed at level $3$.
As above, if $i \neq i'$ then $X_{i}$ and $X_{i'}$ correspond to distinct vertices
even if $X_{i} \cong X_{i'}$.
The construction is recursive, and if $X_{1}$ is a
direct descendant of $X$, then we place $T(X_{1})$ with the root at the vertex
labeled $X_{1}$; since $X_{1}$ is a direct descendant of $X$ its level is
$1$, $2$ or $3$. If the level is $3$ we shift all the levels in $T(X_{1})$ by $+3$.
\bigskip\noindent
Let us discuss an example. Let $X_{3}$ be a hyperbolic manifold with $m+1$ boundary components.
Let $X_{2}$ be double of $X_{3}$ along
$m$ boundary components. Hence $X_{2}$ is a toroidal manifold with $2$ boundary components.
Let $X_{0}$ be the manifold obtained from $X_{2}$
by drilling a hyperbolic knot (which is known to exist,
essentially by Myers~\cite{Myers}).
Hence $X_{0}$ is a hyperbolic manifold with $3$ boundary components.
Now in $T(X_{0})$ there is an edge connecting $X_{0}$ to $X_{2}$,
corresponding to a minimally non hyperbolic\ filling (since we fill only one boundary component, the filling
must be minimal). Next we see two edges from $X_{2}$ to two copies of $X_{3}$.
Thus as we move down $T(X_{0})$ we start with a single hyperbolic manifold with
three boundary components, and later encounter
two hyperbolic manifolds, each with $m+1$ boundary components for an arbitrary $m$
(and, perhaps, other manifolds as well---this may not be all the edges between the levels $0$
and $3$).
As the tree of $T(X_{0})$ contains two copies of $T(X_{3})$ it can be quite big.
We leave it as an exercise to the reader to construct other complicated examples; for instance,
given an integer $m$, construct a hyperbolic manifold $X$ with one
boundary component, so that $T(X)$ admits a directed path of length $m$.
\bigskip\noindent
Our goal is to use $|T(X)|$, the number of vertices in $T(X)$,
as a basis for induction. For that we need:
\begin{prop}
\label{prop:T(X)isFinite}
$T(X)$ is finite.
\end{prop}
\begin{proof}
Let the {\it degree} of a vertex $v$ be the number of direct descendants of $v$ (that is, the number
of edges {\it out} of $v$).
It is easy to see that the degree of every vertex $v$ is finite:
\begin{enumerate}
\item If $v$ corresponds to a hyperbolic manifold $X$, the degree is finite because $X$
admits only finitely many minimally non hyperbolic\ fillings (Proposition~\ref{pro:mnhIsFinite}).
\item If $v$ corresponds to a Seifert manifold or a sol manifold the degree is zero by construction.
\item If $v$ corresponds to a reducible manifold the degree is the number of factors in its prime decomposition
and is therefore finite.
\item If $v$ corresponds to a JSJ manifold the degree is finite by the finiteness of the torus decomposition~\cite{jacoshalen}
and~\cite{Johannson}.
\end{enumerate}
The problem is avoiding an infinite branch. Assume there is such a branch.
Now by construction every edge starting on level $3m$, $3m+1$, or $3m+2$ ends at a level $3m+1$, $m+2$, or
$3m+3$. The vertices at level $3m+3$ that do not correspond to hyperbolic manifolds are leaves; hence
the branch must admit a vertex corresponding to a hyperbolic manifold on every level $3m$.
Let $X_{3m}$ denote this hyperbolic manifold.
We will use the {\it Gromov Norm}; for definition see \cite{GromovBoundedCohomology}.
The Gromov norm has the following properties, proved in~\cite{GromovBoundedCohomology}
and~\cite[Theorem~1]{soma}. Here, $X$ is a compact orientable manifold so that $\partial X$ consist of tori.
\begin{enumerate}
\item If the Gromov norm is non-zero, then it strictly decreases under any filling.
\item The Gromov norm of $X$ equals the sum of the Gromov norms of the components of the
prime decomposition of $X$.
\item The Gromov norm is additive under decomposition along essential tori.
\item The Gromov norm is additive under disjoint union.
\item The Gromov norm is non-negative.
\end{enumerate}
Since $X_{3m+3}$ is obtained from $X_{3m}$ by filling, then (possibly) reducing along
essential spheres and discarding components, and (possibly) decomposing along essential tori
and discarding components, we see that the Gromov norm of $X_{3m+3}$ is strictly smaller than
that of $X_{3m}$. It is well known that the hyperbolic volume is
a constant multiple of the Gromov norm, and so we see that $X_{0},X_{3},X_{6},\dots$
forms a sequence of hyperbolic manifolds with
$\vol[X_{0}] > \vol[X_{3}]> \vol[X_{6}] > \cdots$. But this cannot be, as hyperbolic volumes are well ordered.
\end{proof}
\bigskip\bigskip\noindent
In the following sections, we will use $T(X)$ inductively. The problem is the we are {\it only} dealing with
filling, while non-prime and JSJ manifolds are treated differently on $T(X)$
(reduction along spheres and essential tori, respectively).
Let $\alpha = (\alpha_{1},\dots,\alpha_{n})$ be a
multislope. If there exist a minimally non hyperbolic\ filling $\alpha'$ so that $\alpha' \pf \alpha$
(possibly $\alpha' = \alpha$) we say that {\it $\alpha$ admits a
minimally non hyperbolic\ partial filling}. Note that if $\alpha$ is a non hyperbolic
multislope, which by definition means that $X$ is hyperbolic and $X(\alpha)$ is not, then
$\alpha$ admits a minimally non hyperbolic\ partial filling.
We will need the following proposition; it is useful, for example, when $\alpha$ is a hyperbolic
multislope that admits a minimally non hyperbolic\ partial filling (note that such multislopes do exist;
constructing examples is quite easy).
\begin{prop}
\label{prop:ObtainedByFilling}
Let $X$ be a hyperbolic manifold and
$\alpha$ a multislope of $\partial X$. Suppose that $\alpha$ admits a minimally non hyperbolic\ partial filling $\alpha'$
so that either $X(\alpha')$ is not prime or it is JSJ. Suppose further that
$X(\alpha)$ is irreducible and a toroidal.
Then $X(\alpha)$ is obtained by filling some descendant of $X(\alpha')$ on $T(X)$.
\end{prop}
\begin{proof}
Since $\alpha' \pf \alpha$, it is clear that
$X(\alpha)$ is obtained from $X(\alpha')$ by filling.
Assume first that $X(\alpha')$ is reducible. Then the descendants of $X(\alpha')$
are, by construction of $T(X)$, the factors of the prime decomposition of $X(\alpha')$.
Since $X(\alpha)$ is a prime manifold that is obtained from $X(\alpha')$ by
filling, it is easy to see that in that filling all the components in the prime decomposition
of $X(\alpha')$ become balls except at most one. The proposition follows in this case.
Next assume that $X(\alpha')$ is JSJ. By assumption, $X(\alpha)$ is prime
and a toroidal. Therefore every torus $T$ in $X(\alpha)$ fulfills at least one of the following
three conditions:
\begin{enumerate}
\item $T$ is boundary parallel.
\item $T$ bounds a solid torus.
\item $T$ bounds a knot exterior contained in a ball.
\end{enumerate}
Since $X(\alpha)$ is obtained from $X(\alpha')$ by Dehn filling, $X(\alpha') \subset X(\alpha)$.
Let $T \subset X(\alpha')$ be a torus. Considering $T$ as a torus in $X(\alpha)$ allows us
to endow it with a co-orientation as follows:
\begin{enumerate}
\item If $T$ is boundary parallel, we co-orient it towards the boundary.
\item If $T$ bounds a solid torus, we co-orient it towards the solid torus.
\item If $T$ bounds a knot exterior contained in a ball, we co-orient it towards the knot exterior.
\end{enumerate}
Note that some tori may get both co-orientations (for example, a boundary parallel torus in a solid torus
or an unknotted torus in a ball).
In that event we pick an orientation arbitrarily.
Let $\mathcal{T}$ be the tori of the JSJ decomposition of $X(\alpha')$.
Let $\Gamma$ be the {\it dual graph} to $\mathcal{T}$,
which we defined to be the graph that has one vertex for each component
of the torus decomposition of $X(\alpha')$ and one edge
for each torus $T \in \mathcal{T}$, connecting the vertices that correspond
to the components adjacent to $T$.
Since $X$ is connected so is $X(\alpha')$;
hence $\Gamma$ is connected. Since $X(\alpha)$ is a toroidal and irreducible
it contains no non-separating tori, and is follows that neither does $X(\alpha')$; hence $\Gamma$ contains
no cycles. We conclude that $\Gamma$ is a tree. We endow each
edge of $\Gamma$ with an orientation consistent with the co-orientation of the corresponding torus of $\mathcal{T}$.
Using induction, it is easy to see that $\Gamma$ admits a sink (a vertex connected only to edges that point away from it):
since $\mathcal{T} \neq \emptyset$, $\Gamma$ contains an edge.
Thus $\Gamma$ admits a leaf (a vertex connected to only one other vertex), say $v$. If the edge connected to
$v$ points away from $v$, $v$ is a sink. Otherwise, removing $v$ and the edge attached to it we obtain a tree
with fewer vertices than $\Gamma$.
If the tree obtained contains only one
vertex (and hence no edges), that vertex is a sink of $\Gamma$. Otheriwse,
by induction the tree obtained admits a sink; it is clear that the same vertex is a sink for
$\Gamma$ as well.
Let $X'$ be a component of the JSJ decomposition of $X(\alpha')$ that corresponds to a sink.
By construction of $T(X)$, $X'$ is a direct descendant of $(\alpha')$.
We claim that $X(\alpha)$ is obtained from $X'$ by filling (this allows for the
possibility that $X' \cong X(\alpha)$ and no boundary component is filled).
To see this, let $T$ be a component
of $\partial X'$. We will denote the component of $X(\alpha')$
cut open along $T$ that is disjoint from $X'$ as $Y$.
As above, considering $T \subset X(\alpha)$ we see three cases:
\begin{enumerate}
\item $T$ is parallel to a component of $\partial X(\alpha)$: equivalently,
$Y(\alpha|_{\partial Y}) \cong T^{2} \times [0,1]$. We remove $Y$ and the solid tori
attached to it. The manifold obtained is not changed.
\item $T$ bounds a solid torus outside $X'$: equivalently,
$Y(\alpha|_{\partial Y}) \cong D^{2} \times S^{1}$.
Again we remove $Y$, and in the process of obtaining
$X(\alpha)$ from $X'$ we consider attaching a solid torus to $T$ along the
slope defined by $Y(\alpha|_{\partial Y})$.
\item $Y$ is a knot exterior contained in a ball $D$: if $\partial Y$ is compressible then
$Y \cong D^{2} \times S^{1}$; this was treated in case~(2),
and we assume as we may that this does not happen.
Thus $T$ is essential in $Y(\alpha|_{\partial Y})$, and since $X(\alpha)$ is a toroidal $T$ must compress
in $X(\alpha)$ away from $Y$. Let $D$ be a compressing disk.
It is now easy to see that replacing $Y$ with a solid torus so that the meridian of
the solid torus intersects $\partial D$ exactly once does not change the ambient manifold.
This can be seen as ``unknotting'' $T$ in $D$, see Figure~\ref{fig:unknotting}.
\begin{figure}
\includegraphics[width=3in]{F4}
\caption{Unknotting in a ball}
\label{fig:unknotting}
\end{figure}
\end{enumerate}
Repeating this process on all the components of $\partial X'$ we see that $X(\alpha)$ is
obtained from $X'$ by attaching solid tori.
This completes the proof of Proposition~\ref{prop:ObtainedByFilling}.
\end{proof}
\bigskip\bigskip\noindent
So here is where we stand: if we fill to obtain a prime a-toroidal manifold, Proposition~\ref{prop:ObtainedByFilling}
allows us to travel down $T(X)$ from a vertex at level $3m+1$ or $3m+2$.
Assume, in addition, that
$X(\alpha)$ is not hyperbolic. A manifold corresponding to a vertex labeled $3m$
is Seifert fibered, sol, or a hyperbolic.
The first two cases require direct analysis; in the final
case we are guaranteed to have a minimally non hyperbolic\ filling that allows us to keep going down $T(X)$.
The problem occurs when we want to study hyperbolic fillings. Obviously, one cannot
expect {\it every} filling to admit a minimally non hyperbolic\ partial filling;
this is simply false. Somewhat surprisingly we have the following proposition, that allows us to go down $T(X)$
in certain circumstances; recall that a multislope that does not admit a minimally non hyperbolic\ partial filling is called
totally hyperbolic:
\begin{prop}
\label{prop:UsingT(X)forHyperbolicFilling}
Let $X$ be a hyperbolic manifold and $\epsilon > 0$. Let $\mathcal{A}$ be the set
of multislopes of $\partial X$ so that every $\alpha \in \mathcal{A}$ we have that
$X(\alpha)$ is hyperbolic, and every geodesic in $X(\alpha)$ is longer then $\epsilon$.
Then there are only finitely many totally hyperbolic mutlislopes in $\mathcal{A}$
\end{prop}
\begin{rmk}
Since $\partial X$ may have arbitrarily many components, it is easy to construct examples
of hyperbolic manifolds with infinitely many multislopes $\alpha$ so that $X(\alpha)$
is hyperbolic but does not admit a geodesic shorter than $\epsilon$,
for a fixed $\epsilon>0$. This proposition allows us to study those:
there is a {\it finite set} of totally hyperbolic mulitslopes, and every other multislope in
$\mathcal{A}$ admits a minimally non hyperbolic\ partial filling; these correspond to moving down $T(X)$.
\end{rmk}
\begin{proof}
Denote the components of $\partial X$ by $T_{1},\dots,T_{n}$. We assume as we may that $\mathcal{A}$
is infinite. Let $\alpha^{j} = (\alpha_{1}^{j},\dots,\alpha_{n}^{j})$ be an infinite
set of multislopes in $\mathcal{A}$; we will prove the theorem by showing that
some $\alpha^{j}$ admits a non hyperbolic partial filling.
The remainder of the proof is very similar to the proof of Proposition~\ref{pro:mnhIsFinite}
and we only paraphrase it here.
After subsequencing we may assume that for each $i$ one of the following holds:
\begin{enumerate}
\item For $j \neq j'$, $\alpha^{j}_{i} \neq \alpha^{j'}_{i}$; we assume in addition
that for all $j$, $\alpha_{{i}}^{j} \neq \ensuremath\infty$.
\item For any $j,j'$, $\alpha^{j}_{{i}} = \alpha^{j'}_{i}$ (possibly $\alpha^{j}_{i} = \ensuremath\infty$).
\end{enumerate}
After renumbering we may assume that the constant slopes are $\alpha_{i}^{j}$
for $i > k$ (for some $0 \leq k \leq n$). Since
$\mathcal{A}$ is infinite, $k \geq 1$. For $i > k$, we drop the superscript from $\alpha_{i}$.
Let $\widehat X = X(\ensuremath\infty,\dots,\ensuremath\infty,\alpha_{k+1},\dots,\alpha_{n})$. Then
$X(\alpha^{j}) = \widehat X(\alpha_{1}^{j},\dots,\alpha_{k}^{j})$. If $\widehat X$ were
hyperbolic, then by Thurston's Dehn Surgery Theorem
for large enough $j$, we would obtain a hyperbolic manifold with $k$
geodesics of length less than $\epsilon$; as $k \geq 1$ this violates our assumption.
Thus $\widehat X$ is non-hyperbolic, and
$(\ensuremath\infty,\dots,\ensuremath\infty,\alpha_{k},\dots,\alpha_{n}) \pf (\alpha_{1}^{j},\dots,\alpha_{n}^{j})$
is a non hyperbolic partial filling.
The proposition follows.
\end{proof}
We end this section with the following lemma:
\begin{lem}
\label{lem:TreeOfJSJ}
Let $X$ be a JSJ manifold and $X_{0}$ a connected manifold that is obtained
as the union of a strict subset of the components of
the torus decomposition of $X$.
Then $|T(X_{0})| < |T(X)|$.
\end{lem}
\begin{proof}
If $X_{0}$ is a component of the torus decomposition of $X$ then it corresponds to
a direct descendant of the root of $T(X)$, and clearly $T(X_{0}) \subsetneqq (X)$.
The lemma follows in this case.
If $X_{0}$ is the union of more than one component of the torus decomposition of
$X$, we embed $T(X_{0})$ into $T(X)$ by placing the root of $T(X_{0})$ at the
root of $T(X)$ and using the edges the correspond to the components of the torus
decomposition of $X$ that appear in $X_{0}$. By assumption this is a strict subset
of the components of the torus decomposition of $X$, and hence no all the edges are used.
After this embedding we can view $T(X_{0})$ and a subtree of $T(X)$, and we
see that $T(X_{0}) \subsetneqq (X)$. The lemma follows.
\end{proof}
\section{Cosmetic surgery on $T^2 \times I$}
\label{sec:CosmeticSurgeryOnT2XI}
\bigskip
\noindent
There are two types of theorems proved using $T(X)$. The first asks ``how much can a manifold get twisted
when performing cosmetic surgery'' and the second asks ``how many fillings can
result in a manifold fulfilling such-and-such condition''.
In the next three section we prove theorems of the first type.
Recall that a cosmetic surgery on $L \subset M$ is a surgery with multislope $\alpha$
so that $L(\alpha) \cong M$.
Below we consider cosmetic surgery on links in $T^{2} \times [0,1]$.
Note that $T^{2} \times [0,1]$ gives a natural projection from $T^{2} \times \{1\}$ to $T^{2} \times \{0\}$;
however, after cosmetic surgery, this identification may change. Hence the image of a specific slope in
$T^{2} \times \{1\}$ may give an infinite set after cosmetic surgeries (and this does in fact happen). The theorem
below controls this set, and more generally, the image of a bounded set:
\bigskip\bigskip
\noindent
\begin{thm}
\label{thm:CosmeticSurgeryOnT2XI}
Let $B$ a bounded set of slopes of $T^{2} \times \{1\}$, $L$ be a link in
$T^{2} \times [0,1]$, and $\mathcal{A} = \{\alpha \ | \ L(\alpha) \cong T^{2} \times [0,1]\}$.
For $\alpha \in \mathcal{A}$,
let $B_{\alpha}$ be the set of slopes of $T^{2} \times \{0\}$ that are obtained by projecting $B$
via the natural projection.
Then $\cup_{\alpha \in \mathcal{A}} B_{\alpha}$ is bounded.
\end{thm}
\begin{proof}
We will denote $T^{2} \times [0,1] \setminus N(L)$ as $X$ and $\partial N(L)$ as $\mathcal{T}$.
Note that we may regard $\mathcal{A}$ as a set of multislopes on $\mathcal{T} \subset \partial X$.
The proof is an induction on $|T(X)|$.
\bigskip
\noindent {\bf Assume that $X$ is not prime.} Let $X_{1}$ be the factor of the prime decomposition
of $X$ that contains $T^{2} \times \{0\}$ and $T^{2} \times \{1\}$ (note that both are contained
in the same factor). Then any $\alpha \in \mathcal{A}$ induces $\alpha|_{\partial X_{1}}$,
and $X_{1}(\alpha|_{\partial X_{1}}) \cong T^{2} \times S^{1}$. Moreover, all the other factors of
the prime decomposition of $X$ become balls after filling and therefore do not effect the
identification of $T^{2} \times \{1\}$ with $T^{2} \times \{0\}$. Denote the image of $B$
under the natural projection using the product structure of
$X_{1}(\alpha|_{\partial X_{1}}) \cong T^{2} \times [0,1]$
by $B_{\alpha|_{\partial X_{1}}}$.
Thus for every $\alpha \in \mathcal{A}$,
$B_{\alpha} = B_{\alpha|_{\partial X_{1}}}$; therefore
$\cup_{\alpha \in \mathcal{A}} B_{\alpha} = \cup_{\alpha \in \mathcal{A}} B_{\alpha|_{\partial X_{1}}}$.
Since $|T(X_{1})| < |T(X)|$, $B_{\alpha|_{\partial X_{1}}}$ is bounded by induction.
The proposition follows in this case.
We assume from now on that $X$ is prime.
\bigskip
\noindent {\bf Assume that $X$ is Seifert fibered.}
Fix a Seifert fiberation of $X$ and a multislope $\alpha \in \mathcal{A}$.
If, for some $T_{i} \subset \mathcal{T}$, $\alpha|_{T_{i}}$ is a fiber
then $X(\alpha)$ contains a
sphere that separates $T^{2} \times \{0\}$ from $T^{2} \times \{1\}$, which is impossible since
$X(\alpha) \cong T^{2} \times [0,1]$. Hence the fiberaion of $X$ extends to a finration
of $T^{2} \times [0,1]$. We conclude that $X$ is obtained from $T^{2} \times [0,1]$ by
removing fibers.
Note that up to diffeomorphism the only Seifert fiberation of $T^{2} \times [0,1]$ is annulus cross $S^{1}$.
To see this, simply note that if the base orbifold of a Seifert fibered manifold
has positive genus or positive number of orbifold points then it admits a filling
that is not a lens space. Thus $X$ is obtained from an annulus cross $S^{1}$
by removing a set of $n$ curves that has the form
$\{p_{1},\dots,p_{n}\} \times S^{1}$, showing that $X$
an $n$-times punctured annulus cross $S^{1}$.
Since $X(\alpha)$ results in a Seifert fibered manifold with no exceptional fiber,
the core of the solid torus attached to $T_{i}$ is not an exceptional fiber, and
hence has the form $p_{i}/1$ in the
Seifert notation. Suppose $n>1$. Following Seifert's original work \cite{seifert}, by performing $k$ twists about an annulus
connecting fibers with Seifert invariants $\frac {p_{1}}{q_{1}}$ and
$\frac {p_{i}}{q_{i}}$ the invariants change as follows (for an arbitrary $k \in \mathbb Z$):
$$(p_{1}/q_{1}) \mapsto (p_{1} + k q_{1}/q_{1}) \mbox{ and } (p_{i}/q_{i}) \mapsto (p_{i} - k q_{i}/q_{i}).$$
As in our case $q_{i} = 1$, by choosing $k$ appropriately, we may assume that the filling of $T_{i}$ is
of the form $0/1$, and ignore it (for $i>1$). Thus we have reduced the problem to the
case $n=1$. Let $A \subset X$ be an embedded vertical annulus connecting $T_{1}$ with a curve
on $T^{2} \times \{0\}$ that we will denote as $\gamma$.
By twisting about $A$, we see that the effect of the cosmetic surgery
is the same as $D_{\gamma}^{n}$, an $n$ power of a Dehn twist about $\gamma$, for some $n \in \mathbb Z$.
Hence the image of $B$ after all possible cosmetic surgeries on $L$ is:
$$\{D_{\gamma}^{n}(\beta)\ |\ \beta \in B, n \in \mathbb{Z}\}.$$
By Proposition~\ref{prop:PropertiesOfBoundedSets}~(3), this set is bounded.
We assume from now on that $X$ is prime and not Seifert fibered.
\bigskip
\bigskip
\noindent {\bf Assume that $X$ is a JSJ manifold.} We will denote the tori of the JSJ
decomposition as $\mathcal{F}$. Let $\Gamma$ be the graph dual to the torus decomposition
as defined in~\ref{dfn:jsj}. Since $T^{2} \times [0,1]$
is connected and admits no non-separating tori, $\Gamma$ is a tree.
Note that $\Gamma$ has two (not necessarily distinct) vertices, denoted $v_{0}$
and $v_{1}$, that correspond to the components of the torus decomposition of $X$
that contain $T^{2} \times \{0\}$ and $T^{2} \times \{1\}$. There are two cases
to consider:
\bigskip
\noindent{\bf Case One:} $v_{0} \neq v_{1}$. Let $e$ be an edge on the shortest path
connecting $v_{0}$ and $v_{1}$ and $F \in \mathcal{F}$ the torus corresponding to $e$.
In any filling, $F$ separates $T^{2} \times \{0\}$
from $T^{2} \times \{1\}$. Thus for every $\alpha \in \mathcal{A}$,
we have that $F \subset X(\alpha) \cong T^{2} \times [0,1]$
is isotopic to $T^{2} \times \{1/2\}$. Let $X_{0}$ and $X_{1}$ be the components of
$X$ cut open along $F$ so that $T^{2} \times \{i\} \subset X_{i}$ ($i=1,2$).
We see that the fillings induced by $\alpha \in \mathcal{A}$
fulfill:
$$X_{i}(\alpha|_{\partial X_{i}}) \cong T^{2} \times [0,1].$$
For $i=0,1$, let
$$\mathcal{A}_{i} = \{\alpha|_{\mathcal{F} \cap \partial X_{i}}\ | \ \alpha \in \mathcal{A}\}.$$
Then for any $\alpha \in \mathcal{A}$, $X(\alpha) = X_{0}(\alpha_{0}) \cup X_{1}(\alpha_{1})$,
where here $\alpha_{i} = \alpha|_{\mathcal{F} \cap \partial X_{i}}$.
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})|, |T(X_{1})| < |T(X)|$.
Applying induction to $X_{1}$ we conclude that
$$\cup_{\alpha_{1} \in \mathcal{A}_{1}} B_{\alpha_{1}}$$
is a bounded set of slopes of $F$; we will denote it as $B^{F}$. Next we apply induction to $X_{0}$
and conclude that
$$\cup_{\alpha_{0} \in \mathcal{A}_{0}} B^{F}_{\alpha_{0}}$$
is bounded. It is clear from the discussion above that
$\cup_{\alpha \in \mathcal{A}} B_{\alpha} \subset \cup_{\alpha_{0} \in \mathcal{A}_{0}} B^{F}_{\alpha_{0}}$;
the theorem follows in case one.
\bigskip
\noindent{\bf Case Two:} $v_{0} = v_{1}$, that is, that both $T^{2} \times \{0\}$ and $T^{2} \times \{1\}$
are contained in the same component of the torus decomposition of $X$, say $X_{0}$.
We will denote
the components of $\mbox{cl}(X \setminus X_{0})$ as $X_1,\dots,X_{k}$.
Since $T^{2} \times [0,1]$ does not admit a non separating torus, if
$X_{i} \cap X_{0}$ consists of more than one component (for some $i$) then
$\mathcal{A} = \emptyset$ and there is nothing to prove. We assume as we may
that for every $i$, $X_{i} \cap X_{0}$ is a single torus which we will denote as $T'_{i}$.
Note that $T'_{i}$ is a component of the JSJ decomposition of $X$ and
under the assumptions of case two it cannot be boundary parallel in
$X(\alpha) \cong T^{2} \times [0,1]$.
Hence every $\alpha \in \mathcal{A}$ induces
a filling on every $X_{i}$ fulfilling exactly one of the following conditions:
\begin{enumerate}
\item $X_{i}(\alpha|_{\partial X_{i}}) \cong D^{2} \times S^{1}$.
\item $X_{i}(\alpha|_{\partial X_{i}}) \cong E(K_{i})$, where $K_{i} \subset S^{3}$ is a non
trivial knot and $X_{i}(\alpha|_{\partial X_{i}}) \subset D_{i}$ for some ball $D_{i} \subset X(\alpha)$.
\end{enumerate}
By Lemma~\ref{lem:KnotExteriorsAreDisjoint} we assume as we may that the balls
$D_i \subset X(\alpha)$ are disjointly embedded.
Given $\alpha \in \mathcal{A}$ we construct a multislope $\alpha_{0}$ of $\mathcal{T}_{0}$ as follows:
\begin{enumerate}
\item If $X_{i}(\alpha|_{\partial X_{i}}) \cong D^{2} \times S^{1}$, then $\alpha_{0}|_{T_{i}}$
is the meridian of the solid torus $X_{i}(\alpha|_{\partial X_{i}})$.
\item If $X_{i}(\alpha|_{\partial X_{i}}) \cong E(K_{i})$ we pick a slope that intersects the
meridian of $E(K_{i})$ exactly once (recall Figure~\ref{fig:unknotting}).
\end{enumerate}
Then $X_{0}(\alpha_{0}) \cong X(\alpha)$ and by construction the product structures of
$X_{0}(\alpha_{0})$ and $X(\alpha)$ induce the same projection from the slopes
of $T^{2} \times \{1\}$ to those of of $T^{2} \times \{0\}$.
Let $\mathcal{A}_{0}$ be the set of all multislopes of $X_{0}$ so that
$X_{0}(\alpha_{0}) \cong T^{2} \times [0,1]$. For each $\alpha_{0}\in \mathcal{A}_{0}$
we will denote the image of $B$ under the projection induced by the product structure as
$ B_{\alpha_{0}}$. If $\alpha_{0}$ was constructed as above for some $\alpha \in \mathcal{A}$,
then $B_{\alpha} = B_{\alpha_{0}}$. Thus we see:
$$\cup_{\alpha \in \mathcal{A}} B_{\alpha} \subset \cup_{\alpha_{0} \in \mathcal{A}_{0}} B_{\alpha_{0}}.$$
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})| < |T(X)|$; hence by induction
$\cup_{\alpha_{0} \in \mathcal{A}_{0}} B_{\alpha_{0}}$ is bounded.
The theorem follows in case two.
We assume from now on the $X$ is prime, not Seifert
fibered, and not a JSJ manifold.
\bigskip
\noindent {\bf Assume that $X$ is hyperbolic.} As $T^{2} \times [0,1]$ is non-hyperbolic,
any multislope $\alpha$ with $X(\alpha) \cong T^{2} \times [0,1]$ admits a minimally non hyperbolic\ partial
filling. By Proposition~\ref{pro:mnhIsFinite} $X$ admits only finitely many
minimally non hyperbolic\ multislopes, say denoted by $\alpha_{1},\dots,\alpha_{k}$. For $1 \leq i \leq k$,
we will denote $X(\alpha_{i})$ as $X_{i}$.
Let $\mathcal{A}_{i} = \{\alpha|_{\partial{X}_{i}} \ | \ \alpha \in \mathcal{A}, \ \alpha_{i} \pf \alpha\}$,
that is, $\mathcal{A}_{i}$ consists of the multislopes induced on $X_{i}$ by multislopes $\mathcal{A}$
that admits $\alpha_{i}$ as a partial filling.
For $\alpha_{i} \in\mathcal{A}_{i}$, we will denote the image of $B$ under the
projection induced by the product structure of $X_{i}(\alpha_{i})$ as $B_{\alpha_{i}}$.
Since $X_{i}$ is a direct descendant of the root of $T(X)$, $|T(X_{i})| < |T(X)|$.
By induction $\cup_{\alpha_{i} \in \mathcal{A}_{i}} B_{\alpha_{i}}$ is bounded.
Since every $\alpha \in \mathcal{A}$ admits a minimally non hyperbolic\ partial filling, for every such $\alpha$
there exists $i$ so that
$$X(\alpha) = X_{i}(\alpha_{i}),$$
where here $\alpha_{i} = \alpha|_{\partial X_{i}} \in \mathcal{A}_{i}$. Hence:
$$\cup_{\alpha \in \mathcal{A}} B_{\alpha} = \cup_{i=1}^{k}
\cup_{\alpha_{i} \in \mathcal{A}_{i}} B_{\alpha_{i}}.$$
Since bounded sets are closed under finite union, $\cup_{\alpha \in \mathcal{A}} B_{\alpha}$
is bounded.
This concludes the proof of Theorem~\ref{thm:CosmeticSurgeryOnT2XI}.
\end{proof}
\section{Cosmetic surgery on solid torus}
\label{sec:CosmeticSurgeryOnSolidTorus}
\bigskip
\noindent
Let $V$ be a solid torus and $L \subset V$ a link. In this section we address the following question:
how many slopes one $\partial V$ become the boundary of a meridian disk after cosmetic surgery on $L$?
Before stating the main theorem of this section, we consider some examples. If some component of
$L$ is the core of $V$, then (trivially) the answer is {\it every slope.} Next, let $L' = K'_{1} \cup K'_{2}$
be a two component link, where $K_{1}'$ is a knot that is not a torus knot and admits a non-trivial cosmetic
surgery (see Gabai~\cite{gabai} and Berge~\cite{berge}) and $K_{2}'$ is a core. Assume further that $K'_{2}$
was isotoped to be in a very ``complicated'' configuration with respect to $K_{1}'$ (we allow $K_{2}'$ to pass
through $K_{1}'$ during this isotopy, changing $L'$ but not $K_{1}'$ or $K'_{2}$).
Let $L = K_{1} \cup K_{2}$ be the image of $K_{1}'$ and $K_{2}'$
in the solid torus obtained by cosmetic surgery on $K_{1}'$. Then, due to the isotopy discussed above, we
expect that $K_{2}$ is not a core of the solid torus and certainly $K_{1}$ is not. However, $L$ admits
cosmetic surgeries that realize every slope on the boundary of the solid torus as the boundary of the meridian disk.
The trouble is that although $K_{2}$ is not a core
of the solid torus it becomes a core after cosmetic surgery.
\bigskip\noindent
With this in mind we state the main result of this section; note that condition~(2)
of the theorem is equivalent to requiring that non of the cores of the attached solid tori
is a core of $L(\alpha)$:
\begin{thm}
\label{thm:SlopesOnSolidTorus}
Let $V$ be a solid torus and $L \subset V$ a link. Consider
the set of multislopes $\mathcal{A}$ of $L$ so that for any
$\alpha \in \mathcal{A}$ the following two conditions hold:
\begin{enumerate}
\item $L(\alpha) \cong D^{2} \times S^{1}$.
\item For any $\alpha' \pf \alpha$, $L(\alpha') \not\cong T^{2} \times [0,1]$.
\end{enumerate}
Then the set of slopes on $\partial V$ that bounds a disk in $\{L(\alpha)\ |\ \alpha \in S\}$ is bounded.
\end{thm}
\begin{proof}
Let $X = V \setminus N(L)$ and denote the components of $\partial X$ by
$T,T_{1},\dots,T_{n}$, with $T=\partial X$. We will denote $L(\alpha)$ as
$X(\alpha)$; this is consistent with the notation of subsection~\ref{subsec:notaion}
and should cause no confusion. We induct on $|T(X)|$.
\bigskip
\noindent {\bf Assume $X$ is not prime.} Let $S \subset X$ be an essential separating sphere that
realizes the decomposition $X = X' \# X''$, so that $X'$ is prime and $T \subset X'$
(we are not assuming that $X''$ is prime).
For any $\alpha \in \mathcal{A}$, $X(\alpha) \cong D^{2} \times S^{1}$ which is prime.
Therefore after filling one side of $S$ becomes a ball $D$
and the other becomes a solid torus whose boundary is $T$. We conclude that $\alpha$ induces $\alpha|_{\partial X'}$
and the following conditions hold:
\begin{enumerate}
\item $X'(\alpha|_{\partial X'}) \cong D^{2} \times S^{1}$ and its meridian is the same slope of $T$
as the meridian of $X(\alpha)$.
\item There does not exist $\alpha' \pf \alpha|_{\partial X'}$ so that $X'(\alpha') \cong T^{2} \times [0,1]$.
Otherwise, there would be $\alpha'' \pf \alpha$ so that $X(\alpha'') \cong T^{2} \times [0,1]$,
contradicting our assumptions.
\item $|T(X')| < |T(X)|$: this is immediate from the construction of $T(X)$,
since $X'$ is a direct descendant of the root $X$.
\end{enumerate}
Applying the inductive hypothesis, we conclude that the set of meridians of $X'(\alpha|_{\partial X'})$
for $\alpha \in \mathcal{A}$ is bounded; the theorem follows in this case.
We assume from now on that $X$ is prime.
\bigskip
\noindent {\bf Assume $X$ is Seifert fibered.} Fix a Seifert fiberation on $X$.
First assume that, for some $\alpha \in \mathcal{A}$ and some $i$, $\alpha|_{T_{i}}$
is the fiber; we will denote the meridian disk of the solid torus attached to $T_{i}$
as $D_{i}$.
Then the result of gluing $D_{i}$ to a vertical annulus connecting $\partial D_{i}$
to the regular fiber on $T$ is a meridian disk for $X(\alpha)$. Hence the meridian
of $X(\alpha)$ is the regular fiber. From now on we consider the subset of $\mathcal{A}$
consisting of multislopes for which
$\alpha|_{T_{i}}$ is not a fiber in the fiberation of $X$ (for all $i$). To avoid overly complicated
notation we do not rename $\mathcal{A}$.
Thus the fiberation of $X$ extends to a fiberation of $X(\alpha)$.
Since $X(\alpha) \cong D^{2} \times S^{1}$
its base orbifold is $D^{2}$ with at most one orbifold point.
We see that the base orbifold for $X$ is a punctured disk with at most one orbifold point.
If there is no orbifold point, then there is an index $i$ so that for any $i' \neq i$,
$\alpha|_{T_{i'}}$ intersects the fiber exactly once (possibly, $\alpha|_{T_{i}}$
intersects the fiber once as well).
Define $\alpha' \pf \alpha$ by setting $\alpha'|_{T_{i}} = \ensuremath\infty$
and $\alpha'|_{T_{i'}} = \alpha|_{T_{i'}}$ for all $i' \neq i$.
Then $X(\alpha')$ is a Seifert fiber space over an annulus with no exceptional fibers
and hence $X(\alpha') \cong T^{2} \times [0,1]$, contradicting condition~(2) of the theorem.
We assume as we may that the base orbifold of $X$ has exactly one orbifold point.
Denote the multiplicity of the critical fiber by $p \geq 2$. Then the meridian of
$X(\alpha)$ intersects a fiber on $\partial X$ exactly $p$ times; this gives a
bounded set by Proposition~\ref{prop:PropertiesOfBoundedSets}~(4).
We assume from now on that $X$ is prime and not Seifert fibered.
\bigskip
\noindent {\bf Assume $X$ is a JSJ manifold.} Let $X_{0}$ be the component of the JSJ decomposition
of $X$ that contains $T$ and denote the components of $\partial X_{0} \setminus T$ as $F_{j}$, $j=1,\dots,{m}$. We will denote the
closures of the components of $X \setminus X_{0}$ as $X_{j}$.
To avoid the situation where $X_{j} = \emptyset$, if $F_{j} \subset \partial X$ we push it slightly
into the interior of $X_{0}$ so that $X_{j} \cong T^{2} \times [0,1]$ in that case.
Since $D^{2} \times S^{1}$ admits no non-separating tori, we assume as we may
that $X_{j} \cap X_{j'} = \emptyset$ for $j \neq j'$.
By reordering the indices of $X_{j}$ if necessary we may assume that $F_{j} \subset X_{j}$.
Finally, given $\alpha \in \mathcal{A}$ and $1 \leq j \leq m$, we will denote the components of
$X(\alpha)$ cut open along $F_{j}$ as $X(\alpha)_{j}^{+}$ and $X(\alpha)_{j}^{-}$, with
$\partial X(\alpha)_{j}^{+} = T \cup F_{j}$
and $\partial X(\alpha)_{j}^{-} = F_{j}$. Consider the following subsets
$\mathcal{A}_{j} \subset \mathcal{A}$ (for $j=0,\dots,m$):
\begin{enumerate}
\item $\mathcal{A}_{0}$ consist of all the multislopes $\alpha \in \mathcal{A}$ so that for all $j$,
$X(\alpha)_{j}^{+} \not\cong T^{2} \times [0,1]$.
\item For $1 \leq j \leq m$, $\mathcal{A}_{j}$ consist of all the multislopes $\alpha \in \mathcal{A}$
so that $X(\alpha)_{j}^{+} \cong T^{2} \times [0,1]$.
\end{enumerate}
It is immediate from the definitions that
$$\mathcal{A} = \cup_{j=0}^{m} \mathcal{A}_{j}.$$
We first consider multislopes $\alpha \in \mathcal{A}_{0}$.
By definition of $\mathcal{A}_{0}$, no
$F_{j}$ is boundary parallel in $X(\alpha) \cong D^{2} \times S^{1}$; thus
every torus $F_{j}$ bounds a solid torus
or a non-trivial knot exterior $E(K_{j})$. For each $F_{j}$ that bounds a solid torus,
let $\hat\alpha|_{F_{j}}$ be the slope of $F_{j}$ defined by the
meridian of that solid torus. For each torus $F_{j}$ that bounds a non trivial
knot exterior $E(K_{j})$ we do the following:
by Lemma~\ref{lem:KnotExteriorsAreDisjoint} we may assume that $E(K_{j}) \subset D_{j}$
for disjointly embedded balls $D_{j}$.
We replace every $E(K_{j})$ with a solid torus (which we will denote as $V_{j}$) so that the meridian of $V_{j}$
intersects that meridian of $E(K_{j})$ exactly once.
This does no change the ambient manifold and, since the changes
are contained in balls, the slope of $T$ that is the meridian of the solid
torus $X(\alpha)$ is not changed.
Let $\hat\alpha|_{F_{j}}$ be the slope of $F_{j}$ defined by the meridian of
$V_{j}$. Thus we have defined a slopes $\hat\alpha|_{F_{j}}$ for every $1 \leq j\leq m$;
together they induce a multislope of $\{F_{j}\}_{j=1}^{m}$ which we will denote by $\hat\alpha$.
We claim that $\hat\alpha$ fulfills the following two conditions:
\begin{enumerate}
\item $X_{0}(\hat\alpha) \cong D^{2} \times S^{1}$: this is immediate from the construction.
\item there is no partial filling $\hat\alpha' \pf \hat\alpha$ so that
$X_{0}(\hat\alpha') \not\cong T^{2} \times [0,1]$:
assume, for a contradiction, that such a partial filling $\hat\alpha'$ exists.
Since $T^{2} \times [0,1]$ has two boundary components, $\hat\alpha'$ is
obtained from $\hat\alpha$ by setting the value of $\hat\alpha|_{F_{i}}$ to $\ensuremath\infty$
on exactly one torus $F_{j}$. By the defining condition for $\mathcal{A}_{0}$
this is impossible for values of $j$ for which $X(\alpha)_{j}^{-} \cong D^{2} \times S^{1}$,
and for other values of $j$ this is impossible because $F_{j}$ is contained in
the ball $D_{j} \subset X(\alpha)$.
\end{enumerate}
Thus $\hat\alpha$ satisfies the assumptions of Theorem~\ref{thm:SlopesOnSolidTorus}.
By construction $X_{0}$ corrsponds to a
direct descendant of the root of $T(X)$, and therefore $|T(X_{0})| < |T(X)|$.
By induction, the set of meridians of the solid tori $X_{0}(\hat\alpha)$ (as
$\hat\alpha$ varies over all possible multislope that correspond to multislope
$\alpha \in \mathcal{A}_{0}$) form a bounded set of slopes of $T$, which we will
denote as $B_{0}$. By construction,
the set of meridians of $X(\alpha)$ for $\alpha \in \mathcal{A}_{0}$ is $B_{0}$.
Next fix $1 \leq j \leq n$ and consider $\alpha \in \mathcal{A}_{j}$. By the defining
condition for $\mathcal{A}_{j}$,
$X(\alpha)_{j}^{+} \cong T^{2} \times S^{1}$. Hence
$X(\alpha)_{j}^{-} \cong X(\alpha) \cong D^{2} \times S^{1}$.
Note that $X(\alpha)_{j}^{-}$ is obtained by filling a component of $X$ cut open along $F_{j}$
denoted above as $X_{j}$; the induced filling is given by $\alpha|_{\partial X_{j}}$. We
claim that the following conditions hold:
\begin{enumerate}
\item $X_{j}(\alpha|_{\partial X_{j}}) \cong D^{2} \times S^{1}$: this is immediate from the construction.
\item There is no partial filling $\alpha|_{\partial X_{j}}' \pf \alpha|_{\partial X_{j}}$ with
$X_{j}(\alpha|_{\partial X_{j}}') \cong T^{2} \times [0,1]$: otherwise, there would be a corresponding
partial filling $\alpha' \pf \alpha$ so that
$$X(\alpha') \cong X(\alpha)_{j}^{+} \cup_{F_{j}}
X_{j}(\alpha|_{\partial X_{j}}') \cong T^{2} \times [0,1],$$
contradicting the assumptions of the theorem.
\item $|T(X_{j})| < |T(X)|$: this follows from Lemma~\ref{lem:TreeOfJSJ}.
\end{enumerate}
By induction, the set of
slopes of meridians of $X_{j}(\alpha|_{\partial X_{j}})$ is bounded; we will denote it as $B_{j}'$.
By Theorem~\ref{thm:CosmeticSurgeryOnT2XI}
the set of slopes of $T$ obtained by projecting $B_{j}'$
after all possible cosmetic surgeries on $L \cap X(\alpha)_{j}^{+}$
is bounded; we will it as $B_{j}$. Clearly, the set of meridians on $X(\alpha)$
(for $\alpha \in \mathcal{A}_{j}$) is contained in $B_{j}$.
We have obtained $m+1$ bounded sets, namely, $B_{0},\dots,B_{m}$, so that the meridians of $X(\alpha)$
(for $\alpha \in \mathcal{A}$) are contained in $\cup_{j=0}^{m} B_{j}$. The theorem follows in this case.
We assume from now on that $X$ is prime, not Seifert fibered, and not a JSJ manifold.
\bigskip
\noindent {\bf Assume $X$ is hyperbolic.} By Proposition~\ref{pro:mnhIsFinite}
$X$ admits only finitely many minimally non hyperbolic\ fillings, which we will denote as
$\alpha_{1},\dots,\alpha_{k}$. Fix $1 \leq j \leq k$. If there is no $\alpha \in \mathcal{A}$
for which $\alpha_{j} \pf \alpha$, we set $B_{j} = \emptyset$; otherwise,
any $\alpha \in \mathcal{A}$ for which $\alpha_{j} \pf \alpha$ induces the
multislope $\alpha|_{\partial X(\alpha_{j})}$ on $\partial X(\alpha_{j})$.
We claim that $X(\alpha_{j})$ and $\alpha|_{\partial X(\alpha_{j})}$ fulfill the following four conditions:
\begin{enumerate}
\item $X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})})
\cong D^{2} \times S^{1}$: this is immediate,
as $X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})}) = X(\alpha)$ (we emphasize that
this is equality, not up to diffeomorphism).
\item $\partial X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})}) = T$:
this is immediate, as above.
\item $\alpha|_{\partial X(\alpha_{j})}$ does not admit a partial filling
$\alpha' \pf \alpha|_{\partial X(\alpha_{j})}$ so that
$X(\alpha_{j})(\alpha') \cong T^{2} \times [0,1]$: otherwise the corresponding partial filling
of $\alpha$ would yield $T^{2} \times [0,1]$, violating the second assumption of the theorem.
\item The meridians of the solid tori $X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})})$
and $X(\alpha)$ define the same slope of $T$: this is immediate,
as~(1).
\end{enumerate}
We will denote as $\mathcal{A}_{j} \subset \mathcal{A}$ the set
$$\mathcal{A}_{j} = \{ \alpha|_{X(\alpha_{j})} \ | \ \alpha \in \mathcal{A}, \\ \alpha_{j} \pf \alpha\}.$$
By points~(1)-(3) above, $X(\alpha_{j})$ and $\mathcal{A}_{j}$
fulfill the assumptions of the theorem.
By construction $T(X(\alpha_{j}))$ corresponds to a direct descendant of the root of $T(X)$;
therefore $|T(X(\alpha_{j}))| < |T(X)|$. By induction, the meridians of
$$\{X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})}) \ | \ \alpha \in \mathcal{A}_{j}\}$$
form a bounded set of slopes of $T$,
which we will denote as $B_{j}$. Since $D^{2} \times S^{1}$ is not hyperbolic,
for every $\alpha \in \mathcal{A}$, there is $1 \leq j \leq k$, so that
$\alpha_{j} \pf \alpha$; by point~(4) above the meridian
of $X(\alpha)$ is in $B_{j}$.
We see that
the meridians of $\{X(\alpha) | \alpha \in \mathcal{A}\}$ are
$$\cup_{j=1}^{k} B_{j}.$$
The theorem follows, as the finite union of bounded sets is bounded.
\end{proof}
\bigskip\noindent
Next we prove a proposition about fillings that yield $D^{2} \times S^{1}$; it will be
used in the proof of the ultimate claim in the paper, Proposition~\ref{pro:TheLastCase}.
\begin{prop}
\label{pro:SolidTorusSurgery2}
Let $X$ be a compact orientable connected manifold so that $\partial {X}$ consists of tori.
Denote the components of $\partial X$ by $T,T_{1},\dots,T_{n}$.
Fix $\mathcal{T}$ a non empty subset of $\{T_{1},\dots,T_{n}\}$.
For a multislope $\alpha$ on $T_{1},\dots,T_{n}$, we will denote
the link formed by the cores of the solid tori attached to $\mathcal{T}$ as $\mathcal{L}$.
Let $\mathcal{A}$ be a set of multislopes of $\partial X$ so that every $\alpha \in \mathcal{A}$
satisfies the following conditions:
\begin{enumerate}
\item $X(\alpha) \cong D^{2} \times S^{1}$.
\item $X(\alpha) \setminus \mbox{int}N(\mathcal{L})$ is irreducible.
\item No component of $\mathcal L$ is a core of $X(\alpha)$.
\item $\alpha|_{T} = \ensuremath\infty$.
\end{enumerate}
Then for each $T \in \mathcal{T}$, there exists a bounded set $B_{T}$ of
the slopes of $T$, so that for any $\alpha \in \mathcal A$ there exists $T \in \mathcal{T}$ so that
$$\alpha|_{T} \in B_{T}.$$
\end{prop}
\begin{proof}
We will induct on $|T(X)|$. Parts of the proof are similar to the proof of Theorem~\ref{thm:SlopesOnSolidTorus}
and we will only sketch them here.
\bigskip\noindent
{\bf Assume that $X$ is hyperbolic.} Since $X(\alpha) \cong D^{2} \times S^{1}$ is not hyperbolic, any
$\alpha \in \mathcal A$ factors through a minimally non hyperbolic\ filling. Let $\alpha' \pf \alpha$ be a minimally non hyperbolic\ filling.
If, for some $T \in \mathcal{T}$, $\alpha'|_{T} \neq \ensuremath\infty$, we add $\alpha'|_{T}$ to $B_{T}$.
We assume as we may that for all $T \in \mathcal{T}$, $\alpha'|_{T} = \ensuremath\infty$. Then
$\mathcal T \subset \partial X(\alpha')$. We see that the filling $\alpha|_{\partial X(\alpha')}$
induced by $\alpha$ on $X(\alpha')$ fulfills conditions (1)$\sim$(4) of the proposition; since
$|T(X(\alpha'))| < |T(X)|$, the proposition follows from the inductive hypothesis.
We assume from now on that $X$ is not hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is not prime.} Let $S$ be a sphere embedded in $X$ that realizes the
decomposition $X' \# X''$, where $X'$ is prime and
$T \subset X'$ (we are not assuming that $X''$ is prime).
By condition~(2) of the proposition, $\mathcal T \subset \partial X'$. Let $\mathcal{A}'$
be the restrictions defined by
$$ \mathcal{A}' = \{ \alpha|_{\partial X'} | \alpha \in \mathcal{A}\}.$$
It is easy to see that conditions (1)$\sim$(4) of the proposition hold, and since $|T(X')| < |T(X)|$,
the proposition follows from the inductive hypothesis.
We assume from now on that $X$ is prime and not hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} We first fix the notation that will be used in this case.
Let $X_{0}$ be the component of the torus decomposition of $X$ that contains
$T$. We will denote the components of $\partial X_{0}$ as $T$, $F_{1},\dots,F_{k}$ and the components of
$\mbox{cl}(X \setminus X_{0})$ as $X_{1},\dots,X_{k}$, numbered so that $F_{j} \subset \partial X_{j}$.
To avoid the situation $X_{j} = \emptyset$, if $F_{j} \subset \partial X$ we push it slightly into the
interior so that $X_{j} \cong T^{2} \times [0,1]$ in this case.
Since $D^{2} \times S^{1}$ contains no non-separating tori,
we assume as we may that $X_{j} \cap X_{j'} = \emptyset$ for $j \neq j'$. We will denote as
$\mathcal{F} \subset \{F_{1},\dots,F_{k}\}$ the components $F_{j}$ that bound $X_{j}$ for which
$\mathcal{T} \cap \partial X_{j} \neq \emptyset$.
Given $\alpha \in \mathcal{A}$, we will denote $\alpha|_{\partial X_{j}}$ by $\alpha_{j}$.
(Note that by definition of $\mathcal{F}$, $F_{j} \in \mathcal{F}$ if and only if
$\mathcal{L} \cap X_{j}(\alpha_{j}) \neq \emptyset$.)
Clearly, $\partial X_{j}(\alpha_{j}) = F_{j}$ and $X_{j}(\alpha_{j})$ is either a solid torus or
the exterior of a non trivial knot which we will denote as $E(K_{j})$.
Up-to finite ambiguity, we fix $\mathcal{F}_{\mbox{st}}$ and $\mathcal{F}_{\mbox{k}}$
so that $\mathcal{F} = \mathcal{F}_{\mbox{st}} \sqcup \mathcal{F}_{\mbox{k}}$
and consider the multislopes $\alpha \in \mathcal{A}$ for which
$X_{j}(\alpha_{j}) \cong D^2 \times S^{1}$ whenever $F_{j} \in \mathcal{F}_{\mbox{st}}$ and
$X_{j}(\alpha_{j}) \cong E(K_{j})$ whenever $F_{j} \in \mathcal{F}_{\mbox{k}}$.
To avoid overly complicated notation we do not rename $\mathcal{A}$.
There are two cases to consider:
\medskip\noindent
{\bf Case One: some $X_{j}(\alpha_{j})$ has no core.}
Let $\mathcal{A}_{1} \subset \mathcal{A}$ be defined by requiring that
for some $F_j \in \mathcal{F}_{\mbox{st}}$, no component of $\mathcal{L} \cap X_{j}(\alpha_{j})$
is a core of the solid torus $X_{j}(\alpha_{j})$.
The second assumption of the proposition implies that $\mathcal{L} \cap X_{j}(\alpha_{j})$
is irreducible. By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{j})| < |T(X)|$.
Applying the inductive hypothesis to $X_{j}$ we see that
for each $T \in \mathcal{T} \cap \partial X_{j}$,
there is a bounded set $B_{T}$, so that $\alpha|_{T} \in B_{T}$ for some such $T$.
The proposition follows for $\mathcal{A}_{1}$.
\medskip\noindent
{\bf Case Two: every $X_{j}(\alpha_{j})$ has core.}
Let $\mathcal{A}_{2} = \mathcal{A} \setminus \mathcal{A}_{1}$.
Then for every $F_j \in \mathcal{F}_{\mbox{st}}$ and every $\alpha \in \mathcal{A}_{2}$,
the core of the solid torus attached to one of the components of $\mathcal{T} \cap \partial X_{j}$
is a core of $X_{j}(\alpha_{j})$; we will denote it as $L_{j}$ (there may be more than
one such component; we pick one).
By Lemma~\ref{lem:KnotExteriorsAreDisjoint},
for every $j$ for which $F_{j} \in \mathcal{F}_{\mbox{k}}$,
there exists an embedded ball $D_{j} \subset X_{0}(\alpha_{0})$, so that $X_{j}(\alpha_{j}) \subset D_{j}$
and $D_{j} \cap D_{j'} = \emptyset$ for $j \neq j'$.
Thus the second assumption of the proposition implies that $\mathcal{F}_{\mbox{st}} \neq \emptyset$.
Any $\alpha \in \mathcal{A}_{2}$ induces a multislope on $\partial X_{0} = T,F_1,\dots,F_{k}$,
which we will denote as $\alpha_{0}$, that consists of following slopes:
\begin{enumerate}
\item the meridian of $X_{j}(\alpha_{j})$ (on components $F_{j}$
that bound $X_{j}(\alpha_{j}) \cong D^{2} \times S^{1}$),
\item a slope that intersects the meridian of $X_{j}(\alpha_{j})$ exactly once
(on components $F_{j}$ that bound
$X_{j}(\alpha_{j}) \cong E(K_{j})$, a non trivial knot exterior),
\item $\ensuremath\infty$ (on $T$).
\end{enumerate}
For $F_j \in \mathcal{F}_{\mbox{k}}$, the core of the solid tori attached to $F_{j}$ is an unknot
in $D_{j}$; thus the cores of the solid tori attached to $\cup_{F_j \in \mathcal{F}_{\mbox{\tiny k}}} F_{j}$
form a (possibly empty) unlink, which we will denote as $\mathcal{U}$.
The cores of the solid tori attached to $\cup_{F_j \in \mathcal{F}_{\mbox{\tiny st}}} F_{j}$
form a (non empty) link which we will denote as $\mathcal{L}_{0}$.
We claim that $\mathcal{L}_{0}$ is
irreducible in the complement of $\mathcal{U}$.
Assume, for a contradiction, that $X(\alpha) \setminus \mbox{int}N(\mathcal{L}_{0} \cup \mathcal{U})$
is reducible and let $S$ be a reducing sphere. Since $S$ is disjoint form the cores
of the solid tori attached to $F_{j}$ (for every $F_{j} \in \mathcal{F}$)
we may isotope $S$ out of them. It is now easy to see that $S \subset X(\alpha)$
and is disjoint from $\mathcal{L}$. Since $S$ is a reducing sphere for
$\mathcal{L}_{0} \cup \mathcal{U}$, there are components of
$\mathcal{L}_{0} \cup \mathcal{U}$ in the ball bound by $S$ in $X_{0}(\alpha_{0})$. Clearly,
there are components of $\mathcal{L}$ in the ball bound by $S$ in $X(\alpha)$. Thus $S$ is a reducing
sphere for $\mathcal{L}$, contradicting the second assumption of the proposition.
We assume as we may by Lemma~\ref{lem:TwistingToGetIrreducible} that
the slopes on $\mathcal{U}$ were chosen so that $\mathcal{L}_0$ is irreducible.
We claim that no component of $\mathcal{L}_{0}$ is the core of
$X_{0}(\alpha_{0})$. Assume for a contradiction
that this is not the case and fix $F_j \in \mathcal{F}_{\mbox{st}}$
for which the core of the solid torus attached to $F_{j}$
is a core of $X_{0}(\alpha_{0})$.
Recall that $L_{j}$ is a core of a solid torus attached to a component of
$\mathcal{T} \cap \partial X_{j}$, and is the core of $X_{j}(\alpha_{j})$. Thus
$L_{j}$ is a component of $\mathcal{L}$, and is the core of $X(\alpha)$;
this is impossible as it violates the third assumption of the proposition.
Thus $X_{0}$, $\mathcal{L}_{0}$, and the multislopes induced by $\mathcal{A}_{2}$ satisfy the assumptions of the proposition.
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})| < T(X)|$. Hence by induction,
for every $F_{j} \in \mathcal{F}_{\mbox{st}}$, there is a bounded set of slopes of $F_{j}$,
which we will denote as $B_{F_{j}}$, so that for every multislope $\alpha_{0}$ (induced by some
$\alpha \in \mathcal{A}_{2}$), there is $F_j \in \mathcal{F}_{\mbox{st}}$,
for which $\alpha_{0}|_{F_{j}} \in B_{F_{j}}$. The slope $\alpha|_{\partial N(L_{j})}$
is the projection of $\alpha_{0}|_{F_{j}}$ by the product structure
on $X_{j}(\alpha_{j}) \setminus N(L_{j}) \cong T^{2} \times [0,1]$.
By the $T^{2} \times [0,1]$ Cosmetic Surgery
Theorem~(\ref{thm:CosmeticSurgeryOnT2XI}) the projections of $B_{F_{j}}$
under all possible fillings of $X_{j}$ that yield $T^{2} \times [0,1]$
is a bounded set of slopes of $\partial N(L_{j})$, which we will denote as $B_{j}$.
Thus, for every $\alpha \in \mathcal{A}_{2}$, there exists
$F_j \in \mathcal{F}_{\mbox{st}}$, for which $\alpha|_{\partial N(L_{j})} \in B_{j}$, proving the proposition
in case two.
We assume from now on that $X$ is irreducible and not hyperbolic, and not JSJ.
\bigskip\noindent
{\bf Assume that $X$ is a Seifert fibered space.}
Fix a Seifert fiberation on $X$. We consider three cases, depending
on the intersection of $\alpha|_{T_{i}}$ with slopes defined by the
Seifert fiber on $T_{i}$:
\begin{enumerate}
\item If (for some $T_{i} \in \mathcal{T}$) $\alpha|_{T_{i}}$ is the fiber in the
Seifert fiberation then the intersection number
of $\alpha|_{T_{i}}$ with the fiber is zero.
\item If (for some $T_{i'} \not \in \mathcal{T}$) $\alpha|_{T_{i'}}$ is the fiber in the
Seifert fiberation, then for every $T_{i} \in \mathcal{T}$ the disk obtained by gluing
a vertical annulus connecting $T_{i'}$ to $T_{i}$ with
a meridian disk of the solid torus attached to $T_{i'}$
is a compressing disk for $T_{i}$ and its boundary is a regular fiber. Since
$X(\alpha) \cong D^{2} \times S^{1}$ contains no non separating spheres
or lens space summands, $\alpha|_{T_{i}}$ intersects the regular fiber exactly once.
\item If $\alpha_{T_{i}}$ is not the fiber for any $1 \leq i \leq n$,
then the fiberation on $X$ extends to a fiberation of $X(\alpha)$,
which is a fiberation over
$D^{2}$ with at most one exceptional fiber. The exceptional fiber, if exists, is the core of $X(\alpha)$.
Thus by assumption~(3) of the proposition every component of $\mathcal L$ is a regular fiber, implying that
for every $T_{i} \in \mathcal{T}$, $\alpha|_{T_{i}}$
intersects the fiber in the Seifert fiberation of $X$ exactly once.
\end{enumerate}
We conclude that for every $\alpha \in \mathcal{A}$ there exists $T \in \mathcal{T}$ so that
$\alpha|_{T}$ intersects the fiber at most once.
The proposition follows from Proposition~\ref{prop:PropertiesOfBoundedSets}~(4).
\bigskip\noindent
This completes the proof of Proposition~\ref{pro:SolidTorusSurgery2}.
\end{proof}
\section{Hyperbolic cosmetic surgery: slopes}
\label{section:HyperSurgSlopes}
\bigskip\noindent
In this section we ask ``how much can a hyperbolic manifold get twisted by performing
cosmetic surgery''? (Recall that in this paper by {\it hyperbolic manifold} we mean a
connected compact manifold whose interior
admits a complete finite volume hyperbolic metric.)
Consider the following problem: let $M$ be a hyperbolic manifold so that $\partial M$
is a single torus, $B$ a bounded set of slopes of $\partial M$, and $L \subset M$ a link. If $\alpha$
is a multislope of cosmetic surgery (that is, $L(\alpha) \cong M$), then an identification
of $L(\alpha)$ with $M$ induces a bijection on the slopes of $\partial M$. Our goal is to show that the
union of the images of $B$ under all such bijections is bounded. The theorem below
is stated in terms of fillings (with $X$ corresponding to $M \setminus N(L)$) and is
slightly more general as it allows for more boundary components.
\begin{thm}
\label{thm:CosmeticSurgeryOnM}
Let $M$ be an orientable hyperbolic manifold, $T_{M}$ a component of $\partial M$,
$X$ a compact orientable connected manifold so that $\partial X$ consists of tori that we will denote
as $T,T_{1},\dots,T_{n}$, and $B$ a bounded set of slopes of $T$.
Let $\mathcal{X} = \{(\alpha,f_{\alpha})\}$ be a set of pairs
so that every $(\alpha,f_{\alpha}) \in \mathcal{X}$ satisfies the following conditions:
\begin{enumerate}
\item $\alpha$ is a multislope of $X$.
\item $f_{\alpha}:X(\alpha) \to M$ is a diffeomorphism.
\item $f_{\alpha}$ maps $T$ to $T_{M}$.
\end{enumerate}
For every $x = (\alpha,f_{\alpha}) \in \mathcal{X}$, we will denote the image of $B$ under
the bijection induced by $f_{\alpha}$ from the slopes of $T$ to those of $T_{M}$ as $B_{x}$.
Then $\cup_{x \in \mathcal{X}} B_{x}$ is a bounded set of slopes of $T_{M}$.
\end{thm}
\begin{proof}
We induct on $|T(X)|$.
\bigskip\noindent
{\bf Assume that $X$ is Seifert fibered or sol.} Then $X$ admits no hyperbolic filling.
We assume from now on that $X$ is not Seifert fibered or sol.
\bigskip\noindent
{\bf Assume that $X$ is not prime.} Let $S \subset X$ be a decomposing sphere that realizes
the decomposition $X = X' \cup_{S} X''$, where here $X'$ is a prime manifold containing $T$ (we do not
assume that $X''$ is prime). Then for every $(\alpha,f_{\alpha}) \in \mathcal{A}$
we have
$$X(\alpha) = X'(\alpha|_{\partial X'}) \cup_{S} X''(\alpha|_{\partial X''}).$$
Since $(\alpha) \cong M$ is hyperbolic, $S \subset X(\alpha)$ bounds a ball.
Condition~(3) of the theorem implies that $\alpha|_{T} = \ensuremath\infty$; hence $X''(\alpha|_{\partial X''})$
is a ball. Thus $f_{\alpha}$ induces a diffeomorphism that we will denote as
$$f_{\alpha|_{\partial X'}}:X'(\alpha|_{\partial X'}) \to M.$$
We will denote the set of pairs $\{(\alpha|_{\partial X'},f_{\alpha|_{\partial X'}})\}$
induced by pairs $(\alpha,f_{\alpha}) \in \mathcal{X}$ as $\mathcal{X}'$.
For $(\alpha|_{\partial X'},f_{\alpha|_{\partial X'}}) = x' \in \mathcal{X}'$,
we will denote the image of $B$ under the bijection induced by $f_{\alpha|_{\partial X'}}$
from the slopes of $T$ to those of $T_{M}$ as $B_{x'}$.
Since $X'$ corresponds to a direct descendant of the root of $T(X)$,
$|T(X')| < |T(X)|$. By induction
$$\cup_{x' \in \mathcal{X}'} B_{x'}$$
is bounded set of slopes of $T_{M}$. By construction, every $x' \in \mathcal{X}'$ is induced by
$x \in \mathcal{X}$ for which $f_{\alpha|_{\partial X'}}|_{T} = f_{\alpha}|_{T}$. Thus $B_{x'} = B_{x}$
and therefore
$$\cup_{x \in \mathcal{X}} B_{x} = \cup_{x' \in \mathcal{X}'} B_{x'}$$
is a bounded set of slopes of $T_{M}$. The theorem follows in this case.
We assume from now on that $X$ is prime and not Seifert fibered or sol.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} By Lemma~\ref{prop:ObtainedByFilling},
each $\alpha \in \mathcal{A}$ induces a filling on one of the
components of the torus decomposition of $X$ that yields $M$. Up-to finite ambiguity we
fix one component of the torus decomposition of $X$, that we will denote as $X_{0}$,
and consider only multislopes $\alpha$ that induce a filling on $X_{0}$.
We will use the following notation:
\begin{enumerate}
\item The multislope induced by $\alpha$ on $\partial X_{0}$ will be denoted as $\alpha_{0}$.
\item The closure of the component of $X \setminus X_{0}$ that contains $T$ will be denoted as $X_{1}$
(to avoid the situation $X_{1} = \emptyset$, if $T \subset \partial X_{0}$ we push it slightly into the interior
so that $X_{1} \cong T^{2} \times [0,1]$ in this case).
\item The torus $X_{0} \cap X_{1}$ will be denoted as $F$.
\end{enumerate}
Let $F_{M}$ be a boundary parallel torus in $M$; it follows from the construction in
Lemma~\ref{prop:ObtainedByFilling} that after isotopy of $f_{\alpha}$ if necessary
we may assume that $f_{\alpha}(F) = F_{M}$. We will denote the components of
$M$ cut open along $F_{M}$ as $M_{0}$ and $M_{1}$, where $M_{0}$ is
the component not containing $T_{M}$.
By the construction in Lemma~\ref{prop:ObtainedByFilling}, $f_{\alpha}$ induces a diffeomorphism
$f_{\alpha_{0}}:X_{0}(\alpha_{0}) \to M_{0}$ that maps $F$ to $F_{M}$.
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})| < |T(X)|$. Therefore by induction
the union of the images of $B$ under the bijections induced by $f_{\alpha_{0}}$
is a bounded set of slopes of $F_{M}$ that we will denote as $B_{F_{M}}$.
Every slope of $\cup_{x \in \mathcal{X}} B_{x}$ is obtained from a slope in $B_{F_{M}}$
by projecting using the product structure on $M_{1} \cong T^{2} \times [0,1]$. Since
$M_{1}$ is obtained from $X_{1}$ by filling, by the $T^{2} \times [0,1]$ Cosmetic
Surgery Theorem~(\ref{thm:CosmeticSurgeryOnT2XI}), $\cup_{x \in \mathcal{X}} B_{x}$
is a bounded set of slopes of $T_{M}$.
This completes the proof for JSJ manifolds.
We assume from now on that $X$ is prime and not Seifert fibered, sol, or JSJ.
\bigskip\noindent
{\bf Assume $X$ is hyperbolic.}
Let $\mathcal{X}^{1} \subset \mathcal{X}$ be all pairs $(\alpha,f_{\alpha})$ for which
$\alpha$ is totally hyperbolic. Fix $\alpha$ a totally
hyperbolic filling and let $(\alpha,f_{j})$ be all the elements of $\mathcal{X}^{1}$
that have $\alpha$ as their multislope, and different diffeomorphisms
$f_{j}:X(\alpha) \to M$ (for $j\in J$ for some index set $J$). We will denote
the image of $B$ under the bijection induced by $f_{j}$ between the slopes of $T$
and those of $T_{M}$ as $B_{j}$.
Then $f_{j} \circ f_{1}^{-1}:M \to M$ is a diffeomorphism that sends $T_{M}$ to itself
and $f_{j} = (f_{j} \circ f_{1}^{-1}) \circ f_{1}$. Thus $B_{j} = \phi_{j}(B_{1})$,
where $\phi_{j}$ is a bijection induce by an element of the mapping class group of $M$.
It is straightforward to see that the bijection induced by $\phi_{j}$ is an isometry
of the slopes; hence $\phi_{j}(B_{1})$ is bounded.
Since the mapping class group of hyperbolic manifolds is finite, we see that
$\cup_{j \in J}\phi_{j}(B_{1})$ is bounded as well.
By Proposition~\ref{prop:UsingT(X)forHyperbolicFilling}, $X$ admits
only finitely many totally hyperbolic fillings, and hence
$$\cup_{x \in \mathcal{X}^{1}} B_{x}$$
is a bounded set of slopes.
Next we consider $\mathcal{X}^{2} = \mathcal{X} \setminus \mathcal{X}^{1}$.
By Proposition~\ref{pro:mnhIsFinite}, $X$ admits
only finitely many minimally non hyperbolic\ fillings. We will denote as $\alpha_{1},\dots,\alpha_{k}$
the minimally non hyperbolic\ fillings of $X$ for which $\alpha_{j}|_{T} = \ensuremath\infty$. For
$1 \leq j \leq k$, let $\mathcal{X}_{j}$ be the set of all pairs
pairs $(\alpha,f_{\alpha})$ satisfying the following conditions:
\begin{enumerate}
\item $\alpha$ is a multislope of $X(\alpha_{j})$.
\item $f_{\alpha}:X(\alpha_{j})(\alpha) \to M$ is a diffeomorphism.
\item $f_{\alpha}$ maps $T$ to $T_{M}$.
\end{enumerate}
For $x_{j} =(\alpha,f_{\alpha}) \in \mathcal{X}_{j}$, we will denote the image of $B$
under the bijection induced by
$f_{\alpha}$ between the slopes of $T$ and those of $T_{M}$ as $B_{x_{j}}$.
Since $X(\alpha_{j})$ corresponds to a direct descendant of the root of $T(X)$,
$|T(X(\alpha_{j}))| <|T(X)|$. Thus by induction
$$\cup_{x_{j} \in \mathcal{X}_{j}} B_{x_{j}}$$
is a bounded set of slopes of $T_{M}$.
By definition of $\mathcal{X}^{2}$, for every $x = (\alpha,f_{\alpha}) \in \mathcal{X}^{2}$, $\alpha$ is not minimally non hyperbolic.
Hence $\alpha$ factors through some minimally non hyperbolic\ filling $\alpha_{j}$ (for some $1 \leq j \leq k$),
that is,
$$X(\alpha) = X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j})}).$$
We can view $f_{\alpha}:X(\alpha) \to M$ as a diffeomorphism
$f_{\alpha}:X(\alpha_{j})(\alpha|_{\partial X(\alpha_{j}}) \to M$;
thus we obtain $(\alpha|_{\partial X(\alpha_{j})}, f_{\alpha}) \in \mathcal{X}_{j}$
for which $B_{(\alpha|_{\partial X(\alpha_{j})}, f_{\alpha})} = B_{x}$.
This shows that
$$\cup_{x \in \mathcal{X}^{2}}B_{x} \subset \cup_{j=1}^{k}(\cup_{x_{j} \in \mathcal{X}_{j}}B_{x_{j}}).$$
Thus $\cup_{x \in \mathcal{X}^{2}}B_{x}$ is contained in a finite union of
bounded sets, and hence is itself bounded.
The theorem follows in this final case.
\bigskip\noindent
This completes the proof of Theorem~\ref{thm:CosmeticSurgeryOnM}.
\end{proof}
\section{Hyperbolic cosmetic surgery: radius of injectivity}
\label{section:HyperSurgRadInj}
\bigskip\noindent
Let $X$ be a hyperbolic manifold. A generic filling on $\mathcal{T} \subset \partial X$, with {\it all} the slopes very
long, yields a hyperbolic manifold with at least $|\mathcal{T}|$ short geodesic. However, if $|\mathcal{T}|>1$,
this requires excluding infinitely many multislopes. It is easy to construct examples where
$X$ can be filled to give infinitely many manifolds that violate this rule, for example,
every lens space is obtained by filling the Whitehead link exterior. As another example,
given any hyperbolic manifold $M$, let $K \subset M \# T^{2} \times [0,1]$ be a simple
knot. Then the exterior of $K$ is a hyperbolic manifold
that admits infinitely many distinct multislopes $\alpha^{j}$ so that
$X(\alpha^{j}) \cong M$ for every $j$ without the expected three short geodesics.
In this section we show that although the set of multislopes yielding manifolds without
a short geodesic may be infinite, only finitely many manifolds can be obtained.
\bigskip\noindent
\begin{thm}
\label{thm:HyperbolicFillingShortGeos}
Let $X$ be a compact connected oriented manifold so that $\partial X$ consists of tori. Fix $\epsilon>0$. Then
all but finitely many hyperbolic manifolds that are obtained by filling $X$ admit a geodesic of length
less than $\epsilon$.
If in addition $X$ is hyperbolic, then there are only finitely many totally hyperbolic fillings $\alpha$ on $X$ so
that $X(\alpha)$ does not admit a geodesic of length less than $\epsilon$.
\end{thm}
\begin{proof}
We induct on $|T(X)|$.
\bigskip\noindent
{\bf Assume that $X$ is Seifert fibered or sol.} Then no filling of $X$ yields a hyperbolic manifold.
We assume from now on that $X$ is not Seifert fibered or sol.
\bigskip\noindent
{\bf Assume that $X$ is reducible.} Let $X_{1},\dots,X_{n}$ be the factors of the
prime decomposition of $X$. Then in any filling of $X$ that gives a hyperbolic manifold (say $M$),
exactly one $X_{i}$ fills to give $M$, and every other $X_{i'}$ fills to a ball. Thus every hyperbolic
manifold obtained by filling $X$ is obtained by filling $X_{i}$ for some $i$.
Up-to finite ambiguity we fix a factor $X_{i}$. Since $X_{i}$ corresponds to a direct descendant
of $X$, $|T(X_{i})| < |T(X)|$. By induction there are only finitely
many hyperbolic manifolds obtained by filling $X_{i}$ that do not admit a geodesic of
length less than $\epsilon$. The proposition follows in this case.
We assume from now on that $X$ is prime and not Seifert fibered or a solv manifold.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} Let $X_{1},\dots,X_{n}$ be the components of the
prime decomposition of $X$. By Proposition~\ref{prop:ObtainedByFilling} any hyperbolic
manifold that is obtained from $X$ by filling is obtained by filling some $X_{i}$.
Up-to finite ambiguity we fix a component of the torus decomposition of $X$ which
we will denote as $X_{i}$. By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{i})| < |T(X)|$. By induction there are only finitely
many hyperbolic manifolds obtained by filling $X_{i}$ that do not admit a geodesic of
length less than $\epsilon$. The proposition follows in this case.
We assume from now on that $X$ is prime, not Seifert fibered or sol, and not
JSJ.
\bigskip\noindent
{\bf Assume that $X$ is hyperbolic.} Let $\mathcal{A}$ be an infinite set of multislopes
of $\partial X$ so that $X(\alpha)$ is hyperbolic and does not contain
a geodesic shorter than $\epsilon$ for every $\alpha \in \mathcal{A}$.
(Note that if no such set exists there is nothing to prove.)
We will first establish conclusion~(2) of the theorem by showing that some multislope
$\alpha \in \mathcal{A}$ is not totally hyperbolic.
We will denote the components of $\partial X$ as $T_{1},\dots,T_{n}$.
After subsequencing and reordering if necessary we assume as we may that for some $0 \leq k \leq n+1$ we have:
\begin{enumerate}
\item For every $1 \leq i \leq k$ and every $j \neq j'$, $\ \alpha_{i}^{j} \neq \alpha_{i}^{j'}$
and $\alpha_{i}^{j}\neq\ensuremath\infty$.
\item For every $k+1 \leq i \leq n+1$ and every $j, j'$, $\ \alpha_{i}^{j} = \alpha_{i}^{j'}$.
\end{enumerate}
To avoid overly complicated notation we do not rename $\mathcal{A}$.
Let $\alpha_{0}$ be the restriction $\alpha|_{T_{k+1},\dots,T_{n}}$
for some $\alpha \in \mathcal{A}$ (by construction $\alpha_{0}$ is independent of choice),
$\widehat X = X(\alpha_{0})$ (so $\partial \widehat{X} = T_{1},\dots,T_{k}$),
and $\widehat{\mathcal{A}} = \{ \alpha|_{\partial \widehat{X}} \ | \ \alpha \in \mathcal{A}\}$.
We claim that $\widehat{X}$ is not hyperbolic; assume for a contradiction that it is.
By truncating the cusps of $\widehat X$ we obtain a Euclidean metric on every $T_{i}$ ($1 \leq i \leq k$).
Since $\widehat{\mathcal{A}}$ is infinite and the values
$\{\widehat\alpha|_{T_{i}}\ | \ \widehat\alpha \in {\widehat{\mathcal A}}\ \}$ are distinct, for any $l$ there is
a multislope $\widehat\alpha \in\widehat{\mathcal{A}}$ so that $\widehat\alpha|_{T_{i}}$
is longer than $l$ for all $i$. By Thurston's Dehn
Surgery Theorem, for large enough $l$, $\widehat{X}(\widehat\alpha)$ is hyperbolic and
the cores of the attached solid tori are geodesics of length less
than $\epsilon$, contradicting our assumptions. Thus $\widehat X$ is not
hyperbolic. Since $\mathcal{A}$ is infinite, $k\geq 1$. By condition~(1) above,
for every $\alpha \in \mathcal{A}$, $\alpha_{0}$ is a strict partial filling of $\alpha$.
This shows that $\alpha$ is not totally hyperbolic, establishing the second conclusion
of the theorem.
Let $\mathcal{A}$ be the set of all multislopes
of $\partial X$ so that $X(\alpha)$ is hyperbolic and does not contain
a geodesic shorter than $\epsilon$.
We will denote the set of totally hyperbolic fillings in $\mathcal{A}$ as
$\mathcal{A}_{-}$
and $\mathcal{A} \setminus \mathcal{A}_{-}$ as $\mathcal{A}_{+}$.
Every $\alpha \in \mathcal{A}_{+}$ admits
a minimally non hyperbolic\ partial filling, and the minimally non hyperbolic\ fillings of $X$ correspond to the direct descendants of the root
of $T(X)$; up-to finite ambiguity we fix a direct descendant of the root of $X$ that
we will denote as $X_{i}$. Then $|T(X_{i})| < |T(X)|$. By induction there are only finitely
many hyperbolic manifolds obtained by filling $X_{i}$ that do not admit a geodesic of
length less than $\epsilon$. The theorem follows from this and finiteness of $\mathcal{A}_{-}$
that was established above.
This completes the proof of Theorem~\ref{thm:HyperbolicFillingShortGeos}.
\end{proof}
\section{Cosmetic surgery on $S^{3}$}
\label{section:CosmeticSurgeryOnS3}
\bigskip\noindent
We now to turn to one of the more interesting applications of $T(X)$, concerning
cosmetic surgery on $S^{3}$. Recall that a {\it cosmetic surgery} is $S^{3}$
is a surgery on a link $L \subset S^{3}$ with multislope $\alpha$
so that $L(\alpha) \cong S^{3}$. Note that following examples:
\begin{enumerate}
\item Let $L = K_{1} \cup K_{2}$ be the Whitehead link. Then infinitely many slopes on $K_{1}$ can be completed to
a cosmetic surgery, namely: $1/m$ can be completed to the cosmetic surgery given by $(1/m,1/0)$,
where here and in the examples below we are using the standard meridian--longitude. That
is not a real problem: $\{1/m\}$ is a bounded set.
\item Worse is the Hopf link $H = K_{1} \cup K_{2}$. It is easy to see that $H(p/q,r/s) \cong S^{3}$
if and only if $ps-rq=\pm1$.
Thus {\it every} slope on $K_{1}$ can be completed to a cosmetic surgery.
\item Kawauchi~\cite{kawauchi} constructed a two component link $L = K_{1} \cup K_{2} \subset S^{3}$ that admits a non-trivial
cosmetic surgery. By Teragaito \cite{teragaito} we may assume that $L$, $K_{1}$ and
$K_{2}$ are all hyperbolic (this was also announced by Kawauchi~\cite{kawauchi2}).
For a detailed discussion see the introduction to~\cite{teragaito}.
Let $L' \subset S^{3}$ be the core of the attached solid tori
after this surgery, and let $H' \subset S^{3}$ be the Hopf link. By isotopy of $H'$ (where we allow $H'$ to intersect $L'$)
we place $H'$ in a ``very complicated'' position relative to $L'$. There is a surgery on $L'$ which ``undoes'' the surgery
gets back to $S^{3}$. Denote the image of $H'$ under this surgery by $H = K_{3} \cup K_{4}$.
For any slope $\alpha_{3}$ there exists infinitely many slopes $\alpha_{4}$ so that
$L \cup H(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}) \cong S^{3}$.
However, we expect that $H$ is no longer the Hopf link; in fact, it is quite likely that the components of $H$
are no longer unknotted, as the disks bound by the components of $H'$ are likely to be destroyed by the surgery on $L'$,
and new disks are unlikely to appear.
We do not prove these claims, but in light of this discussion we expect the following to be true:
there exists a four component link in $S^{3}$ (such as $L \cup H = K_{1} \cup K_{2} \cup K_{3} \cup K_{4}$
above) that contains no Hopf sublink (perhaps even no
unknotted components), yet every slope on $K_{3}$ can be completed to a cosmetic surgery.
The moral is this: it is our aim to prove that {\it not} any slope can be competed to a cosmetic surgery,
but one must beware to Hopf links, including those that are invisible in the original link but manifest
themselves after surgery.
\end{enumerate}
\bigskip\noindent
We are now ready to state:
\begin{thm}
\label{thm:cosmeticSurgeryOnS3}
Let $L \subset S^{3}$ be a link and denote its components by $K_{1},\dots,K_{n}$. Let $\mathcal{A}$ be a set
of multislopes of $L$ so that every $\alpha \in \mathcal{A}$ fulfills the following two conditions:
\begin{enumerate}
\item $L(\alpha) \cong S^{3}$.
\item For every $\alpha' \pf \alpha$ with $\alpha'|_{T_{1}} = \ensuremath\infty$, $L(\alpha') \not\cong T^{2} \times [0,1]$.
\end{enumerate}
Then the restrictions $\mathcal{A}_{1} = \{\alpha|_{T_{1}} \ |\ \alpha \in \mathcal{A} \}$ form a bounded set.
\end{thm}
\begin{rmks}
\begin{enumerate}
\item There exist $\alpha' \pf \alpha$ with $\alpha'|_{T_{1}} = \ensuremath\infty$ and $L(\alpha') \cong T^{2} \times [0,1]$
if and only if the cores of the solid tori attached to $\partial N(K_{1})$ and
$\partial N(K_{i})$ form a Hopf link (for aome $2 \leq i \leq n$).
The cores of the solid tori attached along a multislope in $\mathcal{A}$ may, in fact, contain a Hopf sublink $H$;
our assumption only requires that the core of the solid torus attached to
$\partial N(K_{1})$ is not a component of $H$.
\item If there exist $\alpha' \pf \alpha$ with $\alpha'|_{T_{1}} = \ensuremath\infty$ and $L(\alpha') \cong T^{2} \times [0,1]$
then obviously
$\mathcal{A}_{1}$ may contain of all the slopes of $T_{1}$.
\end{enumerate}
\end{rmks}
\begin{proof}[Proof of Theorem~\ref{thm:cosmeticSurgeryOnS3}]
We will denote $\mbox{cl}(S^{3} \setminus N(L))$ as $X$ and $\partial N(K_{i})$ as $T_{i}$.
Although the theorem
was phrased in terms of surgery, we will prove the equivalent statement for fillings of $X$.
We induct on $|T(X)|$.
\bigskip\noindent
{\bf Assume that $X$ is not prime.} Let $X = X' \# X''$, where here $X'$ is the factor of the prime
decomposition of $X$ that contains $T_{1}$.
By renumbering the components of $\partial X$ if necessary we assume as we may that $\partial X_{1} = T_{1},\dots,T_{k}$,
for some $1 \leq k \leq n$. For any multislope $\alpha \in \mathcal{A}$ we have
$$X(\alpha) \cong X'(\alpha|_{\partial X'}) \# X''(\alpha|_{\partial X''}).$$
Thus $X'(\alpha|_{\partial X'}) \cong S^{3} \cong X''(\alpha|_{\partial X''})$.
If, for some $2 \leq i \leq k$,
$$X'(\ensuremath\infty,\alpha_{2},\dots,\alpha_{i-1},\ensuremath\infty,\alpha_{i+1},\dots,\alpha_{k}) \cong T^{2} \times [0,1],$$
then
\begin{eqnarray*}
L(\ensuremath\infty,\alpha_{2},\dots,\alpha_{i-1},\ensuremath\infty,\alpha_{i+1},\dots,\alpha_{n}) &\cong&
X'(\ensuremath\infty,\alpha_{2},\dots,\alpha_{i-1},\ensuremath\infty,\alpha_{i+1},\dots,\alpha_{k}) \# X''(\alpha|_{\partial X''}) \\
&\cong& T^{2} \times [0,1] \# S^{3} \\
&\cong& T^{2} \times [0,1].
\end{eqnarray*}
This contradicts the second assumption of the theorm.
Thus $X'$ and $\mathcal{A}' = \{\alpha|_{\partial X'} | \alpha \in \mathcal{A}\}$
fulfill the assumptions of the theorem. Since $X'$ corresponds to a direct descendant of the root of $T(X)$,
$|T(X')| < |T(X)|$. By induction, $\mathcal{A}_{1}' = \{\alpha'|_{T_{1}} \ |\ \alpha' \in \mathcal{A}' \}$ is bounded.
It is easy to see that $\mathcal{A}_{1} = \mathcal{A}_{1}'$; the theorem follows in this case.
We assume from now on that $X$ is prime.
\bigskip\noindent
{\bf Assume that $X$ is Seifert fibered a sol manifold.} Clearly we may ignore sol manifolds.
If $n=1$ then $L$ is a knot and the result is well known;
assume from now on $n\ge2$. We fix a Seifert fiberation on $X$. Then the fibers on $T_{1}$
define a slope which we denote by $\alpha_{1}^{f}$.
For convenience we will denote $\alpha|_{T_{i}}$ as $\alpha_{i}$.
Define $\mathcal{A}_{f},\ \mathcal{A}_{0}, \ \mathcal{A}_{1} \subset \mathcal{A}$ by:
\begin{enumerate}
\item $\alpha \in \mathcal{A}_{f}$ if $\alpha_{{1}} = \alpha_{1}^{f}$.
\item $\alpha \in \mathcal{A}_{0}$ if for some $2 \leq i \leq n$, $\alpha_{{i}}$ is the fiber on $T_{i}$.
\item $\alpha \in \mathcal{A}_{1}$ if $\alpha \not\in \mathcal{A}_{f} \cup \mathcal{A}_{0}$.
\end{enumerate}
Clearly, $\mathcal{A} = \mathcal{A}_{f} \cup \mathcal{A}_{0} \cup \mathcal{A}_{1}$.
If $\alpha \in \mathcal{A}_{f}$ then $\alpha_{1} \in B_{f}$, where $B_{f}$ is the bounded set of slopes of $T_{1}$
defined by $B_{f} = \{\alpha_{1}^{f}\}.$
For $\alpha \in \mathcal{A}_{0}$, let $D$ be the
disk obtained by attaching a vertical annulus connecting $T_{i}$ and $T_{1}$ to the meridian disk of the solid
torus attached to $T_{i}$ (where here $i$ is as in the definition of $\mathcal{A}_{0}$).
Thus we see that $D$ is a compressing disk for $T_{1}$ and the slope defined by $\partial D$
is $\alpha_{1}^{f}$. Since $X(\alpha_{1},\dots,\alpha_{n}) \cong S^{3}$,
$\Delta(\alpha_{1},\alpha_{1}^{f}) = 1$ (recall that $\Delta$ denotes the geometric intersection
number). Thus $\alpha_{1} \in B_{0}$, where $B_{0}$ is the bounded set of slopes of $T_{1}$ defined by:
$$B_{0} = \{\alpha | \ \Delta(\alpha,\alpha_{1}^{f}) = 1\}.$$
If $\alpha \in \mathcal{A}_{1}$ then
the fiberation of $X$ extends to a fiberation of $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$,
and the fiberation of $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ extends to a fibration of
$X(\alpha) \cong S^{3}$.
Thus $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ is a Seifert fibered
space over $D^{2}$ with at most two exceptional fibers and the cores of the solid tori
attached to $T_{1},\dots,T_{n}$ are fibers.
Assume first that $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ is a Seifert fibered
space over $D^{2}$ with exactly two exceptional fibers. Then $\alpha_{1} \in B_{0}$.
Next assume that $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ is a Seifert fibered
space over $D^{2}$ with at most one exceptional fiber, and the exceptional fiber
(if exists) is the core of the solid torus attached to $T_{i}$;
by renumbering $T_{2},\dots,T_{n}$ if necessary we may assume that
$i=2$. Then $X(\ensuremath\infty,\ensuremath\infty,\alpha_{3},\dots,\alpha_{n}) \cong T^{2} \times [0,1]$,
contradicting our assumption.
Thus we have reduced the proof to the case where $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ is a Seifert fibered
space over $D^{2}$ with exactly one exceptional fiber, and the exceptional fiber
is not the core of a solid torus attached to $T_{i}$ ($2 \leq i \leq n$).
Then the exceptional fiber is contained in $X$ and its multiplicity, which we will denote as $d$,
does not depend on $\alpha \in \mathcal{A}_{1}$.
Since $X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n})$ is a Seifert fibered space over $D^{2}$ with exactly one exceptional fiber,
$X(\ensuremath\infty,\alpha_{2},\dots,\alpha_{n}) \cong D^{2} \times S^{1}$;
we will denoting the slope defined by the boundary of its meridian disk as $\alpha'$. Then
$\Delta(\alpha_{1}^{f},\alpha') = d$. Since $X(\alpha) \cong S^{3}$, we also have that
$\Delta(\alpha',\alpha_{1})=1$.
Thus $\alpha_{1}\in B_{1}$,
where $B_{1}$ is the set of slopes of $T_{1}$ defined by:
$$B_{1} = \{\alpha\ | \ (\exists \alpha')\ \Delta(\alpha_{1}^{f},\alpha') = d \mbox{ and }\Delta(\alpha',\alpha) = 1\}.$$
By applying Proposition~\ref{prop:PropertiesOfBoundedSets}~(4) twice we see that $B_{1}$ is bounded.
Since $\mathcal{A} = \mathcal{A}_{f} \cup \mathcal{A}_{0} \cup \mathcal{A}_{1}$, for any
$\alpha \in \mathcal{A}$, $\alpha_{1} \in B_{f} \cup B_{0} \cup B_{1}$. This completes the proof
for Seifert fibered and sol manifolds.
We assume from now on that $X$ is prime and not Seifert fibered or a slov manifold.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} Let $X_{0}$ be the component of the torus decomposition of $X$
that contains $T_{1}$ and denote the remaining components of $\partial X$ as $\{F_{j}\}_{j=1}^{k}$,
see Figure~\ref{fig:thm10JSJ}.
\begin{figure}
\includegraphics[width=4in]{F3}
\caption{Notation used when $X$ is JSJ}
\label{fig:thm10JSJ}
\end{figure}
Given $\alpha \in \mathcal{A}$, we will denote the components of $X(\alpha|_{T_{2},\dots,T_{n}})$
cut open along $F_{j}$ as follows (see the right side of Figure~\ref{fig:thm10JSJ}):
\begin{enumerate}
\item The component whose boundary is $T_{1} \cup F_{j}$ will be denoted as $X(\alpha)_{j}^{+}$.
\item The component whose boundary is $F_{j}$ will be denoted as $X(\alpha)_{j}^{-}$.
\end{enumerate}
Since $S^{3}$ does not admit a non separating torus, we assume as we may that
$X(\alpha)_{j}^{-} \cap X(\alpha)_{j'}^{-} = \emptyset$ for $j \neq j'$.
There are two cases to consider:
\bigskip\noindent
{\bf Case One.} Let $\mathcal{A}_{1}$ be the multislopes $\alpha \in \mathcal{A}$ for which
$X(\alpha)_{j}^{+} \not\cong T^{2} \times [0,1]$ for all $j$.
Then $\alpha\in \mathcal{A}_{1}$ induces a multislope on $\partial X_{0}$,
which we will denote as $\alpha_{0}$, defined as follows:
\begin{enumerate}
\item $\alpha_{0}|_{T_{1}} = \alpha|_{T_{1}}$.
\item If $X(\alpha)_{j}^{-} \cong D^{2} \times S^{1}$ then $\alpha_{0}|_{F_{j}}$ is the slope
of the meridian of the solid torus $X(\alpha)_{j}^{-}$.
\item If $X(\alpha)_{j}^{-} \neq D^{2} \times S^{1}$ then $X(\alpha)_{j}^{-} \cong E(K_{j})$
for some non trivial knot $K_{j} \subset S^{3}$.
We take $\alpha_{0}|_{F_{j}}$ to be any slope that intersects the meridian of $E(K_{j})$ exactly once.
\end{enumerate}
By Lemma~\ref{lem:KnotExteriorsAreDisjoint}, we may assume that the components $X(\alpha)_{j}^{-}$
in Case~(3) are contained in disjointly embedded balls; hence removing $X(\alpha)_{j}^{-}$ and attaching a solid
torus along $\alpha_{0}|_{F_{j}}$ does not change
$X(\alpha)$ or $X(\alpha)_{j'}^{+}$ (for $1 \leq j' \leq k$), recall
Figure~\ref{fig:unknotting}. Thus $X_{0}$ and the induced slopes
$\{\alpha_{0}\ | \ \alpha \in \mathcal{A}_{1}\}$ satisfy the conditions of the Theorem
(condition~(1) follows from the corresponding assumption for $X$, and condition~(2)
follows from the defining assumption of case one).
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})| < |T(X)|$.
By induction, $\{\alpha_{0}|_{T_{1}} \ | \ \alpha \in \mathcal{A}_{1}\}$ is a bounded set which
we will denote as $B_{1}$. By construction $\alpha_{0}|_{T_{1}} = \alpha|_{T_{1}}$.
Hence $\{\alpha|_{T_{1}} \ | \ \alpha \in \mathcal{A}_{1} \} = B_{1}$ is bounded.
\bigskip\noindent
{\bf Case Two.} Fix $1 \leq j \leq k$. Let $\mathcal{A}_{2,j} \subset \mathcal{A}$ be the multislopes $\alpha \in \mathcal{A}$
for which $X(\alpha)_{j}^{+} \cong T^{2} \times [0,1]$. The definitions immediately imply that the following two conditions
hold:
\begin{enumerate}
\item For any $\alpha \in \mathcal{A}_{2,j}$, $X(\alpha)_{j}^{+}(\alpha|_{T_{1}}) \cong D^{2} \times S^{1}$.
\item $\mathcal{A} = \mathcal{A}_{1} \cup (\cup_{j=1}^{k} \mathcal{A}_{b,j})$.
\end{enumerate}
We will denote the component of $X$ cut open along $F_{j}$ that does not contain $X_{1}$ as $N_{j}$
and $\partial X \cap N_{j}$ as $\mathcal{T}_{j}$; to avoid the situation $N_{j} = \emptyset$,
if $F_{j} \subset \partial X$ we push it slightly into the interior so that $N_{j} \cong T^{2} \times [0,1]$
in this case. Note that $\partial N_{j} = F_{j} \cup \mathcal{T}_{j}$.
Every $\alpha \in \mathcal{A}_{2,j}$ induces the multislope on $\partial N_{j}$,
which we will denote as $\alpha_{j}$, defined by
the slope of the meridian of the solid torus $X(\alpha)_{j}^{+}(\alpha|_{T_{1}})$
on $F_{j}$ and the restriction $\alpha|_{\mathcal{T}}$ on $\mathcal{T}_{j}$.
We show that the following two conditions hold:
\begin{enumerate}
\item $N_{j}(\alpha_j) \cong S^{3}$: by construction, $N_{j}(\alpha_{j}) \cong X(\alpha)$
and by assumption, $X(\alpha) \cong S^{3}$.
\item Let $\alpha_{j}' \pf \alpha_{j}$ be a partial filling for which $\alpha_{j}'|_{F_{j}} = \ensuremath\infty$.
Then $N_{j}(\alpha_{j}') \not\cong D^{2} \times S^{1}$: assume for a
contradiction that $N_{j}(\alpha_{j}') \cong D^{2} \times S^{1}$.
Let $T_{i}$ be the component of $\partial N_{j} \cap X$ for which $\alpha_{j}'|_{T_{i}} = \ensuremath\infty$.
Let $\alpha' \pf \alpha$ be the partial filling giving by setting $\alpha'|_{T_{1}}$ and
$\alpha'|_{T_{i}}$ to $\ensuremath\infty$. Then
\begin{eqnarray*}
X(\alpha') &\cong& X(\alpha)_{j}^{+} \cup_{F_{j}} N_{j}(\alpha_{j}') \\
&\cong& T^{2} \times [0,1] \cup_{F_{j}} T^{2} \times [0,1] \\
&\cong& T^{2} \times [0,1],
\end{eqnarray*}
violating assumption~(2) of the theorem.
\end{enumerate}
Thus the assumptions of the theorem are satisfied by $N_{j}$ and
$\{\alpha_{j} \ | \ \alpha \in \mathcal{A}_{2,j} \}$. By Lemma~\ref{lem:TreeOfJSJ},
$|T(N_{j})| < |T(X)|$. By induction,
$\{\alpha_j|_{F_{j}}\}$ is bounded. For each $\alpha \in \mathcal{A}_{2,j}$,
$\alpha|_{T_{1}}$ is the image of $\alpha_{j}|_{F_{j}}$ under the projection induced by the product structure
$X(\alpha)_{j}^{+} \cong T^{2} \times [0,1]$. By the $T^{2} \times [0,1]$ Cosmetic Surgery
Theorem~(\ref{thm:CosmeticSurgeryOnT2XI}), the union of the images of $\{\alpha_j|_{F_{j}}\}$
under the projections given by all possible filling of $\partial X(\alpha)_{j}^{+} \setminus (T_{1} \cup F_{j})$
for which $X(\alpha)_{j}^{+} \cong T^{2} \times [0,1]$ is a bounded set of slopes of $T_{1}$
that we will denote as $B_{2,j}$. We conclude that $\alpha|_{T_{1}} \in B_{2,j}$.
This completes case two.
\bigskip\noindent
Since $\mathcal{A} = \mathcal{A}_{1} \cup (\cup_{j} \mathcal{A}_{2,j})$ we have
$$\{\alpha|_{T_{1}} \ | \ \alpha \in \mathcal{A} \} \subset B_{1} \cup (\cup_{j}B_{2,j}).$$
The theorem follows for JSJ manifolds, and we assume from now on that $X$ is prime, not Seifert fibered,
sol, or JSJ. Thus $X$ is hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is hyperbolic.} Since $S^{3}$ is not hyperbolic any $\alpha \in \mathcal{A}$
admits a minimally non hyperbolic\ partial filling. Up-to finite ambiguity we fix a minimally non hyperbolic\ filling which we
will denote as $\alpha'$.
If $\alpha'|_{T_{1}} \neq \ensuremath\infty$ then for any $\alpha$ with $\alpha' \pf \alpha$,
$\alpha|_{T_{1}} = \alpha'|_{T_{1}}$ is in the finite (and hence bounded) set $\{\alpha'|_{T_{1}}\}$.
Otherwise, any $\alpha$ with $\alpha' \pf \alpha$ factors through $X(\alpha')$:
$$X \stackrel{\alpha'}\to X(\alpha') \stackrel{\alpha|_{\partial X(\alpha')}}\longrightarrow X(\alpha')(\alpha|_{\partial X(\alpha')}).$$
It is straightforward to see that we can apply induction to $X(\alpha')$ and
$\{\alpha|_{\partial X(\alpha')} \ | \ \alpha \in \mathcal{A} \}$, showing that
$\{\alpha|_{\partial X(\alpha')}|_{T_{1}} \ | \ \alpha \in \mathcal{A} \}$ is bounded.
Thus for any $\alpha$ with $\alpha' \pf \alpha$,
$\alpha|_{T_{1}} = \alpha'|_{T_{1}}$ is in this bounded set.
This completes the proof in this final case.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:SetUpOfProof}
\bigskip\noindent
In this section we prove Theorem~\ref{thm:main} assuming the results of the next three sections.
We decided to present the proof before Sections~\ref{sec:FillingsThatDontFactorThroughM},
\ref{sec:FillingsOfEthatDontFactorThroughTXI}, and~\ref{sec:BothFillingsFactor}
in order to help the reader understand the motivation behind the exact statements proved in those
sections.
\bigskip\noindent
As in the statement of the theorem, let $M$ be a hyperbolic manifold so that
$\partial M$ is a single torus that we will denote as $T$and $V>0$ a fixed number.
Consider $\beta$, a slope on $T$, so that $\lv[M(\beta)] < V$.
By the Structure Theorem of~\cite{rieckyamashita}, there exist
finitely many covers $\phi:X \to E$, with $X$ and $E$ hyperbolic, so that the following
diagram commutes:
\bigskip
\begin{center}
\begin{picture}(200,60)(0,0)
\label{diagram}
\put( 0, 0){\makebox(0,0){$E$}}
\put( 0,50){\makebox(0,0){$X$}}
\put( 0, 40){\vector(0,-1){30}}
\put( 10, 0){\vector(1,0){73}}
\put(100, 0){\makebox(0,0){$(S^3,L)$}}
\put(100, 40){\vector(0,-1){30}}
\put(108,28){\makebox(0,0)}
\put(100,50){\makebox(0,0){$M(\beta)$}}
\put( 10,50){\vector(1,0){75}}
\end{picture}
\end{center}
\bigskip\noindent
Here, the horizontal arrows represent inclusions induced by fillings and the vertical arrows represent covering projections;
$\phi:X \to E$ is an unbranched covering projection between hyperbolic manifolds and $\hat\phi:M \to S^{3}$ is a branched
cover realizing $\lv[M(\beta)]$. Up-to finite ambiguity we fix one cover $\phi:X \to E$. We will denote the components
of $\partial X$ as $T_{1},\dots,T_{n}$.
\bigskip\noindent
{\bf Case One.} We first consider fillings $X \to M(\beta)$ that do not factor though $M$, that is, slopes $\beta$ so that
for some multislope $\alpha$ of $\partial X$
the following two conditions hold:
\begin{enumerate}
\item $X(\alpha) \cong M(\beta)$.
\item There is no $\alpha' \pf \alpha$ so that $X(\alpha') \cong M$.
\end{enumerate}
In Section~\ref{sec:FillingsThatDontFactorThroughM} we prove Theorem~\ref{thm:fillingsThatDontFactor},
showing that the set of slopes $\beta$
that arise in this way, which we will denote as $B_{1}$, is bounded.
We remark that this is a general fact about fillings $X \to M(\beta)$
and does not use the covers $\phi:X \to E$ and $\hat{\phi}:M(\beta) \to S^{3}$.
\bigskip\noindent
{\bf Case Two.}
We next consider fillings $X(\alpha) \cong M(\beta)$ that do factor though $M$, that is, fillings that admit a partial
filling $M$. Up to finite ambiguity, we may assume that the component of $\partial X$
that corresponds to $\partial M$ is $T_{1}$. Thus we are considering multislopes $\alpha$
so that the diffeomorphism $X(\alpha) \to M(\beta)$ induces, by restriction, a diffeomorphism
$$X(\alpha|_{T_{2},\dots,T_{n}}) \to M.$$
Denote the components of $\partial E$ by $T'_{1},\dots,T'_{m}$. By renumbering $T'_{1},\dots,T'_{m}$
if necessary we may assume
that $T'_{1} = \phi(T_{1})$. The diagram above implies that
$\alpha$ induces a multislope of $\partial E$, which we will denote as $\alpha^{E} = (\alpha_{1}^{E},\dots,\alpha^{E}_{m})$.
In Case Two we only consider fillings on $E$ that do not factor through $T^{2} \times [0,1]$; more precisely:
$$(\forall \alpha' \pf \alpha^{E}\mbox{ with } \alpha'|_{T_{1}'} = \ensuremath\infty) \hspace{10pt} E(\alpha') \not\cong T^{2} \times [0,1].$$
In that case, the strategy is as follows: applying the $S^{3}$ Cosmetic Surgery Theorem~(\ref{thm:cosmeticSurgeryOnS3}) we see that the
possibilities for $\alpha_{1}^{E}$ are bounded; the covering projection $\phi:X \to E$ induces a bilipschitz bijection between the
slopes of $T_{1}'$ and those of $T_{1}$ (Lemma~\ref{lem:CorrespondingSlopesFeray}); hence the possibilities for slopes on $T_{1}$ are bounded.
The argument is worked out in detail in
Section~\ref{sec:FillingsOfEthatDontFactorThroughTXI}.
\medskip\noindent
{\bf Case Three.} The last and most exciting case is when the filling of $X$ factors through $M$ and the filling of $E$
factors through $T^{2} \times [0,1]$. The proof in this case is given in Section~\ref{sec:BothFillingsFactor}.
Again, we conclude that $\{\alpha|_{T_{1}}\}$ is bounded.
\bigskip
\bigskip\noindent
Assuming the results of the following sections, we deduce Theorem~\ref{thm:main} as follows:
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let ${\mathcal{A}}$ be the set of all multislopes of $\partial X$ so that
for each $\alpha \in {\mathcal{A}}$ there is a slope $\beta$ of $\partial M$ so that
$X(\alpha) \cong M(\beta)$.
Since $M$ is hyperbolic there is a finite set of slopes of $\partial M$, which we will denote
as $B_{F}'$, so that for any $\beta \not \in B_{F}'$, $M(\beta)$ is hyperbolic and
the core of the attached solid torus is its unique shortest geodesic (Lemma~\ref{lem:HyperbolicDehnSurgery}).
Let $B_{F}$ be the set of slopes of $T$ defined as
$$B_{F} = \{ \beta \ | \ (\exists \beta' \in B_{F}')\ M(\beta) \cong M(\beta')\}.$$
Since $M$ is hyperbolic and $|\partial M|=1$, no manifold is obtained by filling infinitely
many distinct slopes of $\partial M$; hence $B_{F}$ is finite.
We will only consider multislopes $\alpha \in \mathcal{A}$ for which
X$(\alpha) \cong M(\beta)$ for $\beta \not\in B_{F}$.
To void overly complicated notation we will not rename $\mathcal{A}$.
We now consider Cases Two and Three. We will denote as ${\mathcal{A}}_{2,3}$
the multislopes of ${\mathcal{A}}$ in these cases, so that every $\alpha \in \mathcal{A}_{2,3}$
admits a partial filling $\alpha' \pf \alpha$ so that $X(\alpha') \cong M$. Up-to finite ambiguity we may assume
that $\alpha' = \alpha|_{T_{2},\dots,T_{n}}$.
We will denote the set of restrictions $\{\alpha|_{T_{1}}\ | \ \alpha \in {\mathcal{A}}_{2,3}\}$ as $B_{M}'$.
By Propositions~\ref{pro::FillingsOfEthatDontFactorThroughTXI}
and~\ref{pro:TheLastCase} and Lemma~\ref{lem:BoundedSetsOnMathcalT}
(see also Remark~\ref{rmk:LastCase}), $B_{M}'$ is bounded.
Let $\beta$ be a slope
of $\partial M$ so that $M(\beta) \cong X(\alpha)$ for some $\alpha \in {\mathcal{A}}_{2,3}$.
We will consider $X(\alpha)$
as $X(\alpha|_{T_{2},\dots,T_{n}})(\alpha|_{T_{1}})$,
the manifold obtained by filling $X(\alpha|_{T_{2},\dots,T_{n}})$ along slope $\alpha|_{T_{1}}$.
Then we see that $X(\alpha|_{T_{2},\dots,T_{n}}) \cong M$ and
$$X(\alpha|_{T_{2},\dots,T_{n}})(\alpha|_{T_{1}}) \cong M(\beta).$$
Let $f:X(\alpha|_{T_{2},\dots,T_{n}}) \to M$ be a diffeomorphism. Denote the image of
$\alpha|_{T_{1}}$ under $f$ as $\alpha^{M}$ and
the image of $B_{M}'$ under $f_{\alpha|_{T_{2},\dots,T_{n}}}$ as
$B_{M,f}$; note that $\alpha^{M} \in B_{M,f}$. Let $i$ be the isometric
involution on the slopes of $T$ given by Lemma~\ref{lem:MCGisometryOfSlopes}; we will denote
$B_{M,f} \cup i(B_{M,f})$
as $B_{M}$. Clearly $B_{M}$ is bounded and $\alpha^{M} \in B_{M}$.
Since $M(\alpha^{M}) \cong X(\alpha|_{T_{2},\dots,T_{n}})(\alpha|_{T_{1}})$
and $M(\beta)$ are diffeomorphic, by Mostow's Rigidity there is an isometry
$f:M(\alpha^{M}) \to M(\beta)$. Since $\beta \not\in B_{F}$, the cores of the attached solid tori are the shortest geodesics
of $M(\alpha^{M})$ and $M(\beta)$.
Thus $f$ carries the core of the solid torus attached to $M(\alpha^{M})$
to the core of the solid torus attached to $M(\beta)$ and hence by restriction $f$
induces an isometry $M \to M$ that maps $\alpha^{M}$ to $\beta$.
By Lemma~\ref{lem:MCGisometryOfSlopes}, $\alpha^{M} = \beta$
or $\alpha^{M} = i(\beta)$. Since $\alpha^{M} \in B_M$ and $B_{M} = i(B_{M})$,
$\beta \in B_{M}$.
Recall that we denoted the set of slopes of $\partial M$ that is realized in Case One
as $B_{1}$; in Theorem~\ref{thm:fillingsThatDontFactor}
we show that $B_{1}$ is bounded. Combining all the possibilities we see that
$$\beta \in B_{M} \cup B_{1}\cup B_{F}.$$
Theorem~\ref{thm:main} follows.
\end{proof}
\section{Case One: fillings of $X$ that do not factor through $M$}
\label{sec:FillingsThatDontFactorThroughM}
\bigskip\noindent
In this section we consider two manifolds, denoted as $X$ and $M$, where $M$ is a one cusped hyperbolic
manifolds. If $\alpha'$ is a multislope of $X$ so that $X(\alpha') \cong M$, then (trivially) for any slope $\beta$
of $\partial M$ there is a multislope $\alpha$ so that $\alpha' \pf \alpha$ and $X(\alpha) \cong M(\beta)$.
In Theorem~\ref{thm:fillingsThatDontFactor} we show that if one considers only
multislopes $\alpha$ that do not admit
$\alpha' \pf \alpha$ with $X(\alpha') \cong M$,
then the set of slopes of $\partial M$ that give rise to manifolds that are also obtained
by filling $X$ is bounded. This is purely a result about filling and
is independent of the covers considered in this paper. The precise statement is:
\begin{thm}
\label{thm:fillingsThatDontFactor}
Let $X$ be a compact orientable connected manifold so that $\partial X$ consists of tori
and $M$ a hyperbolic manifold with $\partial M$ a single torus.
Let ${\mathcal{A}}$ be a set of multislopes of $\partial X$
and ${\mathcal{B}}$ a set of slopes of $\partial M$
fulfilling the following two conditions:
\begin{enumerate}
\item For every $\beta \in {\mathcal{B}}$ there is a multislope
$\alpha \in {\mathcal{A}}$ so that $X(\alpha) \cong M(\beta)$.
\item For every $\alpha \in \mathcal{A}$ and every $\alpha' \pf \alpha$, $X(\alpha) \not\cong M$.
\end{enumerate}
Then $\mathcal{B}$ is bounded.
\end{thm}
\begin{proof}
By Thurston's Dehn Surgery Theorem (see Lemma~\ref{lem:HyperbolicDehnSurgery}) we may fix an $\epsilon>0$
and a finite set of slopes of $T$, which we will
denote as $B_{f}'$, so that for every $\beta \not\in B_{f}'$ the following
three conditions hold:
\begin{enumerate}
\item $M(\beta)$ is hyperbolic.
\item The core of the attached solid torus, which we will denote as $\gamma$, is a geodesic and $l(\gamma) < \epsilon$.
\item Any geodesic $\delta \subset M(\beta)$ with $l(\delta) < \epsilon$ is a power of $\gamma$.
\end{enumerate}
Since no manifold is obtained by filling along infinitely many distinct slopes of $M$, the set
$B_{f} = \{ \beta \ | \ (\exists \beta' \in B_{f}') \ M(\beta) \cong M(\beta') \}$ is finite.
For the remainder of the proof we
only consider $\beta \not \in B_{f}$. Accordingly, we remove the multislopes $\alpha \in \mathcal{A}$
for which $X(\alpha) \cong M(\beta)$ for $\beta \in B_{f}$. To avoid overly complicated notation we
do not rename $\mathcal{A}$ and $\mathcal{B}$.
\bigskip
\noindent
We induct on $|T(X)|$.
\bigskip\noindent
{\bf Assume that $X$ is Seifert fibered or sol.} Then $X$ cannot be filled to give a hyperbolic manifold.
We assume from now on that $X$ is not Seifert fibered or solv.
\bigskip\noindent
{\bf Assume that $X$ is not prime.} Let $X_{1},\dots,X_{n}$ be the factors of the prime
decomposition of $X$. For every $\alpha \in \mathcal{A}$
we will denote the restriction $\alpha|_{\partial X_{i}}$ as $\alpha_{i}$.
By considering the image of decomposing spheres for $X$ in $X(\alpha)$ we see that
for every $\alpha \in \mathcal{A}$:
\begin{equation}
\label{eq:FillingsThatDontFactor1}
X(\alpha) = X_{1}(\alpha_{1}) \# \cdots \# X_{n}(\alpha_{n}).
\end{equation}
For $1 \leq i \leq n$ we will denote as $\mathcal{A}_{i} \subset \mathcal{A}$ the multislopes $\alpha$ for which
$X_{i}(\alpha_{i}) \cong M(\beta)$ (for some $\beta \in \mathcal{B}$)
and for $i' \neq i$, $X_{i'}(\alpha_{i'}) \cong S^{3}$.
Since $\beta \not \in B_{f}$, $X(\alpha)$ is hyperbolic and hence prime.
Thus by Equation~(\ref{eq:FillingsThatDontFactor1})
$$\mathcal{A} = \cup_{i=1}^{n} \mathcal{A}_{i}.$$
Fix $1 \leq i \leq n$.
Suppose, for a contradiction, that for some $\alpha \in \mathcal{A}_{i}$ there is
$\alpha_{i}' \pf \alpha_{i}$ so that $X_{i}(\alpha_{i}') \cong M$.
Then there is a unique component $T$ of $\partial X_{i}$ for which $\alpha_{i}'|_{T} = \ensuremath\infty$.
Let $\alpha' \pf \alpha$ be the partial filling defined by setting $\alpha'|_{T} = \ensuremath\infty$ and
$\alpha'|_{T'} = \alpha|_{T'}$ for any other component $T'$ of $\partial X$. Then by
Equation~(\ref{eq:FillingsThatDontFactor1}),
\begin{eqnarray*}
X(\alpha') &\cong& X_{i}(\alpha'|_{\partial X_{i}}) \ \# \ (\#_{i' \neq i}X_{i'}(\alpha'|_{\partial X_{i'}})) \\
&\cong& X_{i}(\alpha'_{i}) \ \# \ (\#_{i' \neq i} X_{i'}(\alpha_{i'})) \\
&\cong& M \ \# \ (\#_{i' \neq i}S^{3}) \\
&\cong& M.
\end{eqnarray*}
(Here we used that $ X_{i'}(\alpha_{i'})) \cong S^{3}$, which holds by the definition of $\mathcal{A}_{i}$.)
This contradicts the assumptions of the theorem. Hence $X_{i}$ and
$\{\alpha_{i} \ | \ \alpha \in \mathcal{A}_{i}\}$ fulfill the assumptions of the theorem. Since $X_{i}$
corresponds to a direct descendant of the root of $T(X)$, $|T(X_{i})| < |T(X)|$. We will denote as $B_i$
the set of slopes of $T$ so that for every $\beta \in B_{i}$ there is $\alpha_{i} \in \mathcal{A}_{i}$
with $M(\beta) \cong X_{i}(\alpha_{i})$. By induction, $B_{i}$ is bounded.
As $i$ was arbitrary, we obtain bounded sets $B_{1},\dots,B_{n}$.
Since $\mathcal{A} = \cup_{i=1}^{n} \mathcal{A}_{i}$,
$$\mathcal{B} \subset B_{f} \cup (\cup_{i=1}^{n} B_{i}).$$
The theorem follows for non prime manifolds.
We assume from now on that $X$ is prime and not Seifert fibered or sol.
\bigskip\noindent
{\bf Assume that $X$ is hyperbolic.}
Since there is no lower bound on the length of geodesics in $\{M(\beta) \ | \ \beta \in \mathcal{B} \}$,
Theorem~\ref{thm:HyperbolicFillingShortGeos} does no apply directly;
however, the proof here is similar to the proof of that theorem where more details can be found. We start with:
\medskip\noindent
{\bf Claim.} There are only finitely many totally hyperbolic fillings in $\mathcal{A}$.
\medskip\noindent
When proving the claim we may obviously assume that $\mathcal{A}$ is infinite.
Let $\{ \alpha^{j} \}_{j \in \mathbb N} \subset \mathcal{A}$ be an infinite set.
We will prove the claim by showing that some multislope of $\{ \alpha^{j} \}_{j \in \mathbb N}$
is not totally hyperbolic.
We denote the components of $\partial X$ as $T_{1},\dots,T_{n}$ and $\alpha^{j}|_{T_{i}}$ as $\alpha_{i}^{j}$.
After subsequencing and reordering if necessary we assume as we may that for some $0 \leq k \leq n+1$ we have:
\begin{enumerate}
\item For every $1 \leq i \leq k$ and every $j \neq j'$, $\ \alpha_{i}^{j} \neq \alpha_{i}^{j'}$ and $\alpha_{i}^{j} \neq \ensuremath\infty$.
\item For every $k+1 \leq i \leq n+1$ and every $j, j'$, $\ \alpha_{i}^{j} = \alpha_{i}^{j'}$.
\end{enumerate}
Since $\{ \alpha^{j} \}_{j \in \mathbb N}$ is infinite, $k \geq 1$.
Let $\alpha' = (\ensuremath\infty,\dots,\ensuremath\infty,\alpha^{j}_{k+1},\dots,\alpha^{j}_{n})$; by construction
$\alpha'$ does not depend on $j$.
We claim that $X(\alpha')$ is not hyperbolic. Assume, for a contradiction, that it is.
Then for any $l>0$ we may choose $j$ so that the slopes $\alpha_{1}^{j},\dots,\alpha^{j}_{m}$
are all longer than $l$ (where here the lengths are measured in the Euclidean metrics induced on the
boundary components after some truncation of the cusps). By Thurston's Dehn Surgery Theorem if $l$ is sufficiently large then
$X(\alpha')(\alpha_{1}^{j},\dots,\alpha^{j}_{m})$ is hyperbolic and
admits at least $k$ geodesics that are all shorter than $\epsilon$; since $M(\beta)$
admits only one such geodesic, $k=1$. Thus $X(\alpha')$ and $M$ are one cusped hyperbolic manifolds
that admit infinitely many diffeomorphic fillings. A standard application of Mostow's Regidity shows
that $X(\alpha') \cong M$. Since $\alpha' \pf \alpha^{j}$ this contradicts the second assumption
of the theorem, showing that $X(\alpha')$ is not hyperbolic. Hence $\alpha' \pf \alpha^{j}$
is a non hyperbolic partial filling, showing that $\alpha^{j}$ is not totally hyperbolic.
Hence there are only finitely many minimally non hyperbolic\ fillings, as claimed.
\bigskip\noindent
By the claim there are only finitely many $\alpha \in \mathcal{A}$ that are totally hyperbolic.
We remove these multislopes from $\mathcal{A}$ and remove from $\mathcal{B}$ the finite set
of slopes $\beta$ for which $M(\beta) \cong X(\alpha)$ only for totally hyperbolic fillings;
to avoid overly complicated
notation we do not rename $\mathcal{A}$ and $\mathcal{B}$. Hence every multislope
of $\mathcal{A}$ admits a minimally non hyperbolic\ partial filling. Up-to finite ambiguity we fix one
minimally non hyperbolic\ partial filling of $X$ and denote it as $\alpha'$. By definition of partial filling,
if $\alpha' \pf \alpha$ then $X(\alpha)$ is obtained by filling $X(\alpha')$ along
slopes $\alpha|_{\partial X(\alpha')}$. By assumption, $\alpha$ does not admit
a partial filling that yields $M$; hence the same holds for $\alpha|_{\partial X(\alpha')}$.
Since $X(\alpha')$ corresponds to
a direct descendant of the root of $T(X)$, $|T(X(\alpha'))| < |T(X)|$. By induction
the set of slopes of $T$
$$\{\beta \ | \ M(\beta) \cong X(\alpha')(\alpha|_{\partial X(\alpha')}), \ \alpha \in \mathcal{A}\}$$
is bounded. The theorem follows.
We assume from now on that $X$ is prime and not Seifert fibered, sol, or hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} By Proposition~\ref{prop:ObtainedByFilling} for every
$\beta \in \mathcal{B}$, $M(\beta)$ is obtained by filling some component of the torus
decomposition of $X$. Up-to finite ambiguity we fix such component which we will
denote as $X_{1}$ and consider only slopes $\beta$ (and corresponding multislopes $\alpha$)
so that $M(\beta)$ is obtained by filling $X_{1}$;
to avoid overly complicated notation we do not rename $\mathcal{A}$
and $\mathcal{B}$. Recall that in Proposition~\ref{prop:ObtainedByFilling} for every
$\alpha \in \mathcal{A}$ we constructed a multislope for $X_{1}$, which we will
call the {\it induced} multislope and denote as $\alpha_{1}$, so that $M(\beta) \cong X_{1}(\alpha_{1})$.
We will denote the set of induced multislopes thus obtained
as $\mathcal{A}_{1}$; thus for every $\beta \in \mathcal{B}$
there is $\alpha_{1} \in \mathcal{A}_{1}$ so that $M(\beta) \cong X_{1}(\alpha_{1})$.
Let $\mathcal{A}_{1}^{+}$ be the multislopes $\alpha_{1} \in \mathcal{A}_{1}$ for which there is no
$\alpha_{1}' \pf \alpha_{1}$ so that $X_{1}(\alpha_{1}') \cong M$. Then $X_{1}$ and
$\mathcal{A}_{1}^{+}$ fulfill the assumptions of the theorem. By Lemma~\ref{lem:TreeOfJSJ},
$|T(X_{1})| < |T(X)|$. By induction, the set
$$\{\beta \ | \ (\exists \alpha_{1} \in \mathcal{A}_{1}^{+}) M(\beta) \cong X_{1}(\alpha_{1}) \},$$
which we will denote as $B_{1}^{+}$, is bounded. We will return to $B_{1}^{+}$ at the end of the proof.
We will denote $\mathcal{A}_{1} \setminus \mathcal{A}_{1}^{+}$ as $\mathcal{A}_{1}^{-}$.
Then every $\alpha_{1} \in \mathcal{A}_{1}^{-}$ admits
$\alpha_{1}' \pf \alpha_{1}$ so that $X_{1}(\alpha_{1}') \cong M$. Up-to finite
ambiguity we fix one boundary component of $\partial X_{1}$,
which we will denote as $F$, and consider only the
multislopes $\alpha_{1} \in \mathcal{A}_{1}^{-}$ for which
$\alpha_{1}'|_{F} = \ensuremath\infty$ (in other words, $F \subset \partial X_{1}$
corresponds to $T$ after filling along $\alpha_{1}'$).
Note that $F \not\subset \partial X$, for otherwise
$\alpha_{1}$ would correspond to a multislope $\alpha \in \mathcal{A}$ that admits
$\alpha' \pf \alpha$ for which $X(\alpha') \cong M$, contradicting the assumptions
of the theorem. We will denote the components of $X$ cut open along
$F$ as $X^{+}$ and $X^{-}$, with $X_{0} \subset X^{+}$.
In the remainder of the proof we work directly with the set of multislopes of $\mathcal{A}$ that
induce multislopes of $\mathcal{A}_{1}^{-}$ fulfilling the assumptions above,
which we will denote as $\mathcal{A}^{-} \subset \mathcal{A}$.
Given $\alpha \in \mathcal{A}^{-}$, we will denote the multislope $\alpha|_{\partial X^{+}}$ induced on $\partial X^{+}$
as $\alpha^{+}$ and the multislope $\alpha|_{\partial X^{-}}$ induced on $\partial X^{-}$ as $\alpha^{-}$.
By defining assumption of $\mathcal{A}^{-}_{1}$, $X^{+}(\alpha^{+}) \cong M$
and by Proposition~\ref{prop:ObtainedByFilling} either $X^{-}(\alpha^{-}) \cong D^{2} \times S^{1}$
or $X^{-}(\alpha^{-}) \cong E(K)$ for a non-trivial knot $K \subset S^{3}$. But if the latter occured,
$F$ would be an incompressible torus in $M(\beta)$, which is absurd as $M(\beta)$ is hyperbolic.
Thus $X^{-}(\alpha^{-}) \cong D^{2} \times S^{1}$. We will denote the slope defined by the
meridian of $X^{-}(\alpha^{-})$ on $F$ as $\mu_{F}$ and consider $X(\alpha)$ as $X^{+}(\alpha^{+})(\mu_{F})$,
the manifold obtained by filling $X^{+}(\alpha^{+})$ along slope $\mu_{F}$.
Let $f:X^{+}(\alpha^{+})(\mu_{F}) \to M(\beta)$ be a diffeomorphism. Then,
since we assumed that $\beta \not\in B_{f}$ (recall the definition of $B_{f}$ in the beginning
of the proof), the cores of the solid tori attached to $X^{+}(\alpha^{+})$ and $M$
are the unique shortest geodesics in $X^{+}(\alpha^{+})(\mu_{F})$ and $M(\beta)$.
By Mostow's Rigidity we may assume
that $f$ is an isometry, and thus $f$ carries the core of the solid torus attached to $X^{+}(\alpha^{+})$
to the core of the solid torus attached to $M$. The restriction of
$f$ induces a diffeomorphism which we will denote as $f^{+}:X^{+}(\alpha^{+}) \to M$.
Note that $f^{+}$ maps $\mu_{F}$ to $\beta$.
Turning our attention to the solid torus $X^{-}$, we claim that for every $\alpha \in \mathcal{A}^{-}$,
the induced multislope $\alpha^{-}$ satisfies the following conditions:
\begin{enumerate}
\item $X^{-}(\alpha^{-}) \cong D^{2} \times S^{1}$: this was established above.
\item For any partial filling $\hat\alpha \pf \alpha^{-}$, $X^{-}(\hat\alpha) \not\cong T^{2} \times [0,1]$:
otherwise, let $\alpha' \pf \alpha$ be the partial filling
defined by setting $\alpha'|_{\partial X^{-}} = \hat\alpha$
and $\alpha'|_{\partial X^{+}} = \alpha|_{\partial X^{+}}$. Then we have:
$$X(\alpha') \cong X^{-}(\hat\alpha) \cup_{F} X^{+}(\alpha^{+}|_{\partial X^{+}}) \cong (T^{2} \times [0,1]) \ \cup_{F} \ M \cong M,$$
contradicting the assumption of the theorem.
\end{enumerate}
Thus $X^{-}$ and the multislopes $\mathcal{A}^{-}_{1}$
fulfill the requirements of the Solid Torus Cosmetic Surgery Theorem (\ref{thm:SlopesOnSolidTorus}),
showing that the set of meridians of the solid tori $\{X^{-}(\alpha^{-}) \ | \ \alpha^{-} \in \mathcal{A}^{-}_{1} \}$,
which we will denote as $B^{-}$, is bounded.
By the discussion above, if $X(\alpha) \cong M(\beta)$ then $\beta$ is the image of $\mu_{F} \in B^{-}$
under the diffeomorphism denoted above as $f^{+}:X^{+}(\alpha^{+}) \to M$. We will denote the image
of $B^{-}$ under $f^{+}$ as $B^{-}_{f^{+}}$. By Lemma~\ref{lem:MCGisometryOfSlopes}
there is an isometric involution $i$ on the slopes of $F$ so that
if $g^{+}:X^{+}(\alpha^{+}) \to M$ is a diffeomorphism, the image of $B^{-}$ under
$g^{+}$ is either $B^{-}_{f^{+}}$ or $i(B^{-}_{f^{+}})$.
We will denote $B^{-}_{f^{+}} \cup i(B^{-}_{f^{+}})$ as $B_{M}$.
Clearly, $B_{M}$ is bounded. The theorem follows for JSJ manifolds,
as $\mathcal{B} \subset B_{f} \cup B_{1}^{+} \cup B_{M}$.
\bigskip\noindent
This completes the proof of Theorem~\ref{thm:fillingsThatDontFactor}.
\end{proof}
\section{Case Two: fillings of $E$ that do not factor through $T^{2} \times [0,1]$}
\label{sec:FillingsOfEthatDontFactorThroughTXI}
\bigskip\noindent
In this section we prove the following proposition, that is used in Case Two of the proof of Theorem~\ref{thm:main}:
\begin{prop}
\label{pro::FillingsOfEthatDontFactorThroughTXI}
Let $X$ and $E$ be compact orientable connected manifolds with toral boundary
and $\phi:X \to E$ a branched cover so that $\phi|_{\partial X}:\partial X \to \partial E$
is a cover. Let $\hat{\mathcal{A}}$ be a set of multislopes of $X$ so that every
$\hat\alpha \in \hat{\mathcal{A}}$ induces a multislope on $\partial E$ which we will
denote as $\alpha^{E}$. Fix a component of $\partial X$ (which we will denote we $T_{1}$)
and denote $\phi(T_{1})$ as $T_{1}'$. Suppose that every $\alpha^{E}$ fulfills the following conditions:
\begin{enumerate}
\item $E(\alpha^{E}) \cong S^{3}$.
\item There does not exist $\alpha' \pf \alpha^{E}$ fulfilling the following two conditions:
\begin{enumerate}
\item $\alpha'|_{T_{1}'} = \ensuremath\infty$.
\item $E(\alpha') \cong T^{2} \times [0,1]$.
\end{enumerate}
\end{enumerate}
Then the set of restrictions $\{\hat\alpha|_{T_{1}} \ | \ \hat\alpha \in \hat{\mathcal{A}}\}$ is bounded.
\end{prop}
\begin{proof}
Let $\mathcal{A}_{E}$ be the set multislopes $\{ \alpha^{E} \}$ above.
By the $S^{3}$ Cosmetic Surgery Theorem (\ref{thm:cosmeticSurgeryOnS3}), the set of restrictions
$\{ \alpha^{E}|_{T_{1}'} \ | \ \alpha^{E} \in \mathcal{A}_{E}\}$ is bounded. The set of restrictions
$\{\hat\alpha|_{T_{1}} \ | \ \hat\alpha \in \hat{\mathcal{A}}\}$ is
contained in the image of $\{ \alpha^{E}|_{T_{1}'} \ | \ \alpha^{E} \in \mathcal{A}_{E}\}$
under the bilipschitz bijection induced
by $\phi$ (Lemma~\ref{lem:CorrespondingSlopesFeray}).
The proposition follows.
\end{proof}
\section{Case Three: fillings of $X$ that factor through $M$ and fillings of $E$ that factor through $T^{2} \times [0,1]$}
\label{sec:BothFillingsFactor}
\bigskip\noindent
In this section we tackle Case Three, the final case of Theorem~\ref{thm:main}.
Below we denote as $\mathcal{A}_{3}$ the multislopes in $\mathcal{A}$ that correspond to
this case. After general analysis of the situation, we show that certain conditions must
hold (up-to finite ambiguity). These conditions are summarized in Lemma~\ref{lem:CoresNotInBall}.
Theorem~\ref{thm:main} then follows from Proposition~\ref{pro:TheLastCase} and
Lemma~\ref{lem:BoundedSetsOnMathcalT}.
We begin by fixing our notation.
Let $X$, M$(\beta)$, $E$, $L$, $\phi$, and $\hat\phi$ be as in the
diagram in Section~\ref{sec:SetUpOfProof} (see Page~\pageref{diagram}). We will denote the
multislope of $X$ that corresponds to the filling $X \to M(\beta)$ as $\alpha$ and the
multislope of $E$ that corresponds to $E \to S^{3}$ as $\alpha^{E}$. By construction, $\alpha$
and $\alpha^{E}$ are corresponding multislopes (in the corresponds defined by the restriction of
$\phi:X \to E$ to the boundary; recall Subsection~\ref{subsection:FareyAndCovering}).
Up-to finite ambiguity we fix two components of $\partial E$ which we will
denote as $T_{1}'$ and $T_{2}'$
(note that since $E \to S^{3}$ is assumed to factor through $T^{2} \times [0,1]$, $|\partial E| \geq 2$)
and a component of $\phi^{-1}(T_{1}')$ which we will denote as $T_{1}$.
We will denote the remaining components of $\partial X$ as $T_{2},\dots,T_{n}$ and the remaining components of
$\partial E$ as $T_{3}',\dots,T_{k}'$.
In the two preceding sections we have reduced the proof of Theorem~\ref{thm:main}
to multislopes $\alpha \in \mathcal{A}$ fulfilling
the following conditions:
\begin{enumerate}
\item $X(\alpha) \cong M(\beta)$ (for some slope $\beta$ of $\partial M$).
\item $X(\alpha|_{T_{2},\dots,T_{n}}) \cong M$.
\item $E(\alpha^{E}|_{T_{3}',\dots,T_{k}'}) \cong T^{2} \times [0,1]$.
\end{enumerate}
Thus we obtain the following commutative diagram, where here horizontal
arrows represent inclusions induced by fillings, vertical arrows represent
covering projections, $T^{2} \times [0,1]$ represents $E(\alpha^{E}|_{T_{3}',\dots,T_{k}'})$,
$\phi^{-1}(T^{2} \times [0,1])$ is denoted as $X'$, and $\phi|_{X'}$ is denoted as $\phi'$:
\bigskip
\begin{center}
\begin{picture}(200,60)(0,0)
\put( 0, 0){\makebox(0,0){$E$}}
\put( 0,50){\makebox(0,0){$X$}}
\put( 0, 40){\vector(0,-1){30}}
\put(160, 0){\makebox(0,0){$S^3,L$}}
\put(160, 40){\vector(0,-1){30}}
\put(10,28){\makebox(0,0){$/\phi$}}
\put(70,28){\makebox(0,0){$/\phi'$}}
\put(170,28){\makebox(0,0){$/{\hat{\phi}}$}}
\put(108,28){\makebox(0,0)}
\put(160,50){\makebox(0,0){$M(\beta)$}}
\put(115,50){\makebox(0,0){$M$}}
\put(60,50){\makebox(0,0){$X'$}}
\put( 7,50){\vector(1,0){45}}
\put( 67,50){\vector(1,0){40}}
\put( 123,50){\vector(1,0){23}}
\put(60, 40){\vector(0,-1){30}}
\put(7,0){\vector(1,0){33}}
\put( 90,0){\vector(1,0){55}}
\put(65,0){\makebox(0,0){$T^{2} \times [0,1]$}}
\end{picture}
\end{center}
\bigskip
We will denote the set of all multislopes fulfilling these conditions as $\mathcal{A}_{3} \subset \mathcal{A}$.
For $\alpha \in \mathcal{A}_{3}$, we will denote the restriction $\alpha|_{T_{2},\dots,T_{n}}$ as $\hat\alpha$,
and the set or restriction $\{\hat\alpha \ | \ \alpha \in \mathcal{A}_{3}\}$ as $\hat{\mathcal{A}}$.
Our goal is to show that the set of slopes induced on $T_{1}$ by restricting the multislopes $\mathcal{A}_{3}$
is bounded. However, in the course of the proof we sometimes end up with a bounded set of slopes
of a different component of $\partial X$, say $T_{2}$. It will always be the case that $\phi(T_{2}) =T_{1}'$
or $\phi(T_{2}) = T_{2}'$. If $\phi(T_{2}) = T_{1}'$, then the covering map $\phi$ induces
a bijection between the slopes of $T_{2}$ and those of $T_{1}'$, and a bijection between
the slopes of $T_{1}'$ and those of $T_1$; both bijections are bilipschitz maps
of the Farey graph (Lemma~\ref{lem:CorrespondingSlopesFeray}).
Composing the two bijections, we readily see a bounded set of slopes on $T_{1}$.
If instead $\phi(T_{2}) = T_{2}'$, let $B_{2}$ be the bounded set of slopes of $T_{2}$ and
let $B_{2}'$ be its image (in the slopes of $T_{2}'$) under the bijection induced by $\phi$;
by Lemma~\ref{lem:CorrespondingSlopesFeray}, $B_{2}'$ is bounded.
Since $E(\alpha^{E}|_{T_{3}',\dots,T_{k}'}) \cong T^{2} \times [0,1]$, we can then project $B_{2}'$ from $T_{2}'$
to a set of slopes on $T_{1}'$ using the product structure. As this set depends on $\alpha^{E}$, we denote it as
$B_{1,\alpha^{E}}'$. By the $T^{2} \times [0,1]$
Cosmetic Surgery Theorem~(\ref{thm:CosmeticSurgeryOnT2XI}),
$\cup_{\alpha^{E}} B_{1,\alpha^{E}}'$ is a bounded set of slopes on $T_{1}'$.
Using Lemma~\ref{lem:CorrespondingSlopesFeray} once more, we obtain
a bounded set of slopes on $T_{1}$.
We summarize this:
\begin{lem}
\label{lem:BoundedSetsOnMathcalT}
With the notation above, suppose that for each component $T_{i}$ of $\phi^{-1}(T_{1}' \cup T_{2}')$,
there is a bounded set of slopes of $T_{i}$, that we will denote as $B_{i}$, so that that for each
$\alpha \in \mathcal{A}_{3}$, there is some component $T_{i}$ of $\phi^{-1}(T_{1}' \cup T_{2}')$
so that $\alpha|_{T_{i}} \in B_{i}$.
Then $\{ \alpha|_{T_{1}} \ | \ \alpha \in \mathcal{A}_{3}\}$ is bounded.
\end{lem}
\begin{rmk}
It is unfortunate that this set up does not lend itself well to induction. There are many problems, and
here is perhaps the best example: when considering a JSJ manifold $X$ we apply
Proposition~\ref{prop:ObtainedByFilling} and conclude that $M(\beta)$ is obtained by filling some component of the torus
decomposition of $X$; however, the cover $\phi':X' \to E'$ is nowhere to be found. We must therefore
first identify the essential information that is preserved in the inductive step.
This is done in the following two lemmas:
\end{rmk}
\begin{lem}
\label{lem:X'IsIrreducible}
Any prime factor of $X'$ has at least two boundary components.
\end{lem}
\begin{proof}
If $X'$ is prime then the lemma follows from the fact that $T \times \{0\}$ and
$T \times \{1\}$ (as components of $\partial E(\alpha^{E}|_{T_{3}',\dots,T_{k}'}) \cong T^{2} \times [0,1]$)
have at least
one preimage each. Otherwise, let $S \subset X'$ be a decomposing sphere that realizes the decomposition
$X' = X'' \# X'''$, where $X''$ is a prime factor of $X'$. We will prove the lemma by showing that $|\partial X''| \geq 2$.
\begin{enumerate}
\item Suppose that $|\partial X''| = 0$. Equivalently, $X''$ is closed. Then $X''$ is a prime factor of $M$.
But $M$ is a hyperbolic manifold and $\partial M \neq \emptyset$,
and so it has no closed factors. Thus $|\partial X''| \neq 0$.
\item Suppose that $|\partial X''|=1$. Without loss of generality we may assume that
$\phi'(\partial X'') = T^{2} \times \{0\}$.
We will denote the component of $X'$ cut open along $S$ that
corresponds to $X''$ as $X^{*}$; note that $\partial X^{*}$ has two components: a torus (which we
can naturally identify with $\partial X''$) and a sphere (which we
can naturally identify with $S$). Since $\pi_{2}(T^{2} \times [0,1])$ is trivial,
$\phi'|_{S}:S \to T^{2} \times [0,1]$ can be extended to a map from the closed
ball $D$, which we will denote as $\phi''$. After a homotopy of $\phi'$ if necessary,
we assume as we may that for a sufficiently small fixed $\epsilon>0$
the following three conditions hold:
\begin{enumerate}
\item $\phi''(D) \cap (T^{2} \times [0,\epsilon)) = \emptyset$.
\item $\phi''(D) \cap (T^{2} \times (1-\epsilon,1]) = \emptyset$.
\item $\phi'(X^{*}) \cap (T^{2} \times (1-\epsilon,1]) = \emptyset$.
\end{enumerate}
We paste $\phi'':D \to T^{2} \times [0,1]$ and $\phi':X^{*} \to T^{2} \times [0,1]$ to obtain a map
which we will denote as $\psi:X'' = X^{*} \cup_{S} D \to T^{2} \times [0,1]$.
By conditions~(a) and~(b) above the following holds:
$$\psi|_{\psi^{-1}(T^{2} \times [0,\epsilon))} = \phi'|_{\psi^{-1}(T^{2} \times [0,\epsilon))}.$$
Therefore
$\psi|_{\psi^{-1}(T^{2} \times [0,\epsilon))}:\psi^{-1}(T^{2} \times [0,\epsilon)) \to T^{2} \times [0,\epsilon)$
is a cover and has non-zero degree. On the other hand, $\psi^{-1}(T^{2} \times (1-\epsilon,1]) = \emptyset$ so
$\psi|_{\psi^{-1}(T^{2} \times (1-\epsilon,1])}:\psi^{-1}(T^{2} \times (1-\epsilon,1]) \to T^{2} \times (1-\epsilon,1]$
has degree zero. This contradiction shows that $|\partial X''|\neq1$.
\end{enumerate}
\end{proof}
In light of this lemma, we define:
\begin{notation}
We will denote the prime factor of $X'$
that contains $T_{1}$ as $X''$ and the components of $\partial X'' \setminus T_{1}$
as $\mathcal{T}$. Given $\alpha \in \mathcal{A}_{3}$,
the link in $M$ consisting of the cores of the solid
tori attached to $\mathcal{T}$ will be denoted as $\mathcal{L}_{\alpha}$,
or simply $\mathcal{L}$ when no confusion may arise.
\end{notation}
Next we prove:
\begin{lem}
\label{lem:CoresNotInBall}
$\mathcal{L}$ fulfills:
\begin{enumerate}
\item $\mathcal{L} \neq \emptyset$.
\item $E(\mathcal{L})$ is irreducible.
\end{enumerate}
\end{lem}
\begin{proof}
By Lemma~\ref{lem:X'IsIrreducible}, $X''$
has at least two boundary components.
Hence $\mathcal{T} \neq \emptyset$, and~(1) follows.
If $X'$ is prime that by construction $\mathcal{T} = \partial X' \setminus T_{1}$; hence
$E(\mathcal{L}) \cong X'$ and~(2) follows. Otherwise,
recall the definition of $S$ and $X^{*} \subset X'$ from the construction of $\psi$ above.
Let $X^{**} = \mbox{cl}(X' \setminus X^{*})$. Since $M$ is hyperbolic the $\partial M$ is the
image of $T_{1} \subset X^{*}$, after filling $X^{**}$ we obtain a ball. This shows that
$E(\mathcal{L}) \cong X^{*}$; by constrcution $X^{*}$ is a factor of the prime decomposition
of $X'$ and hence is itself prime.
\end{proof}
\bigskip\noindent
Thus, for every $\alpha \in \mathcal{A}_{3}$, there is $I \subset \{1,\dots,n\}$,
so that the following three conditions are satisfied:
\begin{enumerate}
\item $1 \not\in I \neq \emptyset$.
\item $\{T_{i}\}_{i \in I}$ (which we will denote as $\mathcal{T}$) is a union of components of $\phi^{-1}(T_{1}' \cup T_{2}')$.
\item The cores of the solid tori attached to $\mathcal{T}$form a link (which we will denote as
$\mathcal{L} \subset M$) with an irreducible exterior.
\end{enumerate}
Up-to finite ambiguity we fix $I$ as above and consider only $\alpha \in \mathcal{A}_{3}$
that fulfill these conditions for the fixed index set $I$.
To avoid overly complicated notation we do not rename $\mathcal{A}_{3}$ or $\hat{\mathcal{A}}$.
\bigskip\noindent
We are now ready to state and prove the main proposition of this section:
\begin{prop}
\label{pro:TheLastCase}
Let $M$ be a hyperbolic manifold, $\partial M$ a single torus that we will denote as $T$,
$X$ a compact connected orientable manifold, $\partial X$ tori that we will denote as
$\{T_{i}\}_{i=1}^{n}$, and
$\mathcal{A}_{3}$ a set of multislopes on $\partial X$.
Fix a non empty index set $I \subset \{2,\dots,n\}$; we will denoted $\{T_{i}\}_{i \in I}$ as $\mathcal{T}$.
Denote the link formed by the core of the solid tori attached to $\mathcal{T}$ when filling along $\alpha \in \mathcal{A}$
as $\mathcal{L}_{\alpha} \subset M$ (or simply $\mathcal{L}$ when no confusion may arise).
Assume that any $\alpha \in \mathcal{A}_{3}$ fulfills the following conditions:
\begin{enumerate}
\item $\alpha|_{T_{1}} = \ensuremath\infty$.
\item $X(\alpha) \cong M$.
\item $E(\mathcal{L}_{\alpha})$ is irreducible.
\end{enumerate}
Then for each $i \in I$, there is a bounded set $B_{i}$ of slopes of $T_{i}$,
so that for each $\alpha \in \mathcal{A}_{3}$,
there is an $i \in I$, so that $\alpha|_{T_{i}} \in B_{i}$.
\end{prop}
\begin{rmk}
\label{rmk:LastCase}
By the discussion above (in particular~\ref{lem:CoresNotInBall}),
$X$, $T_{1}$, $\mathcal{T}$, and the multislopes of $\mathcal{A}$ considered in Case Three
of Theorem~\ref{thm:main} satisfy the conditions of the Proposition~\ref{pro:TheLastCase} for some $I$.
By Lemma~\ref{lem:BoundedSetsOnMathcalT},
proving this proposition will complete the proof of Theorem~\ref{thm:main}.
\end{rmk}
\begin{proof}[Proof of Proposition~\ref{pro:TheLastCase}]
We induct on $|T(X)|$.
\bigskip\noindent
{\bf Assume that $X$ is Seifert fibered or sol.} Then no filling of $X$ gives $M$.
We assume from now on that $X$ is not Seifert fibered or sol.
\bigskip\noindent
{\bf Assume that $X$ is hyperbolic.}
Fix $\epsilon>0$ smaller than the length of the shortest geodesic in $M$.
By Theorem~\ref{thm:HyperbolicFillingShortGeos} there are only finitely many
totally hyperbolic fillings on $X$ yielding $M$. Given ${i} \in I$, we will denote the
set of slopes obtained by restricting totally hyperbolic multislopes of $\mathcal{A}$
to $T_{i}$ as $B_{i}^{1}$, whenever the restriction does not equal $\ensuremath\infty$.
Then $B_{i}^{1}$ is finite (and hence bounded).
We will denote as $\mathcal{A}' \subset \mathcal{A}_{3}$ the set
$$\mathcal{A}' = \{\alpha \in \mathcal{A}_{3} \ | \ (\forall i \in I) \ \alpha|_{T_{i}} \not\in B_{i}^{1}\}.$$
From this point on we assume as we may that $\alpha \in \mathcal{A}'$.
Then $\alpha$ is not totally hyperbolic and hence admits a minimally non hyperbolic\ partial
filling. For each $i \in I$, let $B_{i}^{2}$ be the set of restrictions $\{ \alpha_{\mbox{\tiny min}}|_{T_{i}}\}$,
where $\alpha_{\mbox{\tiny min}}$ ranges over all minimally non hyperbolic\ fillings
for which $\alpha_{\mbox{\tiny min}}|_{T_{i}} \neq \infty$.
By Proposition~\ref{pro:mnhIsFinite} $X$ admits only finitely many minimally non hyperbolic\ fillings
and so $B_{i}^{2}$ is finite (and hence bounded).
We will denote as $\mathcal{A}'' \subset \mathcal{A}'$ the set
$$\mathcal{A}'' = \{\alpha \in \mathcal{A}_{3} \ | \ (\forall i \in I) \ \alpha|_{T_{i}} \not\in (B_{i}^{1} \cup B_{i}^{2})\}.$$
From this point on we assume as we may that $\alpha \in \mathcal{A}''$. By deifinition,
any $\alpha \in \mathcal{A}''$ admits a minimally non hyperbolic\ partial filling $\alpha_{\mbox{\tiny min}}$ so that
$\alpha_{\mbox{\tiny min}}|_{T_{i}} = \infty$ for every ${i} \in I$.
Fix a minimally non hyperbolic\ filling $\alpha_{\mbox{\tiny min}}$ for which
$\alpha_{\mbox{\tiny min}}|_{T_{i}} = \infty$ for every ${i} \in I$.
We will denote $X(\alpha_{\mbox{\tiny min}})$ as $X_{1}$ and the
the restrictions to $X_{1}$ of multislopes in $\mathcal{A}''$ that admit a partial filling
$\alpha_{\mbox{\tiny min}}$ as $\mathcal{A}_{\alpha_{\mbox{\tiny min}}}$, that is:
$$\mathcal{A}_{\alpha_{\mbox{\tiny min}}} =
\{\alpha|_{\partial X_{1}} \ | \ \alpha \in \mathcal{A}'' \mbox{ and } \alpha_{\mbox{\tiny min}} \pf \alpha \}.$$
We claim that the following conditions are satisfied:
\begin{enumerate}
\item $|T(X_{1})| < |T(X)|$: this holds since $X_{1}$ corresponds to a direct descendant of the root of $T(X)$.
\item $\mathcal{T} \subset \partial X_{1}$: this follows from the definition of $\mathcal{A}''$,
where we required that $\alpha|_{T_{i}} = \ensuremath\infty$ (for all $i \in I$).
\item For any $\alpha_{1} \in \mathcal{A}_{\alpha_{\mbox{\tiny min}}}$, $\alpha_{1}|_{T_{1}} = \ensuremath\infty$:
it follows from the definitions that $\alpha_{1}|_{T_{1}} = \alpha|_{T_{1}}$;
hence by the first assumption of the proposition, $\alpha|_{T_{1}} = \ensuremath\infty$.
\item For any $\alpha_{1} \in \mathcal{A}_{\alpha_{\mbox{\tiny min}}}$, $X_{1}(\alpha_{1}) \cong M$:
by definition, $X_{1}(\alpha_{1}) = X(\alpha_{\mbox{\tiny min}})(\alpha_{1}) = X(\alpha)$.
By the second assumption of the proposition, $X(\alpha) \cong M$.
\item For any $\alpha_{1} \in \mathcal{A}_{\alpha_{\mbox{\tiny min}}}$,
$E(\mathcal{L}_{\alpha_{i}})$ is irreducible, where here
$\mathcal{L}_{\alpha_{1}}$ denotes the link formed by the cores of the solid tori attached to
$\mathcal{T} \subset \partial X_{1}$: it is straightforward to see that
$E(\mathcal{L}_{\alpha_{1}}) = E(\mathcal{L})$.
By the third assumption of the proposition, $E(\mathcal{L})$ is irreducible.
\end{enumerate}
By~(2)--(5), $X_{1}$, $T_{1}$, $\mathcal{T}$, and $\mathcal{A}_{\alpha_{\mbox{\tiny min}}}$ fulfill
the assumptions of the proposition. By~(1) we may apply induction, showing that
for every $i \in I$, there is a bounded set of slopes of $T_{i}$, that we will denote as $B_{i,\alpha_{\mbox{\tiny min}}}^{3}$,
so that for each $\alpha_{1} \in \mathcal{A}_{\alpha_{\mbox{\tiny min}}}$, there is an ${i} \in I$,
so that $\alpha_{1}|_{T_{i}} \in B_{i,\alpha_{\mbox{\tiny min}}}^{3}$.
With the notation of the preceding paragraph, every $\alpha \in \mathcal{A}''$ admits a
minimally non hyperbolic\ filling $\alpha_{\mbox{\tiny min}}$
and $\alpha|_{T_{i}} = \alpha_{1}|_{T_{i}}$ for every $i \in I$. Hence for some $i \in I$,
$\alpha|_{T_{i}} \in B_{i,\alpha_{\mbox{\tiny min}}}^{3}$.
By Proposition~\ref{pro:mnhIsFinite}, $X$ admits only finitely many minimally non hyperbolic\ fillings. Hence the set
$$B_{i}^{3} = \bigcup_{\alpha_{\mbox{\tiny min}}} B_{i,\alpha_{\mbox{\tiny min}}}^{3}$$
is bounded. The proposition follows by setting
$$B_{i} = B_{i}^{1} \cup B_{i}^{2}\cup B_{i}^{3}.$$
We assume from now on that $X$ is not Seifert fibered, sol, or hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is not prime.} Let $X_{1}$ be the factor of the prime decomposition
of $X$ that contains $T_{1}$; say $X = X_{1} \# X_{1}'$ (we are not assuming that
$X_{1}'$ is prime). Then any $\alpha \in \mathcal{A}$ induces the multislopes
$\alpha_{1} = \alpha|_{\partial X_{1}}$ and $\alpha_{1}' = \alpha|_{\partial X'_{1}}$
on $\partial X_{1}$ and $\partial X_{1}'$, respectively.
Since $X_{1}(\alpha_{1}) \# X_{1}'(\alpha_{1}') = X(\alpha) \cong M$ and $M$ is hyperbolic,
the following conditions hold:
\begin{enumerate}
\item Since $T_{1} \subset \partial X_{1}$, $X_{1}(\alpha_{1}) \cong M$ (with $\alpha_{1}|_{T_{1}} = \ensuremath\infty$).
\item $X_{1}'(\alpha_{1}') \cong S^{3}$.
\end{enumerate}
By construction the reducing sphere that gives the decomposition $X = X_{1} \# X_{1}'$ is disjoint from $\mathcal{L}$.
Since $E(\mathcal{L})$ is irreducible, no component of it is contained in $X_{1}'(\alpha_{1}')$;
equivalently, $\mathcal{T} \subset \partial X_{1}$.
It is easy to see that $X_{1}$, $T_{1}$, $\mathcal{T}$, and $\{\alpha_{1} \ | \ \alpha \in \mathcal{A} \}$ fulfill the
assumptions of the proposition.
Since $X_{1}$ corresponds to a direct descendant of the root of $T(X)$, $|T(X_{1})| < |T(X)|$.
By induction, for each ${i} \in I$,
there is a bounded set of slopes $B_i$ of $T_{i}$, so that for each $\alpha \in \mathcal{A}$,
there is an ${i} \in I$ with $\alpha_{1}|_{T_{1}} \in B_{i}$. Since for all $i \in I$, $\alpha|_{T_{i}} = \alpha_{1}|_{T_{i}}$,
the proposition follows in this case.
We assume from now on that $X$ is prime and not Seifert fibered, sol, or hyperbolic.
\bigskip\noindent
{\bf Assume that $X$ is JSJ.} Let $X_{0}$ be the component of the torus decomposition of $X$ that contains $T_{1}$.
Denote the components of $\partial X_{0} \setminus T_{1}$ as $\{F_{j}\}_{j=1}^{k}$. Denote the component of
$\mbox{cl}(X \setminus X_{0})$ containing $F_{j}$ as $X_{j}$, see Figure~\ref{fig:F1}.
\begin{figure}
\includegraphics[height=2.5in]{F1}
\caption{The notation when $X$ is JSJ (the shaded region is $X_{j}^{c}$)}
\label{fig:F1}
\end{figure}
To avoid the situation where $X_{j} = \emptyset$, if $F_{j} \subset \partial X$ we push it slightly into the interior of $X$
so that $X_{j} \cong T^{2} \times [0,1]$ in that case. Since $M$ is hyperbolic
it admits no non separating tori; thus we assume as we may that $X_{j} \neq X_{j'}$
for $j \neq j'$, for otherwise no filling of $X$ yields $M$. By renumbering if necessary,
we assume as we may that $\partial X_{j}$ contains a component of $\mathcal{T}$ exactly when $j \leq m$,
for an appropriately chosen $m$.
For $\alpha \in \mathcal{A}$ we will denote the restriction $\alpha|_{\partial X_{j}}$ as $\alpha_{j}$
(by definition $\alpha_{j}|_{F_{j}} = \ensuremath\infty$). Since $X(\alpha) \cong M$ is hyperbolic, every
torus in $X(\alpha)$ is either boundary parallel ({\bf A} below),
or bounds a solid torus ({\bf B} and {\bf C} below), or bounds a non trivial knot exterior in a ball ({\bf D} below).
Thus for every $1 \leq j \leq k$ exactly one of the following holds:
\begin{description}
\item[Case~{\bf A}] $X_{j}(\alpha_{j}) \cong M$ and
$\mbox{cl}(X(\alpha) \setminus X_{j}(\alpha_{j})) \cong T^{2} \times [0,1]$.
\item[Case~{\bf B}] $X_{j}(\alpha_{j}) \cong D^{2} \times S^{1}$ and no component of $\mathcal{L} \cap X_{j}(\alpha_{j})$
is a core of $X_{j}(\alpha_{j})$.
\item[Case~{\bf C}] $X_{j}(\alpha_{j}) \cong D^{2} \times S^{1}$
and some component of $\mathcal{L} \cap X_{j}(\alpha_{j})$, which we will denote as $K_{j}$,
is a core of $X_{j}(\alpha_{j})$.
\item[Case~{\bf B}] $X_{j}(\alpha_{j}) \cong E(K_{j})$ for a non-trivial knot $K_{j} \subset S^{3}$
and $X_{j}(\alpha_{j}) \subset D_{j}$
for some ball $D_{j} \subset X(\alpha)$.
\end{description}
\bigskip
\noindent We first consider the following:
\bigskip\noindent
{\bf Case~{\bf A} happens for some $j \leq m$.}
Fix $j$ ($1 \leq j \leq m$) and let $\mathcal{A}_{j}' \subset \mathcal{A}$ be
$$\mathcal{A}_{j}' = \{\alpha \in \mathcal{A} \ | \ X_{j}(\alpha_{j}) \cong M \mbox\}.$$
Note that for any $\alpha \in \mathcal{A}$, if $F_{j}$ is as in Case~{\bf A} above then
$\alpha \in \mathcal{A}_{j}'$. We will denote the set of restrictions
$\{\alpha_{j} \ | \ \alpha \in \mathcal{A}_{j}' \}$ as $\mathcal{A}_{j}$.
Let $\mathcal{T}_{j} = \mathcal{T} \cap X_{j}$ and $\mathcal{L}_{j} = \mathcal{L} \cap X_{j}(\alpha_{j})$;
equivalently, $\mathcal{L}_{j}$ is the link formed by the cores of the solid tori attached to $\mathcal{T}_{j}$.
Since $j \leq m$, $\mathcal{T}_{j} \neq \emptyset$.
If, for some $\alpha_j \in \mathcal{A}_{j}$, $\mathcal{L}_{j}$ were reducible then any reducing sphere for
$X_{j}(\alpha_{j}) \setminus \mbox{int}N(\mathcal{L}_{j})$
would be a reducing sphere for $\mathcal{L}$; this contradicts the assumptions of the proposition.
Hence $X_{j}$, $F_{j}$, $\mathcal{T}_{j}$ and $\mathcal{A}_{j}$ fulfill the assumptions of the proposition.
By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{j})|<|T(X)|$.
By induction on each component $T$ of $\mathcal{T}_{j}$,
there is a bounded set of slopes that we will denote as $B_{T}$,
so that for each $\alpha_{j} \in \mathcal{A}_{j}$,
there is a component $T$ of $\mathcal{T}_{j}$,
so that $\alpha_{j}|_{T_{i}} \in B_{T}$. It follows that for every $\alpha \in \mathcal{A}_{j}'$,
$\alpha|_{T} \in B_{T}$ for some $T$; the proposition follows in this case.
We may consider from now on only multislopes from
$\mathcal{A} \setminus (\cup_{j \leq m} \mathcal{A}_{j}')$ (and in particular, we
assume as we may that Case~{\bf A} above
does not happen for $j \leq m$). To avoid overly complicated notation we do not rename $\mathcal{A}$.
\bigskip\noindent
Next we consider the following:
\bigskip\noindent
{\bf Case~{\bf A} happens for some $j \geq m+1$.} Fix $j$ ($m+1 \leq j \leq k$).
We will denote $\mbox{cl}(X \setminus X_{j})$ as $X_{j}^{c}$
and the restriction $\alpha|_{\partial X_{j}^{c}}$ as $\alpha_{j}^{c}$.
Since $j \geq m+1$, $\mathcal{T} \subset \partial X_{j}^{c}$.
Let $\mathcal{A}_{j}' \subset \mathcal{A}$ be the set:
$$\mathcal{A}_{j}' = \{\alpha \in \mathcal{A} \ | \ X_{j}^{c}(\alpha^{c}_{j}) \cong T^{2} \times [0,1]\}.$$
Note that for any $\alpha \in \mathcal{A}$, $F_{j}$ is as in Case~{\bf A} above if and only if
$\alpha \in \mathcal{A}_{j}'$.
Given $\alpha \in \mathcal{A}'_{j}$ we will denote the restriction $\alpha|_{X_{j}^{c}}$
as $\alpha^{c}_{j}$ (by definition $\alpha_{j}^{c}|_{F_{j}}$ and $\alpha_{j}^{c}|_{T_{1}}$ are $\ensuremath\infty$).
We will denote the set of restrictions $\{\alpha_{j}^{c} \ | \ \alpha \in \mathcal{A}_{j}' \}$
as $\mathcal{A}_{j}^{c}$.
Thus we have a manifold $X_{j}^{c}$ so that $\mathcal{T} \subset \partial X_{j}^{c}$ and
multislopes $\mathcal{A}_{j}^{c}$ so that for every $\alpha_{j}^{c} \in \mathcal{A}_{j}^{c}$,
$X_{j}^{c}(\alpha_{j}^{c}) \cong T^{2} \times [0,1]$
and $\partial X_{j}^{c}(\alpha_{j}^{c}) = T_{1} \cup F_{j}$.
Up-to finite ambiguity we fix $J \subset \{1,\dots,k\}$,
and denote as $\mathcal{A}_J^{c} \subset \mathcal{A}_{j}^{c}$ the multislopes
for which $X_{j'}(\alpha_{j'}) \cong E(K_{j'}) \subset D_{j'}$ for a non-trivial knot
$K_{j'} \subset S^{3}$ and a ball $D_{j'} \subset X(\alpha)$
if and only if $j' \in J$.
By Lemma~\ref{lem:KnotExteriorsAreDisjoint} we may assume that the balls
$\{D_{j'}\}_{j' \in J}$ are disjointly embedded.
\begin{figure}[h]
\includegraphics[height=2.5in]{F2}
\caption{$Y'(\alpha_{Y'})$}
\label{fig:F2}
\end{figure}
Let $Y'$ be the closure of $X_{j}^{c} \setminus (\cup_{j' \in J} E(K_{j'}))$ (see Figure~\ref{fig:F2};
in that figure $\{F_{j'}\}_{j' \in J}$ are the two tori on the top right).
By restriction, any $\alpha_{j}^{c} \in \mathcal{A}_{J}^{c}$ induces
a slope on each component of
$\partial Y' \setminus (T_{1} \cup F_{j} \cup (\cup_{j' \in J} F_{j'}))$.
For every $j' \in J$, we pick a slope on $F_{j'}$ that intersects the meridian of $E(K_{j'})$ exactly once.
(There are infinitely many way to do this; we will exploit this flexibility soon
when appealing to Lemma~\ref{lem:TwistingToGetIrreducible}.)
Denote the multislope obtained as $\alpha_{Y'}$, and note that $Y'(\alpha_{Y'}) \cong T^{2} \times [0,1]$
and $\partial Y'(\alpha_{Y'}) = T_{1} \cup F_{j}$. We will denote $\mathcal{T} \cap \partial Y'$
as $\mathcal{T}_{Y}$, the link formed by the cores of the solid tori attached to $\mathcal{T}_{Y}$
as $\mathcal{L}_{Y}$, and the link formed by the cores of the solid tori attached to $\cup_{j' \in J}F_{j'}$
as $\mathcal{U}'$ (in Figure~\ref{fig:F2}, $\mathcal{U}'$ is red).
Since the components of $\mathcal{U}'$ are unknots contained in the disjointly
embedded balls $D_{j'}$, $\mathcal{U}'$ is an unlink.
Finally assume, for a contradiction, that $\mathcal{T}_{Y} = \emptyset$.
Then $\mathcal{L} \subset \cup_{j' \in J} D_{j'}$;
this contradicts the assumption that $E(\mathcal{L})$ is irreducible,
showing that $\mathcal{T}_{Y} \neq \emptyset$.
If $Y'(\alpha_{Y'}) \setminus \mbox{int}N(\mathcal{L}_{Y} \cup \mathcal{U}')$ is irreducible we denote
$Y'$ as $Y$, $\mathcal{U}'$ as $\mathcal{U}$, and $\alpha_{Y'}$ as $\alpha_{Y}$.
Otherwise, let $S$ be a reducing sphere realizing the decomposition of
$Y'(\alpha_{Y'}) \setminus \mbox{int}N(\mathcal{L}_{Y} \cup \mathcal{U}')$ as $Y'' \# Y'''$, where $Y''$ is irreducible and
$T_{1} \subset Y''$ (in Figure~\ref{fig:F2}, $S$ is green).
Note that $Y'(\alpha_{Y'}) \setminus \mbox{int}N(\mathcal{L}_{Y} \cup \mathcal{U}')$ is obtained
from $Y'$ by filling $\partial Y' \setminus (T_{1} \cup F_{j} \cup \mathcal{T}_{Y} \cup (\cup_{j' \in J} F_{j'}))$
(in Figure~\ref{fig:F2} the components that are {\it not} filled are $T_{1}$, $F_{j}$, the reds,
and the blues).
Since $\mathcal{T} \subset \mathcal{T}_{Y} \cup (\cup_{j' \in J} \partial X_{j'})$,
$\mathcal{L} \cap S = \emptyset$.
We consider $S$ as a sphere in
$$Y'(\alpha_{Y'}) \cup_{F_{j}} X_{j}(\alpha_{j}) \cong T^{2} \times [0,1] \ \cup_{F_{j}} \ M \cong M.$$
Since $M$ is hyperbolic, $S$ bounds a ball in $ Y'(\alpha_{Y'}) \cup_{F_{j}} X_{j}(\alpha_{j})$
which we will denote as $D$.
Since $T_{1} \subset Y''$, $D = Y'''(\alpha_{Y'}|_{Y'''})$.
Clearly, $\mathcal{T}_{Y} \cap D = \emptyset$, for otherwise $S$ would be a reducing sphere for
$E(\mathcal{L})$. We will denote as $Y$ the manifold obtained from $Y'$ by filling
$(\cup_{j' \in J} F_{j'}) \cap D$ along the multislope induced by $\alpha_{Y'}$.
We will denote the multislope $\alpha_{Y'} | _{\partial Y}$ as $\alpha_{Y}$
and $\mathcal{U}' \setminus (\mathcal{U}' \cap D)$ as $\mathcal{U}$.
By construction $\mathcal{T}_{Y} \subset \partial Y$, and therefore
we may consider $\mathcal{L}_{Y}$ as the link in $Y(\alpha_{Y})$
formed by the cores of the solid tori attached to $\mathcal{T}_{Y}$. Then the following two conditions hold:
\begin{enumerate}
\item $\mathcal{U}$ is an unlink.
\item $Y(\alpha|_{Y}) \setminus \mbox{int}N(\mathcal{L}_{Y} \cup \mathcal{U}) \cong Y''$ and hence
is irreducible.
\end{enumerate}
We choose the slopes of $\mathcal{U}$,
as we know we may by Lemma~\ref{lem:TwistingToGetIrreducible},
so that $E(\mathcal{L}_{Y})$ is irreducible.
\begin{lem}
\label{lem:no-core}
For every $\alpha_{Y} \in \mathcal{A}_{Y}$, there is a slope $\alpha'$ of
$F_{j}$, so that $Y(\alpha_{Y})(\alpha')$ satisfies the following condition:
\begin{enumerate}
\item $Y(\alpha_{Y})(\alpha') \cong D^{2} \times S^{1}$.
\item $\mathcal{L}_{Y} \subset Y(\alpha_{Y})(\alpha')$ is irreducible.
\item No component of $\mathcal{L}_{Y}$ is a core of $Y(\alpha_{Y})(\alpha')$ .
\end{enumerate}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:no-core}]
Fix $\alpha_Y \in \mathcal{A}_{Y}$.
By construction, $Y(\alpha_{Y}) \cong T^{2} \times [0,1]$;
hence for any slope $\alpha'$,
$Y(\alpha_Y)(\alpha') \cong D^{2} \times S^{1}$.
Thus~(1) is satisfied by any slope $\alpha'$ of $F_{j}$.
Since the exterior of $\mathcal{L}_{Y}$ as a link in $Y(\alpha_{Y})$ is
irreducible, by Hatcher, there is a finite set of slopes of $F_{j}$, which we will denote as $B_{f}$,
so that for any slope $\alpha' \not\in B_{f}$, the exterior of $\mathcal{L}_{Y}$
as a link in $Y(\alpha_{Y})(\alpha')$ is irreducible.
For~(3) we fix a component $K$ of $\mathcal{L}_{Y}$. Let $[K]$ denote the homology
class represented by $K$ in $H_{1}(Y(\alpha_{Y});\mathbb Z)$ ($[K]$ is only defined up to sign).
We consider two possibilities:
\begin{description}
\item[{$[K]$} is not primitive] Then by Lemma~\ref{lem:CoresOfSolidTori}, $K$ is not a core of
of $Y(\alpha_{Y})(\alpha')$ for any slope $\alpha'$; we set $B_{K} = \emptyset$.
\item[{$[K]$} is primitive] By Lemma~\ref{lem:CoresOfSolidTori}, if $K$ is a core
of $Y(\alpha_{Y})(\alpha')$ then $[K]$ and $[\alpha']$ generate
$H_{1}(Y(\alpha_{Y}))$. We will denote as $B_{K}$ the set of slopes
of $F_{j}$ that correspond to homology classes fulfilling this condition. It is easy to see that
$B_{K}$ has diameter $2$ in the Farey graph of the slopes of $F_{j}$.
\end{description}
Since $B_{f} \cup (\cup_{K} B_{K})$ is a finite union of bounded sets it is itself bounded;
hence its complement is not empty. The Lemma~\ref{lem:no-core} follows by picking
$\alpha' \not\in B_{f} \cup (\cup_{K} B_{K})$.
\end{proof}
For each $\alpha_{Y} \in \mathcal{A}_{Y}$ we pick a slope $\alpha'$ of $F_{j}$
satisfying the conditions of Lemma\ref{lem:no-core}.
By Proposition~\ref{pro:SolidTorusSurgery2}, on every component $T$
of $\mathcal{T}_{Y}$, there exists a bounded set $B_{T}$, so for that every $\alpha_{Y} \in \mathcal{A}_{Y}$,
there is a component $T$ of $\mathcal{T}_{Y}$, so that $\alpha_{Y}|_{T} \in B_{T}$.
Given any $\alpha \in \mathcal{A}$, we construct
$\alpha_{j}^{c} \in \mathcal{A}_{J}^{c}$ and $\alpha_{Y}$ as above.
The proposition follows in this case since $\alpha|_{T} = \alpha_{j}^{c}|_{T} = \alpha_{Y}|_{T} \in B_{T}$.
We may consider from now on only multislopes from
$\mathcal{A} \setminus (\cup_{j \geq m+1} \mathcal{A}_{j}')$.
Thus from now on we will only consider multislopes for which Case~{\bf A}
does not happen for any $j$.
To avoid overly complicated notation
we do not rename $\mathcal{A}$.
\bigskip\noindent
Next we consider the following:
\bigskip\noindent
{\bf Case~{\bf B} happens for some $j \leq m$.} Fix $1 \leq j \leq m$.
We will denote as $\mathcal{A}_{j}' \subset \mathcal{A}$
the multislopes $\alpha \in \mathcal{A}$ for which
$X_{j}(\alpha_{j}) \cong D^{2} \times S^{1}$ and no component
of $\mathcal{L} \cap X_{j}(\alpha_{j})$ is a core.
Note that Case~{\bf B} occurs if and only if $\alpha \in \mathcal{A}_{j}$.
We will denote the set of restrictions $\{\alpha_{j} \ | \ \alpha \in \mathcal{A}_{j}' \}$ as $\mathcal{A}_{j}$
and $\mathcal{T} \cap X_{j}$ as $\mathcal{T}_{j}$.
Since $j \leq m$, $\mathcal{T}_{j} \neq \emptyset$.
Given $\alpha_{j} \in \mathcal{A}_{j}$,
we will denote the link formed by the cores of the solid tori attached
to $\mathcal{T}_{j}$ as $\mathcal{L}_{j}$. It is easy to see that
if $X_{j}(\alpha_{j}) \setminus \mbox{int}N(\mathcal{L}_{j}$) were reducible
then $E(\mathcal{L})$ would be reducible,
contradicting the third assumption of the proposition. By the assumption of Case~{\bf B}, no
component of $\mathcal{L}_{j}$ is a core of the solid torus $X_{j}(\alpha_{j})$.
Therefore by Proposition~\ref{pro:SolidTorusSurgery2}, for each component $T$ of $\mathcal{T}_{j}$,
there is a bounded set of slopes of $T$ which we will denote as $B_{T}$, so that for each $\alpha_{j} \in \mathcal{A}_{j}$,
there is a component $T$ of $\mathcal{T}_{j}$ so that $\alpha_{j}|_{T} \in B_{T}$.
The proposition follows in this case since for any $\alpha \in \mathcal{A}_{j}'$,
$\alpha|_{T} = \alpha_{j}|_{T}$.
We may consider from now on only multislopes from
$\mathcal{A} \setminus (\cup_{j \leq m} \mathcal{A}_{j}')$.
To avoid overly complicated notation
we do not rename $\mathcal{A}$.
\bigskip\noindent
We have reduced the proof to the following:
\bigskip\noindent
{\bf Cases~{\bf A} never happens and Case~{\bf B} never happens for $j \leq m$.}
Consider $J_1,J_2 \subset \{1,\dots,k\}$ fulfilling the following conditions:
\begin{enumerate}
\item $\emptyset \neq J_{1} \subset \{1,\dots,m\}$.
\item $J_{1} \cap J_{2} = \emptyset$.
\item $\{1,\dots,m\} \subset J_{1} \cup J_{2}$.
\end{enumerate}
Let $\mathcal{A}_{J_{1},J_{2}} \subset \mathcal{A}$ be the multislopes $\alpha$ fulfilling the following conditions:
\begin{enumerate}
\item For every $j \in J_{1}$, $X_{j}(\alpha_{j}) \cong D^{2} \times S^{1}$.
\item For any $1 \leq j \leq k$, $X_{j}(\alpha_{j}) \cong E(K_{j})$ (for a non-trivial knot $K_{j} \subset S^{3}$)
if and only if $j \in J_{2}$.
\end{enumerate}
Using the fact the Case~{\bf A} does not happen (that is, every $F_{j}$ bounds a solid torus or
a knot exterior contained in a ball) and irreducibility of $\mathcal{L}$ and
Lemma~\ref{lem:KnotExteriorsAreDisjoint} (which together imply that $J_{1} \neq \emptyset$),
it is easy to see that for any $\alpha \in \mathcal{A}$ there is a
choice of $J_{1},J_{2}$ as above for which $\alpha \in \mathcal{A}_{J_{1},J_{2}}$; thus
$$\mathcal{A} = \bigcup_{J_{1},J_{2}} \mathcal{A}_{J_{1},J_{2}}.$$
Up-to finite ambiguity we fix $J_1,J_2 \subset \{1,\dots,k\}$ fulfilling the conditions above
and consider only multislopes from $\mathcal{A}_{J_{1},J_{2}}$.
Fix a multislope $\alpha \in \mathcal{A}_{J_{1},J_{2}}$.
By Lemma~\ref{lem:KnotExteriorsAreDisjoint}
we may fix disjointly embedded balls $\{ D_{j}\}_{j \in J_{2}}$ so that $E(K_{j}) \subset D_{j}$.
Since Case~{\bf B} does not happen for $j \leq m$, for every $j \in J_1$, at least one component of
$\mathcal{L}$ is a core of $X_{j}(\alpha_{j})$; we choose one and denote it as $K_{j}$.
We will denote $\cup_{j \in J_{1}} K_{j}$ as $\mathcal{L}_{1}$ and $\cup_{j \in J_{1}} T_{j}$
as $\mathcal{T}_{1}$.
The multislope $\alpha$ induces a multislope on $\partial X_{0}$
as follows: for$j \not\in J_{2}$, the slope induced on $F_{j}$ is the meridian of solid torus
$X_{j}(\alpha_{j})$.
For $j \in J_{2}$, the slope induced on $F_{j}$ is any slope that intersects
the meridian of $E(K_{j})$ once (we will exploit this flexibility soon, when appealing
to Lemma~\ref{lem:makingL1irreducible}). We will denote the multislope induced by $\alpha$
on $\partial X_{0}$ as $\alpha_{0}$. By construction
$X_{0}(\alpha_{0}) \cong M$. For $j \in J_{1}$, we will denote the core
of the solid torus attached to $F_{j}$
as $K_{j}$ and the link formed by the cores of the solid tori attached to $\mathcal{T}_{1}$
as $\mathcal{L}_{1}$; no confusion should arise from this notation, as $K_{j}$
and $\mathcal{L}_{1}$ are isotopic to the knots and link denote that way previosly.
We will denote the link formed by the solid tori attached to $\cup_{j \in J_{2}} F_{j}$
as $\mathcal{U}$. By construction, the components of $\mathcal{U}$
are unknots embedded in the the disjoint balls $D_{j}$, and hence
$\mathcal{U}$ is an unlink.
In order to apply Lemma~\ref{lem:TwistingToGetIrreducible} we need to know that
$\mathcal{L}_{1}$ is irreducible in the complement of $\mathcal{U}$;
this is not quite the case, but we can obtain this by considering only some of the
components of $\mathcal{U}$. To that end we prove:
\begin{lem}
\label{lem:makingL1irreducible}
Suppose $S$ is a reducing sphere for $\mathcal{L}_{1}$ in the complement of
$\mathcal{U}$. Then $S$ bounds a ball $D \subset X_{0}(\alpha_{0})$
so that $D \cap \mathcal{L}_{1} = \emptyset$ and $D$ contains
at least one component of $\mathcal{U}$.
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:makingL1irreducible}]
Let $S$ be a reducing sphere
for $\mathcal{L}_{1}$ in $X_0(\alpha_{0}) \setminus N(\mathcal{U})$;
equivalently, $S$ is a reducing sphere for
$X_0(\alpha_{0}) \setminus N(\mathcal{U} \cup \mathcal{L}_{1})$.
Fix $1 \leq j \leq m$;
note that either $j \in J_{1}$ or $j \in J_{2}$. If $j \in J_{2}$,
then by construction $S$ is disjoint from $F_{j}$. If $j \in J_{1}$,
then $S \cap K_{j} = \emptyset$ (since $K_{j} \subset \mathcal{L}_{1}$).
Since $K_{j}$ is a core of the solid torus attached to $F_{j}$, we may isotope $S$ out of
this solid torus without intersecting $K_{j}$. After performing this isotopy
(if necessary) for each $1 \leq j \leq m$, the reducing sphere $S$
is disjoint from $F_{j}$ for every $1 \leq j \leq m$. Since $X_{0}(\alpha_{0}) \cong X(\alpha)$,
we may consider $S$ as
a sphere in $X(\alpha)$, where we see that $\mathcal{L} \cap S = \emptyset$.
Hyperbolicity of $X(\alpha)$ and irreducibility of $E(\mathcal{L})$ imply that
$S$ bounds a ball $D \subset X(\alpha)$ so that $\mathcal{L} \cap D = \emptyset$.
It follows that $X_{j}(\alpha_{j}) \cap D = \emptyset$ for $1 \leq j \leq m$;
therefore $\mathcal{T}_{1} \cap D = \emptyset$.
Hence $S$ bounds a ball in $X_{0}(\alpha_{0})$ that is disjoint from
$\mathcal{T}_{1}$ and hence from $\mathcal{L}_{1}$.
On the other hand, $S$ does not bound a ball in $X_0(\alpha_{0}) \setminus N(\mathcal{U})$;
hence $D$ must contain at least one component of $\mathcal{U}$.
This completes the proof of Lemma~\ref{lem:makingL1irreducible}.
\end{proof}
Let $D$ be as in Lemma~\ref{lem:makingL1irreducible}. We remove the components of
$\mathcal{U} \cap D$ from $\mathcal{U}$; to avoid overly complicated notation we do not rename $\mathcal{U}$.
We repeat this process as long as we can; it terminates since the number of components of $\mathcal{U}$ is reduced.
When it does, $\mathcal{L}_{1}$ is irreducible in the complement
of the unlink $\mathcal{U}$. By Lemma~\ref{lem:TwistingToGetIrreducible}
we may change the slopes $\alpha_{0}|_{\mathcal{U}}$
so that the exterior of $\mathcal{L}_{1}$ is irreducible.
To avoid overly complicated notation we do not rename $\alpha_{0}$
We will denote as $\mathcal{A}_{0}$ the multislopes induced on $\partial X_{0}$ by
multislopes of $\mathcal{A}_{J_{1},J_{2}}$ via the procedure described above.
We see that $X_{0}$, $T_{1}$, $\mathcal{T}_{1}$, and $\mathcal{A}_{0}$
fulfill the assumptions
of the proposition. By Lemma~\ref{lem:TreeOfJSJ}, $|T(X_{0})| < |T(X)|$. By induction, for each component
$F_{j}$ of $\mathcal{T}_{1}$,
there is a bounded set of slopes of $F_{j}$ which we will denote as
$B_{F_{j}}$, so that for each $\alpha_{0} \in \mathcal{A}_{0}$,
there is some component $F_{j}$ of $\mathcal{T}_{1}$ with $\alpha_{0}|_{F_{j}} \in B_{F_{j}}$.
Given $j\in J_{1}$, let $S \neq F_{j}$ be a component of $\partial X_{j}$.
Let $\beta$ be a multislope of $\partial X_{j}$ so that $\beta|_{F_{j}}$
and $\beta|_{S}$ are both $\ensuremath\infty$ and $X_{j}(\beta) \cong T^{2} \times [0,1]$.
We will denote as $B_{\beta}$ the projection of $B_{F_{j}}$ to the slopes of $S$
induced by the product structure on $X_{j}(\beta)$. By the
$T^{2} \times [0,1]$ Cosmetic Surgery Theorem (\ref{thm:CosmeticSurgeryOnT2XI}),
the set $\cup_{\beta} B_{\beta}$ is bounded (where here the union is taken over all possible multislopes
$\beta$ as above; if there is no such multislope then $\cup_{\beta} B_{\beta} = \emptyset$).
We will denote $\cup_{\beta} B_{\beta}$ as $B_{S}$.
Given $\alpha \in \mathcal{A}_{J_{1},J_{2}}$, let $\alpha_{0}$ be the induced multislope
on $X_{0}$ as above. Let $F_{j}$ be the component for which $\alpha_{0}|_{F_{j}} \in B_{F_{j}}$.
Recall that $K_{j}$ is a core of a solid torus attached to $X_{j}$ which is also a core
of $X_{j}(\alpha_{j})$. Let $S$ be the component of $\partial X_{j}$ that corresponds
to $K_{j}$. By definition of $\alpha_{0}|_{F_{j}}$, it is the projection of $\alpha|_{S}$
under the natural projection give by the product structure on $X_{j}(\alpha_{j}) \setminus N(K_{j})$.
Thus $\alpha|_{S} \in B_{S}$. This show that for every $\alpha \in \mathcal{A}_{J_{1},J_{2}}$,
there is a boundary component $S$, so that $\alpha|_{S} \in B_{S}$. Since $B_{S}$ are bounded,
this completes the proof of Proposition\ref{pro:TheLastCase}.
\end{proof}
\nocite{*}
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,742 |
\section{Introduction}
\label{sec:intro}
{The importance} of video compression has come to the fore over the past decade driven by the tension between the huge quantities of video content consumed everyday and the bandwidth available to transmit it. This challenge has been addressed through the creation of new video coding standards, the latest activity being by the Joint Video Exploration Team (JVET), who published the first version of H.266/Versatile Video Coding (VVC) \cite{s:VVC1} in 2020. Compared to its predecessor, H.265/High Efficiency Video Coding (HEVC), VVC has achieved up to 50\% performance improvement through the adoption of numerous sophisticated coding tools, in particular with improved support for formats with high spatial resolutions, high dynamic range and spherical content. Alongside VVC, the Alliance for Open Media (AOMedia) also published its first video coding format, AOMedia Video (AV1) in 2018, which has been reported to offer comparable coding performance to VVC \cite{j:Chen1}.
Machine learning, especially based on deep convolutional neural networks (CNNs), has being increasingly applied in the context of video compression and has achieved promising results both when used in conjunction with conventional coding algorithms and in the form of new end-to-end architectures. In addition to conventional normative coding tools, deep learning can also been employed at the video decoder as a post-processing stage to further reduce noticeable artefacts and enhance the visual quality of compressed content. For the state-of-the-art coding standards, such as VVC and AV1, most existing learning-based post-processing approaches can only offer evident improvement for less efficient coding configurations (e.g. intra coding), and the employed networks are normally trained to minimise average absolute pixel distortions rather than explicitly to improve perceptual quality.
Based on the Generative Adversarial Network (GAN) paradigm, we propose a novel CNN-based post-processing approach with an extension that achieves improved perceptual reconstruction quality. This approach has been evaluated on the VVC Test Model (VTM) 4.0.1 and on AV1 libaom 1.0.0, with results showing consistent improvement on standard JVET test sequences for different QP values based on different quality measurements. We have further analysed the computational complexities of different CNN structure variants and correlated them with overall coding gains.
This paper is a comprehensive extension of our previous work \cite{c:Zhang30}, which solely focused on the PSNR driven optimization of VVC compressed content. The primary differences are summarized below:
\begin{itemize}
\item The CNN model used for post-processing has been extensively upgraded with a new GAN-based perceptual training strategy, which can significantly improve the perceptual quality of the final reconstructed content compared to \cite{c:Zhang30}.
\item This CNN-based post processing approach has been also trained and evaluated on AV1 compressed results (alongside VVC in \cite{c:Zhang30}) and achieved similar coding gains.
\end{itemize}
In the remainder of the paper, we first survey the prior work in the field of deep video compression and, in particular, describe learning-based post-processing approaches. Secondly, we present our proposed CNN-based post-processing approach, describing the network architecture and how it was trained and evaluated. We then summarize and discuss the experimental results, highlighting the performance improvements over standard video codecs. Finally we conclude the paper and outline possible future work.
\section{PRIOR WORK}
\label{sec:review}
\subsection{Deep video compression}
In the past few years, machine learning, in particular deep neural networks, has been increasingly applied to image and video compression, demonstrating significant potential compared to conventional coding methods. Learning-based video coding algorithms can be classified into two primary groups: new end-to-end network architectures and those that enhance individual conventional coding tools.
Machine learning has been employed to refine existing coding tools within a conventional coding framework, including intra and inter prediction, transformation, quantization and in-loop filtering \cite{j:Liu3}. New coding tools have also been developed with the support of neural networks, such as CNN-based spatial resolution and bit depth adaptation \cite{j:Zhang12}. Moreover, the classic hybrid video coding framework has been challenged by new deep network architectures, which enable end-to-end training and optimization \cite{c:Balle,c:Rippel}. This latter approach often employs a general rate distortion framework with non-linear transforms, which are based on convolutional filters and nonlinear activation functions. Although these solutions show great promise as an alternative to conventional codecs, their performance cannot still compete with the latest standardized video codecs, including VVC and AV1.
\subsection{CNN-based post processing}
Compression processes often introduce various visible artefacts such as blocking mismatches, banding and blurring, especially when large quantization steps are employed. These unpleasant distortions can be mitigated by filtering the reconstructed frames. When this enhancement process is performed outside of the encoding loop (generally after decoding), it is referred to as post-processing.
In standardized codecs, filters have also been designed for use within the encoding loop to reduce compression artefacts. VVC employs three different types in-loop filters including de-blocking filters (DBF), sample adaptive offset (SAO) and adaptive loop filters (ALF) \cite{s:VVC1}. In AV1, in addition to deblocking filters, there are two other filtering operations: constrained directional enhancement filtering and loop restoration filtering \cite{j:Chen1}.
CNNs are now also playing an important role in image restoration (including super-resolution), and these approaches can also be employed for post-processing of compressed video content to improve the overall reconstruction quality. Although various researchers have implemented CNN-based post-processing approaches in the context of HEVC and VVC \cite{j:Ma2,j:Zhao3,r:JVETN0254,r:JVETO0079}, most of these can only achieve coding gains for All Intra configurations, which offer lower coding efficiency compared to Random Access configurations based on hierarchical B frame structures. In addition, all the employed CNN models in these approaches were trained to optimize a simple loss function based on pixel distortions ($\ell 1$ or $\ell 2$ loss), which can lead to over-smoothed reconstruction results.
\section{PROPOSED ALGORITHM}
\label{sec:algo}
{\bf Figure \ref{fig:workflow}} shows a high level coding workflow with a CNN-based post-processing module. This section focuses on the structure of the employed CNN architecture, and the details of network training and evaluation.
\begin{figure}[htbp]
\centerline{\includegraphics[width=1.01\linewidth]{framework.pdf}}
\caption{A typical coding workflow with a CNN-based post-processing module.}
\label{fig:workflow}
\end{figure}
\subsection{The CNN architecture}
\begin{figure*}[htbp]
\centerline{\includegraphics[width=1.01\linewidth]{CNN.pdf}}
\caption{The employed GAN architecture, comprising generator and discriminator stages}
\label{fig:gan}
\end{figure*}
The CNN architecture used in this work is illustrated in {\bf Figure \ref{fig:gan}}.
\subsubsection{The generator network} is a modified version based on the generator (SRResNet) of SRGAN \cite{c:Ledig}, and this has been previously employed by the authors in video compression systems based on spatial resolution and bit depth resampling \cite{j:Zhang12}. This network takes a 96$\times$96 RGB (compressed) image block as input and produces an image block with the same format, targeting to its corresponding original (uncompressed) counterpart.
Residual blocks (RB) form the basic unit in this network, which contains two convolutional layers and a parametric Rectified Linear Unit (PReLU) activation function in between them. A skip connection is used between the input of each RB and the output of its second convolutional layer. The number of residual blocks is configurable and was set to 16 in this work.
The input of the network is connected to these successive RBs through a convolutional layer (also with a ReLU). Between the network output and the output of the last RB, there is also a convolution layer (output layer) followed by a Tanh activation function. An additional skip connection is used between the output of the input layer and the output of the last RB. A long skip connection is also employed between the input the first RB and the output of the output layer to produce the final output.
\subsubsection{The discriminator} used in this work is similar to that in SRGAN \cite{c:Ledig}, which takes the output of the generator (\textit{fake}) and compares to its corresponding original (\textit{real}). This network consists of one input layer (with a Leaky ReLU), seven identical convolutional layers, and two dense layers. Each of the convolutional layers is followed by a batch normalization layer and a leaky ReLU activation function. After the second dense layer, a Sigmoid activation function is employed to output a probability to predict how much the quality of the \textit{real} image block is perceptually better than the \textit{fake} one.
The parameters used in each convolutional layer including kernel sizes, feature map numbers and stride values are shown in Figure \ref{fig:gan}.
\subsection{Training database}
The training material is essential for learning-based algorithms. We need to ensure the training content is diverse and covers various texture types in order to achieve good model generalisation and avoid potential over-fitting problems. To train the employed network, we have selected 432 uncompressed video sequences from a publicly available training database, BVI-DVC \cite{j:Zhang17}, which was designed specifically for deep video compression. All these sequences have the same frame rate of 60 frames per second, YCbCr 4:2:0 format, and with four different spatial resolutions including 3840$\times$2160, 1920$\times$1080, 960$\times$540, and 480$\times$270. We have encoded these 432 original sequences using VVC VTM 4.0.1 and AV1 libaom 1.0.0 with the coding configurations summarized in \textbf{Table \ref{tab:cfg}}.
\begin{table*}[ht]
\caption{The coding configuration employed for VVC VTM and AV1 libaom.}
\centering
\begin{tabular}{c | p{1.6cm} | p{10.7cm}}
\toprule
{Codec} & {Version} & {Configuration parameters} \\
\midrule
VVC VTM & 4.0.1 & Random Access configuration \cite{s:JVETCTC}. IntraPeriod=64, GOPSize=16, QP=22, 27, 32, 37, 42.\\
\midrule
AV1 libaom& 1.0.0-5ec3e8c (02/05/2020)& {-{}-}i420 {-{}-}psnr {-{}-}usage=0 {-{}-}verbose {-{}-}cpu-used=0 {-{}-}threads=0 {-{}-}profile=0 {-{}-}width=\$w {-{}-}height=\$h {-{}-}input-bit-depth=10 {-{}-}bit-depth=10 {-{}-}fps=\$fps/1001 {{-{}-}passes=1} {{-{}-}kf-max-dist=64 {-{}-}kf-min-dist=64 }{-{}-}drop-frame=0 {-{}-}static-thresh=0 {-{}-}arnr-maxframes=7 {-{}-}arnr-strength=5 {{-{}-}lag-in-frames=19} {-{}-}aq-mode=0 {-{}-}bias-pct=100 {-{}-}minsection-pct=1 {-{}-}maxsection-pct=10000 {-{}-}auto-alt-ref=1 {-{}-}min-q=0 {-{}-}max-q=63 {{-{}-}max-gf-interval=16} {-{}-}min-gf-interval=4 {-{}-}frame-parallel=0 {-{}-}color-primaries=bt709 {-{}-}end-usage=q {-{}-}sharpness=0 {-{}-}undershoot-pct=100 {-{}-}overshoot-pct=100 {-{}-}tile-columns=0 {-{}-}cq-level=\{32, 43, 55, 63\} {w/o} {-{}-}enable-fwd-kf=1\\
\bottomrule
\end{tabular}
\label{tab:cfg}
\end{table*}
For each codec, the reconstructed video frames for each QP value and their corresponding originals were randomly selected and segmented into 96$\times$96 colour image blocks (after converting to the RGB space from YCbCr 4:2:0). We have also rotated selected image blocks to further improve data diversity. As a result, for each video codec and QP group, there are over 100,000 image blocks pairs (compressed and original).
\subsection{Training strategy}
We trained this network using two different methodologies: (i) only train the generator using $\ell 1$ loss (mean absolute difference) (ii) jointly train both the generator and the discriminator based on perceptually-inspired loss functions. The CNN models obtained by these two training methods are used to post-process VVC and AV1 compressed content in the evaluation experiments, and their results are compared in the next section.
\subsubsection{$\ell 1$ loss}
is firstly employed to train the network (generator only) using the material generated for each QP group and codec. This results in a total number of nine CNN models for different evaluation scenarios:
\small
\begin{equation}
\left\{
\begin{array}{l r}
\mathrm{CNN}_\mathrm{VVC,QP22}, \ \text{if} & \mathrm{QP_{eval}}\leq 24.5 \\
\mathrm{CNN}_\mathrm{VVC,QP27}, \ \text{if} & 24.5<\mathrm{QP_{eval}}\leq 29.5 \\
\mathrm{CNN}_\mathrm{VVC,QP32}, \ \text{if} & 29.5<\mathrm{QP_{eval}}\leq 34.5 \\
\mathrm{CNN}_\mathrm{VVC,QP37}, \ \text{if} & 34.5<\mathrm{QP_{eval}}\leq 39.5 \\
\mathrm{CNN}_\mathrm{VVC,QP42}, \ \text{if} & \mathrm{QP_{eval}}> 39.5
\end{array}
\right.
\end{equation}
\begin{equation}
\left\{
\begin{array}{l r}
\mathrm{CNN}_\mathrm{AV1,QP32}, \ \text{if} & \mathrm{QP_{eval}}\leq 37.5 \\
\mathrm{CNN}_\mathrm{AV1,QP43}, \ \text{if} & 37.5<\mathrm{QP_{eval}}\leq 49 \\
\mathrm{CNN}_\mathrm{AV1,QP55}, \ \text{if} & 49<\mathrm{QP_{eval}}\leq 59 \\
\mathrm{CNN}_\mathrm{AV1,QP63}, \ \text{if} & \mathrm{QP_{eval}}> 59
\end{array}
\right.
\end{equation}
\normalsize
Here $\mathrm{QP_{eval}}$ represents the base QP value employed in the evaluation phase for the two different codecs, and $\mathrm{CNN}_{c,q}$ is the CNN model trained for different codecs (VVC or AV1) and QP values.
\subsubsection{Perceptual loss functions}
have been employed to train the whole GAN architecture following a two stage training strategy. This was initially designed to train the Relativistic GAN for image generation \cite{jolicoeur2018relativistic}, and has also been used to train the CNN models for spatial resolution and bit depth up-sampling \cite{c:ZHang31,c:Zhang29}. In the first stage, the generator is trained separately using the multi-scale structural similarity index (MS-SSIM) \cite{c:mssim} as the loss function. The trained generator model is employed as the starting point when the generator and discriminator are trained together in the second phase.
The generator is trained using a combined loss function, $\mathcal L_\mathrm{gen}$ in the second stage:
\begin{equation}
\mathcal L_\mathrm{gen} =\mathcal L_{\rm SSIM}+\alpha \cdot \mathcal L_{\ell1} + \beta \cdot \mathcal L_G^{a}
\label{eq:gen}
\end{equation}
in which $\mathcal L_{\rm SSIM}$ stands for the SSIM \cite{j:ssim} loss (1-SSIM) between the generator output and the target, while $\mathcal L_{\ell1}$ is the $\ell 1$ loss between them. $\mathcal L_G^{a}$ is defined as the adversarial loss for the generator:
\begin{small}
\begin{equation}
\begin{split}
{\mathcal L_G^{a}}=&-E_{I_r}[{\rm ln}(1-({\rm Sig}(O_d(I_r)-E_{I_f}[O_d(I_f)])))]\\
&-E_{I_f}[{\rm ln}({\rm Sig}(O_d(I_f)-E_{I_r}[O_d(I_r)]))]
\label{eq:alg}
\end{split}
\end{equation}
\end{small}
Here $I_r$ and $I_f$ are denoted as the \textit{real} and \textit{fake} image blocks respectively. $E_{I_r}[\cdot]$ represents the mean operation for all the \textit{real} (\textit{fake} if $I_r$ is replaced by $I_f$) image blocks, and $O_d(\cdot)$ is the output of the discriminator. `Sig' represents the Sigmoid function.
For the discriminator, the loss function $\mathcal L_D$ is given by (\ref{eq:1}):
\begin{small}
\begin{equation}
\begin{split}
{\mathcal L_D}=&-E_{I_r}[{\rm ln}({\rm Sig}(O_d(I_r)-E_{I_f}[O_d(I_f)]))]\\
& -E_{I_f}[{\rm ln}(1-({\rm Sig}(O_d(I_f)-E_{I_r}[O_d(I_r)])))]
\label{eq:1}
\end{split}
\end{equation}
\end{small}
\subsubsection{The training configuration} is summarized as follows. We implemented all networks based on the TensorFlow 1.8.0 framework, and set the learning rate and weight decay to 0.0001 and 0.1 (for every 100 epochs) respectively for both training stages. The total number of training epochs are 200. We have used Adam optimisation algorithm during the training with the hyper parameters of $\beta_1=0.9$ and $\beta_2=0.999$. The two weights $\alpha$ and $\beta$ in equation (\ref{eq:gen}) are set up to 0.025 and \num{5e-3} respectively.
\subsection{Evaluation operation}
In the evaluation stage, when we use the trained CNN models to enhance compressed video frames, each frame is segmented into 96$\times$96 overlapping image blocks with an overlap of 4 pixels. These blocks are converted to RGB color space as the input of CNN models, and the output image blocks are then aggregated in the same way to form the final video frame. Here only the generator network is used in the evaluation (the discriminator is for training only).
\section{RESULTS AND DISCUSSION}
\label{sec:results}
\begin{table*}[htbp]
\centering
\small
\caption{The compression performance of the proposed method benchmarked on the original VVC VTM 4.0.1. Negative BD-rate values indicate coding gains.}
\begin{tabular}{l ||r |r ||r |r ||r |r ||r |r }
\toprule
Method & \multicolumn{4}{c||}{VTM-PP ($\ell 1$ loss)} &\multicolumn{4}{c}{VTM-PP (Perceptual loss)}\\
\midrule Metric&\multicolumn{2}{c||}{PSNR} &\multicolumn{2}{c||}{VMAF} &\multicolumn{2}{c||}{PSNR} &\multicolumn{2}{c}{VMAF}\\
\midrule QP Range & {H-QPs} & {L-QPs} & {H-QPs} & {L-QPs}&{H-QPs} &{L-QPs} &{H-QPs} &{L-QPs}\\
\midrule Class-Sequence & BD-rate & BD-rate & BD-rate & BD-rate & BD-rate& BD-rate&BD-rate&BD-rate\\
\midrule
A1-Campfire&-3.3\% & -2.3\% &-5.6\% & -4.6\% &+0.2\%&-0.4\%&-10.4\%& -10.7\%\\
A1-FoodMarket4&-2.6\% & -2.0\% &-3.8\% & -3.0\% &-0.0\%&+0.1\%&-8.4\%& -7.4\%\\
A1-Tango2&-3.3\% & -2.9\% &-3.4\% & -3.0\% &-1.1\%&-0.6\%&-7.8\%& -9.3\%\\
\midrule A2-CatRobot1&-5.2\% & -5.2\% &-4.6\% & -4.4\% &-0.6\%&-1.1\%&-14.4\%&-17.8\% \\
A2-DaylightRoad2&-6.0\% & -7.1\% &-6.8\% & -7.2\% &-1.1\%&-2.1\%&-19.5\%& -23.4\%\\
A2-ParkRunning3&-0.8\% & -0.4\% &-2.3\% & -0.2\% &+2.1\%&+2.4\%&-11.7\%& -11.1\%\\
\midrule \textbf{Class A}&-3.5\% & -3.3\% &-4.4\%& -3.7\% &-0.1\%&+0.3\%&-10.1\%&-13.3\%\\
\midrule B-BQTerrace&-2.2\% & -1.0\% &-6.1\% & -1.1\% &+2.0\%&+0.3\%&-23.3\%& -28.8\%\\
B-BasketballDrive&-3.4\% & -3.1\% &-1.8\% & +2.7\%&-0.9\%&-0.9\%&-7.9\%& -8.7\%\\
B-Cactus&-3.4\% & -3.0\% &-5.1\% & -4.4\% &+0.2\%&-0.2\%&-15.8\%&-17.1\%\\
B-MarketPlace&-2.6\% & -2.3\% &-4.8\% & -4.0\%&+1.2\%&+0.3\%&-17.7\%& -18.2\%\\
B-RitualDance&-3.8\% & -3.5\% &-4.6\% & -2.6\% &-1.1\%&-1.2\%&-11.7\%&-11.2\%\\
\midrule \textbf{Class B}&-3.1\% & -2.6\% &-4.5\% & -1.9\%&+0.3\%&-0.3\%&-15.3\%&-16.8\%\\
\midrule C-BQMall&-5.6\% & -5.6\% &-4.7\% & -6.8\% &-2.1\%&-2.5\%&-14.3\%&-13.6\%\\
C-BasketballDrill&-3.9\% & -3.6\% &-3.8\% & -2.8\% &-1.4\%&-1.6\%&-12.3\%&-11.7\%\\
C-PartyScene&-4.1\% & -4.3\% &-5.9\% & -4.1\% &-0.4\%&-1.4\%&-16.1\%&-13.8\%\\
C-RaceHorses&-3.1\% & -2.1\% &-3.4\% & +1.2\% &+0.2\%&-0.4\%&-11.9\%&-10.4\%\\
\midrule \textbf{Class C}&-4.2\%& -3.9\% &-4.5\% & -3.1\% &-0.9\%&-1.5\%&-13.7\%&-12.4\%\\
\midrule D-BQSquare&-8.7\% & -9.6\% &-10.1\% & -11.6\%&-4.0\%&-4.5\%&-16.6\%& -19.5\%\\
D-BasketballPass&-6.1\% & -5.6\% &-5.4\% & -4.0\% &-3.0\%&-2.8\%&-9.8\%&-8.1\%\\
D-BlowingBubbles&-3.7\% & -3.8\% &-4.8\% & -3.8\% &-0.5\%&-1.3\%&-16.1\%&-14.5\%\\
D-RaceHorses&-4.8\% & -4.2\% &-5.2\% & -1.0\% &-1.6\%&-1.9\%&-11.8\%&-10.5\%\\
\midrule \textbf{Class D}&-5.8\% & -5.8\% &-6.4\% & -5.1\% &-2.3\%&-2.6\%&-13.6\%&-13.2\%\\
\midrule \midrule \multirow{2}{*}{\textbf{Overall}}&-4.0\% & -3.8\% &-4.9\% & -3.4\% &-0.6\%&-1.2\%&-13.5\%&-14.2\%\\
\cmidrule{2-9}& \multicolumn{2}{c||}{BD-rate$=$-3.9\%}& \multicolumn{2}{c||}{BD-rate$=$-4.2\%}&\multicolumn{2}{c||}{BD-rate$=$-0.9\%}& \multicolumn{2}{c}{BD-rate$=$-13.9\%}\\
\bottomrule
\end{tabular}
\label{tab:results_vtm}
\end{table*}
\begin{table*}[htbp]
\centering
\small
\caption{The compression performance of the proposed method benchmarked on the original AV1 1.0.0. Negative BD-rate values indicate coding gains.}
\begin{tabular}{l || M{2cm} ||M{2cm} ||M{2cm} ||M{2cm}}
\toprule
Method & \multicolumn{2}{c||}{AV1-PP ($\ell 1$ loss)} & \multicolumn{2}{c}{AV1-PP (Perceptual loss)} \\
\midrule Metric&{PSNR} &{VMAF} & PSNR & VMAF \\
\midrule Class-Sequence & BD-rate & BD-rate & BD-rate & BD-rate \\
\midrule
A1-Campfire&-5.0\% &-7.3\% & -1.1\% & -11.8\% \\
A1-FoodMarket4&-4.0\% &-8.9\% & -1.1\% & -9.6\% \\
A1-Tango2&-4.6\% &-8.2\% & -1.9\% & -10.7\% \\
\midrule A2-CatRobot1&-5.9\% &-5.9\% & -1.9\% & -15.0\% \\
A2-DaylightRoad2&-7.7\% &-5.0\% & -3.1\% & -17.2\% \\
A2-ParkRunning3&-1.9\% &-2.2\% & +0.4\% &-11.3\%\\
\midrule \textbf{Class A}&-4.9\% &-6.3\% & -1.5\%& -12.6\% \\
\midrule B-BQTerrace&-4.5\% &+1.4\% & -1.3\% & -10.2\% \\
B-BasketballDrive&-6.0\% &-3.0\% & -2.9\% &-7.1\% \\
B-Cactus&-3.8\% &-3.3\% &-0.7\%&-11.9\%\\
B-MarketPlace&-3.0\% &-4.5\% & -0.5\% &-17.4\%\\
B-RitualDance&-4.7\% &-5.0\% & -2.3\% &-11.1\% \\
\midrule \textbf{Class B}&-4.4\% &-2.9\%& -1.5\%& -11.5\% \\
\midrule C-BQMall&-6.3\% &-2.4\% & -2.9\% &-9.4\%\\
C-BasketballDrill&-6.8\% &-2.4\% & -3.1\% &-9.9\%\\
C-PartyScene&-7.8\% & +2.3\% &-3.3\%&-9.1\%\\
C-RaceHorses&-4.1\% & -3.8\% &-1.8\% & -8.4\% \\
\midrule \textbf{Class C}&-6.3\% &-1.6\% & -2.8\% &-9.2\% \\
\midrule D-BQSquare&-16.1\% &+11.2\% & -8.4\% & -4.5\% \\
D-BasketballPass&-7.0\% &-3.8\% & -3.9\% &-8.2\% \\
D-BlowingBubbles&-6.2\% &+2.0\% & -2.8\% &-9.7\% \\
D-RaceHorses&-5.6\% &-3.0\% & -3.0\% &-7.7\%\\
\midrule \textbf{Class D}&-8.7\% &+1.6\%& -4.5\% & -7.5\% \\
\midrule \midrule \textbf{Overall}& -5.8\% &-2.7\% & -2.4\% & -10.5\% \\
\bottomrule
\end{tabular}
\label{tab:results3_av1}
\end{table*}
The proposed post-processing approach has been utilised to enhance both VVC and AV1 compressed content. We have used all 19 test sequences from the JVET CTC standard dynamic range (SDR) testset. None of these sequences were included in the CNN training database.
VVC compressed content was generated using VTM 4.0.1 with the Random Access configuration. AV1 libaom 1.0.0 was used to produce AV1 content, with similar coding parameters to those for VVC. The employed configurations for both codecs are summarised in Table \ref{tab:cfg}. The tested QP values are 22, 27, 32, 37 and 42 for VVC, and 32, 43, 55 and 63 for AV1.
The final video quality was evaluated using two assessment methods, Peak Signal-to-Noise ratio (PSNR) and Video Multimethod Assessment Fusion (VMAF) \cite{w:VMAF}. PSNR is the most commonly used quality metric in the video compression community, while VMAF is a machine learning based metric, which combines multiple existing quality metrics and a video feature using a Support Vector Machine regression approach. Compared to PSNR, VMAF has been reported to provide more accurate prediction of perceptual quality. The performance of the proposed approach has been compared with two original codecs using the Bj{\o}ntegaard Delta rate measurements (BD-rate) \cite{r:Bjontegaard}. For VVC, the compression performance is evaluated for both low (22-37) and high QP (27-42) ranges.
In order to further benchmark the performance of the proposed algorithm, another two state-of-the-art CNN-based post-processing approaches, denoted as JVET-N0254 \cite{r:JVETN0254} and JVET-O0079 \cite{r:JVETO0079}, are also compared here in the context of VVC (for low QP range only). Results of both are based on the RA configuration and were submitted to MPEG JVET meetings as VVC proposals.
\subsection{Compression performance}
\textbf{Table \ref{tab:results_vtm}} and \textbf{\ref{tab:results3_av1}} summarize the compression performance of the proposed method when it is applied to VVC and AV1 compressed content. For $\ell 1$ trained CNNs, we note that the average bit-rate savings according to PSNR are 3.9\% and 5.8\% against the original VVC and AV1 respectively. If the perceptual quality metric, VMAF, is used to assess video quality, the coding gains are 4.2\% over VVC and 2.7\% over AV1. When we use perceptual loss function trained models for post-processing, the coding gains appear much more significant based on the assessment of VMAF -- 13.9\% and 10.5\% over VVC and AV1 respectively. We have also compared the proposed method with two JVET proposals, JVET-N0254 \cite{r:JVETN0254} and JVET-O0079 \cite{r:JVETO0079}, in \textbf{Table \ref{tab:vtmcompare}}, where the $\ell 1$ trained CNNs provides superior enhancement performance to these two works for all resolution classes according to PSNR.
The rate-quality (for both PSNR and VMAF) curves for four test sequences with various spatial resolutions, \textit{DaylightRoad2}, \textit{RitualDance}, \textit{RaceHorses} and \textit{BasketballPass}, are also shown in \textbf{Figure \ref{im:curves_vtm}} and \textbf{\ref{im:curves_av1}}.
\begin{table}[htbp]
\centering
\caption{Comparison between the proposed method and two existing CNN-based PP approaches for VVC (low QP range).}
\begin{tabular}{l || M{1cm} || M{1cm}|| M{1.15cm}}
\toprule
\multirow{2}{*}{Sequence (Class)} & {O0079} \cite{r:JVETO0079}&N0254 \cite{r:JVETN0254} &{Proposed Method} ($\ell 1$)\\
\cmidrule{2-4}
\centering
& BD-rate (PSNR) & BD-rate (PSNR) & BD-rate (PSNR)\\
\midrule
\centering
\textbf{Class A (2160p)}& -1.3\%&-1.7\%
&\textbf{-3.5\%}\\
\midrule \textbf{Class B (1080p)} &-1.5\%
& -1.1\% &\textbf{-3.1\%}\\
\midrule \textbf{Class C (480p)} &-3.3\%&
-1.4\% &\textbf{-4.2\%}\\
\midrule \textbf{Class D (240p)} &-5.0\%& -1.4\% &\textbf{-5.8\%}\\
\midrule \textbf{Overall} &-2.6\%& -1.4\% &\textbf{-4.0\%}
\\\bottomrule
\end{tabular}
\label{tab:vtmcompare}
\end{table}
\begin{figure*}[h]
\centering
\centering
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{A2DaylightRoad2_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{B1RitualDance_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{C1RaceHorses_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{D1BasketballPass_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{A2DaylightRoad2_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{B1RitualDance_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{C1RaceHorses_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{D1BasketballPass_VMAF.pdf}}
\end{minipage}
\caption{Rate-PSNR and Rate-VMAF curves for four selected test sequences based on the VVC VTM 4.0.1}
\label{im:curves_vtm}
\end{figure*}
\begin{figure*}[ht]
\centering
\centering
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{A2DaylightRoad2_AV1_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{B1RitualDance_AV1_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{C1RaceHorses_AV1_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{D1BasketballPass_AV1_PSNR.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{A2DaylightRoad2_AV1_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{B1RitualDance_AV1_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{C1RaceHorses_AV1_VMAF.pdf}}
\end{minipage}
\begin{minipage}[b]{0.235\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{D1BasketballPass_AV1_VMAF.pdf}}
\end{minipage}
\caption{Rate-PSNR and Rate-VMAF curves for four selected test sequences based on the AV1 1.0.0.}
\label{im:curves_av1}
\end{figure*}
\subsection{Subjective comparison}
\textbf{Figure \ref{fig:perceptual1}} and \textbf{\ref{fig:perceptual2}} provide subjective comparisons between the reconstructed frames generated by the original VVC/AV1, $\ell 1$ trained CNNs and perceptual loss function trained models. We can observe that the reconstructed blocks of the proposed method (for both $\ell$1 and perceptually trained models) exhibit fewer noticeable blocking artefacts compared to the anchor codecs. In addition, the CNN models trained using perceptual loss functions produce results with slightly more textural detail and higher contrast than those generated by $\ell$1 trained networks.
\begin{figure*}[ht]
\centering
\scriptsize
\centering
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_orig.png}}
(a) Original \\ \ \
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_anchor.png}}
(b) VTM 4.0.1, QP=42 \\ (VMAF=62.5)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_L1.png}}
(c) VTM-PP: $\ell$1, QP=42 \\ (VMAF=64.0)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_GAN.png}}
(d) VTM-PP: GAN, QP=42 \\ (VMAF=66.2)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_orig.png}}
(e) Original \\ \ \
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_anchor.png}}
(f) VTM 4.0.1, QP=42 \\ (VMAF=70.4)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_L1.png}}
(g) VTM-PP: $\ell$1, QP=42 \\ (VMAF=72.2)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_GAN.png}}
(h) VTM-PP: GAN, QP=42 \\ (VMAF=74.3)
\end{minipage}
\caption{Example blocks of the reconstructed frames for the anchor VTM 4.0.1 and the proposed approach. These are from the 91st and 320th frames of \textit{Cactus} and \textit{BQMall} sequences respectively. We can note that the results produced by the perceptual loss function trained models exhibit more textural detail and higher contrast than those generated by $\ell$1 trained networks (the difference is more evident within the regions with the stamps in \textit{Cactus}, and the areas with the bags and the white skirt in \textit{BQMall}).}
\label{fig:perceptual1}
\end{figure*}
\begin{figure*}[ht]
\centering
\scriptsize
\centering
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_AV1_orig.png}}
(a) Original \\ \ \
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_AV1_anchor.png}}
(b) AV1, QP=63 \\ (VMAF=75.2)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_AV1_L1.png}}
(c) AV1-PP: $\ell$1, QP=63 \\ (VMAF=76.4)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{cactus_AV1_GAN.png}}
(d) AV1-PP: GAN, QP=63 \\ (VMAF=78.1)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_av1_orig.png}}
(e) Original \\ \ \
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_av1_anchor.png}}
(f) AV1, QP=63 \\ (VMAF=83.7)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_av1_L1.png}}
(g) AV1-PP: $\ell$1, QP=63 \\ (VMAF=84.9)
\end{minipage}
\begin{minipage}[b]{0.245\linewidth}
\centering
\centerline{\includegraphics[width=1\linewidth]{mall_av1_GAN.png}}
(h) AV1-PP: GAN, QP=63 \\ (VMAF=86.2)
\end{minipage}
\caption{Example blocks of the reconstructed frames for the anchor AV1 and the proposed approach. These are from the 120th and 320th frames of \textit{Cactus} and \textit{BQMall} sequences respectively. The results produced by the perceptual loss function trained models exhibit more textural detail and higher contrast than those generated by $\ell$1 trained networks (the difference is more evident within the regions with the stamps in \textit{Cactus}, and the areas with the bags and the white skirt in \textit{BQMall}).}
\label{fig:perceptual2}
\end{figure*}
\subsection{Complexity analysis}
The relative computational complexity of the proposed method, benchmarked on the original VVC and AV1 codecs, is presented in \textbf{Table \ref{tab:complexity}}. We have used a shared cluster computer at the University of Bristol, to execute all the computations. This computer contains multiple notes with 2.4GHz Inter CPUs, 138GB RAM and NVIDIA P100 GPU devices. We note that the average decoding complexity is 56.3 and 70.0 times compared to the original VVC VTM 4.0.1 and AV1 respectively, due to the employment of CNN-based post-processing at the decoder. This is for a configuration with 16 residual blocks in the generator.
\begin{table}[htbp]
\centering
\caption{Relative Complexity of the proposed method benchmarked on original VVC and AV1 decoders.}
\begin{tabular}{l|| M{2cm}|| M{2cm}}
\toprule
\multirow{1}{*}{Hose Codec} & \multicolumn{1}{c||}{VVC} & \multicolumn{1}{c}{AV1}\\
\midrule
\textbf{Class A (2160p)}&18.6$\times$&23.2$\times$\\
\midrule \textbf{Class B (1080p)} &36.8$\times$&38.3$\times$\\
\midrule \textbf{Class C (480p)} &76.5$\times$&83.3$\times$\\
\midrule \textbf{Class D (240p)} &116.9$\times$&166.7$\times$\\
\midrule \textbf{Average} &56.3$\times$&70.0$\times$
\\\bottomrule
\end{tabular}
\label{tab:complexity}
\end{table}
Moreover, we have further investigated the relationship between the number of residual blocks and compression performance. \textbf{Figure \ref{fig:bd}} shows the coding gains (in terms of PSNR) and algorithm relative complexity using CNN models with different numbers of residual blocks (N=4, 8, 12 and 16) to process VVC VTM QP 42 compressed content. Again the relative complexity here is benchmarked on the original VVC decoder. We can observe that when the number of residual blocks (N) increases, the PSNR gain relative to the original VVC content increases in a linear fashion.
\begin{figure}[htbp]
\centering
\centering
\begin{minipage}[b]{0.45\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{figures/C1.pdf}}
\end{minipage}
\begin{minipage}[b]{0.45\linewidth}
\centering
\centerline{\includegraphics[width=1.1\linewidth]{figures/C2.pdf}}
\end{minipage}
\caption{(right) PSNR gains for different number of residual blocks. (left)
Relative complexity for different number of residual blocks.}
\label{fig:bd}
\end{figure}
\section{CONCLUSIONS}
\label{sec:conclusions}
In this paper, we presented a CNN-based post-processing approach, which achieves evident and consistent coding gains over standardized video codecs, VVC and AV1. The employed CNN model was trained using both $\ell 1$ and perceptually inspired methodologies. We would like to recommend future work focusing on computational complexity reduction and further improvement on the training methodology.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
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Q: Storage management in C I wrote the program to find out about memory management in C. But I noticed something strange about this program: Why is the variable "a" in memory location 6487580 and not in memory location 6487576.
Because if a pointer is 8 bytes, then it should only reach up to this address. And then the variable "a" would have space there.
#include <stdio.h>
#include <stdlib.h>
int main(){
int a=10;
int *ptr;
ptr=&a;
int **ptrptr;
ptrptr=&ptr;
printf("adress:\n");
printf("Adress a: %i\n",&a);
printf("Sizeof a: %i\n",sizeof(a));
printf("Adress ptr: %i\n",&ptr);
printf("Sizeof ptr: %i\n",sizeof(ptr));
printf("Adress ptrptr: %i\n",&ptrptr);
printf("Sizeof ptrptr: %i\n",sizeof(ptrptr));
return 0;
}
A: Memory alignment. It is likely that the computer you're running on can retrieve data from memory faster or more efficiently if it's stored on a doubleword (8 byte) boundary, or at least the compiler is programmed to make that assumption. If you examine the addresses in binary you'll note they all end in 00, putting them on a doubleword boundary.
Thus, ptrptr is stored on a doubleword boundary at the low-order location and takes 8 bytes. ptr is stored next in memory, and occupies the next 8 bytes. Then 4 bytes are skipped (6487576-6487579) so that a will be aligned on a doubleword boundary at 6487580.
Your compiler may have flags which instruct it to align variables on word or doubleword boundaries - or, conversely, to ignore such alignment concerns. Consult your local documentation for such information.
| {
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Velika Bršljanica – wieś w Chorwacji, w żupanii bielowarsko-bilogorskiej, w mieście Garešnica. W 2011 roku liczyła 228 mieszkańców.
Przypisy
Miejscowości w żupanii bielowarsko-bilogorskiej | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,374 |
{"url":"https:\/\/msp.org\/gt\/2015\/19-6\/b12.xhtml","text":"#### Volume 19, issue 6 (2015)\n\n 1 T Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013) 2855 MR3190300 2 J\u00a0A Behrstock, Asymptotic geometry of the mapping class group and Teichm\u00fcller space, Geom. Topol. 10 (2006) 1523 MR2255505 3 F Bonahon, Geodesic laminations on surfaces, from: \"Laminations and foliations in dynamics, geometry and topology\" (editors M Lyubich, J\u00a0W Milnor, Y\u00a0N Minsky), Contemp. Math. 269, Amer. Math. Soc. (2001) 1 MR1810534 4 M\u00a0R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999) MR1744486 5 J\u00a0F Brock, The Weil\u2013Petersson metric and volumes of $3$\u2013dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495 MR1969203 6 J Brock, H Masur, Coarse and synthetic Weil\u2013Petersson geometry: Quasi-flats, geodesics and relative hyperbolicity, Geom. Topol. 12 (2008) 2453 MR2443970 7 J Brock, H Masur, Y Minsky, Asymptotics of Weil\u2013Petersson geodesic, I: Ending laminations, recurrence, and flows, Geom. Funct. Anal. 19 (2010) 1229 MR2585573 8 J Brock, H Masur, Y Minsky, Asymptotics of Weil\u2013Petersson geodesics, II: Bounded geometry and unbounded entropy, Geom. Funct. Anal. 21 (2011) 820 MR2827011 9 P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkh\u00e4user (1992) MR1183224 10 R\u00a0D Canary, D\u00a0B\u00a0A Epstein, P\u00a0L Green, Notes on notes of Thurston, from: \"Fundamentals of hyperbolic geometry: Selected expositions\" (editors R\u00a0D Canary, D Epstein, A Marden), London Math. Soc. Lecture Note Ser. 328, Cambridge Univ. Press (2006) 1 MR2235710 11 G Daskalopoulos, R Wentworth, Classification of Weil\u2013Petersson isometries, Amer. J. Math. 125 (2003) 941 MR1993745 12 B Farb, A Lubotzky, Y Minsky, Rank-$1$ phenomena for mapping class groups, Duke Math. J. 106 (2001) 581 MR1813237 13 A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Ast\u00e9risque 66\u201367, Soc. Math. France (1979) 284 MR568308 14 D Gabai, Almost filling laminations and the connectivity of ending lamination space, Geom. Topol. 13 (2009) 1017 MR2470969 15 S Hensel, P Przytycki, R\u00a0C\u00a0H Webb, $1$\u2013slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. 17 (2015) 755 MR3336835 16 J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221 MR523212 17 E Klarreich, The boundary at infinity of the curve complex, preprint (1999) 18 C Leininger, A Lenzhen, K Rafi, Limit sets of Teichm\u00fcller geodesics with minimal non-uniquely ergodic vertical foliation, preprint (2015) arXiv:1312.2305 19 G Levitt, Foliations and laminations on hyperbolic surfaces, Topology 22 (1983) 119 MR683752 20 H Masur, Extension of the Weil\u2013Petersson metric to the boundary of Teichm\u00fcller space, Duke Math. J. 43 (1976) 623 MR0417456 21 H Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992) 387 MR1167101 22 H\u00a0A Masur, Y\u00a0N Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999) 103 MR1714338 23 H\u00a0A Masur, Y\u00a0N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 MR1791145 24 B Modami, Prescribing the behavior of Weil\u2013Petersson geodesics in the moduli space of Riemann surfaces, J. Topol. Anal. 7 (2015) 543 MR3400125 25 K Rafi, A characterization of short curves of a Teichm\u00fcller geodesic, Geom. Topol. 9 (2005) 179 MR2115672 26 K Rafi, Hyperbolicity in Teichm\u00fcller space, Geom. Topol. 18 (2014) 3025 MR3285228 27 K Rafi, S Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009) 2141 MR2507116 28 C Series, The modular surface and continued fractions, J. London Math. Soc. 31 (1985) 69 MR810563 29 S\u00a0A Wolpert, Geometry of the Weil\u2013Petersson completion of Teichm\u00fcller space, Surv. Differ. Geom. 8, International Press (2003) 357 MR2039996 30 S\u00a0A Wolpert, Families of Riemann surfaces and Weil\u2013Petersson geometry, CBMS Regional Conference Series in Mathematics 113, Amer. Math. Soc. (2010) MR2641916","date":"2023-01-29 10:01:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.720229983329773, \"perplexity\": 2979.291301524866}, \"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\/1674764499710.49\/warc\/CC-MAIN-20230129080341-20230129110341-00644.warc.gz\"}"} | null | null |
Speeches from the 2008 Democratic National Convention
Robert Casey, Jr.
Remarks to the 2008 Democratic National Convention
I'm honored to stand before you as Governor Bob Casey's son and a proud supporter of Barack Obama. Pennsylvania is home to some of the hardest-working, toughest, most decent people in America.
For eight years, the people of Pennsylvania have been hit hard by the Bush-Cheney economy, an economy that favors the powerful and leaves everyone else to fend for themselves. We've seen our jobs disappear overseas, our wages go down and the price we pay at the pump skyrocket to record highs. We've been hit hard, but we're ready to fight back, and we're ready for a president who will fight for us. That's why I am proud to support Barack Obama for President of the United States.
In a time of danger around the world and economic trouble here at home, I know that Barack Obama will lead us, heal us and help us rebuild the country we love. I know this because I know Barack Obama. I have seen how he inspires people, including my four daughters, to believe that the failures of the past will soon give way to the change we need. I have seen his leadership up close in the Senate, bridging partisan divides and finding common ground. And I have seen him carry those same leadership skills off the floor of the Senate and into cities and towns all across Pennsylvania.
I traveled with Barack by bus and train across our state, from Pittsburgh to Paoli, from Johnstown to Downingtown. He was equally at home talking football with Jerome Bettis and Franco Harris as he was with talking jobs with the folks on the shop floor of the Erie Bolt Company, or talking sports with the guys at the bar at Sharky's in Latrobe.
Everywhere Barack went, people who may have been asking who this guy was ended up seeing what I saw: a husband, a father of two daughters and a man of deep faith. Everywhere we went, the people of Pennsylvania gave him the highest praise they give anyone: He's one of us too.
And Pennsylvania couldn't be prouder of our native son, Joe Biden from Scranton. No one knows us better than Joe.
After eight years of a president who lets the oil companies and the Washington lobbyists call the shots, I say it's about time we had a president and vice president who really know us. We are joined tonight by another great champion of working people, someone with whom I've worked on early childhood education; someone who conducted her campaign with rare grace under real pressure; a senator who has worked to bring our party and our country together: Hillary Rodham Clinton. When she endorsed Barack, Senator Clinton called upon us to "do all we can to help elect Barack Obama the next President of the United States."
Traveling around Pennsylvania, and looking around this room, I have no doubt that is exactly what we're going to do. So now let us work together, with a leader who, as Lincoln said, appeals to the better angels of our nature. Barack Obama and I have an honest disagreement on the issue of abortion. But the fact that I'm speaking here tonight is testament to Barack's ability to show respect for the views of people who may disagree with him.
I know Barack Obama. And I believe that as president, he'll pursue the common good by seeking common ground, rather than trying to divide us. We are strongest when we are together. And there has never been a more important time to devote ourselves to common purpose.
The people of Pennsylvania can't afford four more years of Bush-Cheney economics, and with John McCain, that's exactly what we'd get. John McCain calls himself a maverick, but he votes with George Bush 90 percent of the time. That's not a maverick. That's a sidekick.
The Bush-McCain Republicans inherited the strongest economy in history and drove it into a ditch. They cut taxes on the wealthiest of us and passed on the pain to the least of us. They ran up the debt, gave huge subsidies to big oil companies, and now they're asking for four more years.
How 'bout four more months? We can't afford four more years of deficit and debt, drift and desperation. Not four more years. Four more months. And we can't afford another president who will veto children's health insurance for 10 million children, or who will keep senior citizens from seeing the doctors they trust. Not four more years. Four more months.
Governor Casey used to say that the ultimate question for those in public office is this: what did you do when you had the power? Barack Obama and Joe Biden will use that power to help the folks on the shop floor of the Erie Bolt Company, the guys at Sharky's, and the millions of Americans just like them, struggling but ready to fight back. We know they will because as Pennsylvanians know, Joe Biden is one of us. And Barack Obama is one of us too.
<<Back to the 2008 Democratic National Convention Page
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package au.net.ritterwolf.jmu.ansi;
import static au.net.ritterwolf.jmu.ansi.Color.WHITE;
import static au.net.ritterwolf.jmu.ansi.Weight.NORMAL;
public class State {
private Weight weight;
private Color color;
private boolean underline;
private boolean inverse;
public State() {
weight = NORMAL;
color = WHITE;
underline = false;
inverse = false;
}
public Weight getWeight() {
return weight;
}
public void setWeight(Weight weight) {
this.weight = weight;
}
public Color getColor() {
return color;
}
public void setColor(Color color) {
this.color = color;
}
public boolean isUnderline() {
return underline;
}
public void setUnderline(boolean underline) {
this.underline = underline;
}
public boolean isInverse() {
return inverse;
}
public void setInverse(boolean inverse) {
this.inverse = inverse;
}
@Override
public String toString() {
return "State [weight=" + weight + ", color=" + color + ", underline="
+ underline + ", inverse=" + inverse + "]";
}
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + ((color == null) ? 0 : color.hashCode());
result = prime * result + (inverse ? 1231 : 1237);
result = prime * result + (underline ? 1231 : 1237);
result = prime * result + ((weight == null) ? 0 : weight.hashCode());
return result;
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
State other = (State) obj;
if (color != other.color)
return false;
if (inverse != other.inverse)
return false;
if (underline != other.underline)
return false;
if (weight != other.weight)
return false;
return true;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,576 |
\section{Introduction}
The journal \textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \LaTeX.
The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
This document, \verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.
This is not a general guide on how to use \LaTeX, of which many excellent examples already exist.
We particularly recommend \textit{Wikibooks \LaTeX}\footnote{\url{https://en.wikibooks.org/wiki/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.
Alternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.
For guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\footnote{\label{foot:itas}\url{http://www.oxfordjournals.org/our_journals/mnras/for_authors/}}.
Only technical issues with the \LaTeX\ class are considered here.
\section{Obtaining and installing the MNRAS package}
Some \LaTeX\ distributions come with the MNRAS package by default.
If yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \TeX\ Archive Network\footnote{\url{http://www.ctan.org/tex-archive/macros/latex/contrib/mnras}} (CTAN).
The files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \LaTeX\ distribution), or used temporarily by placing them in the working directory for your paper.
To use the MNRAS package, simply specify \verb'mnras' as the document class at the start of a \verb'.tex' file:
\begin{verbatim}
\documentclass{mnras}
\end{verbatim}
Then compile \LaTeX\ (and if necessary \bibtex) in the usual way.
\section{Preparing and submitting a paper}
We recommend that you start with a copy of the \texttt{mnras\_template.tex} file.
Rename the file, update the information on the title page, and then work on the text of your paper.
Guidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\ref{foot:itas}}$.
Note that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).
If a paper is accepted, it is professionally typeset and copyedited by the publishers.
It is therefore likely that minor changes to presentation will occur.
For this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.
Papers must be submitted electronically via the online submission system; paper submissions are not permitted.
For full guidance on how to submit a paper, see the instructions to authors.
\section{Class options}
\label{sec:options}
There are several options which can be added to the document class line like this:
\begin{verbatim}
\documentclass[option1,option2]{mnras}
\end{verbatim}
The available options are:
\begin{itemize}
\item \verb'letters' -- used for papers in the journal's Letters section.
\item \verb'onecolumn' -- single column, instead of the default two columns. This should be used {\it only} if necessary for the display of numerous very long equations.
\item \verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.
\item \verb'referee' -- \textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.
\item \verb'galley' -- \textit{(deprecated)} no running headers, no attempt to align the bottom of columns.
\item \verb'landscape' -- \textit{(deprecated)} sets the whole document on landscape paper.
\item \verb"usenatbib" -- \textit{(all papers should use this)} this uses Patrick Daly's \verb"natbib.sty" package for citations.
\item \verb"usegraphicx" -- \textit{(most papers will need this)} includes the \verb'graphicx' package, for inclusion of figures and images.
\item \verb'useAMS' -- adds support for upright Greek characters \verb'\upi', \verb'\umu' and \verb'\upartial' ($\upi$, $\umu$ and $\upartial$). Only these three are included, if you require other symbols you will need to include the \verb'amsmath' or \verb'amsymb' packages (see section~\ref{sec:packages}).
\item \verb"usedcolumn" -- includes the package \verb"dcolumn", which includes two new types of column alignment for use in tables.
\end{itemize}
Some of these options are deprecated and retained for backwards compatibility only.
Others are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.
If you want to include any other packages, see section~\ref{sec:packages}.
\section{Title page}
If you are using \texttt{mnras\_template.tex} the necessary code for generating the title page, headers and footers is already present.
Simply edit the title, author list, institutions, abstract and keywords as described below.
\subsection{Title}
There are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').
Enter them with \verb'\title[]{}' like this:
\begin{verbatim}
\title[Running head]{Full title of the paper}
\end{verbatim}
The full title can be multiple lines (use \verb'\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\le~45$ characters on a single line.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Authors and institutions}
Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.
If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.
For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.
Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.
\subsection{Abstract and keywords}
The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.
Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.
\section{Sections and lists}
Sections and lists are generally the same as in the standard \LaTeX\ classes.
\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
\subsection{Subsection}
\subsubsection{Subsubsection}
\paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.
Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Lists}
Two forms of lists can be used in MNRAS -- numbered and unnumbered.
For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.
For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.
\section{Mathematics and symbols}
The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.
Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.
\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:
\begin{verbatim}
\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb'
\section{Introduction}\label{sec:intro}
With the Atacama Large Millimetre Array (ALMA) telescope having reached full operation, the field of proto-planetary discs is undergoing a rapid observational expansion. Thanks to the order of magnitude improvement in spatial resolution, we now have the possibility of resolving the signatures of planets in formation in these discs, in this way transforming planet formation into an observational field.
The most striking result of these observations is the ubiquity of annular structures, colloquially described as gaps and rings. While some discs do show alternative structures like spirals or crescents \citep[e.g.,][]{2013Sci...340.1199V,2015ApJ...812..126C,2016Sci...353.1519P,2018ApJ...853..162B,2018A&A...619A.161C}, most of the discs observed at high resolution are characterised by axisymmetric structures. The prevalence of axisymmetric structures was already clear from the publication of several high-resolution observations targeting individual sources \citep{Brogan2015,2016PhRvL.117y1101I,2016ApJ...820L..40A,2017A&A...597A..32V,2017A&A...600A..72F,2017ApJ...840...23L,2018A&A...610A..24F,2018MNRAS.475.5296D,2018ApJ...866L...6C} and from the survey in Taurus \citep{2018ApJ...869...17L}, but recently it was made even clearer by the publication of DSHARP \citep{2018ApJ...869L..41A,2018ApJ...869L..42H}, a homogeneous high resolution survey of 20 discs. All the 18 single disc systems show annular structure; only a minority also exhibit additional structure superimposed on the background annular structure, with 3 showing spirals and 2 showing crescents.
Planets naturally create annular structures in discs \citep[e.g.,][]{2004A&A...425L...9P,2012A&A...545A..81P,Picogna2015,Rosotti2016,2017MNRAS.469.1932D,2018ApJ...869L..47Z} and are therefore the leading explanation for these structures. There are however also alternative interpretations. An intriguing idea is that these observed structures do not correspond to real features in the disc surface density, but they are caused by opacity changes \citep{2015ApJ...806L...7Z,2016ApJ...821...82O,2017ApJ...845...68P,2017A&A...600A.140S}. In this view the change in opacity should happen at the locations of snowlines, where the most abundant molecules change from the solid to the gas phase, triggering compositional changes in the dust. Recent work \citep{2018ApJ...869L..42H} however has put this idea into question since the location of most of the observed gaps do not correspond to the predicted location of the snowlines. In addition, some gaps do correspond to physical structures in the gas surface density, as shown by gas emission line profiles \citep{2016PhRvL.117y1101I,2017A&A...600A..72F} and kinematically derived rotation curves \citep{2018ApJ...860L..12T,2018ApJ...868..113T}. Some gaps \citep{2018ApJ...869L..48G} are so deep that they cannot be accounted for by opacity variations, and they must correspond to a real depletion in surface density. Still open instead is the possibility that these structures are created by the interplay between magnetohydrodynamics (MHD) and dust dynamics \citep[e.g.,][]{2015A&A...574A..68F,2018A&A...609A..50D}. In a similar way to planets, MHD processes can also alter the gas surface density and its pressure profile, causing a variation in the dust radial velocity and therefore its surface density. This possibility has received less attention than the planet hypothesis and it is currently less clear how to distinguish between the two.
Rather than contributing directly to this debate, in this paper we will focus on the planet hypothesis and explore its consequences. One of the fundamental questions in this case is what is the range of masses of the putative planets. Broadly speaking, on the upper end of the planet mass range we can exclude in most cases that these planets are gas giants of several Jupiter masses. These planets tend to create non-axisymmetric structures like spiral arms and crescents (see for example the gallery of simulated observations in \citealt{2018ApJ...869L..47Z}) and allow for very little passage of dust through the planet orbit \citep{2012ApJ...755....6Z}, depleting most of the disc interior to the planet location. As such, they are more commonly invoked to explain the few discs that show prominent spiral arms \citep{2015MNRAS.451.1147J,2016ApJ...816L..12D,2017ApJ...839L..24M,2018MNRAS.474L..32J} or the so-called ``transition discs'' \citep[e.g.,][]{2015A&A...580A.105P,2016MNRAS.459L..85D}, rather than the gapped discs (though with the recent high-resolution observations the distinction between the two categories is becoming blurred). On the lower end of the mass range instead, it is well known \citep{Lin1993,Crida2006} that there is a minimum threshold mass needed to open a gap. In the \textit{gas} case, it is very well known that this depends on the disc aspect ratio and viscosity; in general, gap opening requires higher-mass planets if the disc is thicker (i.e., hotter) and more viscous. ALMA observations however probe the \textit{dust}. Because it is easier to open gaps in dust than in gas, this mass threshold is \textit{quantitatively} different (lower) than for the conventional gas case. Ultimately, however, the mass threshold should not change \textit{qualitatively} since the the dust morphology is set by the underlying gas profile. In particular, the gas radial pressure gradient determines the dust radial drift velocity, and in turn its surface density \citep{2012ApJ...755....6Z,Rosotti2016}.
While the dependence of the gap-opening mass on the disc parameters has been extensively studied in previous works, a less explored aspect is the dependence on the stellar mass. Since the gaps are of dynamical origin, the real underlying parameter is the planet-star mass ratio, not the absolute planet mass. Targeting lower mass stars could then open the exciting possibility of detecting planets significantly lower in mass. Since the threshold mass around a solar mass star is typically in the super-Earth regime \citep{Rosotti2016}, in principle the sensitivity around an ultracool dwarf should be comparable to an Earth mass. Conceptually, this is a similar motivation to the study of exoplanets around mature low mass stars with the conventional techniques of transits and radial velocity, that has led, for example, to the discovery of the TRAPPIST1 system \citep{2016Natur.533..221G}. However, the dependence on the aspect ratio mentioned above also needs to be taken into account. It is well known \citep[e.g.,][]{2012A&A...539A...9M,2013A&A...554A..95P} that discs around low mass stars and brown dwarves are geometrically thicker than those around solar mass stars, as a consequence of the reduced gravitational potential\footnote{In general, this is is more important than the fact that these discs are colder due to the fainter central star. We will discuss this in the detail in the rest of the paper.}. This effect makes it \textit{harder} to open gaps around lower mass stars, in the opposite direction to what we have described before.
Determining which of the two effects is dominant is the purpose of this paper. Building on the disc-planet interaction dusty simulations presented by \citet{Rosotti2016}, in this work we will explore the dependence of the gap-opening mass on the stellar mass. This paper is observationally focused and our definition of gap opening is therefore an observable gap \textit{in the dust}. A key aspect of this work is that we compute the disc temperature rather than leaving it as a free parameter (as commonly done in hydrodynamic simulations).
This paper is structured as follows. We explain our methodology in section \ref{sec:methods} and present our results in section \ref{sec:results}. We then discuss the implications of our results for observations targeting low mass stars and for the planet hypothesys for the origin of gaps in section \ref{sec:discussion} and finally draw our conclusions in section \ref{sec:conclusions}.
\section{Methods}\label{sec:methods}
Our methodology consists of running hydrodynamical simulations of the gas and dust dynamics with the code FARGO to study how these components respond to the presence of a planet.
We then use the radiative transfer code RADMC-$3$D\footnote{\url{http://www.ita.uni-heidelberg.de/dullemond/software.radmc-3d/}} to compute the disc temperature and generate synthetic images.
Finally, we use the CASA tool to simulate realistic ALMA observations.
We detail this work flow in the following sections \ref{ssec:hydrosim}, \ref{ssec:RT} and \ref{ssec:CASA}.
One particular aspect to note is that for the hydrodynamical simulations the disc temperature, typically parameterised through the disc aspect ratio $h/r$, is a free parameter.
This is particularly important because the aspect ratio has a major impact on the minimum gap opening planet mass (MGOPM).
To run realistic hydrodynamical simulations, it is thus necessary to know how the disc aspect ratio varies as a function of stellar mass and disc radius.
To this end, we perform a preliminary set of calculations with RADMC-$3$D, not containing any planet, that we describe in section \ref{sssec:dischr}.
\subsection{Hydrodynamical Simulations}\label{ssec:hydrosim}
The simulations we present in this paper use a custom version of the \textsc{fargo-3d} code \citep{2016ApJS..223...11B}, modified to include dust dynamics as described in \citet{Rosotti2016}; we refer the reader to that paper for more details on the dust algorithm. Briefly, dust is described as a pressure-less fluid, evolving because of gravity, gas drag and diffusion. We implement gas drag using a semi-implicit algorithm that automatically reduces to the short-friction time approximation for tightly coupled dust (so that the timestep does not become vanishingly small) and to an explicit update for loosely coupled grains. For the diffusion, we use a diffusion coefficient equal to the shear viscosity coefficient of the gas (in other words, the Schmidt number is 1).
We use 2D cylindrical coordinates and dimensionless units in which the orbital radius of the planet ($r_p$) is at unity, the unit of mass is that of the central star and the unit of time is the inverse of the Kepler-frequency of the planet. The inner radial boundary of our grid is at 0.5 $r_p$ and the outer boundary at $3 r_p$; we use non reflecting boundary conditions at both boundaries. While \citet{Rosotti2016} fixed the dust density to its initial value at the inner boundary, here we allow the dust density to drop below this value (see also \citealt{Meru2018}). The resolution is 450 and 1024 uniformly spaced cells in the radial and azimuthal direction, respectively. The planet is kept on a circular orbit whose orbital parameters are not allowed to vary (see \citealt{Meru2018} for a study on the effects of migration). The surface density profile follows $\Sigma \propto r^{-1}$; since we fix the planet orbital parameters the value of the normalisation constant is arbitrary. Finally, in this paper we use the $\alpha$ prescription of \citet{ShakuraSunyaev} for what concerns the viscosity and we assume $\alpha = 10^{-3}$.
The dynamics of the dust depends on the magnitude of the acceleration induced by gas drag, i.e. $a=\Delta v/t_s$, where $\Delta v$ is the relative velocity between the gas and the dust $t_s$ is the stopping time which depends on the grain properties and chiefly on the grain size. This is typically expressed in units of the local Keplerain time and called Stokes number $St=t_s/\Omega_K^{-1}$. To keep our simulations scale-free, each dust species in our simulation has a constant Stokes number. Once the scaling parameters of the disc have been chosen, it is then possible to convert the Stokes numbers to physical grain sizes as we explain in the next section. We use 5 dust sizes, with Stokes numbers (logarithmically spaced) ranging from $2 \times 10^{-3}$ to 0.2.
As mentioned before, in this paper the aspect ratio plays an important role. Therefore, we run a grid of models with aspect ratios of 0.025, 0.033, 0.04, 0.05, 0.066, 0.085 and 0.1 at the planet location; we discuss in section \ref{sssec:dischr} the link with the physical separation of the planet from the star. For any chosen normalisation, the aspect ratio in the disc varies in a power-law fashion with a flaring index of 0.25.
The goal of this paper is to study, for any planet location, the minimum gap opening planet mass. For this reason, for every different aspect ratio we run simulations with different planet masses, which for a star of 1 $M_\odot$ correspond to planet masses of 2.5, 4, 8, 12, 20, 60 and 120 $M_\oplus$. Note that we do not run the full range of planet masses for every aspect ratio, because in some cases it is already obvious that a planet of a given mass is able (or not) to open a gap. We then re-use the grid of simulations when considering different stellar masses, but note that, because in any simulation the planet-star mass ratio is fixed, this leads to different absolute planet masses.
\subsection{Radiative Transfer} \label{ssec:RT}
\paragraph*{Basic Disc Properties}\label{sssec:discprop}
In order to simulate images of discs around stars of different masses we construct disc models by adopting basic disc properties and scaling laws for the mass and radius of the disc with stellar mass.
All discs were assumed to have a fixed gas to dust ratio of $100:1$.
In these calculations we are interested in the disc temperature, which is mostly set by the small, well-coupled grains; therefore we do not take into account dust settling.
The unperturbed surface density profile of each disc was assumed to be azimuthally symmetric and inversely proportional to radius:
\begin{equation} \label{eqn:sigma}
\Sigma(r, \phi) = \Sigma(r) \propto \frac{1}{r},
\end{equation}
the same as the relationship adopted by the hydrodynamical simulations discussed in section \ref{ssec:hydrosim}
\subsubsection{Scaling Relations}\label{sssec:scalings}
The scaling relation between protoplanetary disc mass ($M_{d}$) and stellar mass ($M_{*}$) is well constrained by observations:
\begin{equation}
\label{eqn:Md_M*}
M_{d} \, \propto \, M_{*}^{1.3}.
\end{equation}
These observations (of the Chamaeleon I star forming region) correct for the variation in stellar luminosity, and therefore disc temperature, with stellar mass \citep{Pascucci2016}.
Several power laws describing the scaling relationship between protoplanetary disc radius ($R_{d}$) and disc mass have been proposed (for example, by \cite{Tazzari2017} and \cite{Tripathi2017}, among others).
For this work we adopt the scaling derived from temperature corrected data \citep{Andrews2018}.
Combined with equation \ref{eqn:Md_M*}, this gives:
\begin{equation}
R_{d} \, \propto \, M_{*}^{0.6}.
\end{equation}
Typical values for the outer disc radii and disc mass for a $1 \, \mathrm{M_{\sun}}$ mass star were adopted as the constants of proportionality, giving the following scaling relations.
\begin{equation}
\label{eqn:discmass_starmass}
M_{d} = \bigg(\frac{M_{*}}{\mathrm{M_{\sun}}} \bigg)^{1.3} \, 0.01 \, \mathrm{M_{\sun}}
\end{equation}
\begin{equation}
\label{eqn:discradius_starmass}
R_{d} = \bigg(\frac{M_{*}}{\mathrm{M_{\sun}}} \bigg)^{0.61} \, 100 \, \mathrm{AU}
\end{equation}
In reality, observations show some spread around these average values, but we neglect this to reduce the number of free parameters in our simulations. For what concerns the outer radius, this serves only as a guide to know how large the average disc (for a given stellar mass) is; the results we will present in the following sections contain the necessary information to know MGOPM at large radii in case the radius of an individual disc is larger than the average value. For what concerns the disc mass, it should come as a caveat that there are instead physical effects that we are neglecting: namely, the variation of the mid-plane disc temperature with the disc surface density (which scales as $\Sigma^{-1/4}$, see appendix \ref{sec:append_temp}) and the variation of the Stokes number with surface density (although the grain size might also depends on the disc surface density, e.g. \citealt{Birnstiel2012}).
\subsubsection{Pre-main Sequence Evolutionary Models}\label{sssec:stardiscparam}
The properties of the central star, specifically the mass, radius and effective temperature, are required by the radiative transfer code.
These properties were extracted from a series of pre-main sequence evolutionary tracks from \citet{Siess2000}, and are shown in table \ref{tab:stellarprop}.
For all discs considered in this work the age of the system was assumed to be $10^{6}$ years.
We comment on the choice of the pre-main sequence evolutionary tracks in section \ref{sec:robustness} and \ref{sec:spread}.
\begin{table}
\centering
\begin{tabular}{| c | c | c | c |}
\hline
$\mathbf{ M_{*} \, \big[ M_{\sun} \big] } $ & $\mathbf{ L_{*} \, \big[ L_{\sun} \big] } $ & $\mathbf{ R_{*} \, \big[ R_{\sun} \big] } $ & $\mathbf{T_{eff} \, \big[ K \big]}$ \\
\hline
$1.0$ & $2.33$ & $2.62$ & $4278$ \\
\hline
$0.7$ & $1.72$ & $2.54$ & $4024$ \\
\hline
$0.3$ & $0.69$ & $2.32$ & $3360$ \\
\hline
\end{tabular}
\caption{The stellar properties (mass $\big( M_{*} \big) $, luminosity $ \big( L_{*} \big)$, radius $ \big( R_{*} \big)$, and effective temperature $ \big( T_{eff} \big)$) used in the radiative transfer code RADMC-$3$D, obtained from the pre-main sequence evolutionary models at an age of $10^{6} \, \mathrm{years}$. }
\label{tab:stellarprop}
\end{table}
\subsubsection{Model Disc Aspect Ratio Profiles}\label{sssec:dischr}
The 3D temperature profiles of a series of systems comprising a central star and an empty (containing no planets) model disc were obtained using the default RADMC-$3$D model \verb+ppdisk+.
Mid-plane temperature profiles were calculated for stellar masses of $M_{*} = 0.3$, $0.7$ and $1.0 \, \mathrm{M_{\sun}}$, with the relevant stellar properties (luminosity, radius and effective temperature) obtained from the pre-main sequence evolutionary models discussed in section \ref{sssec:stardiscparam}.
A power law fit was applied to these temperature profiles and used to calculate the aspect ratio profiles for each system.
These temperature profiles are shown in figure \ref{fig:Temp_Scaling}.
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/T_r.pdf}
\caption{Mid-plane temperature profiles calculated for the three stellar masses using the RADMC$3$D \texttt{ppdisk} model, assuming the stellar parameters given in table \ref{tab:stellarprop}.
The profile produced by this model is shown as a solid line, and the power law fit to each profile is shown as a dashed line. }
\label{fig:Temp_Scaling}
\end{figure}
The aspect ratio is given by the ratio of pressure scale height, $h$, to radial position within the disc, $r$:
\begin{equation}
\label{eqn:aspectratio}
\frac{h}{r} = \frac{c_{s}}{v_{k}} = \sqrt{\frac{k_{B} T}{\mu m_{p}} \frac{r}{G M_{*}}},
\end{equation}
where $c_{s}$ is the isothermal sound speed, given by $c_{s} = \sqrt{\frac{k_{B} T}{\mu m_{p}}}$ where $T$ is the temperature in the disc and $\mu = 2.3$ is the mean molecular weight.
The locations in each disc at which the aspect ratio is equal to the values for which hydrodynamical simulation data were extracted and are shown in table \ref{tab:aspectratio}.
The aspect ratio at the location of the planet in the hydrodynamical simulation sets the semi-major axis of the planet in the radiative transfer model, and additionally sets the length scale of the data.
The relationship between temperature at a given position within the disc and stellar mass was investigated using linear regression and found to scale approximately as:
\begin{equation}
\frac{T}{\mathrm{K}} \propto \bigg( \frac{M_{*}}{\mathrm{M_{\odot}}} \bigg)^{0.15}
\label{eqn:T_M}
\end{equation}
We emphasise that the above scaling only holds in the case of a specific assumption about the stellar mass luminosity relation; for the particular case used here, $L \propto M_\ast^{1.07}$ when measured between $0.3$ and $1.0 M_\ast$.
\begin{table*}
\centering
\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
Semi-Major Axis [AU] & \multicolumn{7}{|c|}\textbf{{Aspect Ratio}}\\
\hline
\textbf{Stellar Mass} $\big[ \mathrm{\mathbf{M_{\sun}}} \big]$ & $\mathbf{0.025}$ & $\mathbf{0.033}$ & $\mathbf{0.04}$ & $\mathbf{0.05}$ & $\mathbf{0.066}$ & $\mathbf{0.085}$ & $\mathbf{0.1}$ \\
\hline
$ \mathbf{0.3} $ & - & $1.1$* & $1.9$ & $3.6$ & $8.4$ & $17.9$ & $29.5$ \\
\hline
$ \mathbf{0.7} $ & $1.3$* & $2.9$ & $5.2$ & $10.1$ & $23.2$ & $50.3$ & - \\
\hline
$ \mathbf{1.0} $ & $2.5$ & $5.3$ & $9.7$ & $19.8$ & $36.2$ & $75.7$ & - \\
\hline
\end{tabular}
\caption{The semi-major axis (in $\mathrm{AU}$) corresponding to the aspect ratio of the hydrodynamical simulations, for the three different stellar masses considered in this project. Situations where the aspect ratio profile of the disc does not encompass the simulation aspect ratio are marked by a dash (-).
The values marked by asterisks correspond to semi-major axes for which no radiative transfer simulations were run, due to their extreme proximity to the inner edges of the discs in question. }
\label{tab:aspectratio}
\end{table*}
\subsubsection{Conversion of Hydrodynamical Simulation Data} \label{sssec:conversion}
\paragraph*{Extrapolation}
We use a $3$D spherical coordinate system with $N_{r} = 256$ logarithmically spaced points, $N_{\theta} = 100$ points distributed linearly in the three intervals $[0, \frac{\pi}{3}]$, $[\frac{\pi}{3}, \frac{2 \pi}{3}]$, $[\frac{2 \pi}{3}, \pi]$, as $N_{\theta} = \{ 10,80,10 \} $, and $N_{\phi} = 200$ grid points linearly spaced from $0$ to $2 \pi$ in the azimuthal direction.
The first six radial cells of the hydrodynamical simulation data were excluded as they show artefacts caused by the inner boundary condition before the hydrodynamical simulation data is mapped onto this grid.
In some cases the simulation data does not cover the entire extent of the model disc and in these situations the surface density was extrapolated out to the edges of the disc.
\paragraph*{Interpolation}
The dust population was modeled as ten logarithmically spaced grain size bins between $a = 10^{-5} \, \mathrm{cm}$ and $0.1 \, \mathrm{cm}$, with a size distribution described by:
\begin{equation}
\label{eq:size_dist}
\frac{\mathrm{d}N}{\mathrm{d}a} \propto a^{-3.5}.
\end{equation}
The opacities of the dust grain populations are calculated from the grain size and the mass absorption coefficients.
The mass absorption coefficients used in this work were calculated using Mie theory, using the optical properties of astronomical silicates from \citet{Weingartner2001}.
In order to compute the surface density of each dust species we compute the Stokes number, $St$, from the normalised gas surface density. Assuming that all particles are in the Epstein regime, the Stokes number obeys:
\begin{equation}
\label{eq:Stokes}
St = t_{s} \Omega = \frac{\pi}{2} \frac{a \rho_{d}}{\Sigma_{g}}.
\end{equation}
We then use the result of the hydrodynamical simulations to interpolate the surface density linearly in terms of Stokes number.
If the calculated Stokes number is smaller than the smallest value for which there is a hydrodynamical simulation ($St = 2 \times 10^{-3}$) then the dust is assumed to follow the gas surface density distribution.
The largest grains in the model discs considered typically have $St \sim 0.0037$ at the inner edge, and $St \sim 0.4$ at their outer edge, so the hydrodynamical simulations provide sufficient coverage for the surface density of these particles to be constructed.
The mass in each dust grain size bin was scaled according to equation \ref{eq:size_dist}.
The $3$D density profiles are calculated from the surface density as:
\begin{equation}
\label{eq:3Ddens}
\rho (r) = \frac{\Sigma (r, \phi) }{\sqrt{2 \pi} H(r)} \mathrm{e}^{\frac{-z^{2}}{2 H(r)^{2}}},
\end{equation}
where $z = r \cos(\theta)$ is vertical height within the disc and $H (r)$ is the pressure scale height as a function of position in the disc, calculated as:
\begin{equation}
\label{eqn:H}
H (r) = h_{0} \bigg( \frac{r}{r_{0}} \bigg)^{0.25} r,
\end{equation}
where $h_{0}$ is the reference aspect ratio taken at $r_{0}$, the planet location.
\subsubsection{RADMC-$3$D Parameters}
The radiative transfer simulations and image generation were carried out using RADMC-$3$D.
All images were calculated at a wavelength of $850 \, \mathrm{\mu m}$, equivalently a frequency of $353 \, \mathrm{GHz}$ which corresponds to ALMA band 7.
The radiative transfer simulations used $2 \times 10^{7}$ photons, and $1 \times 10^{7}$ photons were used for the image generation.
We find that this number of photons is sufficiently high to show little noise in the resulting images.
\subsection{Gap Analysis} \label{sssec:gapanalysis}
There have been several different methods for characterising gap properties proposed in the literature (\citet{DeJuanOvelar2013} and \citet{Akiyama2016}).
In this work we modify the definition for depth described by \citet{Rosotti2016}, as described here.
The data from the simulated images was averaged azimuthally, to give a radial surface brightness profile, $S_{\nu}(r)$.
All discs were assumed to be at a distance of $140 \, \mathrm{pc}$ and face on.
If the surface brightness profile shows an obvious gap feature then there is no need for further analysis.
In some cases there is a less distinctive feature visible and therefore a more robust definition of whether a gap exists is needed.
In order to determine the detectability of a gap a linear fit in log-log space was applied to a region of the surface brightness profile near the feature to calculate a background surface brightness profile, $S_{\nu,b}(r)$.
The depth of a potential gap is defined to be:
\begin{equation}
\label{eqn:depth}
Depth = \bigg{|} \frac{S_{\nu}(r_\mathrm{gap}) - S_{\nu, b}(r_\mathrm{gap})}{S_{\nu, b}(r_\mathrm{gap})} \bigg{|}
\end{equation}
where $r_\mathrm{gap}$ is the location of the gap (where the difference between the real surface brightness profile and the background profile was greatest).
We define gaps as detectable if $ Depth \geq 0.1$, i.e. if the decrease in surface brightness is greater than $10\%$.
This method is illustrated in figure \ref{fig:20_066_Gap_properties}, for the case of a system containing a $20 \, \mathrm{M_{\earth}}$ mass planet in the disc around a $1 \, \mathrm{M_{\sun}}$ mass star at semi-major axis of $36.2 \, \mathrm{AU}$.
The depth of the gap is marked and the absolute change in surface brightness was found to be greater than the limit adopted, therefore the feature was defined as a gap.
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/Depth_Analysis.pdf}
\caption{The top plot shows the azimuthally averaged surface brightness profile extracted from the simulation of a $20 \, \mathrm{M_{\earth}}$ mass planet at a semi-major axis of $36.2 \, \mathrm{AU}$ in the disc around a $1 \, \mathrm{M_{\sun}}$ star.
There is a feature visible at approximately $40 \, \mathrm{AU}$ but it is not definitively a gap.
The bottom plot shows the results of the analysis carried out on the normalised surface brightness profile, with the depth of the gap marked by a black arrow.
This corresponds to a depth of $\sim 13 \% $ and so the gap is defined as detectable. }
\label{fig:20_066_Gap_properties}
\end{figure}
\subsection{Simulated observations}\label{ssec:CASA}
We use images produced by the radiative transfer code to generate simulated observations of the system, using the Common Astronomy Software Application\footnote{\url{http://casa.nrao.edu/index.shtml}} (CASA) v5.1.2-4.
The \verb+simobserve+ task was used to simulate the observed visibilities, from which the simulated observations were produced using the \verb+simanalyse+ task.
The full $12 \, \mathrm{m}$ array was used in configuration $24$, which gave a resolution at $850 \, \mathrm{\mu m}$ of approximately $0.025 \arcsec$.
We assume an integration time of $6 \, \mathrm{hours}$, and use the full bandwidth of $7.5 \, \mathrm{GHz}$.
Noise was introduced using the \verb+tsym-atm+ parameter, with the value for the precipitable water vapour, $0.913 \, \mathrm{mm}$, representative of typical observing conditions.\footnote{\url{https://almascience.eso.org/proposing/sensitivity-calculator}}
As before, all discs are assumed to be at a distance of $140 \, \mathrm{pc}$.
A gap was defined as detectable in the simulated images in the same way as for radiative transfer images, described in section \ref{sssec:gapanalysis}.
\section{Results}\label{sec:results}
\subsection{A Single Representative System}
The methodology described in section \ref{sec:methods} is illustrated here for the case of a $20 \, \mathrm{M_{\earth}}$ mass planet around a $ 1 \, \mathrm{M_{\sun}}$ star at the location where the aspect ratio is $0.05$, which corresponds to a semi-major axis of $19.8 \, \mathrm{AU}$.
The FARGO surface density data for this simulation is shown in figure \ref{fig:FARGO_Sig}.
From this figure it can be seen that the dust surface density profiles are largely azimuthally symmetric, with the exception of a thin spiral feature that is more pronounced in the gas and dust species with low Stokes numbers.
Previous work by \cite{2015MNRAS.451.1147J} has suggested that, even for extremely massive planets, spiral features may be challenging to observe in continuum images as they are narrow and have low contrast.
As a consequence, we use azimuthally averaged profiles of both surface density and image surface brightness in the remainder of this work without further detailed consideration of any asymmetric features.
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{plots/FARGO_Sigma.png}
\caption{Surface density plots of the FARGO simulation data for an example disc, in this case for a $20 \, \mathrm{M_{\earth}}$ mass planet for a $1 \, \mathrm{M_{\sun}}$ mass star, and an aspect ratio at unit radius of $0.05$.
The surface density profiles for the gas and the five different Stokes numbers are shown.
The planet opens a shallow gap in the gas, which is deeper and wider for increasing Stokes number.
Spiral features are also visible, and are more pronounced in the gas and dust species with low Stokes numbers.
In the dust species with the largest Stokes number the planet opens a hole, which extends from the location of the planet to the inner edge of the disc (at half unit radius). }
\label{fig:FARGO_Sig}
\end{figure*}
The image produced by RADMC-$3$D is shown in figure \ref{fig:Image}, and the corresponding simulated observation is shown in figure \ref{fig:Sim_Obs}.
The gap created by the planet can clearly be seen, as can the bright ring produced outside the location of the planet.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{plots/image.png}
\caption{The radiative transfer image of a model disc around a $1 \, \mathrm{M_{\sun}}$ star containing a $20 M_{\earth}$ mass planet at an aspect ratio of $0.05$, corresponding to a semi-major axis of $19.8 \, \mathrm{AU}$. }
\label{fig:Image}
\end{figure}
\begin{figure*}
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=1.\textwidth]{plots/Simulated_Observation.png}
\caption{Simulated observation in ALMA band $7$, at $850 \, \mathrm{\mu m}$, of a model disc around a $1 \, \mathrm{M_{\sun}}$ star containing a $20 M_{\earth}$ mass planet at an aspect ratio of $0.05$, and therefore a semi-major axis of $19.8 \, \mathrm{AU}$.
The disc is assumed to be face on at a distance of $140 \, \mathrm{pc}$.
It should be noted that the colour scale in this image is not the same as that used in figure \ref{fig:Image}}
\label{fig:Sim_Obs}
\end{minipage}\hfill
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.83\textwidth]{pdfplots/CASA_Surface_Brightness.pdf}
\caption{Azimuthally averaged surface brightness profiles produced from the radiative transfer image and simulated observation of a model disc around a $1 \, \mathrm{M_{\sun}}$ star containing a $20 M_{\earth}$ mass planet at an aspect ratio of $0.05$, and therefore a semi-major axis of $19.8 \, \mathrm{AU}$.}
\label{fig:casa_Sb}
\end{minipage}
\end{figure*}
The corresponding azimuthally averaged surface brightness profiles extracted from these images are shown in figure \ref{fig:casa_Sb}.
Both show a clear decrease in surface brightness at the location of the planet below the background level. We can thus conclude that this planet is gap opening without the need for further analysis. The thermal noise can be more clearly seen in the outer portion of the disc, where the surface brightness is low.
\subsection{Summary of $1 \, \mathrm{M_{\sun}}$ Mass Star Case}\label{ssec:1Ms}
For a disc around a $1 \, \mathrm{M_{\sun}}$ mass star (for which $R_{d} = 100 \, \mathrm{AU}$), images at $850 \, \mathrm{\mu m}$ were simulated for discs containing planets with masses of $2.5$, $4$, $8$, $12$, $20$, $60$ and $120 \, \mathrm{M_{\earth}}$ at six orbital radii between $2.5$ and $75.7 \, \mathrm{AU}$.
Radiative transfer images and simulated observations were generated and analysed as described in sections \ref{ssec:RT}, \ref{sssec:gapanalysis} and \ref{ssec:CASA}.
\paragraph*{Radiative Transfer Images}
For a given semi-major axis more massive planets create more obvious features within the disc.
This is illustrated in figure \ref{fig:hr05_Sb}, which shows the azimuthally averaged surface brightness profiles for three discs containing planets of different masses at $19.8 \, \mathrm{AU}$.
The most massive planet, with a mass of $20 \, \mathrm{M_{\earth}}$, produces a large decrease in surface brightness, as well as a bright ring outside the location of the planet.
These same features can be seen but are much less pronounced for the smaller planet, with a mass of $12 \, \mathrm{M_{\earth}}$, and no obvious feature can be seen at all for the least massive, $8 \, \mathrm{M_{\earth}}$ mass planet.
For a planet of given mass, the features produced are more prominent for planets located at smaller semi-major axes, as illustrated in figure \ref{fig:12Me_Sb}.
This is due to the lower aspect ratio at smaller semi-major axes, for which a lower planet mass is required to open a gap.
This plot shows the azimuthally averaged surface brightness profiles produced from the simulations of three discs containing $12 \, \mathrm{M_{\earth}}$ mass planets at different semi-major axes.
A deep gap and prominent bright ring is produced by the planet at $9.7 \, \mathrm{AU}$, while a smaller gap is produced by the planet at $19.8 \, \mathrm{AU}$.
The planet furthest out, at $36.2 \, \mathrm{AU}$, produces no visible feature at all.
\begin{figure*}
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.85\textwidth]{pdfplots/hr05_Surface_Brightness.pdf}
\caption{Azimuthally averaged surface brightness profiles showing the effect of varying the planet mass for a fixed semi-major axis.
These profiles were generated from the images produced by RADMC-$3$D of three model discs around a $1 \, \mathrm{M_{\sun}}$ star containing three different mass planets ($8$, $12$ and $20 \, \mathrm{M_{\earth}}$) at an aspect ratio of $0.05$, and therefore a semi-major axis of $19.8 \, \mathrm{AU}$. More massive planets create more notable feature in the surface brightness profile.}
\label{fig:hr05_Sb}
\end{minipage}\hfill
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.85\textwidth]{pdfplots/Mp12_Surface_Brightness.pdf}
\caption{ Illustrative plot showing the effect of varying the planet semi-major axis for a fixed planet mass. The surface brightness profiles shown are for the case of a $12 \, \mathrm{M_{\earth}}$ mass planet around a $1 \, \mathrm{M_{\sun}}$ star, but at three different locations. The perturbation induced by the planet becomes stronger at smaller radii, as a result of the lower disc aspect ratio. }
\label{fig:12Me_Sb}
\end{minipage}
\end{figure*}
\paragraph*{Simulated Observations}
For each case in which a detectable gap was opened a simulated ALMA band $7$ observation was generated, as described in section \ref{ssec:CASA}.
From these we recover the same trends discussed in the images.
\paragraph*{Results}
The results for all simulations performed for the $1 \, \mathrm{M_{\sun}}$ case are summarised in figure \ref{fig:Summary_10}, which distinguishes between three different results: $1)$ no gap is present in the radiative transfer image, $2)$ a gap is detected in the radiative transfer image but is not visible in the simulated observation, $3)$ a gap is detectable in both the radiative transfer image and the simulated observation.
This summary figure shows that gaps are opened for small semi-major axes and/or large planet mass.
We find that the noise introduced by simulating an ALMA observation has little effect on the detectability of gaps.
A more important effect is the finite resolution of the simulated observations.
For planets with semi-major axes of $5.3 \, \mathrm{AU}$ or less, the gap that was defined as detectable in the radiative transfer image is no longer visible in the data extracted from the simulated observation.
The ALMA configuration we used gave an angular resolution of $0.025 \arcsec$, which at a distance of $140 \, \mathrm{pc}$ is approximately $ 5 \, \mathrm{AU}$, explaining this result.
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/Summary_1Ms.pdf}
\caption{A summary of the detectability of gaps created by planets within the disc around a $1 \, \mathrm{M_{\sun}}$ star.
A red cross indicates that no detectable gap was produced.
A purple cross indicates that a detectable gap was visible in the radiative transfer image, but not in the simulated observation.
A green circle indicates a detectable gap that was visible in both the radiative transfer image and the simulated observation.
The grey shaded region indicates the $M_{min} \propto r^{0.75}$ fit supported by these results, which is in line with the theoretical arguments we present in section \ref{ssec:expected_scalings}. }
\label{fig:Summary_10}
\end{figure}
\subsection{Changing Stellar Mass}
For a disc around a $ 0.7\, \mathrm{M_{\sun}}$ mass star ($R_{d} = 81 \, \mathrm{AU}$), observations at $850 \, \mathrm{\mu m}$ were simulated for discs containing planets at six orbital radii between $2.9$ and $50.3 \, \mathrm{AU}$.
The images generated by the radiative transfer code and the simulated observations were analysed as discussed in sections \ref{sssec:gapanalysis} and \ref{ssec:CASA} with the results shown in figure \ref{fig:Summary_07}. \\
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/Summary_07Ms.pdf}
\caption{A summary of the detectability of features created by planets within the disc around a $0.7 \, \mathrm{M_{\sun}}$ mass star. See figure \ref{fig:Summary_10} for an explanation of colours and symbols.}
\label{fig:Summary_07}
\end{figure}
For a disc around a $ 0.3\, \mathrm{M_{\sun}}$ mass star ($R_{d} = 48 \, \mathrm{AU}$), observations were simulated for discs containing planets at orbital radii between $1.9$ and $29.5 \, \mathrm{AU}$.
The images generated were analysed as discussed in sections \ref{sssec:gapanalysis} and \ref{ssec:CASA} with the results are shown in figure \ref{fig:Summary_03}.
In both of these systems, as for the $1 \, \mathrm{M_{\sun}}$ case, the gaps at small semi-major axes visible in the radiative transfer images are not visible in the simulated observations.
Inspection of these simulated images suggests qualitatively that it is harder to open gaps in discs around lower mass stars.
For example, the minimum planet mass required to produce a visible gap at $\sim 10 \, \mathrm{AU}$ in the disc around a $1 \, \mathrm{M_{\sun}}$ mass star is approximately $5 - 6 \, \mathrm{M_{\earth}}$, but in the disc around a $0.3 \, \mathrm{M_{\sun}}$ mass star it is approximately $8 \, \mathrm{M_{\earth}}$.
We discuss this result in section \ref{ssec:comparison}.
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/Summary_03Ms.pdf}
\caption{A summary of the detectability of features created by planets within the disc around a $0.3 \, \mathrm{M_{\sun}}$ mass star. See figure \ref{fig:Summary_10} for an explanation of colours and symbols.}
\label{fig:Summary_03}
\end{figure}
\section{Dependence on the stellar mass}
\label{sec:discussion}
We now investigate the variation in the MGOPM with, most importantly, the stellar mass, and also with semi-major axis within the disc. To do this we assume that the dependence on these two parameters is a power law and we fit the results presented in sections \ref{sec:results} to derive the exponents. We also present analytic scaling arguments and compare these to our results. We then discuss the robustness of our results, considering especially the effect of the luminosity spread, and finally discuss the observational implications.
\subsection{Results of the numerical simulations} \label{ssec:comparison}
The exponent of the power law relating MGOPM and semi-major axis is compatible with a value of $0.75$, and therefore we use this value in the following analysis. We will show in section \ref{ssec:expected_scalings} that this value is in agreement with theoretical arguments.
Figures \ref{fig:Summary_10}, \ref{fig:Summary_07} and \ref{fig:Summary_03} show the fits for the three different stellar masses considered by this work. We also show as a gray shaded region the allowed range of normalisation constants that are in agreement with our results. In all cases this power law provides a plausible fit.
The range of allowed normalisation constants is small for the $0.3 \, \mathrm{M_{\sun}}$ mass star, and larger for the $0.7$ and $1 \, \mathrm{M_{\sun}}$ mass stars.
In general the relation between MGOPM and semi major axis for a given stellar mass can be expressed as:
\begin{equation}
\begin{aligned}
& & & & 1.41 \leq \, &A_{0.3 \mathrm{M_{\sun}}} \leq 1.47 \\
\frac{M_{pl,m}}{\mathrm{M_{\earth}}} &= A_{M_{*}} \bigg( \frac{r_{pl}}{\mathrm{AU}} \bigg)^{0.75} & &\mathrm{where} & 0.98 \leq \, &A_{0.7 \mathrm{M_{\sun}}} \leq 1.31 \\
& & & & 0.85 \leq \, &A_{1.0 \mathrm{M_{\sun}}} \leq 1.2
\label{eqn:minplanetmass}
\end{aligned}
\end{equation}
Addressing the main motivation behind this paper, from these fits it can be seen that $A_{M_{*}}$ \textit{increases} with decreasing stellar mass. This means that the fact that these discs are geometrically thicker is the dominant effect. We will show this more formally in the next section \ref{ssec:expected_scalings}. The scaling of the MGOPM with stellar mass is consistent with:
\begin{equation}
\begin{aligned}
A_{M_{*}} &\propto M_{*}^{\alpha} & &\mathrm{where} & -0.45 \leq \, &\alpha \leq -0.14
\end{aligned}
\end{equation}
In summary, the MGOPM is larger at greater semi-major axes for fixed stellar mass, and at fixed semi-major axes is larger for lower mass stars.
Using a representative value of $A_{M_{*}} \approx 1 \, M_{*}^{-0.33}$, the MGOPM can be expressed as a function of stellar mass and planet semi-major axis as:
\begin{equation}
\frac{M_{pl,m}}{\mathrm{M_{\earth}}} \approx 1 \,\left( \frac{M_{*}}{\mathrm{M_{\sun}}} \right)^{-0.33} \left( \frac{r_{pl}}{\mathrm{AU}} \right)^{0.75}
\label{eqn:summary_Mmin}
\end{equation}
The relation given in equation \ref{eqn:summary_Mmin} is illustrated in figure \ref{fig:summary}. The plot also shows the resolution of the simulated observations, $\sim 5 \, \mathrm{AU}$. In addition, while this paper focuses on ALMA, for reference we plot also the resolution that may be achieved using the ngVLA\footnote{https://science.nrao.edu/futures/ngvla} at $3 \, \mathrm{mm}$ ($5$ milliarcsec, which at a distance of $140 \, \mathrm{pc}$ corresponds to a distance of $0.7 \, \mathrm{AU}$). See \citet{Ricci2018} for a dedicated study of ngVLA capabilities in detecting planet-formed gaps.
It can be seen how, at least on average, around a lower mass star a smaller part of the disc can be resolved by ALMA (e.g., for a 0.3 $M_\odot$ star the average outer radius is 30 au, i.e. the dynamical range in radius is a factor of 6). In this region, the MGOPM is approximately that of Neptune (15-20 $M_\oplus$). Around solar mass stars, the negative scaling with stellar mass means that the sensitivity improves, and the detection limit becomes $\sim 5 M_\oplus$.
\begin{figure}
\centering
\includegraphics[width = 0.4\textwidth]{pdfplots/summary.pdf}
\caption{A summary of the minimum gap opening planet mass (MGOPM) as a function of planet semi-major axis and stellar mass. The black line is the average size of a disc for the given stellar mass.
The estimated resolution of the ALMA configuration used ($\sim 5 \, \mathrm{AU}$) is shown as a red dashed line.
The estimated resolution of $3 \, \mathrm{mm}$ observations using the ngVLA ($\sim 0.7 \, \mathrm{AU}$) is shown as a blue dashed line. It can be seen that the MGOPM increases with semi major axis and decreases with stellar mass.
}
\label{fig:summary}
\end{figure}
\subsection{Analytical expectations for the MGOPM scaling relations} \label{ssec:expected_scalings}
While the MGOPM we consider in this paper is quantitatively different from the conventional gap opening criterion in the gas (\citealt{Crida2006}; see e.g. \citealt{2014ApJ...782...88F,2015MNRAS.448..994K} for recent developments), the formation of a dust gap still requires a perturbation in the gas. For this reason, although the amplitude of the perturbation is different, we can assume that the two criteria should scale in the same way with the disc parameters. A similar assumption was also made by \citet{2017MNRAS.469.1932D} following the results of \citet{Rosotti2016}, see for example the blue area in their Figure 2. In this section we validate this assumption, by showing that it accounts for the scalings found in the simulations we have run. For what concerns the normalization, instead, we must rely on numerical simulations.
Proto-planetary discs are characterised by relatively low values of the viscosity and high values of the aspect ratio. Therefore, it is safe to assume that the most stringent criterion for gap opening is the pressure criterion, rather than the viscous one \footnote{This is different in the regime of low viscosity $\alpha \lesssim 10^{-4}$, where the \citet{Crida2006} criterion is no longer applicable \citep[e.g.,][]{2013ApJ...769...41D}.}. For this reason in the rest of the section we will consider only the former (see also discussion in \citealt{Rosotti2016}). Following standard arguments \citep{Lin1993}, an estimate of the expected scaling of the MGOPM $(M_{pl,m})$ with semi-major axis and stellar mass can be obtained by equating the Hill radius,
\begin{equation}
r_{H}= r \Big( \frac{M_{pl,m}}{3M_{*}} \Big)^{1/3},
\label{eqn:hill_r}
\end{equation}
of the planet with the pressure scale height $(H)$. This gives the scaling condition that:
\begin{equation}
\frac{M_{pl,m}}{M_{*}} \propto \bigg( \frac{H}{r} \bigg)^{3}.
\label{eqn:Mmin}
\end{equation}
For fixed stellar mass, recalling the expression for the aspect ratio given by equation \ref{eqn:H}, this gives:
\begin{equation}
\frac{M_{pl,m}}{M_{\earth}} \propto \bigg( \frac{r}{\mathrm{AU}} \bigg)^{0.75}.
\label{eqn:Mmin_r}
\end{equation}
And for a given position in the disc, using equation \ref{eqn:T_M}, the aspect ratio is expected to scale as:
\begin{equation*}
\frac{H}{r} \, \propto \, \sqrt{\frac{T(M_{*})}{\mathrm{K}} \bigg( \frac{M_{*}}{\mathrm{M_{\sun}}}\bigg)^{-1}}
\, \propto \, \bigg( \frac{M_{*}}{\mathrm{M_{\sun}}}\bigg)^{-0.425}
\end{equation*}
So, using equation \ref{eqn:Mmin}, the MGOPM, for fixed position within the disc is expected to scale with stellar mass as:
\begin{equation}
\label{eq:Mmin_Ms}
\frac{M_{pl,m}}{\mathrm{M_{\oplus}}} \, \propto \, \bigg( \frac{h}{r_{pl}} \bigg)^{3} \bigg( \frac{M_{*}}{\mathrm{M_{\sun}}}\bigg) \propto \bigg( \frac{M_{*}}{\mathrm{M_{\sun}}} \bigg)^{-0.275}
\end{equation}
Combining these equations \ref{eqn:Mmin_r} and \ref{eq:Mmin_Ms} gives an expression for the expected scaling of the MGOPM with both planet semi-major axis and stellar mass:
\begin{equation}
\frac{M_{pl,m}}{\mathrm{M_{\oplus}}} \, \propto \, \bigg(\frac{r}{\mathrm{AU}}\bigg)^{0.75} \bigg(\frac{M_{*}}{\mathrm{M_{\sun}}} \bigg)^{-0.275}
\label{eqn:Mmin_comb}
\end{equation}
Comparing equations \ref{eqn:minplanetmass} and \ref{eqn:Mmin_comb}, we conclude that there is excellent agreement between the results of our simulations and the analytical arguments. As mentioned, in our analysis we have assumed that the exponent of the scaling of the MGOPM with radius is $0.75$, because it gives a good description of the results. The range of exponent values describing the scaling with stellar mass are consistent with the value of $-0.275$ predicted in section \ref{ssec:expected_scalings}.
\subsection{Robustness of the results to changes in the stellar mass luminosity relationship} \label{sec:robustness}
We have demonstrated that, for a particular choice of the relationship
between stellar mass and luminosity as detailed in Table \ref{tab:stellarprop} (based on the
pre-main sequence evolutionary tracks of \citet{Siess2000} at an age of
1Myr), the minimum mass of planets that can be detected at a given radius
in the disc is a decreasing function of stellar mass (equation \ref{eqn:Mmin_comb}).
Thus planet detection via structure in submm images is apparently
harder in the case of lower mass stars.
This contrasts strongly with the situation encountered in the case of other
planet detection methods. For example, in the case of radial velocity
methods, the detectable planet mass scales linearly with stellar mass,
whereas for rocky planets (i.e., with a roughly constant density) detected by the transit method it scales as $M_\ast^3$.
We now consider if there is any plausible stellar mass luminosity relation
that could result in a positive dependence of minimum detectable planet mass on stellar mass. Let us assume that, for some mass-luminosity relation, the
scaling of temperature with stellar mass is $T \propto M_\ast^a$ (by analogy
with equation (\ref{eqn:T_M})). Proceeding as in the previous section we obtain that
\begin{equation}
M_{pl,m} \propto M_\ast^{(3a-1)/2}.
\label{eq:mpl_mstar}
\end{equation}
implying that a positive dependence of detectable planet mass on stellar mass would correspond to $a > 0.33$. The temperature (at a given radius) mostly depends on the stellar luminosity, but can also depend on the stellar mass. Therefore, we can parametrise this dependence with the form $T \propto L^b M^c$. If we consider a mass luminosity relation of the form $L \propto M^d$, we obtain
\begin{equation}
a=c+bd
\end{equation}
To measure the values of the parameters $b$ and $c$, we have run another grid of radiative transfer simulations of the same type of section \ref{sssec:dischr}, i.e. discs that contain no planets, in which we explored the effect of varying the luminosity of the central star while keeping the other parameters fixed (note the difference from the calculations in section \ref{sssec:stardiscparam}, where we also varied the stellar and disc properties). From these simulations, we find that for a fixed location within the disc, the disc temperature $T_{\rm disc}$ scales with stellar luminosity approximately as
\begin{equation}
\frac{T_{\rm disc}}{\mathrm{K}} \propto \bigg( \frac{L_{*}}{\mathrm{L_{\odot}}} \bigg)^{0.2};
\label{eqn:Tdisk_L}
\end{equation}
we show the results of our calculations in figure \ref{fig:midT_fixedr}. Therefore, $b=0.2$. The exponent is slightly flatter than the 1/4 one would naively expect from energy argument; we interpret this as due to the dependence of the Planck mean opacity with temperature (see appendix \ref{sec:append_temp}).
For what concerns the values of $c$, from the radiative transfer grid we deduce a value of $c \sim 0.04$; in an alternative way, we can deduce the value of $c = -0.06$ from the fact that $a=0.15$ (equation (\ref{eqn:T_M})) for the case $d=1.07$ (appropriate to the Siess et al isochrone at 1 Myr). This means that the explicit dependence on the stellar mass is a small effect, confirming that the temperature is mainly set by the stellar luminosity.
Neglecting $c$, we obtain that a positive dependence of minimum detectable planet mass on stellar mass ($a > 0.33$)
requires $d>1.65$; alternatively, the limiting value is $1.45$ for the case of $c=0.04$. This value is higher than that predicted by the \citet{Siess2000} tracks, in accord with the results of the previous sections. However, for another widely used set of pre-main sequence tracks, the models by \citet{Baraffe2015}, the temperature-luminosity relation is in general steeper; for reference, we find a value of 1.47 at 1 Myr. This value corresponds exactly to the limiting case we identified before. To inspect this case more closely, we extracted the values of the disc temperature from the radiative transfer grid at the stellar luminosities predicted by the \citet{Baraffe2015} tracks. In line with the arguments above, in this case we find a value of $a=0.32$, implying that for the \citet{Baraffe2015} tracks the dependence of MGOPM with stellar mass is essentially flat (see \autoref{eq:mpl_mstar})
\begin{figure}
\centering
\includegraphics[width = 0.45\textwidth]{pdfplots/T_L_fixed_r.pdf}
\caption{The variation in mid-plane temperature at fixed radial position with stellar luminosity, shown for the three stellar masses and four different locations. From these radiative transfer calculations we deduce a scaling $T_{\rm disc} \propto L_{*}^{0.2}$, slightly flatter than the $L_{*}^{1/4}$ one might naively expect from energy arguments, as a result of the dependence of the Planck mean opacity with temperature (see appendix \ref{sec:append_temp}).}
\label{fig:midT_fixedr}
\end{figure}
We thus conclude that, depending on the stellar track used, MGOPM could become flat with stellar mass. Nevertheless, we can also conclude that MGOPM \textit{does not improve} towards low stellar masses, in contrast to other planet detection techniques. Therefore, the robust result of this paper is that there is no benefit in terms of planet mass sensitivity when observing discs around lower mass stars.
\subsection{Effect of the luminosity spread} \label{sec:spread}
Up to now we only considered a single luminosity for each stellar mass. In reality, it is well known that in star forming regions stars of the same mass exhibit a wide range of luminosities, a phenomenon colloquially called "luminosity spread", possibly due to the stellar accretion history \citep{2011ApJ...738..140H,2012ApJ...756..118B,2018MNRAS.474.1176J} or an age spread. Since the stellar luminosity affects the disc temperature and therefore MGOPM, we need to quantify how this effect changes the conclusions of this paper.
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{pdfplots/MGOPM_Ls_apl.pdf}
\caption{A summary of the minimum gap opening planet mass (MGOPM) as a function of planet semi-major axis and stellar luminosity, for the three values of stellar mass considered.
The luminosity values used for each stellar mass are shown by a red solid line; we show with the red dashed lines the typical range of variation (2-$\sigma$, i.e. 95 per cent of the sources) encountered in observations.
The average disc size for each mass is shown by a black dashed line.
The estimated resolution of the ALMA configuration used ($\sim 5 \, \mathrm{AU}$) is shown as a black dotted line.
It can be seen that the MGOPM increases with both semi major axis and stellar luminosity.
}
\label{fig:MGOPM_Ls_apl}
\end{figure*}
To this end, we consider equation \ref{eqn:Mmin} for fixed stellar mass and semi-major axis; we obtain that:
\begin{equation}
\frac{M_{pl,m}}{M_{\earth}} \propto \bigg( \frac{T_{\rm disc}}{\mathrm{K}} \bigg)^{1.5}.
\label{eqn:Mmin_T}
\end{equation}
In the previous section we have shown that $T_{\rm disc} \propto L_\ast^{0.2}$. Combining these gives us an expected power law index for the scaling of the MGOPM with stellar luminosity of $0.3$, and so we expect the scaling with both planet semi-major axis and stellar luminosity to be:
\begin{equation}
\frac{M_{pl,m}}{\mathrm{M_{\oplus}}} \, \propto \, \bigg(\frac{r}{\mathrm{AU}}\bigg)^{0.75} \bigg(\frac{L_{*}}{\mathrm{L_{\sun}}} \bigg)^{0.3}.
\label{eqn:Mmin_Ls_apl}
\end{equation}
This predicted MGOPM for three values of the stellar mass are shown in figure \ref{fig:MGOPM_Ls_apl}.
To quantify the importance of the luminosity spread, the last ingredient we need is an estimate of how much the stellar luminosity can vary for a given stellar mass. To quantify this, we have collected the samples presented in the recent X-shooter spectral surveys of Chameleon I \citep{Manara2017} and Lupus \citep{Alcala2017}. We extracted from the two samples the luminosity and stellar mass (note that \citealt{Alcala2017} reports three different values for the stellar mass depending on the model used; here we use only the value derived from the models of \citealt{Baraffe2015} since this is the only one employed by \citealt{Manara2017} and, as discussed by these authors, there is little difference between the models in deriving the stellar mass) and then fitted them with a power-law using the widely-used package \texttt{linmix} \citep{Kelly2007}. The result of the fit reports a 1-$\sigma$ spread of 0.36 dex. To show this on figure \ref{fig:MGOPM_Ls_apl}, we have indicated with the dashed lines the 2-$\sigma$ spread around the average value.
In itself, the effect of the luminosity spread can be significant: as an illustrative example, a 0.3 $M_\odot$ star with a luminosity that is 2-$\sigma$ below the average has a similar MGOPM (or even smaller) to a 1 $M_\odot$ star with an average luminosity. At the same time, we point out that the luminosity is a quantity that can easily be estimated from optical observations and allows one to correct the estimate of MGOPM for a specific disc.
\subsection{Observational implications}
In this paper we have presented scaling relations of the MGOPM with stellar mass, luminosity and planet orbital radius. These relations, summarised in figure \ref{fig:MGOPM_Ls_apl}, can be readily used when interpreting high-resolution imaging of discs to set a lower limit on the masses of the putative planets responsible for annular structures.
As a caveat, in this paper we employed a value of the viscous parameter $\alpha = 10^{-3}$. This value was chosen because it is lower than the current upper limits set by direct measurements of the turbulence \citep[e.g.,][]{2018ApJ...856..117F}, but is still in a reasonable range to account for the observed accretion rates onto young stars without invoking other mechanisms for angular momentum transfer, such as disc winds. The precise value of the MGOPM will depend on the value of $\alpha$, but in general we do not expect the trends that we present here to depend on the value of $\alpha$. Another caveat is that we neglected the effect of dust back-reaction on the gas \citep[e.g.,][]{2018ApJ...868...48K,2018ApJ...854..153W,2019arXiv190910526D}, though this is unlikely to change MGOPM since it becomes relevant only in presence of a strong dust accumulation. This requires a planet well above gap-opening mass, at least for the Stokes numbers we simulate here; the situation might change in presence of significantly larger Stokes numbers.
Consideration of how the MGOPM varies as a function of stellar mass is
of considerable interest for the interpretation of the incidence of structure in submm disc images in different stellar mass ranges . From radial velocity surveys it is evident that
giant planets (loosely defined as being more massive than Neptune) at distances up to several au are rarer around lower mass stars \citep{2008PASP..120..531C,2010PASP..122..905J,2013A&A...549A.109B,2014ApJ...791...91C,2015ARA&A..53..409W}\footnote{There is indication \citep{2013A&A...549A.109B,2013ApJ...767...95D,2015ApJ...814..130M}, both from transit and radial velocity surveys, that super-Earths in the innermost au are in fact \textit{more} abundant around low mass stars than around solar.}. While ALMA surveys of disc sub-structure do not overlap in spatial scales with those probed by radial velocity surveys of mature planet populations, it is nevertheless of interest to discover if the incidence of young planets at large radii is also lower in
low mass stars than in higher mass counterparts.
Our study shows that using canonical relationships between stellar mass and luminosity, the opening of gaps around low mass stars is harder, or just as difficult. We note that this depends on the distribution of masses and luminosities present in the population; figure \ref{fig:MGOPM_Ls_apl} also shows, based on data from Lupus and Chamaeleon, that the difference in MGOPM due to the luminosity spread can cancel out the effect due to the stellar mass. Thus future studies of the relative incidence of discs with substructure as a function of stellar mass need to be interpreted with care.
With knowledge of the stellar luminosity on a source by source basis, Figure \ref{fig:MGOPM_Ls_apl} can be used to assess whether planet formation at young ages and large radii is indeed disfavoured in the vicinity of lower mass stars.
At the moment, sufficient high resolution imaging data do not exist to make this test; most of the observations of discs around very low mass stars \citep{2014ApJ...791...20R,2016A&A...593A.111T,2016ApJ...819..102V,2018AJ....155...54W} have low spatial resolution.
Encouragingly, some high-resolution observations are taking place, e.g. \citet{2018A&A...615A..95P} for a 0.1-0.2 $M_\odot$ star.
The samples of \citet{Andrews2018} and \citet{2019ApJ...882...49L} also contain a few low-mass stars, though at the moment there is no correlation between the sub-structure properties with the stellar properties \citep{2018ApJ...869L..42H}.
At the moment it is difficult to say whether this is due to selection biases (for example \citealt{Andrews2018} targeted the brightest discs), the low number statistics (even combined, there are only a handful of stars in these samples below 0.5 $M_\odot$) or if it is physical.
Imaging these discs might seem harder because they are in general significantly fainter than around solar-mass stars, since the disc sub-mm flux strongly correlates with the host stellar mass \citep{Pascucci2016}.
However, it should be kept in mind that the disc size also correlates with the stellar mass; in fact, the disc surface brightness is almost constant \citep{Tripathi2017,Andrews2018} across the disc population.
Because interferometers like ALMA are sensitive to surface brightness, rather than absolute flux, the prospect to image discs around low mass stars looks encouraging.
Future observations will thus provide the datasets necessary to test how the incidence of planets at 10s of AU in young systems depends on the mass of the central star.
\section{Conclusions}
\label{sec:conclusions}
In this paper we have investigated the planet gap opening mass, defined as relevant for ALMA continuum observations (i.e., in the dust, rather than in the gas), across stellar masses and for different distances from the star. We have highlighted how the dependence on the stellar mass is the net result of the competition between the two different effects: on one hand, gap opening depends on the planet-stellar mass ratio, favouring gap opening by lower mass planets around low mass stars. On the other hand, discs around low mass stars are geometrically thicker due to the reduced gravity, making gap opening more difficult due to the increased pressure forces.
We have shown that if we assume a dependence of stellar luminosity on stellar mass appropriate to the \citet{Siess2000} isochrones at $1$ Myr, the latter effect is more important than the former; in this case we would therefore predict that the gap opening mass \textit{decreases} with stellar mass and that planet induced structure should therefore be more readily detectable in the case of more massive stars. For the \citet{Baraffe2015} tracks, the two effects almost exactly cancel each other; it is therefore a robust conclusion that there is no benefit in looking for planets around low mass stars.
The gap opening mass also increases with the distance from the star, as expected in a flaring disc.
We provide a simple scaling relation (see Eq. \ref{eqn:summary_Mmin} and figure \ref{fig:summary}) that expresses the gap opening mass as a function of orbital radius and stellar mass, where $A_{M_{*}}$ is the gap opening mass in Earth masses at a distance of $1$ au. This relation can readily be used in the interpretation of observations and is applicable at angular distances from the star that exceed the beam size.
However the detailed interpretation of future imaging results needs to take into account the actual stellar
luminosities in the observed sample, since the luminosity spread at a given mass introduces significant differences for individual discs. In general the stellar luminosity of each source will also be known and
we also provide relations to take this into account when estimating the gap opening mass, see Eq. \ref{eqn:Mmin_Ls_apl} and figure \ref{fig:MGOPM_Ls_apl}.
Planets are often held responsible for the annular structures now ubiquitously observed in proto-planetary discs and future surveys will determine how the incidence of such structures depends on stellar mass. Our study has provided the framework within which the results of such surveys should be interpreted.
\section*{Acknowledgements}
We thank an anonymous referee for their constructive criticism that significantly improved this paper. This work has been supported by the DISCSIM project, grant
agreement 341137 funded by the European Research Council under
ERC-2013-ADG and also by the European Union's Horizon
2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 823823 (DUSTBUSTERS). This work used the DIRAC Shared Memory Processing system at the University of Cambridge, operated by the COSMOS Project at the Department of Applied Mathematics and Theoretical Physics on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/J005673/1, STFC capital grant ST/H008586/1, and STFC DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National E-Infrastructure. This work is part of the research programme VENI with project number 016.Veni.192.233, which is (partly) financed by the Dutch Research Council (NWO).
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THE
_Old — Fashioned_
FRUIT GARDEN
Copyright © 1989, 2012 by Jo Ann Gardner
Originally published in Canada by Nimbus Publishing
All Rights Reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018.
Skyhorse Publishing books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.
Skyhorse® and Skyhorse Publishing® are registered trademarks of Skyhorse Publishing, Inc.®, a Delaware corporation.
www.skyhorsepublishing.com
10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data available on file.
ISBN: 978-1-61608-621-3
Printed in China
This fruit of my labor is dedicated to Jigs,
beloved partner in all things.
To those who joy in their work,
this old earth laughs with them.
––Samuel Fraser, 1924
**_Contents_**
Author's Notes on a New Edition
Acknowledgments
Foreword
Introduction
A Short Course in Fruit Preserving
Rhubarb
Recipes
Strawberries
Recipes
Raspberries
Recipes
Red Currants
Recipes
Gooseberries
Recipes
Black Currants
Recipes
Elderberries
Recipes
Citron Melon
Recipes
An Assortment of Basics
Tree Fruits and Wild Fruits
Recipes
Appendix
Bibliography
Index
**_Author's Notes_** **_on a New Edition_**
Over forty years ago, in 1971, we moved ourselves, our four children, and a Noah's Ark of animals to a remote farm at the northeastern tip of Nova Scotia, to Cape Breton Island. We wanted to own a piece of earth where we could grow our food and keep some poultry and livestock, as we had been doing at rented farms. We were tired of planting strawberries and not being around to harvest them because the landlord sold the place where we were living, but with the money we saved we hadn't been able to afford anything suitable in the northeastern U.S. where Jigs and I had always lived.
We had no idea, really, of what we would be up against in Cape Breton: long, gray, maritime winters of freezing and thawing; heavy clay soil, slow to warm and leached of nutrients; and wind . . . oh, the wind, enough to blow our little greenhouse roof off, not once, but three times. I know, because I helped, in a roaring wind, to nail it back down.
I haven't mentioned income, because for the first five years, we hadn't any. We lived on our savings, and we lived a very lean life. After the first year, the bad roads shook up our old truck so much we abandoned it and used our team of horses wherever we needed to go, not very far. I called our backlands farm the No Capitol of the World: no vehicle, no running water (Jigs, ever creative, tells me it did run . . . out of the hand pump), and no phone. We did have electricity, but not a very sound system and we did not tax it much. We heated and cooked solely with wood and we had no freezer.
But there is no reason to be sorry for those years. We learned so much. Not only about Canada, and Cape Breton and its folk culture, of which we were sublimely ignorant, but about ourselves and how far we could stretch our abilities. In 1978 when the youngest of our children left the farm, I found myself to be the only other hand on staff. With no prior experience (I graduated from Bennington, majoring in drama and literature), with pitchfork in hand, standing on the hay wagon that Jigs drove around the field, I learned to build tons of hay each summer and helped to unload them in the barn. I pulled at the other end of a cross-cut saw to cut down huge trees for firewood . . . when Jigs pointed in the direction I was to run after the tree cracked, you can be sure I ran fast. I learned to spread manure by moonlight with the team of horses after a day in the woods, a day that also included barn chores (milking, cleaning out pens, feeding, and watering) and cooking three big meals a day.
Those were also the years of enormous fruit harvests. Early on, we had built two log cabins for guests to bring in some income, and before breakfast, they might pick buckets of gooseberries or baskets and baskets of raspberries that had to be turned to account as fast as I could manage. With no freezer, I adapted older recipes to suit my circumstances; readers may appreciate how I dispensed with topping and tailing gooseberries. And somehow I managed not only to preserve the fruit harvest but produce superior products. What I learned became the source of inspiration for _The Old-Fashioned Fruit Garden_ , the first edition of this book, published in 1989.
A lot has happened in the intervening years. In 2001, for one thing, we moved back to the States, to the foothills of the Adirondacks, in New York's Champlain Valley. We live now in a converted one-room schoolhouse with 10, instead of 100 acres of land, and we keep flocks of poultry. We still maintain large gardens that mostly feed us, we raise as much fruit as we can, and we still enjoy foraging in the wild. The principles that unconsciously drove us—simplicity, economy, and self-reliance—are still in practice, imprinted on our souls. Looking back, I see that we were green when green was just a color, but we didn't live 'green' on principle. Our lives grew out of our preferences. We followed a path of our own desire. Whatever we did and however we have lived, for better or for worse, there is no pretension about it. It just _is_.
A lot has happened, too, in the field of fruit development. There are many new varieties (more correctly, cultivars) among traditional June-bearing, everbearing, and day-neutral strawberries, for instance, so that the home gardener has the possibility of harvesting more than one crop a season; the same is true of raspberries, among the fall-bearing type. And blackberries are coming along with similar characteristics.
Among fruit trees, there are more semi-dwarf types that take up far less room than standards, and there are even some that can be grown in tubs. Blueberries, a beautiful bush in flower, whose leaves turn bright scarlet in the fall, have been the subject of breeding and selection to find ever more beautiful and productive varieties.
In general, the breeding trend is toward fruits that are not only productive, disease and insect resistant, and adaptable to different conditions, but ones that have ornamental value. Landscaping with edibles is definitely a trend among home gardeners.
Today, a significant number of gardeners are interested in growing less common fruits. As Tim Malinich, Extension Educator of the Lorain County Extension Office in Ohio told me, "Ten years ago, if I did a talk and mentioned elderberries or currants a few people would have had experience or interest. Today, a significant number of gardeners either have experience with or are willing to work with less common fruits such as currants, elderberries, and gooseberries. In fact, it is becoming almost trendy to grow them." Their health benefits, too (along with blueberries), have been in the news, altogether giving these fruits a higher profile.
This new edition of _The Old-Fashioned Fruit Garden_ takes into account all these trends—in fruit breeding and interest in the less common fruits—and offers one of the most comprehensive collections of recipes and directions for preserving and using currants (including a hefty section on black currants), gooseberries, and elderberries, for these were the very fruits that were the backbone of our Cape Breton gardens. What was old-fashioned is now new. I have added quince to the mix, an old-fashioned tree that seems to me elegant enough to grow just for looks, but I do include recipes.
Along with the many recipes for making jellies and jams with no commercial pectin added (these are now called "artisan" products), the hallmark of this book, I have added new ones like Ginger-Peach Jam; more accompanying dairy products like Yogurt and Yogurt Cheese Spread that you can make even without a cow tethered in your front yard; and directions for making fruit sherberts, sorbets, and frozen fruit yogurts (the freezer is modestly featured in this edition). On a hot summer's day, you may want to make my Black Currant Sorbet and serve it, as I suggest, embellished with a fresh-picked, edible Zebrina flower ( _Malva sylvestris_ Zebrina), a lovely rose-pink with purple stripes.
To meet the current high standards for home fruit preserving, I have updated directions to include the latest information from the current _Ball's Blue Book Guide to Preserving_ (2011).
The new Appendix takes into account the Internet (I have also included phone numbers wherever possible). There you will find sources for contacting your nearest Cooperative Extension office; finding the nearest U-Pick operation if you don't have your own garden (you could also consider bartering from someone who does); where to find uncommon fruit plants and heirloom seeds for growing citron ("the preserving melon"); Pyola, an organic insecticide that works; where to find canning equipment and the latest new pectins that don't take a lot of sugar or any (you can adapt my recipes); cheese-making supplies to make all the dairy products I discuss in the book: and the hand tools to make truly artisan fruit products. I have not forgotten Canadian readers and have included Canadian sources, too.
I am thrilled to offer a new generation of fruit growers and preservers, the novice and the experienced, this new edition of _The Old-Fashioned Fruit Garden._ I trust that those who loved the old edition will find a place for the new one, too, on the shelves of their gardening/cookbook library.
Westport, New York, 2012
**_Acknowledgments_**
I wish to thank the Horticultural Research Institute of Ontario at Vineland Station, the Agriculture Canada Research Station at Kentville, Nova Scotia, and the Agriculture Canada Research Station at Buctouche, New Brunswick, for providing me with information on the status of black currants in North America.
I am also indebted to the following institutes in the United States: the Connecticut Agricultural Experiment Station, the Missouri Cooperative Extension Service, the New York State Agricultural Experiment Station, the New York State Department of Environmental Conservation, the Rutgers Extension Service, the Maine Department of Conservation, the University of Alaska Agricultural Experiment Station, the University of Minnesota Agricultural Extension Service, the University of Rhode Island Cooperative Extension, the University of Wisconsin Extension Program, the United States Department of Agriculture, and the departments of agriculture of Michigan, New Hampshire, New Jersey, Oregon, and Vermont.
I would also like to thank Carol Makielski, formerly of the Makielski Berry Farm & Nursery in Michigan, and Michael McConkey of Edible Landscaping in Virginia for encouraging me in my promotion of black currants. I am indebted to the North American Fruit Explorers and the Brooklyn Botanical Garden Information Service for their help in this research.
I am grateful to those unnamed generations of cooks from whose creative genius I have freely borrowed, as well as to a few friends and guests whose recipes have enriched my repertoire.
I owe thanks to _Rural Delivery, Farmstead, Country Journal,_ and _Harrowsmith,_ which have previously published some of the material in this book in a different form.
I am thankful to Dorothy Blythe of Nimbus Publishing for her interest in a gardener's cookbook and to Nancy Robb for her editorial comments and suggestions. Any inconsistencies or idiosyncracies are my own.
_Jo Ann Gardner_ Orangedale, N.S., 1989
To the above, I add thanks to Tim Malinich, Extension Educator, Horticulture, of the Lorain County Extension Office, for sharing his views on plant breeding trends, trials of new blackberry cultivars, the status of black currants in his state, and more; and to Judy French of the Cornell Cooperative Extension office in Westport, New York, for her assistance. Special thanks to Lois Rose, Nell Gardner, Jeanne Leblanc, Bob Miller, Rob Demuro, and Russell Studebaker for their help in various ways, all important to me.
I am indebted to Jenn McCartney at Skyhorse for her patience in guiding me through the process of this new edition, and to Jigs, my partner-in-all-things, and an invaluable advisor on all literary points.
J. A. G., 2012
**_Foreword_**
I made my first jelly, wild grape, in Madison, Wisconsin, in the late summer of 1956. Exactly why, I cannot remember. That summer, we had our first garden, I was beginning to learn about wildflowers, and I was interested in making some of the things we usually bought, such as soap. Perhaps the recollection of my mother hanging a jelly bag from a broom handle suspended between two dining-room chairs helped inspire me.
I suppose that having seen some grapes on one of my wildflower rambles, I must have looked up a recipe. Luckily, this turned out to be a great jelly. If our first product had been ordinary or dull, our jam-and jelly-making careers might have developed more slowly.
I remember grape-picking expeditions with my wife, Jo Ann, and our one-year-old son, Seth: I'm in a tree dropping bunches of wild grapes to Jo Ann, who is tucking them into Seth's baby carriage. Then the three of us are wandering through an overgrown, abandoned quarry, picking grapes from vines on crumbling stone walls.
I have another wild-grape memory, of quite a different sort; I'm crouching under a big bush on a side street in Madison, picking grapes from a vine growing over the bush. A man stops beside the bush, peers under, and asks me what I'm doing. He turns out to be an acquaintance who runs a bookstore downtown. While I crouch in the green shade under the bush and he stands on the sidewalk in the bright sun, we have a small, friendly debate that, although I did not know it then, I would have over and over again, and not just about jams and jellies, either.
He thought that making my own jelly was foolish because (1) I could buy it in a store for less than the cost of making it; (2) the time spent picking and processing grapes could be used to earn more money than my jelly would be worth; and (3) crouching under a bush in broad daylight in a modern city was really, well, nutty looking.
That was the first time I engaged in that debate, and I'm afraid I did not give effective answers, although I do remember pointing out that I preferred to spend my time picking wild fruit than earning money. I also said that I could not buy superior jelly anywhere. Now, however, having been through this argument scores of times, I know other things to say: my time is as valuably spent picking wild fruit and making excellent nutritious preserves as it is doing anything else; I do not want to give up a significant portion of my life for a wage so that I can buy a jar of sweetened chemicals; when we make our own food, we _know_ it is good; and, finally, I have learned not to worry about being nutty looking. There are thousands of people walking around loose today doing worse-looking things than crouching under a bush and picking wild grapes.
For some years, when I was a teacher and had summers off, I did the jam and jelly making, as well as all the other canning and preserving, for our growing family. In the early 1960s, we began to sell some jams and jellies. But when we started farming and founded a school, there was a radical reorganization of the division of labor in our family. Jo Ann took over my kitchen tasks, and all our preserves improved. She had more mouths to feed, larger gardens to process, as well as meat, produce, and dairy products to sell. In 1971, we moved to Cape Breton. Not only did we have a farm to run, we also began to rent log cabins to visitors. So Jo Ann had more meals to cook, and they had to be first rate.
When I come in from the stables or fields nowadays, I am, at most, Jo Ann's assistant, sometimes washing up after butter making or drawing off whey from cheese or packing away case after case of jam. It is just as well. As our many guests and customers will testify, Jo Ann is a good cook: intelligent, hard working, and sensible. I, on the other hand, usually give in to anarchic tendencies in the kitchen, disregarding directions in a fine frenzy of creative disaster. But I will not go into my mistakes. Follow Jo Ann's directions exactly, and you will make excellent jams, jellies, sasses, juices, and fresh-fruit preserves.
_Jigs Gardner_
Orangedale, N.S., 1989
**_Introduction_**
When was the last time you tasted some absolutely terrific homemade jam, the kind that made you want to try making it, too? I do not mean freezer jam or some concoction made with Jell-O or commercial pectin and loads of sugar. I mean the real thing: jam made with fresh berries that are mashed, heated with just enough sugar to bring out the fresh-fruit flavor, brought to a boil, and cooked for no more than 15 minutes (usually less), until the mixture has thickened.
Jam making, like other kinds of fruit preserving, has become almost a lost art superseded by the mass production and marketing of fruit products in the supermarket. At one time, 100, 75, even 50 years ago, the fruit garden, like the vegetable garden, supplied a great part of the family's food needs. Many backyards and almost all farms had fruiting shrubs, a rhubarb patch, and a variety of apple trees to provide fruit for baking, cooking, drying, or storing. Few women in these households did not know how to turn their garden fruits into an astonishing number of concoctions— jams, jellies, preserves, conserves, marmalades, sauces, juices, wines, vinegars, dried fruits—astonishing when compared with the paucity of products offered in today's markets. Have you ever tried to buy red currant sauce or gooseberry preserves—or fresh gooseberries, for that matter?
The "plucky housewife" of the turn of the century preserved from necessity because processed foods were not readily available.
Home preserving was a serious business and one in which the homemaker took great pride. There was a lot of skill involved in choosing varieties of fruits for different purposes. Take pears, for instance: the Winter Nellis was known for storing; the Pound, for eating; and the Bartlett, for flavor. How many women, or men, today know one variety from another or can even find each in the marketplace?
Jam making is probably the last vestige of home fruit preserving, but the skills that were once an absolute necessity for turning out successful products have been neglected in the almost universal desire to emulate mass-produced store-bought varieties. For the unskilled, foolproof methods such as using commercial pectin and the freezer guarantee a "successful" product every time.
What are the qualities of store-bought preserves? In jam, they include a heavy sweet taste, an overfirm set, an absence of real fruit flavor—in sum, a general lack of all distinction. Unfortunately, the word _homemade_ is no longer synonymous with all the superior qualities. No doubt, among the home-processed fruit products of the past, there were a lot of duds: burned jams, runny jellies, moldy preserves. But when the household itself depended on and processed all the fruit for the family, quality—the kind associated with the best of homemade—counted.
There may be some merit in the Victorian notion that tending a garden, watching plants grow, has an ennobling influence. But there are other compelling reasons for creating your own fruit garden, that is, an old-fashioned garden that can supply you and your family with almost all your fruit needs.
First, there is the reason of economy: fruit and fruit products can be expensive for a growing family. If a small fruit garden can supply products as good, or better, as the supermarket at much less expense, it makes a lot of sense to consider making even small plots productive. Wild fruits should not be overlooked: abandoned orchards can yield truckloads of apples for cider, wild grapes can make an incomparable jelly, staghorn sumac can supply a tart lemonlike flavor for juice.
Second, the quality of fruit bought in a store cannot compare to the quality of fruit from a garden. Store-bought fruit is not so fresh; consider as well the handling and shipping and the heavy use of pesticides. Have you ever compared the taste of a California strawberry out of season with one just picked from your own patch? Have you ever tasted raspberry juice, jam, or ice cream made from real raspberries, not from a concentrate or "flavor" whipped up in a food chemist's lab? You cannot even find gooseberry or black currant products in most ordinary stores; even the fancy gourmet products are inferior to those you can make yourself from the harvest of old-fashioned fruiting shrubs. To avoid making products with a store-bought taste, however, you must follow the simple preserving methods described later in this book.
Third, there is the important consideration of energy use and conservation. Do you really think that small is beautiful, that less is more, that technology should be carefully, not wantonly, used, that resources should be husbanded, not squandered, that we must learn to live in harmony with the natural world?
The old-fashioned fruit garden can point the way by showing you how to achieve energy efficiency through the best use of simple technology. Only those fruits most suitable to one's environment are grown; they are harvested for particular uses and processed by the simplest means to achieve a superior product.
Is it absolutely necessary to "pop" food in the freezer or "toss" it into the blender when other techniques less demanding of energy, less destructive to the land, are accessible? Before World War II, people used noninstant powdered milk, knives for cutting and chopping, hand food mills for puréeing, and rotary beaters for beating.
I do not advocate going back in time, but we must begin to think about how we use technology. The care with which we do this will be reflected in a higher quality of life, a life more in tune with the natural world rather than antagonistic toward it. Growing and processing your own fruit, taking as little from the land as possible while creating the conditions for continual renewal and regrowth, is immensely satisfying and ecologically sound. "The land," wrote Liberty Hyde Bailey, the great American horticulturist, "is the cemetery of the ages and the resurrection of all life."
Technology constantly changes, and just as there have been many improvements made by plant breeders in creating vigorous, disease-resistant fruit varieties, so has modern industry produced superior equipment and techniques for home processing. But I leave the electric dehydrators and similar devices to the do-it-yourself sophisticates. The technology I have in mind really _is_ simple and within reach of most people.
Canning and jelly jars with vacuum-sealing snap lids and screw bands are a great improvement over older kinds of jars and closures and ensure successful preserving if directions are carefully followed.
Whatever they cost, they will pay for themselves in the first season. Paraffin has replaced the brandy-soaked writing paper and layers of cotton batting once used to prevent molding [ _note_ : paraffin in no longer recommended]. The water-bath canner, itself an improvement over more cumbersome pots and boiling-water methods, has been superseded by the steam canner in my kitchen; it not only reduces the amount of water and fuel needed but saves time, too. The steam juicer has eliminated the need for using the traditional jelly bag for extracting juice, though the latter works perfectly well and is less expensive.
With improvements in preserving equipment, less sugar is needed to keep fruit products sound; old-fashioned preserving tended to be lavish in its use of sweeteners. Perhaps surprisingly, the freezer has only a limited role in preserving fruit. Why reap the benefits of sun-ripened fruit and then store that fruit in a freezer for an unlimited time when simple techniques will not only suffice but also produce a better product? Canned blueberries vacuum sealed at their best will keep a long time if stored in a cool place away from light and heat; they will be just as delicious as when they were first packed and sealed in their own juices.
All the fruits described in _The Old-Fashioned Fruit Garden,_ except, of course, the wild fruits, were grown on less than half an acre of land in poor northern soil. I have selected small fruits—strawberries, raspberries, red and black currants, gooseberries, elderberries, and citron, as well as rhubarb—because these do well almost anywhere, are easy for the novice to grow, and are sometimes overlooked. Tree-fruit cultivation is more demanding of soil, climate, and space than small-fruit cultivation. As there are already many excellent books and pamphlets on the subject, I have limited myself to uncommon preserving and cooking recipes in the chapter on wild and tree fruits. If you live in a warmer climate, there are many more fruits, such as figs, cherries, and grapes, that could be included in a small garden. But no matter what fruits you choose to grow, if you intend to have an old-fashioned fruit garden, you must follow the principles of energy efficiency that begin with the way the soil is prepared for planting and continue through to the last step of processing.
The chapters deal with one fruit each, and they are arranged in the order the fruits are harvested, beginning with rhubarb and ending with citron; the chapter on wild and tree fruits follows. I chose to devote individual chapters to each member of the genus _Ribes_ —red currants, gooseberries, and black currants. All three have been out of favor for decades, partly because they are host to a fungus called white-pine blister rust, which kills white pines but does not damage the carriers. For most of this century, there have been restrictions against those fruits in the United States; since 1966, with the lifting of a federal ban, each state has regulated their sale. There have been no similar regulations in Canada.
In some states, laws have either been relaxed or not enforced for two reasons: the white pine has declined in commercial value, and the regulations have been ineffective, as _Ribes_ species grow abundantly in the wild. Nevertheless, the black currant, which often shoulders all the blame for the disease, continues to be maligned as a result of a general lack of knowledge about its special qualities. It is about time someone in North America rescued this fruit from oblivion. I hope that interested gardeners in the United States will demand a reappraisal of this fruit from the state agencies that are charged with regulating its growth. Perhaps plant breeders and nurserymen will then take up the challenge to make this fruit safe for all to grow, available to everyone who wishes to include it in his garden. There are indications that this is beginning to happen, with more breeding programs being conducted at agricultural research stations and by private researchers.
I have not, however, included a chapter on Josta, a gooseberry-black currant hybrid. Hardy and disease resistant, the Josta was bred in Germany. Its bush is tall and tolerates a wide variety of soil conditions, producing dense clusters of deep-purple berries that can be eaten fresh. Josta bushes are available in Canada and the United States (see Appendix). Although reports are mixed, Josta is worth a try, and suggests more possibilities. Use in similar ways to gooseberries or black currants.
In each chapter, there are guidelines for planting, cultivating, and harvesting the fruit. These are general and meant as a guide, not a blueprint. More detailed directions for fruit growing suitable to your area can be obtained, usually in pamphlet form, from your local cooperative extension office or department of agriculture. You will also learn the fruit varieties appropriate for your area and the ways to deal with your particular environment.
The recipes in this book represent our favorite ways to use fruit, both fresh and processed, the result of years of preserving; some recipes are adaptations of ones in books, and these sources are cited in the bibliography. Each preserving recipe is arranged to coincide with the ripeness of the fruit as it is harvested. Jelly recipes that call for a mixture of ripe and underripe berries precede recipes for juice or wine, where the ripest, juiciest berries are required. Every once in a while, I have inserted accompaniments within the preserving recipes. Black currant jelly, for instance, goes great with cream cheese, so the two recipes appear together. Following the preserving recipes in each chapter, there are "cooking" recipes. In addition, one chapter focuses on basic recipes that allow for the substitution of different fruits. No more frantic searching through cookbooks, with sticky fingers, to find another way to use your bumper crop of strawberries or black currants.
I do not deny that planting, harvesting, and processing an old-fashioned fruit garden is a labor, but I think that once you have tasted your own red currant sauce on your morning pancakes, eaten a bowl of fresh-fruit ice cream, or made what everyone is bound to consider the world's best strawberry jam, you will feel well rewarded. Even spending a day or two picking the fully laden black currant bushes, if you are lucky enough to be able to buy them, can be a respite from more worldly activities. You may also gain much satisfaction from rediscovering the virtues of simplicity. That's a lot to get from a small fruit garden.
THE
_Old — Fashioned_
FRUIT GARDEN
**Berry juice**
**_A Short Course_** **_in Fruit Preserving_**
I really didn't know how to boil an egg when I got married in the early 1950s. I certainly knew nothing about fruit preserving. All our fruit products—and all our fruit, for that matter—came straight from the supermarket.
With a growing family and a limited income, however, I learned quickly. Necessity is a good teacher. My husband, Jigs, was the leader in the preserving operation because he had had some experience. It was not long, though, before I understood terms such as _boiling-water bath_ and _snap lids._ By the time I was left in sole command, as a result of our expanded farming, I had to contend with a large fruit garden and no help at all.
Not only that. As a full partner in the farm, I spent as much if not more time in the field as in the kitchen. At the height of our fruit season, I could be found making square tonloads of hay on our horse-drawn wagon.
Under such conditions, I streamlined our fruit preserving, the benefits of which I pass on to you. The technology I use remains, with few exceptions, simple, a testimony to the truth I learned some time ago: there is no correlation between using high technology and producing high quality. In fact, technology often gets in the way.
**Equipment**
When making jam, jelly, or sass, use a 2-gallon (8-L) wide-mouth stainless steel pot. The wide mouth allows for fast evaporation, which makes for quick-setting products. Stainless-steel pots also cook evenly, so they are worth the extra expense. Large wide-mouth enamel preserving pots are good for making preserves, juices, and cooking fruit for drying, all of which call for large batches. Tin or copper pots are not recommended because they discolor and flavor fruit products; aluminum pots are too thin. Other necessities for fruit preserving are a large enamel water-bath canner which comes with a rack and a tight-fitting cover, or a steam canner; 1-qt (1-L), 1-pt (500-mL), and 1/2-pt (250-mL) canning jars and small jelly jars with matching snap lids and screw bands; cheesecloth for a jelly bag, or a steam juicer. _Note: old-timers were the great recyclers, using any jar that will take new snap lids and old screw bands, and re-using vacuum-sealed jars. But this practice, like others (sealing with hot paraffin, sealing jams, jellies, syrups, etc. without processing), is not regarded as safe._ Other utensils include wooden spoons, a small metal spoon, a potato masher, wooden stamper, or chopper (I use an old-fashioned egg chopper) for preparing fruit for jam-making, a jar lifter and kitchen tongs, large screens, trays, and cookie sheets, a long-handled ladle and fork, a food mill, and a wide-mouth funnel.
**Terms and Methods**
Do not be intimidated by these terms and the descriptions of methods. Considering they provide you with the key to mastering home fruit preserving, they are worth reading carefully.
BOILING-WATER BATH A method of processing homemade fruit products in a water-bath canner so that they do not spoil, that is, become moldy or ferment. _It is now recommended that all homemade fruit products should be processed in a boiling-water bath_. Before processing, seal filled canning jars with snap lids and screw bands. Be sure to wipe the sealing edge of the jar to remove any debris that might cause an imperfect seal, then set the snap lid in place, and screw on the metal band, tightening it all the way.
Always process a full load of jars at a time, using water-filled jars to fill the empty spaces. Place bottled-and-sealed fruits on a rack and then lower the rack into the water-bath canner, partly filled with hot water. Add extra water as necessary, to bring the water level to 1 to 2 inches (2.5 to 5 cm) above the tops of the jars. Also make sure that there is about 1 inch (2.5 cm) of space between the jars to allow the water to circulate. Cover the pot and bring the water to a rolling boil. Count the processing time from this moment. At the end of the processing time, wait 5 minutes before removing cover. Lift the jars out of the water with a jar lifter and cool them on a rack or towel in a draft-free place.
Check canning jars in 12 to 24 hours after processing to make sure they are vacuum sealed. After processing, as the jars are cooling, the lids will be sucked down and become concave and will _snap,_ making a vacuum seal. If the lids fail to snap, they will remain convex, and these preserves should be reprocessed or refrigerated and used immediately. After 24 hours, the screw bands can be removed and reused.
The steam canner, not to be confused with the pressure canner, is a luxurious alternative to the more traditional boiling-water bath. Follow the manufacturer's directions. These usually specify putting 2 qt (2 L) water in the bottom part of the steam canner and counting the processing time from when steam starts coming out of two holes at the base of the cover.
One benefit of the steam canner is that you can process as little as 1 pt (500 ml) at a time. Another is that it doesn't require nearly as much water as a boiling water bath. Having lived with water scarcity for so many years, the steam canner was a blessing. Directions may call for screw bands to be not fully tightened before processing. Steam canners are more commonly used in Europe, though they are available (see Appendix).
COLD PACK A term for filling canning jars with raw fruit that is then covered with boiling water or syrup. Then the jars are processed in a boiling-water bath or steam canner. The cold-pack method works best with soft fruits like raspberries, which lose their shape when precooked.
EXTRACTION A procedure for cooking fruit in order to extract its juices. To make an extraction, mash the fruit, add water or not, cook it until tender, and strain through a jelly bag. The steam juicer, also commonly used in Europe, has eliminated the more time-consuming process of making an extraction. Nonetheless, steam juicers should be reserved for making large quantities. Place the fruit in the top part of the pot. On heating, juice collects in the bottom, ready for jarring or for use in jelly making. Steam juicers are available (see Appendix), but they are rather expensive.
HEADROOM The amount of air left between the top of the canned fruit and the lid. Follow directions, but do not worry excessively about this. The fruit inevitably settles, so allow for this when filling jars.
HOT PACK A term for filling canning jars with precooked hot fruit. I prefer this method to the cold-pack one. Precooked fruit, unlike raw fruit, does not shrink when it is processed in a boiling-water bath. A half-filled jar of preserves is a sorry sight, and the preserves may not keep well.
JELLY BAG A sack for straining fruit when making an extraction. Jelly bags can be made from any porous material, but one of the best is cheesecloth. Buy about 1 yd (1 m) of fine cheesecloth and cut a double layer large enough to be draped in the pot receiving the juice. Using clothespins, secure the material around the lip of the pot and then pour the cooked fruit into the cheesecloth. Gather the four corners of the cheesecloth into pairs of two and tie them together to make a bag that can be suspended from any pole, hook, or nail. Place the receiving pot underneath to catch the dripping juice.
PECTIN The substance in fruits that causes them to jell when combined and cooked with sugar. The amount of pectin in fruits varies: elderberries and rhubarb are low in pectin; strawberries, medium; raspberries, a little higher on the scale; currants, gooseberries, apples, and quince very high. Slightly underripe fruit has more pectin than fully ripe fruit: the substance declines in strength as the fruit ripens and decomposes. Citrus fruits, high in pectin, are added to other marmalades to help them jell as well as give them a piquant taste. Commercial pectins are made from citrus fruit. Traditional pectins, in either crystal or liquid form, require a certain amount of sugar to work (usually a lot). Low-sugar or no sugar-needed pectin requires the addition of unsweetened fruit juice. Instant pectin is for making no-cook jam that must be stored in the freezer (freezer jam).
SCALDED JARS Sterilized jars. Hot scalded jars should always be ready to receive the cooked fruit. First, wash the jars and lids in hot, soapy water and rinse them well. Submerge as many jars as possible into a deep pot of hot, almost boiling water; put the lids in a smaller, shallow pot of hot water. Allow them to simmer until they are ready to be filled. Spear each jar with a long-handled fork and drain over the pot as it is being removed from the water. If you are handy with the fork, you can retrieve the lids the same way. Otherwise, use kitchen tongs.
SHEETS OFF A SPOON A killer for the beginner. When the jelly or jam has cooked at a rolling boil for about 10 minutes or less, begin testing for the jelling point—that is, the point at which the mixture begins to hold its shape when cooled. Use a metal kitchen spoon, not a wooden one, for this test.
Scoop a little of the hot jelly or jam onto the spoon, cool it quickly by blowing across it, then slowly pour the mixture back into the pot, pouring from the _side_ of the spoon. If it runs off as water, the mixture is far from ready; if it runs off in two drops, it is as thick as syrup and is close to done; if the two drops have merged and the mixture wrinkles slightly and slides as a whole off the side of the spoon, the jelling point has been reached. Jelly slides and sheets; jam tends to fall in globs. Once you are experienced, you will be able to spot the jelling point of jelly by the look of the boiling mixture: thick and light-colored foaming bubbles.
A selection of preserves
**Steps for Canning and Preserving**
The following are general guidelines for understanding the basic procedures involved in making various commonly used fruit products. Once you understand them, you will be able to amend and adapt recipes to suit your situation. In the meantime, consult the preserving recipes I offer, as the details may differ from these descriptions. In addition, you'll find other treasures not discussed here—marmalade, wine, chutney, and butter, for example— because they are more complicated, vary more from recipe to recipe, or are less common.
Here I must also say a few words about the substitution of honey for sugar in fruit preserving. Honey does not combine well with natural pectin and for that reason should not be used in jam or jelly making if a firm set is desired. Furthermore, honey, with its distinctive taste, tends to mask fruit flavors. Sugar, on the other hand, used here in the minimum amounts required for setting, brings out and enhances fruit flavors. Honey, though, does combine well with liquid, so it could be added to juices or to syrups used for preserving whole fruits. Let your preference be your guide.
**Jelly**
Jelly is made from juice extracted from mashed and heated berries that are cooked with sugar. The jelled mixture should be clear, bright in color, and firm yet tender enough to spread easily on bread. Undercooked jelly is runny, and it becomes moldy.
Overcooked jelly is tough, treacly, and strong tasting. The secret of superior jelly lies in using fruits high in natural pectin so that an excessive amount of sugar is not needed for setting. Remember, the faster the jelly or jam sets, the more the fruit flavor is retained, so cook _small_ batches in a _large_ pot. Note: freezer stored fruit works well for jelly-making.
**_Steps:_** Sort berries. Use a combination of about two-thirds of ripe berries and one-third of underripe. (Imperfect berries are okay as long as there is no decay.) Rinse quickly in cold water. Avoid over-handling. Drain.
Place fruit in a large preserving pot, mash with a wooden stamper, and add water if necessary. (Strawberries and raspberries need little or no water; gooseberries and currants may be almost covered with water. The amount added will determine the strength of the juice, and the more concentrated it is, the quicker it will jell.)
Cover pot and simmer fruit until juice runs freely, stirring occasionally to prevent sticking.
Strain hot mixture through a jelly bag [p. 6] and let drip for several hours or overnight. Squeeze bag gently.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Bring to a boil, covered. Stir in sugar with a long-handled wooden spoon. Add equal amount of sugar to juice. If fruit is high in pectin, use 3/4 cup (175 mL) sugar to each cup (250 mL) juice.
Bring to a boil again. Boil, uncovered, for about 10 minutes, skimming froth as necessary. _If the mixture threatens to boil over, toss a small piece of butter into the pot._
Begin testing for jelling point. When a small amount of mixture sheets off a metal spoon [p. 6-7], remove from heat.
Let jelly subside, stir, and skim if desired. Pour into hot scalded jelly jars [p. 6], leaving 1/4 inch (5 mm) headroom, wipe jar edges, and seal at once with snap lids and screw bands. Process for 10 minutes in a boiling-water bath or steam canner, or according to recipe. Wait 5 minutes until removing the pot or steam canner cover. Store sealed jars in a cool, dark, dry place.
Note: If jelling point is not reached after 20 minutes of rapid boiling, _stop_ cooking, remove pot from heat, and pour mixture into hot scalded jars to use as sass (facing page).
**_Yield:_** 2 qt (2 L) fruit makes about four 1/2-pt (250-mL) jars.
**Jam**
Jam is mashed, partially mashed, or chopped whole fruit cooked with sugar until the mixture thickens—it should spread on a piece of bread without dribbling over the edges. Superior jam has a fresh-fruit flavor, is not overly sweet, is a bit chunky in texture and, of course, has a beautiful bright color. To achieve such a product every time, cook the fruit as soon as possible after it has been picked, include a healthy sprinkling of underripe fruit in the harvest to ensure quick setting, and cook the fruit in small batches in a large pot. Note: freezer stored fruit works very well for jam-making if the fruit was picked in prime condition and quickly frozen.
**_Steps:_** Sort fruit. Use a combination of two-thirds of ripe berries and one-third of underripe. Imperfect, bruised berries are okay as long as there is no decay. Rinse berries quickly in cold water. Avoid overhandling. Drain.
Measure 1 qt (1 L) fruit and place in a large stainless-steel pot. _Partially_ mash berries with a potato masher or wooden stamper, and coarsely chop strawberries or other fruit, to make a chunkier jam. If berries are tough—such as black currants and gooseberries— add a little water or as directed.
Cover and bring berries to simmering point, stirring once or twice to prevent sticking. Stir in sugar with a long-handled wooden spoon. If not following a recipe, add equal amount of sugar to fruit. If fruit is high in pectin, use 3/4 cup (175 mL) sugar to 1 cup (250 mL) fruit.
Bring mixture to a rolling boil. Stirring occasionally, boil rapidly, uncovered, for no more than 10-15 minutes or until mixture thickens and begins to cling slightly to bottom of pot. A small amount should fall in a glob off the side of a metal spoon (see Sheets Off a Spoon, p. 6-7). If mixture threatens to boil over toss in a small piece of butter.
Remove jam from heat. Let subside, stir, and skim if desired. Pour into hot scalded [p. 6] 1/2-pt (250-mL) or 1-pt (500-mL) jars, leaving 1/4 inch (5 mm) headroom, wipe jar edges, and seal at once with snap lids and screw bands. Process in a boiling-water bath or steam canner for 15 minutes or according to recipe. Wait 5 minutes before removing cover. Store sealed jars in a cool, dark, dry place.
**_Yield:_** 1 qt (1 L) fruit makes 1 1 /2 to 2 pt (750 mL to 1 L) jam.
**Sass**
I will be frank. _Sass,_ a New England colloquial term for all the vegetables in the garden (from "garden sass" or "sauce") was in the beginning, nothing more than failed jam or jelly, which, I and a zillion other cooks discovered, makes fine toppings for pancakes, ice cream, etc. To make sass on purpose, use any jam or jelly recipe, reduce the amount of sugar, and cook the mixture until it only thickens, not jells. Sass making is great for beginners: it is a foolproof process. There is no need to cook in small batches, no need to know the crucial jelling point, and the result is always delicious.
**_Steps:_** Sort and wash fruit as for jam [p. 9]. (Some recipes call for ripe fruit, allowing for overripe or damaged fruit as well.)
In a large stainless-steel pot, partially mash 2 qt (2 L) berries. Cover and bring to simmering point, stirring as necessary. Or make an extraction [p. 5] as for jelly, bringing 2 qt (2 L) juice to a boil, covered.
Stir in sugar. If following a jam or jelly recipe, use one-half to three-quarters the amount called for. Bring mixture to a boil and boil, uncovered, for 10 to 20 minutes or until mixture thickens.
Remove sass from heat and let subside. Stir. Pour into hot scalded jars [p. 6], leaving about 1/2 inch (1 cm) headroom, wipe jar edges, and seal at once with snap lids and screw bands. Process canning jars in a boiling-water bath [p. 4] or steam canner for 20 minutes or according to recipe. Wait 5 minutes before removing cover. Store sealed jars in a cool, dark, dry place.
**_Yield:_** _2_ qt (2 L) fruit or juice makes about 4 pt (2 L) sass.
**Preserves**
Preserves are whole fruit preserved in a sugar or honey syrup. They need not be nearly so sweet as the store-bought variety. Home preserves are just sweet enough to bring out the natural fruit flavor. They can be eaten straight out of the jar or with cream; they can be poured over cakes, mixed with different fruits; their juice can be used for gelatin.
**_Steps:_** Sort fruit. Use perfectly ripe, _not_ overripe, fruit. Rinse quickly in cold water. Drain.
Follow one of these procedures:
(a) Layer fruit and sugar (1/2 to 1 cup/125 to 250 mL sugar to each qt/L fruit) consecutively in a large preserving pot. Cover and let stand for several hours or overnight to draw out juices. Slowly bring mixture to a boil, stirring as necessary to prevent sticking.
(b) In a large preserving pot, add sugar to fruit, cover, and heat until simmering, stirring as necessary.
(c) In a large stainless-steel pot, make a syrup of 2 or 3 cups (500 or 750 mL) sugar to 1 qt (1 L) water. If using 2 qt (2 L) fruit or less, add fruit to syrup. Otherwise, add syrup to fruit in a large preserving pot. Bring to simmering point, stirring as necessary.
Remove preserves from heat. Ladle into hot scalded jars [p. 6], leaving 1/2 inch (1cm) headroom, wipe jar edges, and seal at once with snap lids and screw bands. Process in a boiling-water bath or steam canner for 20 minutes or according to recipe. Wait 5 minutes before removing cover. Store sealed jars in a cool, dark, dry place.
Note: To can fruit without sugar, precook fruit in a little water, ladle into hot scalded jars with boiling water, adjust lids, and process for 15 minutes in a boiling-water bath or steam canner.
To can fruit with honey, substitute honey for half the sugar called for in the recipe. Use scant measure or 1/4 cup (50 mL) less honey than half the amount of sugar.
**_Yield:_** 2 to 3 lb (1 to 1.5 kg) fruit makes 1 qt (1 L) processed; 1/2 to 1 cup (125 to 250 ml) sugar syrup covers 1 qt (1 L) fruit.
**Dried Berries**
It is not necessary to have any special equipment for drying fruit, whether drying berries or making apple rings or fruit leathers, although I have seen dandy home-rigged dehydrators (commercial ones make a lot of noise and usually dry only a small amount at a time). Why use electricity when summer sun, air drying, or the heat left over from baking will do just fine? Berries can be dried in the sun, but those simmered first in a sugar or honey syrup keep their flavor, color, and shape better. The syrup can then be bottled and used as a base for juice or sauce. Eat dried berries as they are or use them in baking.
**_Steps:_** Sort berries. Use only firm, ripe, less-seedy berries such as black currants, blueberries, or elderberries. Rinse quickly in cold water. Drain.
Using 3 cups (750 mL) fruit to 1 cup (250 mL) sugar or 3/4 cup (175 mL) honey, layer berries and sugar consecutively in a large preserving pot. Cover and let stand overnight to draw out juices.
Bring mixture to a boil, reduce heat, and simmer for 15 minutes, stirring occasionally to prevent sticking.
Remove from heat. Skim out berries or pour mixture through a strainer. Reserve juice.
Put berries on paper-lined trays— heavy brown shopping bags work well. Using a long-handled fork, spread out berries evenly.
Place trays in direct sunlight. If insects are a problem, cover berries with a single layer of cheesecloth.
The next day, change paper, stir berries, and continue stirring daily until they are as dry as raisins, about 3 to 4 days in good sun. Store berries in glass jars or crocks; they will keep almost indefinitely.
**_Variation:_** To make juice as a by-product, place reserved juice back on stove. Bring to a boil and pour into hot scalded jars [p. 6], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process 15 minutes or according to recipe. When serving, dilute with water to taste.
**_Yield:_** 2 qt (2 L) berries makes about 1 qt (1 L) dried.
**Juice**
Juice is one of the easiest ways to store an abundant harvest of overripe fruit. Homemade juice is superior to commercial varieties in every way, in fresh-fruit flavor especially. And many kinds of juice are difficult if not impossible to find on the market. Also, homemade juice is useful for making gelatin, adding to punch, and flavoring frosting, ice cream, sorbets and ices. When serving as a drink, dilute it with water to taste.
**_Steps:_** Sort fruit. Rinse quickly in cold water. Drain.
In a large preserving pot, combine fruit and water, using 1/2 cup (125 mL) water to each qt (L) fruit. Make an extraction [p. 5] as for jelly. Measure juice.
Bring juice to a boil, covered. If no particular recipe is used, stir in 1/2 cup (125 mL) sugar (or a little less honey) to every qt (L) juice. Bring juice to a boil again, reduce heat, and simmer, uncovered, for 5 minutes.
Remove juice from heat. Pour into hot scalded jars [p. 6], leaving 1/2 inch (1 cm) headroom, wipe jar edges, and seal at once with snap lids and screw bands. Process canning jars in a boiling-water bath [p. 4] or steam canner for 15 minutes or according to recipe. Store sealed jars in a cool, dark, dry place.
**_Yield:_** 2 to 3 qt (2 to 3 L) fruit makes 1 qt (1 L) juice.
**Leather**
Talk about simple technology! You may want to buy a wood-burning cookstove just to make leather, although a just warm gas or electric oven will work, too. I dry my leathers in a just-warm oven after baking, on the surface of the stove away from the heat, or underneath the stove, a source of steady warmth as any cat will tell you. As a result, I am able to turn literally bushels of fruit into a delicious, natural confection. Any fruit can be used—soft fruits, leftovers, odd bits and pieces at the end of the season. This is a good way to use fruit too good to discard but unsuitable for processing or eating fresh. All sorts of combinations are possible, too: raspberry and strawberry; red currant and gooseberry; raspberry and red currant; apple and cranberry, blueberry, pear, peach, or plum. Pulp the fruit, spread it thinly in a pan, and dry it completely. The finished product is chewy and tasty. Also, use leathers in fruit dishes: just cut them into small pieces and soak them in water. Below are general steps; directions may differ according to individual recipes.
**_Steps:_** Chop or mash fruit to a pulp, removing seeds and pits. Use a blender, if desired, to make a smooth purée. For hard fruits, cut up and place in a large preserving pot. Add a little water and bring fruit to a boil. Simmer until soft, adding sugar or honey to taste if desired, then put fruit through a food mill.
Spread pulp evenly about 1/8 to 1/4 inch (3 to 5 mm) thick on cookie sheets lined with heavy plastic wrap—cut-up freezer bags work well.
Set trays in oven after heat has been turned off or at a temperature of 120° to 140°F (50° to 60°C). Leave oven door open partway.
After several days, when leather is dry on top, lift it carefully from plastic. Remove plastic and turn leather over to dry other side. Set trays back in oven until leather feels as dry and as pliable as its namesake. (You can also dry it until brittle. Both are usable.)
When cool, roll up large pieces of leather in a plastic bag and store in a large jar or crock. If leather becomes damp, remove lids. (To store brittle leather, break it into small pieces and place in jars, crocks, or plastic bags. In plastic bags, it will soften a bit.) True leather will keep for years. Our apple leather, still good, is over twenty years old. You can appreciate how jerky, a sort of dried meat leather from raw, marinated meat, was once such a reliable source of sustenance on the frontier.
**_Yield:_** 1 1/2 to 2 1/2 lb (750 g to 1.25 kg) fruit makes two average-sized cookie sheets of leather.
**_Rhubarb_**
Rhubarb and the small fruits are the backbone of the home fruit garden. All are hardy, all are easy to grow under a wide range of conditions, all are productive in a small area, and almost all survive some neglect. Rhubarb, though strictly speaking a perennial vegetable, is the first fruit we harvest. We look forward to it in the early spring, watching the ground closely for some signs of the emerging tips that will become the first stalks, the most tender of the new season. Processed fruit is fine for the winter, but in late May, when the pile of stored cabbages gives out and the apple bin is empty, we hanker for something truly fresh. No doubt we could rush the rhubarb season by covering a plant or two with plastic to warm the ground and speed the growth of stalks, but we prefer to let nature take its course.
We could not think of any fruit garden without thinking of at least a half-dozen clumps of rhubarb, for pies and stewing and marmalade and juice all season—at least until midsummer. If you enjoy the special flavor of this old-fashioned fruit, then you must plant at least a few clumps, for it makes only a brief appearance in the marketplace. Just six plants, with their broad leaves, make an attractive and impenetrable hedge in our kitchen garden.
Rhubarb is easy to grow. It needs a southern exposure, good drainage, and plenty of fertilizer (manure or compost) and moisture—ingredients essential to maintaining healthy and abundant leaf growth, the key factor in the production of large, healthy stalks. Rhubarb is especially suited to northern climates. The cold winter gives the plant a season of rest, and then it can put all its energy into stalk production in the spring.
**Planting**
Rhubarb plants are not hard to find, either at a friend's or at a local nursery. Two plants will, when established, yield plenty of stalks for one person. Most gardeners like to have two types: a red one for its color and a green one for its many uses. Valentine, Ruby, and McDonald are popular varieties, but sometimes, as we do, people have excellent unknown types taken from a neighbor's garden.
Grow plants from pieces taken from the crowns of dormant plants, each piece having two or three buds. (The crown is the nubby part that is above ground and attached to the roots below.) In the northeast, plant rhubarb in the spring to ensure sufficient root growth before cold weather and freezing temperatures set in. In areas where soil drainage is a problem, grow the plants on a slope, preferably one with a southern exposure.
To plant, make holes about the depth of the blade of a shovel and the width of 2 shovelfuls. The holes should be approximately 1 yd (1 m) apart, as the roots will eventually spread that distance laterally. Water the hole and refill it with a shovelful of well-rotted compost and then a layer of topsoil. Set the piece of rhubarb crown into this mixture, making sure the buds are 2 to 3 inches (5 to 7.5 cm) below the soil line. Firmly tamp the soil around the plant and water it. Weed the plant well all summer, and if the weather is dry and windy, surround the new growth with a layer of mulch.
**Cultivating**
In the second season, after rhubarb shoots have appeared, weed each plant thoroughly. Then fertilize and mulch as follows: spread 3 to 4 shovelfuls of well-rotted compost around the plant, but _do not_ cover the crown; cover the compost completely with cardboard, worn-out nonsynthetic carpet, or any nonporous decomposable material; put a fairly thick layer of straw, sawdust, or whatever mulching material is available on top of the paper to hold it down. This fertilizer-mulch not only provides the roots with food for new plants but also helps conserve moisture for the growing leaves and stalks. In time, all the layers will break down to form a friable humus around each plant. The fertilizer-mulch technique is particularly valuable in areas where the soil is poor and the growing conditions are difficult.
In the fall, when the plants are dormant, throw a shovelful of manure over each plant; fresh manure is okay, and if it contains straw, so much the better. In the spring, when new growth begins, this light mulch will give some protection against late frosts and will help promote early harvests by giving extra nourishment to the soil. With the spring fertilizer-mulch and the fall dressing of manure, special winter protection is unnecessary, even in harsh northern climates.
To get early production in the spring, make a little hotbed around each plant. Place a box, open at both ends, over the emerging tips and heap fresh, strawy horse manure around the sides of the box. Also cover the top of the box. Harvest growing stalks through the open sides, but when warmer weather arrives, remove the box and let growth occur naturally. (A piece of plastic draped over the tips and held in place with a few clumps of manure will bring the same results. Remove the plastic as the stalks grow.)
It is important to pull seed stalks whenever you see them. They have white plumelike flowers, and they take away from the plant's ability to produce stems, the only edible parts of rhubarb. They are attractive, though, so if you have a large enough patch, you can leave a few fluffy white blooms for a great distant accent.
**Harvesting**
Always _pull_ rhubarb stalks, never cut them—cutting leaves open wounds that encourage disease. Grasp each stalk near the base of the plant, twist it slightly in one direction, and pull. Cut off the leaves, which contain a poisonous concentration of oxalic acid, and throw them onto the compost heap. We lay them around the rhubarb plants as we harvest the stalks, to discourage weeds and add to the mulch that is already there.
In the northeast, the rhubarb season extends from late May to mid-July, depending on the varieties. Stalks may be lightly harvested from new plants in the second year of growth and for 3 to 5 weeks in the third year. Thereafter, if the plants are healthy, harvesting can continue until stalk production has noticeably slowed down. A good plan is to harvest all the usable stalks (the large ones) from each plant as they are produced, turning them into sundry products. Generally, the all-red variety is tender early in the season but quickly turns stringy; the greenish-red one is good for all-season production; and the all-green large-stalked kind, often found in abandoned farmyard patches, is best harvested late in the season. After harvesting, plants need a one- or two-week rest, depending on local growing conditions, to regrow sufficient stalks for another picking. Remember, each plant should have a good growth of leaves left at the end of the season for future stalk production.
**Preserving, Canning, Freezing, and Cooking**
When the stalks are still tender, early in the season, it is the best time to start preserving rhubarb. Canned rhubarb is easy. Just layer the cut up fruit with sugar, heat until fruit is tender, then pour into jars and process a short time. Freezing tender rhubarb stalks to use later with strawberries is easy too.
**Canned Rhubarb**
A MIXTURE OF THE ALL-RED AND GREENISH-RED VARIETIES MAKES AN ATTRACTIVE PRESERVE THAT CAN BE USED IN PIES OR EATEN WITH SWEET CREAM AND CANNED STRAWBERRIES; THE DRAINED JUICE CAN BE USED IN GELATIN.
Makes about 7 qt (7 l)
**15 lb rhubarb (7.5 kg)**
**6 1/2 cups sugar (1.625 L)**
Cut early tender stalks into 1-inch (2.5-cm) pieces. For fast cutting, assemble 3 or 4 stalks, lay them flat on a board, and with a sharp knife, karate length of stalks.
Measure cut fruit. Using 1/2 cup (125 mL) sugar to each qt (L) rhubarb, layer fruit and sugar consecutively in a large preserving pot until pot is no more than half full. Cover and let stand for 3 to 4 hours, allowing sugar to draw out juices. Slowly heat mixture to boiling point and boil for 30 seconds, stirring often.
Remove mixture from heat. Ladle into hot scalded canning jars [p. 6], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process 15 minutes in a boiling-water bath [p. 26] or steam canner.
Cut up fruit without sugar, pack it in a bag or freezer container, and freeze until needed. Then it is time for marmalade: just simmer cut-up fruit with sugar and oranges—delicious.
Continue eating bowlfuls of stewed rhubarb and rhubarb pies for as long as you can, but one July day you will know that from then on it is rhubarb-juice time. Good for winter breakfasts, as well as for hot summer days, this juice, made with oranges and lemons, preserves rhubarb flavor in a refreshing way. Rhubarb wine uses end-of-the-season stalks, too. Review Short Course [p. 3–14] before preserving.
**Rhubarb Marmalade I**
THIS IS AN OLD, UNUSUAL RECIPE ADAPTED FROM AN ANCIENT COOKBOOK, ONE OF THE TREASURES OF MY COLLECTION. ACTUALLY, MY HUSBAND, JIGS, DISCOVERED IT WHEN SEARCHING FOR A WAY TO PRESERVE HUNDREDS OF POUNDS OF RHUBARB HARVESTED FROM AN ABANDONED FARMYARD PATCH. WITH THIS RECIPE, YOU CAN COOK FAIRLY LARGE BATCHES AT A TIME, EVEN STIRRING WITH AN ARCHETYPAL LONG-HANDLED WOODEN SPOON. USE TENDER STALKS.
Makes 7 1/2 pt (3.75 L)
**8 juicy oranges**
**4 lb sugar (2 kg)**
**5 lb rhubarb (2.5 kg)**
Remove peel, in quarters, from oranges and set aside. Divide oranges into sections, removing seeds, and cut sections into small pieces. Place orange pieces in a large preserving pot.
Prepare peel. Put quartered orange peel in a saucepan, cover with cold water, and boil until tender. Drain and cool. Scrape off white part of skin with a small sharp knife (or slither it off in one quick motion). Using kitchen scissors, cut peel into tiny pieces. Set aside.
Cut rhubarb into 1/2-inch (1-cm) pieces; add to orange pieces. Cover fruit, bring to a boil, uncover, and boil for 30 minutes, conscientiously stirring bottom of pot with a long-handled wooden spoon to prevent scorching. Stir in sugar and prepared peel. Cook slowly for about 2 hours or until mixture thickens, stirring as necessary.
Remove marmalade from heat. Stir. Pour into hot scalded jars [p. 6], leaving 1/4 inch (5 mm) headroom, and seal with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Swisher Family Never-Fail Pie Crust**
MIKE SWISHER'S DUTCH ANCESTORS SETTLED IN VIRGINIA WHERE THEY WERE FRUIT FARMERS. THE SUCCESS OF THIS UNIQUE HEIRLOOM RECIPE (TESTED IN OUR KITCHEN) IS THE COMBINATION OF A COLD PASTE WORKED INTO THE COLD BUTTER-FLOUR MIXTURE; PART LARD OR SHORTENING WORKS TOO. FOR A 2-CRUST 9-INCH (23 CM) PIE.
**2 cups all-purpose flour, sifted (500 mL)**
**1 tsp salt (5mL)**
**¾ cup + 1 Tbsp unsalted butter, chilled (177 mL + 15 mL)**
**¼ cup water, iced (50 mL)**
Reserve 6 T (88 mL) of the flour and mix it with the ice water to make a paste. Blend cold butter with remainder of flour. When blended completely add flour paste and stir with a fork or work in with finger. Work dough only until it coheres—as little as possible. Form into a ball and chill while you make the filling. Cut dough in half and roll out bottom crust. Fill pie, roll out top crust, and bake at 400º F (200º C) for 10 minutes, then at 350º F (180º C) for about 45 minutes or until filling is bubbling.
Roll out any leftover dough and spread with cinnamon and sugar. Roll up and bake in a 400º F (200º C) oven for 5 to 10 minutes or until lightly browned. These are called Rollies in our family and are absolutely delicious.
To make baked pie shells, prick bottom crust and bake at 400º F (200º C) for 15 minutes.
**Rhubarb-Strawberry Jam**
IF YOU STILL HAVE TENDER STALKS OF RHUBARB WHEN YOUR STRAWBERRIES RIPEN OR FROZE SOME TENDER STALKS, TRY THIS OLD-FASHIONED JAM.
Makes about 2 pt (1 L)
**1 qt hulled strawberries (1 L)**
**1 cup water (250 mL)**
**1 lb rhubarb (500 g)**
**4 cups sugar (1 L)**
In a large stainless-steel pot, lightly mash or chop strawberries. Cut rhubarb into 1/2-inch (1-cm) pieces; add to strawberries. Mix. Add water, cover, and simmer mixture until fruit is soft. Stir in sugar and bring mixture to a rolling boil. Stirring as necessary, boil, uncovered, for about 15 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 6] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**English Muffins**
ONCE YOU HAVE MADE RHUBARB MARMALADE, YOU MUST MAKE YOUR OWN ENGLISH MUFFINS. YOU WILL NEVER BUY THEM AGAIN. BEFORE TOASTING THESE MUFFINS, TEAR THEM APART WITH A FORK TO GIVE THEM THEIR CHARACTERISTIC ROUGH TEXTURE. THIS WAY, THEY CAN SOAK UP MORE BUTTER BEFORE THE MARMALADE IS ADDED.
Makes 1 dozen
**1/4 tsp ginger (1 mL)**
**1 tsp salt (5 mL)**
**1 tsp sugar (5 mL)**
**1 tsp sugar (5 mL)**
**1 Tbsp dry yeast (15 mL)**
**3/4 cup cold water (175 mL)**
**1/4 cup warm water (50 mL)**
**4 cups unbleached white flour (1 L)**
**1/2 cup milk (125 mL)**
**3 Tbsp shortening (50 mL) Cornmeal**
In a large bowl, sprinkle ginger, 1 tsp (5 mL) sugar, and yeast over warm water to help the yeast dissolve faster, especially in a drafty kitchen! Meanwhile, in a pot, scald milk, melting shortening in it at same time. Stir in salt and 1 tsp (5 mL) sugar. Remove from heat and add cold water, cooling mixture to warm.
Add cooled milk mixture to dissolved yeast. Add 2 cups (500 mL) flour. Mix well. Mix in remaining flour or whatever amount needed to make a bread-like dough that is a bit stickier. Turn dough out onto a lightly floured board and knead until smooth and elastic. Return to bowl and grease top of dough. Cover bowl with a clean towel and place in a warm spot to allow dough to rise.
When dough has doubled in size, punch it down and cut into 12 equal pieces. Roll each piece into a ball. Flattening each piece a bit, dip each one, on all sides, in a bowl of cornmeal. Place muffins on a board sprinkled with cornmeal and let rise slightly.
Heat an ungreased pancake griddle on top of stove. When hot, cook muffins until browned on both sides.
**Freezing Rhubarb**
With the acquisition of a freezer late in life, I discovered that the tender stalks of the early harvest freeze well for later use with strawberries or other fruits (rhubarb combines well with blueberries) or for stewing or baking. Just cut up as you would for fresh, stewed rhubarb, and place in a zip-lock freezer bag or container. No sugar needed to keep in fresh condition.
**Rhubarb Juice**
MAKE THIS JUICE WITH TOUGH, STRINGY STALKS. IN WINTER, POUR IT OVER CLEAN FRESH SNOW. IN SUMMER, POUR IT OVER CHIPPED ICE, ADDING A SPRIG OR TWO OF MINT, OR USE IT TO FLAVOR AND SWEETEN ICE TEA (TWO-THIRDS TEA, ONE-THIRD JUICE). ADD TO PUNCH AT ANY SEASON. WHEN SERVING, DILUTE IT WITH WATER TO TASTE.
Makes about 3 1/2qt (3.5 L)
**5 lb rhubarb (2.5 kg)**
**3 qt water (3 L)**
**2 lemons**
**1 1/2 cups sugar (375 mL)**
**2 oranges**
Cut rhubarb into 1-inch (2.5-cm) pieces and cut up oranges and lemons, skin and all. Place fruit in a large preserving pot and add water. Cover and simmer until fruit is soft. Strain through a jelly bag [p. 6] and let drip for several hours or overnight.
Bring juice to a boil, covered, stir in sugar, and bring to a boil again. Boil, uncovered, for 5 minutes.
Remove juice from heat. Pour into hot scalded canning jars [p. 6], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process for 15 minutes in a boiling-water bath [p. 26] or steam canner.
**Rhubarb Sherbert**
DELICIOUS AND REFRESHING. REVIEW AN ASSORTMENT OF BASICS [P. 139–146] FOR GENERAL COMMENTS REGARDING USING AN ICE CREAM MACHINE.
Makes about 1 1/2 qt (1.5 L)
**3 cups unsweetened rhubarb juice (750 mL)**
**2 cups milk (500 mL)**
**1 cup sugar (250 mL)**
Pre-mix juice, sugar, and milk in ice cream machine canister, cover with freezing cover and chill for faster freezing. When ready to begin freezing, turn on the machine, set canister in ice bucket, pour in water and layer ice and salt as directed. Remove freezing cover, insert paddle, and put mixing cover in place, adding more water according to directions. Turn on machine and add more salt and ice as directed when ice begins to melt. Continue freezing until sorbet is the consistency you want. This should take about 20-25 minutes or less. Remove canister, stir to blend if necessary, store in freezer for an hour or more, then you may remove sherbert to a plastic freezer container and return to freezer.
**Rhubarb Wine**
IF AGED FOR SEVERAL SEASONS, THIS WINE TASTES LIKE A GOOD SHERRY. TRY IT CHILLED OR WITH CRUSHED ICE. USE TOUGH, STRINGY RHUBARB.
Makes 3 to 4 qt (3 to 4 L)
**5 lb rhubarb (2.5 kg)**
**1 Tbsp wine or baking yeast (25 mL)**
**1 lemon**
**1 gallon boiling water (4 L)**
**3 lb sugar (1.5 kg)**
Finely chop clean rhubarb and lemon and place in a 2-gallon (8-L) crock or plastic bucket. Add boiling water and cover. Let stand for 3 days, stirring 3 times a day. Strain and reserve juice.
Dissolve yeast in a little warm water. Add sugar and yeast to juice, stirring in well. Cover mixture and let stand in a warm place for about 1 month.
Siphon wine into 1-gallon (4-L) glass jugs, using wads of cotton batting for stoppers. When wine stops fermenting, or stops bubbling, siphon into l-qt (l-L) bottles and seal with caps or corks. Store in a cool, dark, dry place. If sealed with corks, store lying down.
_Note:_ One of the best fruit wines, this is a sweet wine until well aged. For a drier wine, reduce sugar to 2 cups (500 mL).
**Rhubarb Marmalade II**
THIS IS ANOTHER OLD RECIPE, FROM MY LADYE'S COKE BOOK, PUBLISHED IN 1924 BY THE BRATTLEBORO, VERMONT, WOMAN'S CLUB. THIS SMALL BOOK COVERS EVERY ASPECT OF COOKING, AS WELL AS HOUSEHOLD HINTS SUCH AS HOW TO USE EGG SHELLS FOR CLEANING BOTTLES, HOW TO MEND KID GLOVES, AND HOW TO MAKE SOAP.
Makes about 1 1/2 pt (750 mL)
**2 cups unsweetened stewed rhubarb (500 mL)**
**1 cup orange juice (250 mL)**
**4 cups sugar (1 L)**
**1 cup sliced orange (250 mL)**
In a large stainless-steel pot, simmer together fruit and juice, uncovered, for 1 hour. Stir in sugar and continue simmering for about 30 minutes or until mixture thickens.
Remove marmalade from heat. Stir. Pour into hot scalded jars [p. 6], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Rhubarb Bread**
THIS IS A QUICK BREAD. USE TENDER STALKS.
**3/4 cup honey (175 mL)**
**1 1/2 cups diced rhubarb (375 mL)**
**1/2 cup cooking oil (125 mL)**
**1 egg**
**1/2 tsp vanilla (2 mL)**
**1/2 cup chopped nuts or sunflower seeds**
**(125 mL)**
**1 cup buttermilk (250 mL)**
**2 cups unbleached white flour (500 mL)**
**1/2 cup whole wheat flour (125 mL)**
Topping:
**1/2 cup brown sugar (125 mL)**
**1 tsp cinnamon (5 mL)**
**1 tsp salt (5 mL)**
**1 1/2 Tbsp butter (20 mL)**
In a large bowl, mix together honey, oil, and egg; beat well. In a separate bowl, dissolve baking soda in buttermilk and set aside.
Combine white and whole-wheat flours. Add salt. Add flour and buttermilk alternately to honey-oil mixture, stirring after each addition. Stir in rhubarb, vanilla, and nuts or sunflower seeds. Pour batter into a large greased loaf pan or two small ones.
Prepare topping. Mix together brown sugar, cinnamon, and butter in a small bowl. Sprinkle over bread. Bake bread in a 325°F (160°C) oven for about 1 hour or until a sharp knife or cake tester inserted in center comes out clean.
_Note:_ If you do not have buttermilk, mix together 1/2 cup (125 mL) milk and 1/2 cup (125 mL) yogurt or sour cream.
**Stewed Rhubarb**
RHUBARB SHOULD BE STEWED EARLY IN THE SEASON, BUT IF IT IS STEWED LATE, FOLLOW THE ALTERNATIVE METHOD BELOW.
Serves 4
**1 lb rhubarb (500 g) Sugar**
Cut tender stalks into 1-inch (2.5-cm) pieces and place in a stainless-steel pot. Add a little water, cover and, stirring to prevent scorching, boil rhubarb until tender, only a few minutes. Remove from heat. While still warm, add sugar to taste.
_Note:_ An alternative way to make stewed rhubarb is to layer cut-up stalks with sugar, using 1/2 cup (125 mL) sugar to each qt (L) rhubarb. Cover and let stand for 2 hours. Bring to a boil and simmer until fruit is soft, stirring occasionally.
_Variation:_ To make _Rhubarb Fool,_ purée stewed rhubarb and mix with sweet cream.
**Rhubarb Pie**
THE EGG MAKES THIS RHUBARB PIE A LITTLE DIFFERENT, QUITE RICH, BUT NOT OVERLY SWEET. USE ONLY TENDER STALKS.
**2 1/2 cups diced rhubarb (625 mL)**
**1 egg, beaten**
**Dash salt**
**1 cup sugar (250 mL) Pastry (see below)**
**2 Tbsp flour (25 mL)**
In a large bowl, mix together diced rhubarb, sugar, flour, beaten egg, and salt. Put filling in an unbaked 9-inch (23-cm) pie shell and cover with a top crust. Prick top crust a few times with a fork. Bake pie in a 375°F (190°C) oven for about 50 minutes or until inner juices boil over a bit.
**Rhubarb Upside-Down Cake**
THE ORANGE-CINNAMON GLAZE COMBINES WELL WITH THE SWEETENED RHUBARB TO GIVE THIS DESSERT A DISTINCTIVE FLAVOR. USE TENDER STALKS.
**2 heaping cups diced rhubarb (500 mL)**
**1/2 tsp salt (2 mL)**
**1/4 cup shortening or lard (50 mL)**
**2/3 cup sugar (150 mL)**
**1 Tbsp flour (15 mL)**
**1 egg, beaten**
**1 tsp grated orange rind (5 mL)**
**3 Tbsp milk (50 mL)**
**1 tsp cinnamon (5 mL)**
**1 cup flour, sifted (250 mL)**
Glaze:
**2 Tbsp sugar (25 mL)**
**2 Tbsp orange juice (25 mL)**
**2 tsp baking powder (10 mL)**
**1 Tbsp sugar (15 mL)**
Arrange rhubarb in a lightly greased 9-inch (2.5-L) square baking dish. In a bowl, mix together 2/3 cup (150 mL) sugar, 1 Tbsp (15 mL) flour, orange rind, and cinnamon. Sprinkle over rhubarb, stirring lightly with a fork to distribute sugar mixture.
Sift together 1 cup (250 mL) flour, 2 Tbsp (25 mL) sugar, baking powder, and salt. Cut in shortening or lard. Mix in beaten egg and milk. Drop batter by spoonfuls over rhubarb mixture. Bake cake in a 350°F (180°C) oven for 25 minutes.
Meanwhile, prepare glaze. Mix together 2 Tbsp (25 mL) orange juice and 1 Tbsp (15 mL) sugar. Drizzle over baked cake and return to oven for 15 minutes.
_Note:_ To make with canned rhubarb [p. 41], substitute 1 qt (1 L) drained canned rhubarb for fresh and omit 2/3 cup (150 mL) sugar.
**_Strawberries_**
Although an aura of romantic nostalgia surrounds the wild North American strawberry, the backyard gardener can be well satisfied with establishing and maintaining a patch of the cultivated type. The cultivated strawberry is a hybrid of two wild species, the American _Fragaria virginiana_ and the South American _Fragaria chiloensis,_ bred for vigor, large fruit, and disease resistance.
Some insist that wild strawberries have a more intense flavor than the domesticated. But there are so many varieties with differing degrees of sweetness and taste that it does not seem at all compromising to have a splendid patch of cultivated strawberries in one's own backyard, ready to eat fresh with cream, top a shortcake, or turn into jam.
Strawberries were first cultivated in 1840, and it is not surprising that since then they have grown in popularity with home gardeners. Commercial production has increased as well, even though the wild species are uniquely flavored and it is a pleasant pastime to roam favorite meadows and hillsides in search of the small bright-red berries.
There is, of course, one all-important difference to keep in mind: wild strawberries require only picking; cultivated strawberries, on the other hand, require faithful attention. In fact, of all the fruit you are likely to grow in a small fruit garden, strawberries require the most work. Once the essentials of successful production are understood, however, maintenance, while at times laborious, is not difficult. The reward, a bountiful supply of delicious fruit, is worth the effort.
The essence of sustained high-quality strawberry production can be simply stated: strawberries _must_ be weed free. For the gardener, this means getting down on hands and knees in the early spring and pulling out _each_ weed from among the plants and between the rows. Otherwise, the roots of the weeds become inextricably intertwined with the many strands of the growing strawberry plant. This is a serious business, one to which you can devote whole days.
If you want your strawberry patch to be more than a one-season affair, the following plan, based on single-row cultivation, not matted-row, is guaranteed to bring good results.
**Planting**
If the area available for your strawberries is small, stick to one variety; if not, try two to see which one you prefer. There are an endless number of strawberry varieties to choose from among June bearers (the traditional type). Everbearers produce an early and late crop, produce few runners, and can be grown where summer temperatures are consistently in the high range. Day neutral types (true everbearers) are hardier, produce better quality fruit and few runners; plants should be replaced every other year or even every year to maintain high productivity. The everbearing and day neutral strawberries are best for planting in a strawberry jar or space-saving pyramid. Consult your local Cooperative Extension office for detailed information about the best varieties of strawberries for your area (see Appendix for locating the office nearest you). Your choice also depends, of course, on your taste. We prefer the June bearing Bounty for its intensely sweet flavor, but commercial growers don't because this strawberry is not as large as other varieties. In any event, it is best to get plants (consisting of crowns and roots) from local nurseries rather than from a friend, as nursery-grown stock, with its good root system, is cultivated specially for the purpose of starting new plants.
Twenty-five plants are plenty to start with: under reasonable conditions, one plant will produce at least 1 qt (1 L) strawberries. It is important to start small with strawberries. Learn their culture before getting in too deep because there is nothing more discouraging than facing a large bed of weedy strawberries in the spring. Then it's easy to give up and say, "The weeds took over." If you plan sensibly, the weeds will never have a chance.
Choose a well-drained sunny spot for your strawberry bed, either a level area or a slight, not steep, incline. Enrich sandy or clay soil with a lot of organic matter. It's a good idea to cultivate the site a season in advance to help prevent the growth of weeds. Plow or dig the plot by hand in the spring, then work in a good layer of well-rotted compost or manure. The following spring, before planting, smooth the area, making a fine, level surface. Although strawberries are usually planted (and sold) in the spring, an early-fall planting can be successful if there is sufficient moisture to help establish the plants. A dry windy spell means death to new plants.
To plant, hoe rows about 4 to 5 ft (1.25 to 1.5 m) apart. Mark the rows with string, and with a trowel, dig shallow holes about 18 to 24 inches (45 to 60 cm) apart all along the rows. Water each hole, then set in the strawberry plant, spreading out the roots to comfortably fit the space. Firmly tamp the earth around the roots to eliminate air pockets, and make sure the crown of the plant _is exactly_ at soil level—correctly set crowns bring hardier, more vigorous plants. While planting, keep the roots of the unplanted strawberry crowns in a bucket of water. If left out in the air, they could easily dry out and wilt.
When all the plants have been set in the ground, lay down a heavy organic mulch between the rows, the second most essential requirement for maintaining high-quality strawberry production. Mulch conserves moisture, helps eliminate weeds, adds nutrients, and improves the texture of the soil. Mulch softens the soil, making spring weeding easier.
We used a mulch made up of two layers of material: a thick nonporous bottom layer of old paper and a 2-inch (5-cm) top layer of eel grass (gathered from Nova Scotia beaches), straw, or rotted sawdust. Eel grass and rotted sawdust also keep slugs away from strawberries during cool, damp weather. With this heavy mulch between the rows, no winter protection was necessary.
If the weather is dry during the week after planting, water the strawberry plants daily until they are established.
**Cultivating**
Gardeners are usually advised to pick the flowers of first-year strawberry plants to prevent the setting of fruit. The plant, it is believed, needs to put all its energy into maintaining vigorous growth. But it's a matter of choice: we have found that particularly vigorous varieties are not harmed by letting them set fruit the first year. In any case, once the first-year plants are established, in the early spring of their second year, pull out every weed around the plants. (Use a trowel to dig out the roots of the weeds.) Then fill in any spaces left from winter kill with runner plants that have spread in and between the rows. Do not allow the plants in the rows to set more than 2 to 3 runners apiece (simply dig them up), but save some plants to enlarge the patch. This method of cultivation will produce a series of solid narrow rows, much easier to harvest than wide matted rows that have been allowed to fill in with runners. Plants in the single-row method tend to be more vigorous and produce larger berries.
Then there is little to do but wait for the berries and watch for the reappearance of any weeds. Once early-spring weeding has been done and the mulch refurbished or replaced, little weeding is necessary. It is a good idea, though, to root out weeds the first few times you harvest berries. That will probably be sufficient until the end of the season, when you should weed once more.
After the harvest is over, mow the patch, setting the mower blade high to avoid cutting the crowns of the plants. Mowing gets rid of dead leaves and helps rejuvenate the plants by encouraging new leaf growth, essential for maintaining vigorous plants. New growth before the fall gives the plants a better chance of surviving harsh winter conditions.
The third year of growth is the strawberry plant's last season of bearing. For this reason, section your plot into first-, second-, and third-year plants so that there are always strawberry-bearing plants, as well as new replacer plants. Select replacer plants from the extra runners in the established rows. Use only the best-quality plants— the first two runners are considered the best— young and healthy, with a good root system. Introduce new commercial stock eventually, though, because rows established exclusively with homegrown replacements will "run out"—that is, the plants will decline in vigor and their berries in size. A good plan is to buy new stock every fourth year, after the end of the three-year cycle of every newly established section of the strawberry bed.
**Harvesting**
In the northeast, the strawberry season usually lasts three weeks, beginning on our farm in early July. When the plants begin to bear heavily, pick strawberries every other day, in the morning. If you pick them while a little underripe—not fully red—store them in a cool basement for 24 hours. At the end of that time, they will be fully ripe and ready to eat fresh or to make into canned preserves. For jam making, pick a mixture of ripe and underripe fruit and use it right away or store in the freezer as directed. When picking, carry two strawberry baskets down each row—one for large perfect berries, the other for culls, or odd-shaped, damaged, or small berries. Fill the perfect-berry box to almost overflowing because the contents will sink slightly. Use up the culls right away and never hull cultivated strawberries until you are ready to use them.
**Preserving, Canning, Freezing, And Cooking**
The easiest and everyone's favorite way to preserve the essence of strawberries is in jam. Culls make fine jam, but, as already mentioned, use them up right away, for they deteriorate faster than perfect fruit. Canned strawberries are also delicious, and because a lot of berries can be processed at once, canning is a good way to preserve a large quantity of fruit. Do not believe contemporary cookbooks that tell you canned strawberries are not worth doing. Freezing chopped strawberries, mixed with a relatively small amount of sugar, preserves the color and flavor of the fruit, for which there are many uses. Strawberry juice and strawberry sass are luxuries, but why not try them? You can afford to be extravagant with your own strawberry patch—your reward for weeding, and keeping the plants in top condition. More simple but just as satisfying are strawberries and cream. Too often strawberries are gussied up in all sorts of dishes, usually difficult and time consuming to prepare. Having said that, however, I do recommend trying some of the fancier cooking recipes. Review Short Course [p. 3–14] before preserving. And on a warm summer evening, with friends gathered around, why not drop a large, perfect strawberry to soak in everyone's glass of champagne?
**The World's Best Strawberry Jam**
STRAWBERRY JAM IS REALLY THE REASON FOR THIS BOOK, BECAUSE THROUGH MAKING IT, I DISCOVERED THE PRINCIPLES OF GOOD JAM MAKING WITHOUT ADDED PECTIN, AND FRUIT PRESERVING IN GENERAL. OVER THE YEARS, THIS JAM HAS BEEN THE STAR OF STARS, SOMETHING SPECIAL TO EVERYONE WHO HAS TASTED IT. YET THE RECIPE IS SO SIMPLE YOU MAY QUESTION ITS VALIDITY. HARVEST RIPE AND UNDERRIPE BERRIES AND COOK THEM QUICKLY IN SMALL BATCHES, AND I GUARANTEE A STRAWBERRY JAM AS YOU HAVE NEVER TASTED BEFORE.
Makes 1 1/2 pt (750 mL)
**4 cups hulled strawberries (1 L)**
**2 1/2 cups sugar (625 mL)**
In a large stainless-steel pot, chop berries enough to produce a chunky, but manageable, spread. Partially thawed frozen and chopped berries work very well. Cover and heat until simmering, stirring as necessary. Stir in sugar and bring to a rolling boil. Stirring occasionally, boil, uncovered, for about 10 minutes or less with prime fruit, until mixture thickens and begins to cling to bottom of pot. This jam should sheet off a metal spoon [p. 6–7].
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 6] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in the boiling-water bath [p. 26] or steam canner for 15 minutes.
**Cracked-Wheat Cereal**
THERE'S NO NEED TO TELL YOU HOW TO USE STRAWBERRY JAM, BUT YOU MAY NOT HAVE THOUGHT OF TRYING A SPOONFUL IN YOUR MORNING CEREAL.
Serves 2 to 3
1 cup medium coarse-ground wheat berries (250 mL)
2 cups hot water (500 mL)
1 scant tsp salt (5 mL)
In a saucepan, stir ground wheat berries into hot water. Add salt, cover, and bring to a boil. Reduce heat and simmer for about 15 to 20 minutes or until all water is absorbed. Ladle into bowls and add jam and cream to taste.
**Canned Strawberry Preserves**
THESE PRESERVES ARE NOT OVERLY SWEET—GREAT IN A WINTER STRAWBERRY-SHORTCAKE OR IN BAVARIAN CREAM. USE FIRM RIPE WHOLE BERRIES.
Makes 3 1/2 qt (3.5 L)
6 qt hulled strawberries (6 L)
3 cups sugar (750 mL)
In a large preserving pot, layer strawberries and sugar consecutively. Cover and let stand in a cool place for 5 to 6 hours.
Heat mixture slowly, uncovered. Stir occasionally but be careful not to bruise fruit. Bring to boiling point and remove from heat.
Ladle preserves into hot scalded canning jars [p. 6], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process jars in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Freezing Strawberries**
Chunky Strawberry Sauce
To make a chunky sauce that can be used to top shortcake, mix with other fruits, canned or frozen, or to stir into yogurt or cereal, cut up hulled fruit and for every 1 qt (1 L) stir in 1/2 cup (125 mL) sugar. I use a hand food processor which is very easy to use, clean, and enables me to process a lot of strawberries in a short time (see Appendix for sources). Leave fruit/sugar mixture at room temperature about 10 minutes to bring out juices. Pour into freezer containers and freeze.
**Frozen Whole Strawberries**
To freeze whole, hulled strawberries if you're too busy to make them into jam right away, simply put them in a freezer bag. Freezing does not adversely affect their jam making properties.
**Strawberry Sass**
MANY YEARS AGO, IN NEW ENGLAND, "GARDEN SASS," A CORRUPTION OF THE ENGLISH TERM _GARDEN SAUCE,_ REFERRED TO ALL THE VEGETABLES RAISED IN THE GARDEN. MY FAMILY HAS CORRUPTED THE TERM STILL FURTHER TO MEAN "FRUIT SAUCE." USE DEAD-RIPE STRAWBERRIES, THOUGH OVERRIPE AND DAMAGED ONES ARE FINE.
Hulled strawberries
Sugar
In a large preserving pot, simmer strawberries, covered, with a little water. Mash berries and strain through a jelly bag [p. 6]. Let drip for several hours or overnight. Measure juice and cook 2 qt (2 L) at a time in a large stainless-steel pot. Cover, bring to a boil, and stir in 3 cups (750 mL) sugar to each qt (L) juice. Bring to a boil again and cook, uncovered, until desired thickness is reached, about 15 minutes.
Remove sass from heat, let subside, and stir. Pour into hot scalded jars [p. 6], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Strawberry Juice**
THE FLAVOR AND COLOR OF THIS JUICE IS ALMOST UNBELIEVABLE. USE OVERRIPE STRAWBERRIES, AND WHEN SERVING, DILUTE WITH WATER TO TASTE.
Hulled, mashed strawberries
Sugar or honey
In a large preserving pot, combine strawberries and water, using 1/2 cup (125 mL) water to each qt (L) mashed fruit. Simmer, covered, until juice runs freely and strawberries are tender. Strain through a jelly bag [p. 6] and let drip for several hours or overnight.
Measure juice. Heat, and stir in 1/2 cup (125 mL) sugar (or a little less honey) to every qt (L) juice. Bring to a boil and let boil, uncovered, for 5 minutes.
Remove juice from heat. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Strawberry Leather**
We learned to make leathers from a wonderful friend in her 80s. She is remarkably thrifty, not to mention active. There is no such thing as wasted produce at her home. Use overripe or damaged fruit as long as the flavor is not impaired.
In a large preserving pot, mash hulled strawberries to a pulp. Spread pulp 1/4 inch (5 mm) thick on cookie sheets lined with heavy plastic wrap. Set pans in sun or in a 120°F (50°C) oven. When pulp is completely dry on one side, about 3 days, turn over and dry other, removing plastic.
When leathers can be lifted off pans in sheets, they are done. Cool leathers on a cake rack, cut into pieces, and store in plastic bags or any covered container. Check containers occasionally for mold. Or dust leathers with icing sugar and stack in layers with wax paper in between.
**Strawberries and Cream**
We are aficionados of this simple dish and have put a lot of thought and care (though not much work) into its preparation. You can, of course, buy different kinds of cream—sweet, whipping, or sour— and if you are lucky enough to own a cow, preferably a Jersey, "the cream cow," you can make your own first-rate products. But do not just cut up strawberries and serve them with some sort of cream. That's a common dish. Here's our version.
Prepare strawberries. Hull and cut up ripe strawberries and place in a bowl. Sprinkle with a little sugar, cover, and let stand at room temperature for 1 hour to draw out juices.
Prepare sweet cream. Chill fresh milk for at least 24 hours. Skim cream from milk and pour over bowl of prepared strawberries. One gallon (4 L) fresh milk makes about 1 qt (1 L) sweet cream.
_Variations:_ To make _whipping cream,_ chill fresh milk for 24 hours. Skim cream from milk. Place cream and a beater in a bowl and chill for several hours. Whip cream and add a dollop to a bowl of prepared strawberries.
To make _sour cream,_ let sweet cream stand, covered, at room temperature for 24 hours. Add vinegar to taste.
To make _crème fraiche_ , stir _whipping cream_ into _sour cream_ to taste.
**Devonshire Cream**
DEVONSHIRE CREAM IS NOT ACTUALLY MEANT TO BE EATEN WITH FRESH STRAWBERRIES. REAL DEVONSHIRE CREAM IS THICK ENOUGH TO SPREAD ON HOT SCONES—THE WAY THE BRITISH EAT IT—AND IS LAVISHLY TOPPED WITH STRAWBERRY JAM. BUT IF YOU MAKE YOUR OWN, EITHER FROM FRESH MILK OR BY THINNING CREAM CHEESE WITH SWEET CREAM, YOU CAN MAKE IT AS THICK AS YOU CHOOSE.
Makes 1 1/2 qt (1.5 L)
Let 2 gallons (8 L) fresh whole milk stand, covered, at room temperature for 6 to 12 hours, depending on how thick and tangy you want cream.
In a large pot, heat milk slowly, uncovered, until surface barely begins to wrinkle. _Do not boil milk; do not stir it while it is heating._
Remove milk from heat, cover, and chill for 12 to 24 hours, depending on desired thickness. Spoon off thick cream and chill until ready to serve.
**Yogurt**
WHILE ON THE SUBJECT OF DAIRY PRODUCTS TO ACCOMPANY FRUIT, IT SEEMS LIKE A GOOD TIME TO INTRODUCE YOGURT, FRUIT'S NATURAL PARTNER. SERVE WITH STRAWBERRIES OR OTHER FRUIT ANY WAY YOU LIKE.YOU CAN'T GO WRONG.
1 qt milk, raw or pasteurized (1 L)
4 Tbsp culture from your preferred yogurt (58 mL)
Heat milk to 180° F (82° degrees C). Let cool to 116° F (47° C). Pour through strainer into an insulated tub (I use a Yogotherm—see Appendix under Cheesemaking Supplies). Stir in culture, stirring well, cover tightly, and leave undisturbed about 4 hours. Remove yogurt container and chill overnight. Alternatively you can stir culture into a little of the cooled milk, then stir this into the rest of the milk, pour through a strainer and stir in as directed.
**Greek Yogurt and Yogurt Cheese Spread**
Strain yogurt through two thicknesses of cheesecloth until the desired consistency. For Greek-style yogurt, let drain until thicker than regular yogurt but not as thick as yogurt cheese. For cheese, let drain until the consistency of a soft spread. Great with strawberry jam and all jams and jellies.
**Strawberry Frosting**
FRESH-STRAWBERRY FROSTING IS A NICE WAY TO MARK THE END OF THE HARVEST WHEN JUST A FEW BERRIES WILL GO A LONG WAY. THIS IS ALWAYS A BIG HIT, ESPECIALLY ON A SIMPLE CAKE.
Frosts 1 medium-sized cake
1/4 tsp salt (1 mL)
1 tsp vanilla (5 mL)
3 Tbsp butter (50 mL)
2 1/2 cups sifted icing sugar (625 mL)
1/4 cup crushed ripe strawberries (50 mL)
Halved perfect strawberries
In a large bowl, blend together salt and butter. In a separate bowl, combine crushed strawberries and vanilla. Add icing sugar and crushed berries alternately to butter. Stir after each addition. Add only enough berries to achieve desired consistency.
Spread frosting over top and sides of cake. Garnish with halved perfect strawberries and let stand for several hours.
_Note:_ In winter, substitute strawberry juice [p. 56] or 1 to 2 Tbsp (15 to 25 mL) strawberry preserves [p. 54–55] for fresh berries.
**Two-Egg Cake**
THIS IS A GOOD CHOICE FOR STRAWBERRY FROSTING. ALSO USE IT AS A BASE FOR STRAWBERRY SHORTCAKE.
2 cups sifted cake flour (500 mL)
1 tsp vanilla or other flavoring (5 mL)
2 1/2 tsp baking powder (22 mL)
1/4 tsp salt (1 mL)
2 eggs, separated
1/2 cup shortening (125 mL)
2/3 cup milk (150 mL)
3/4 to 1 cup sugar (275 to 250 mL)
Sift together flour, baking powder, and salt. In a bowl, cream shortening. Beat in sugar, and cream until fluffy. Add vanilla.
In a small bowl, beat egg yolks well; beat into shortening mixture. Add flour and milk alternately to shortening-egg mixture, beating smooth after each addition.
In a separate bowl, beat egg whites until stiff; fold into cake batter. Pour batter into a greased 8-inch x 12-inch (3-L) cake pan or standard angel-food cake pan. Bake in a 350°F (180°C) oven for about 50 minutes.
_Note:_ To assemble strawberry shortcake, cut cake into squares and split each square in half. Ladle desired strawberry topping onto bottom layer. Cover with other piece of cake and garnish with more strawberry mixture and with whipped cream [p. 58].
**Strawberry Tart**
1 qt ripe strawberries (1 L)
2 to 4 Tbsp flour or cornstarch (25 to 75 mL)
1 cup sugar (250 mL)
1/4 tsp salt (5 mL)
1 1/4 cups cold water (300 mL)
1 Tbsp lemon juice (optional) (15 mL) Baked pie shell [p. 46]
Whipped cream [p. 58]
Hull strawberries, setting aside some for garnish. Cut up remaining berries, place in a pot, and stir in sugar, salt, and lemon juice. In a bowl, dissolve flour or cornstarch in cold water; add to strawberries. Stirring well, bring mixture to a boil, reduce heat, and simmer until thick. Remove from heat, cool, stirring occasionally, and chill.
Pour chilled filling into a baked 9-inch (23-cm) pie shell or 6 baked tart shells and top with whipped cream. Garnish with whole perfect strawberries.
**Strawberry Bavarian Cream**
HULL STRAWBERRIES AND SET ASIDE A HANDFUL OF PERFECT ONES FOR DECORATING. IN A BOWL, LIGHTLY MASH REMAINING STRAWBERRIES AND STIR IN SUGAR. COVER AND LET STAND AT ROOM TEMPERATURE FOR 1 HOUR.
Serves 6
1 qt ripe strawberries (1 L)
1/2 cup boiling water (125 mL)
3/4 cup sugar (175 mL)
2 cups whipping cream [p. 58] (500 mL)
2 Tbsp gelatin (25 mL)
1/4 cup cold water or strawberry preserves juice [p. 56] (50 mL)
Meanwhile, in a saucepan, sprinkle gelatin over cold water or preserves juice; let stand for 5 minutes. Stir to dissolve gelatin. Add boiling water and heat mixture over low heat, stirring until all gelatin granules are dissolved. Cool, stir in crushed fruit, and chill until jelled.
In a bowl, whip cream; fold into chilled gelatin-strawberry mixture. Pour Bavarian cream into a cold wet mold, bread pan, or individual custard cups. Garnish with a handful of strawberries, whole or halved, and chill for several hours or until firm.
_Note:_ To make with strawberry preserves [p.54–55], substitute 1 qt(l L) drained preserves for fresh strawberries and omit sugar.
To make Bavarian-cream tarts, pour chilled Bavarian cream into 6 baked tart shells or small baked pie shells [p. 46]. Garnish with fresh fruit and serve.
**Viennese Chocolate Torte**
RECOMMENDED FOR FANS OF OLD WORLD CONFECTIONS, THIS DESSERT IS OUTRAGEOUS. SAVE IT FOR A BIG CELEBRATION, FORGET ABOUT CALORIES AND EXPENSE, AND GLORY IN THE MIXTURE OF FRUIT AND CHOCOLATE.
6-oz pkg semisweet chocolate chips or bits (175 g)
1/2 pt heavy cream, whipped (p. 57) (250 mL)
4 eggs, separated
1/3 cup sugar (75 mL)
1/3 cup grated nuts (75 mL)
Frosting:
Sweetened crushed strawberries or strawberry jam [p. 54]
6 sq semisweet chocolate
1/2 cup butter (125 mL)
In the top of a double boiler, melt chocolate chips over boiling water; let cool. Meanwhile, in a bowl, beat together egg yolks and sugar. _Beat!_ Add nuts and melted chocolate chips.
In a separate bowl, beat egg whites until stiff. _Stiff!_ Fold into chocolate mixture. Pour batter into a greased 9-inch (2.5-L) spring-form pan. Bake in a 325°F (160°C) oven for 25 to 30 minutes. Cool completely.
Place cake on a serving platter and chill. Spread crushed strawberries (no juice) or strawberry jam over top of cake. Spread whipped cream over tops and sides. Chill cake thoroughly.
Meanwhile, prepare frosting. Place squares of semisweet chocolate and butter in top of a double boiler. Melt over boiling water, beating well. Cool to lukewarm and spoon over top of chilled cake.
**Strawberry-Rhubarb Parfait**
I'VE ALWAYS BEEN ATTRACTED TO PARFAIT GLASSES AND HANKERED TO FILL THEM WITH SOMETHING . . . SIMPLE. ONCE YOUR PANTRY SHELVES ARE FILLED WITH DELICIOUS FRUIT PRODUCTS, INFINITE COMBINATIONS ARE POSSIBLE. IF THE TENDER STALKS OF RHUBARB ARE NO LONGER AVAILABLE, MAKE STEWED RHUBARB FROM THE CUT-UP FROZEN STALKS [P. 45].
Stewed rhubarb
Whipped cream
Chunky strawberry sauce
In the parfait glass, layer stewed rhubarb, whipped cream, and chunky strawberry sauce in that order; repeat until glass is filled. Top with a perfect strawberry. Chill and serve. Don't forget the long-handled spoons.
**_Raspberries_**
It's odd, but in an era of relative abundance, many look back with nostalgia to the days before convenience foods, supermarkets, and super highways. Many long for the time when they could buy locally produced, freshly picked raspberries by the quart, as many as they liked, for preserving or for eating fresh with lots of _real,_ thick sweet cream, the top cream poured off the pasteurized milk delivered in glass bottles by the local milkman.
In 1858, in just one small area of New York State, 1 million pints of fresh raspberries were shipped to market. By the 1960s, commercial raspberry production had declined drastically. Now fresh raspberries are seldom seen on the market, and when they do appear, they are so expensive that buying enough for a breakfast bowlful seems like the height of extravagance.
The decline in commercial raspberry production coincided with the development of modern agriculture, which is based on mechanization, centralization of production, and mass marketing. Raspberry production thrived when agriculture used primarily hand labor. A good picker could harvest 200 lb (100 kg) fruit a day; five pickers could harvest an acre every two days. The fruit was then quickly shipped to nearby markets, essential for successful raspberry production because the delicate berries deteriorate rapidly in transit. Now the areas of cultivation are so far away from market that the produce is usually sold processed rather than fresh. As for top cream and the milkman, forget it.
Of course, picking wild raspberries is an alternative, but cultivated raspberries are superior in size and flavor. Moreover, cultivated plants are prolific: anyone with enough backyard space to plant 12 canes can expect to harvest 10 to 15 qt (10 to 15 L) a season— _if_ the plants are well cared for. This means supplying the plants with plenty of moisture and enriched well-drained soil, and keeping the patch free of weeds; if perennial weeds take hold, the patch is doomed. These conditions, though not difficult, must be met every year to ensure a bountiful harvest. Neglect will result in fewer, smaller berries, weak, diseased plants, and eventually death of the whole planting. If planted and cared for conscientiously, a patch will last almost indefinitely; and by choosing two different varieties, the season can be extended. Berries by the bowlful will be standard breakfast fare—for generous amounts of thick cream, tether a Jersey cow in your yard—and there will be lots left over for preserving.
**Planting**
The good news is that there are numerous varieties of raspberries to choose from to suit your particular needs and climate, not only traditional red berries, but purple, black, even yellow. The development of fall-bearing types has opened up new possibilities for a late crop of berries, too, but they need to be managed to produce a full late crop in the north (see below). No matter which types you buy, select a sunny spot out of the wind if possible, but not a frost pocket. A slope with a northern exposure is often recommended because it conserves moisture, so important for high-quality production. Also, such a site is not so susceptible to winter freezing and thawing, which causes plants to heave. If a northern exposure is not possible, however, plant canes wherever drainage conditions are good.
Prepare the spot a season ahead by plowing or digging up the earth and enriching it with compost or manure, as recommended for strawberry production. Raspberry canes tolerate a wide range of soils, but, in addition to good drainage, they must be supplied with nutrients. In northern areas, plant canes in the spring rather than the fall so that the roots have time to establish themselves before winter.
To plant, mark rows about 4 ft (1.25 m) apart, using string; two to four 40-ft (12-m) rows are more than adequate for the average family, ensuring a good supply of fresh berries. Along the rows, make holes at 2-ft (60-cm) intervals: push a spade about 6 inches (15 cm) into the soil, and work it back and forth until there is an opening about 2 inches (5 cm) wide. This spacing is a little closer than ordinarily recommended. Close planting will provide just enough shade to help conserve moisture, which encourages the production of larger berries.
Trim each cane to about 6 inches (15 cm). Water each hole and then plant each cane, firmly tamping the dirt around the roots with your feet. In a dry spell, water the plants regularly until they show new growth.
**Cultivating**
Successful cultivation involves yearly fertilizing, mulching, pruning, and topping. It also entails making and adjusting supports for keeping the canes in neat rows.
In the late fall or winter, following the spring planting, fertilize the raspberries by heaping well-rotted compost or manure on the dormant plants. The next spring, the first season of bearing, put mulch between the rows. Lay down a thick layer of paper and cover it with several inches of _rotted_ sawdust or wood chips. This mulch will reduce weeds, conserve moisture and, not least of all, make it easier to walk up and down the rows during harvesting. When the time comes, you will see that this sawdust carpet is a blessing.
Because raspberry plants are biennial, fruiting always occurs in the second year. After fruiting, canes that have produced berries must be pruned from the plants, either by hand or with small clippers. In places where winters are severe, with lots of wind and blowing snow, it is best to wait until early spring to remove old canes because they will help protect young, tender canes from whipping. It is easy to distinguish old canes from new: old ones are tan and brittle, and they break easily. After they are removed, lay them down between the rows as added mulch.
Each spring, renew the paper-sawdust mulch and remove suckers, or secondary shoots, that have grown between the rows. Allow each row to fill out about 12 inches (30 cm), removing overcrowded plants and weak canes. Top off tall canes with hand clippers so that berries can be picked conveniently.
Make supports for the patch. Pound a 6-ft (2-m) wooden or metal post into the ground at the beginning, middle, and end of each row. Attach two 18-inch (45-cm) crosspieces to each post, one about 2 ft (60 cm) from the ground, the other about 4 ft (1.25 m). On the left end of the top crosspiece at the beginning of the row, attach strong cord or wire. String the cord to the center post, wrap it around the left end of the top crosspiece, pulling the cord taut. String the cord to the third post and wrap it securely around the left end of the corresponding crosspiece. Pull the cord around to the right end of that crosspiece and proceed up the other side of the row. Repeat the process for the bottom crosspieces. The raspberry canes will then be tightly penned in.
Tuck all canes inside these supports each spring; ruthlessly remove ones growing outside. Tighten the cords when necessary. Wooden posts will likely need replacing eventually, but if you treat the part below ground with creosote, as fence posts are, they will last at least 10 years, depending on the climate.
Fall-bearing raspberries are vigorous plants. If left on their own, wintered-over canes will produce fruit in summer, but the fall crop will be reduced. To get a full crop of fall berries, cut back canes after frost or at the end of winter, before the new growing season begins. The following season, as new plants grow up, remove the outside suckers and thin plants to about 6 inches (15 cm) apart. The remaining plants will put all their effort into producing a later, more plentiful crop of berries, by September and October in the north.
**Harvesting**
Harvesting is the culmination of all your work, literally the fruit of your labor. In northern climates, the raspberry season usually begins in mid-to late July, and picking the large moist berries will become a pleasant morning occupation. Use 1-pt (500-mL) containers for picking to prevent the berries on the bottom from being squashed. Unless you plan to eat raspberries for breakfast, wait until the sun has dried the dew: raspberries keep better if they are not damp. (For this reason, use rained-on fruit for cooked dishes, sauces, or juice.) Do not forget to pick the berries closest to the ground, as this well-shaded fruit often grows the largest. And, of course, do not pick raspberries with their stems. It's not easy to do, but I have seen it done.
**Preserving, Canning, Freezing, And Cooking**
Surely, jam is the most popular raspberry product. Raspberries contain plenty of natural pectin, so when they are mixed with the right amount of sugar, they make a firm jam. Canned raspberries, like canned strawberries, are simple to make. The berries break down somewhat during processing, but the result is still delicious. Berries frozen with a little sugar maintain their shape better. Red currants will be ripening while raspberries are still plentiful, so do not be afraid to mix the two, in preserves, juices, jams—whatever you like. (Check the chapter on red currants for more ideas.) Raspberry juice and wine is for the end of the harvest, when you have made enough jam and preserves. Raspberries are splendid in baking, too: in cakes, tarts, squares, or puddings.
No matter what you choose to do with your harvest, however, do it without delay! Raspberries deteriorate almost as soon as they are picked. You can refrigerate them for two or three days, but the quality does not compare to that of newly picked fruit. Review Short Course [p. 3–14] before preserving. Use the directions and recipes below for black raspberries. But keep in mind that these are seedier.
**Raspberry Jam**
THE BASIC APPROACH TO JAM MAKING, OUTLINED IN THE RECIPE THE WORLD'S BEST STRAWBERRY JAM, HOLDS TRUE FOR RASPBERRY JAM AS WELL. USE A MIXTURE OF RIPE AND UNDERRIPE RASPBERRIES.
Makes 1 1/2 to 2 pt (750 mL to 1 L)
1 qt raspberries 1 L
3 scant cups sugar (750 mL)
In a large stainless-steel pot, lightly mash raspberries. Simmer, covered, until bubbling. Stir in sugar and bring to a boil. Stirring as necessary, boil, uncovered, for about 10 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Raspberries Canned in Red Currant Juice**
THIS IS THE EASIEST AND MOST DELICIOUS WAY TO PRESERVE WHOLE RASPBERRIES. USE FIRM JUST-RIPE RASPBERRIES AND RIPE RED CURRANTS.
Makes about 6 qt (6 L)
4 qt red currants with stems (4 L)
8 qt raspberries (8 L)
12 cups sugar (3 L)
Prepare red currant juice. In a large preserving pot, combine red currants with a little water. Mash currants, cover, and simmer until currants lose color and juice runs freely. Stir often to prevent sticking. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Bring juice to a boil, uncovered, and stir in sugar. Reduce heat and cook mixture slowly for 20 minutes. Gently stir in raspberries and bring mixture to boiling point.
Remove preserves from heat. Ladle into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process 15 minutes in a boiling-water bath [p. 26] or steam canner.
**Plain Canned Raspberry Preserves I**
EVEN IF YOU CAN'T PRESERVE YOUR ABUNDANT HARVEST OF RASPBERRIES IN RED CURRANT SYRUP, YOU CAN PRESERVE THEM IN A LIGHT SUGAR-AND-WATER SYRUP. USE FIRM RIPE RASPBERRIES.
Makes 5 to 6 qt (5 to 6 L)
Light syrup:
8 qt raspberries (8 L)
4 cups sugar (1 L)
2 qt water (2 L)
Prepare syrup. In a large stainless-steel pot, combine sugar and water. Stir constantly and bring mixture to a boil.
Place raspberries in a large preserving pot. Add syrup. Keeping pot on simmer, let berries sit in syrup until heated through.
Remove preserves from heat. Ladle into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process jars in a boiling-water bath [p. 26] or steam canner. Process 1-pt (500-mL) jars for 15 minutes, 1-qt (1-L) jars for 20.
_Note:_ To make with honey, reduce sugar to 1 1/2 cups (375 mL) and add 1 1/2 cups (375 ml.) honey.
**Plain Canned Raspberry Preserves II**
RASPBERRIES PRESERVED THIS WAY ARE NOT QUITE SO FIRM AS THOSE PRESERVED IN SYRUP, BUT THEY ARE AS FLAVORFUL. IF YOU HAVE A LOT OF BERRIES, USE SEVERAL POTS FOR LAYERING THE FRUIT AND SUGAR SO THE FRUIT DOESN'T BREAK DOWN TOO MUCH WHEN HEATED. USE FIRM RIPE RASPBERRIES.
Makes 3 to 4 qt (3 to 4 L)
6 qt raspberries (6 L)
Sugar
In a large preserving pot, layer raspberries and sugar consecutively until pot is no more than half full. Use 1/2 to 1 cup (125 to 250 mL) sugar to each qt (L) raspberries. Cover and let stand overnight to draw out juices.
Heat mixture slowly, stirring gently with a long-handled wooden spoon to prevent scorching. Bring to boiling point.
Remove preserves from heat. Ladle into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process jars in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Freezing Raspberries**
Use 1/2 cup sugar (125 mL) for each quart (1L) of berries. Place berries in a freezer container, add the sugar, gently shaking it in to coat the fruit. To freeze them for making into jam later in the season, spread the berries on a cookie sheet. Place it flat in the freezer. When the berries are frozen, put them in a freezer container without sugar and freeze.
**Raspberry Sass**
SOME PEOPLE PREFER THIS ON PANCAKES TO ANYTHING ELSE, EVEN MAPLE SYRUP. RASPBERRIES FOR SASS CAN BE A LITTLE LESS FIRM THAN THOSE USED FOR JAM OR FOR CANNED PRESERVES.
Makes about 1 ½ qt (1.5 L)
2 qt raspberries (2 L)
4 cups sugar (1 L)
In a large stainless-steel pot, heat raspberries, covered, until juice begins to run. Stir in sugar and bring to a boil. Reduce heat and simmer, uncovered, for about 10 minutes or until mixture thickens.
Remove sass from heat. Stir. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Raspberry-Red Currant Sass**
USE QUITE RIPE RASPBERRIES FOR THIS EXTRAVAGANT SASS. YOU WILL NEVER TASTE ANYTHING LIKE IT.
Makes 2½ to 3 qt (2.5 to 3 L)
1 qt red currants with stems (1 L)
4 qt raspberries (4 L)
2 cups (an approximation, based on the mL amount provided) water (500 mL)
Sugar
Prepare red currant juice. In a large stainless-steel pot, combine red currants and water. Mash currants, cover, and simmer until white, stirring often. Strain through a jelly bag [p. 28] and let drip for several hours or overnight. Measure 2 cups (500 mL) juice.
In a separate pot, crush raspberries. Cover with currant juice and let stand for 10 to 15 minutes. Bring to a boil and simmer for 20 minutes. Stir in sugar, bring mixture to a boil again, and boil, uncovered, for 5 minutes.
Remove sass from heat and let subside. Stir. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p.28] or steam canner for 10 minutes.
**Raspberry Juice**
THIS JUICE HAS AN INCREDIBLE COLOR AND FLAVOR. WHEN SERVING RASPBERRY JUICE, DILUTE IT WITH WATER TO TASTE. USE DEAD-RIPE AND OVERRIPE RASPBERRIES.
Makes 2 to 3 qt (2 to 3 L)
6 qt raspberries (6 L) Sugar
2 cups water (500 mL)
In a large preserving pot, mash raspberries thoroughly. Add water to prevent scorching. Stirring frequently, simmer raspberries, covered, until juice runs freely. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice. Cover and bring to a boil. Stir in 1/2 to 1 cup (125 to 250 mL) sugar to each qt (L) juice. Bring to a boil again, reduce heat, and simmer, uncovered, for 5 minutes.
Remove juice from heat. Pour into hot scalded jars [p. 26], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 28] or steam canner for 15 minutes.
_Variation:_ To make _raspberry-red currant juice,_ substitute red currants with stems for half the raspberries. (Actually, any proportion is fine.) The blending of these two fruits, of the sweet and the tart, results in a distinctive, refreshing flavor that can be found only in your kitchen or in a fancy specialty shop.
**Raspberry-Flavored White Wine**
A SHORT CSUT TO WINE MAKING. PICK RIPE OR OVERRIPE RASPBERRIES–– _NOT_ RAINED-ON, AS THEY ENCOURAGE MOLD—AND CHOOSE A FAVORITE WINE.
Makes about 3 qt (3 L)
4 to 5 qt raspberries (4 to 5 L)
2 qt white wine (2 L)
2 cups sugar (500 mL)
In a large preserving pot, mash raspberries. Strain through a jelly bag [p. 28]. Measure 1 qt (1 L) juice and place in a container. Add sugar, stir, cover, and let stand for 3 days.
In a separate container, combine juice and white wine. Bottle and seal at once. Let stand at least a week before tasting.
**Raspberry-Jam Squares**
THE FRESH FLAVOR OF THE JAM GIVES THESE SQUARES THEIR DISTINCTION.
Makes 15 squares
2 cups flour (500 mL)
1 /4 cup ice water (50 mL)
1/2 tsp salt (2 mL)
1 pt raspberry jam [p. 67] (500 mL)
2/3 cup shortening or lard (150 mL)
1 egg yolk
Sift together flour and salt. With a pastry blender, cut in half the shortening until mixture forms peas. Cut in remaining shortening. Gradually sprinkle cold water on dough, mixing at same time. Handle dough as little as possible.
Grease an 8-inch x 12-inch (3-L) baking pan. Press half the dough onto bottom of pan. Cover evenly with jam.
Roll out remaining dough on a board, adding flour if necessary to make a pliable dough, and place on top of jam. Lightly beat egg yolk and brush over top crust. Prick crust a few times with a fork. Bake in a 350°F (180°C) oven for 35 minutes or until crust is lightly browned. Cool and cut into squares.
_Note:_ If you do not have raspberry jam, many other kinds will make good substitutes. Red currant jam [p. 81] or gooseberry jam [p. 93] are particularly good, as is Gooseberry-Red Currant Bar-le-Duc [p. 92].
**Raspberry Pudding, or Rothe Gruetze**
THIS IS A EUROPEAN VARIATION OF MENNONITE _MOOS,_ OR FRUIT SOUP.
Serves 6
4 cups ripe raspberries (1 L)
1/3 cup cornstarch (75 mL)
4 cups ripe red currants (1 L)
1 1/2 cups sugar (375 mL)
4 cups cold water (1 L)
Whipped cream [p. 58]
In a large pot, combine raspberries, red currants, and cold water. Bring to a boil and boil until fruit is tender, about 5 minutes. Mash berries and strain, reserving juice. In a cup, dissolve cornstarch in a little water.
Add sugar to reserved juice. Bring mixture to a boil and add cornstarch. Stirring constantly, simmer until thick. Pour into custard cups and chill. Serve with whipped cream.
**Raspberry Spritz**
RESERVE THIS RECIPE FOR CHRISTMAS, NEW YEAR'S, OR A SPECIAL OCCASION.
Makes 15 squares
2 cups butter, softened (500 mL)
2 tsp vanilla (10 mL)
1 cup sugar (250 mL)
4 cups flour (1 L)
1 egg
Raspberry jam [p. 67]
1 tsp salt (5 mL _)_
In a large bowl, cream together butter and sugar. Beat in egg, salt, and vanilla. Gradually add flour, mixing until dough is formed.
Press slightly more than half the dough onto an ungreased cookie sheet, spreading to 1/4 inch (5 mm) thick. Generously cover with jam.
Roll out remaining dough on a board. Using a star-shaped cookie cutter, cut shapes. Place stars on top of jam, about 3 to a row. Bake in a 400°F (200°C) oven for 15 minutes or until stars are golden brown. Cool and cut into star-shaped squares.
Raspberry Pudding
**Raspberry Slump**
SLUMP, LIKE GRUNT, IS A NEW ENGLAND TERM FOR A PUDDING-LIKE DESSERT. IT IS DIFFICULT TO SORT OUT ALL THESE PUDDINGS AND DEFINE THEIR DIFFERENCES. NONETHELESS, GRUNTS ARE USUALLY STEAMED FOR SEVERAL HOURS; SLUMPS ARE OFTEN COOKED ON TOP OF THE STOVE. THIS SLUMP, HOWEVER, IS NEITHER STEAMED NOR COOKED. IT IS BAKED. SERVE IT HOT WITH THICK SWEET CREAM OR THINNED DEVONSHIRE CREAM [P. 57].
Serves 4
1 qt ripe raspberries (1 L)
1 1/2 tsp baking powder (7 mL) Handful ripe red currants
1/2 tsp salt (2 mL)
1 3/4 cups sugar (425 mL)
2 Tbsp melted butter (25 mL)
1 cup flour (250 mL)
1/2 cup milk (125 mL)
Place raspberries and red currants in a buttered baking dish and sprinkle with 1/4 cup (50 mL) sugar.
In a large bowl, mix together 1 1/2 cups (375 mL) sugar, flour, baking powder, salt, melted butter, and milk until a batter is formed.
Pour batter over raspberries and smooth out to cover. Bake in a 350°F (180°C) oven for 35 minutes or until juice starts to bubble over crust.
**Chocolate Cake Supreme**
3/4 cup butter (175 mL)
1 tsp baking powder (5 mL)
2 eggs
Dash salt
1 tsp vanilla (65 mL)
1/3 cup cocoa (75 mL)
2 cups flour (500 mL)
1 cup cold water (250 mL)
1 1/2 cups sugar (375 mL) Raspberry jam [p. 67]
1 tsp baking soda (5 mL) Whipped cream [p. 58]
Ripe raspberries
In a large bowl, cream butter. Beat in eggs one at a time and add vanilla. Sift together flour, sugar, baking soda, baking powder, salt, and cocoa. Add sifted ingredients and cold water alternately to butter-egg mixture. Beat after each addition.
Pour batter into a greased 9-inch (3-L) angel-food cake pan. Bake in a 350°F (180°C) oven for about 30 minutes or until a knife or cake tester inserted near center comes out clean.
Cool cake. Cut into 3 layers and spread a thick coating of jam between each and on top of cake. Garnish each piece with whipped cream and ripe raspberries.
**Super Chocolate Roll**
5 eggs, separated
1 to 2 tsp vanilla (5 to 10 mL)
1 cup sifted icing sugar (250 mL)
Raspberry jam [p. 67]
3 Tbsp cocoa (50 mL)
Whipped cream [p. 58]
Line a shallow 10-inch x 14-inch (2-L) cake pan with foil, and grease foil with butter.
In a large bowl, beat egg whites into soft peaks. Add icing sugar 1 Tbsp (15 mL) at a time, beating constantly. Using a whisk, beat in cocoa.
In a separate bowl, beat egg yolks. With a rubber spatula, fold yolks into egg-cocoa mixture. Add vanilla to taste. Pour batter into prepared pan, spreading evenly. Bake in a 350°F (180°C) oven for about 20 minutes. Remove from oven and spread a damp dish towel over cake.
Sprinkle a dry dish towel with icing sugar. When cake has cooled slightly, remove damp cloth and turn cake onto icing-sugar towel. Carefully loosen corners of foil and tear it off. Roll up cake in icing-sugar towel and cool completely.
Unroll cake and cover top with a thick layer of jam. Without towel, roll up cake again, jam side in. Cover jellyroll completely with whipped cream. Chill for about 2 hours before serving
**_Red Currants_**
Like raspberries, red currants did not survive the transition from "old-fashioned" farming to high-technology "agribiz." These juicy berries deteriorate rapidly in transit, they are harder to pick than other small fruits, and they grow best in cold climates, so they are not so adaptable as, say, strawberries. Altogether, they do not suit modern agriculture or mass marketing.
It is hard to believe that red currants were so highly prized in the early 20th century. Nearly every garden had at least a red currant bush or two, and local markets sold thousands of bushels of currants each year. No decent housekeeper neglected to store enough berries for the annual jelly making. What well-stocked cupboard did not include jars of the clear red jelly, rich in pectin and vitamin C, esteemed as much for gracing a roast as for soothing a fever or a burn? You can look for a jar of this heretofore treasured jelly on the supermarket shelf, but you look in vain.
Fashions are cyclical, however, and once more red currants are being looked on with favor by the home gardener. If they do not suit modern agricultural techniques, they are more than suitable for the backyard fruit garden. Red currant shrubs are attractive in any season, they bear well, and their produce can be turned into a number of sought-after products. Gardeners need have only a few shrubs, for they alone will provide enough berries to make the legendary jelly, as well as other old-time favorites.
One word of caution. Red currants, along with black currants and gooseberries, belong to the genus _Ribes_ and therefore may carry the fungus white-pine blister rust. If there are white pines on your property or in your area, plant currants 1000 ft (300 m) away from stands of the trees. Check with the local Cooperative Extension office to find out if there are any restrictions against _Ribes_ in your area (see Appendix; also for mail order sources for this specialty fruit).
**Planting**
Buy two or three plants—perhaps two varieties, an early-and later-ripening one. Red Lake, for example, is a commonly grown variety, and two bushes will produce 6 to 12 qt (6 to 12 L) currants. Like many small fruits, red currants do best if the shrubs are planted on a slope with a northern exposure, providing just the right amount of sun and shade and drainage for bountiful fruiting. They thrive particularly well in rich clay loam.
Plant in early spring, when bushes are dormant. Trim plants to 4 to 6 inches (10 to 15 cm) in height and put them in a bucket of water while preparing holes. With a spade, make holes about 5 ft (1.5 m) apart and deep enough to accommodate the roots of the plants. If you have several plants, dig the holes in rows about 6 to 10 ft (2 to 3 m) apart. Do not be surprised by the gaps in this spacing. The small rootstocks will, in a few years' time, spread out into handsome bushes that will fill in the empty areas.
Add a shovelful of well-rotted compost to each hole, water the holes, and set in the plants, carefully placing the roots over a small mound of soil and firmly tamping down the earth around them. If there is a dry spell following planting, water the plants regularly until they show signs of new growth.
**Cultivating**
Spread a 2-ft (60-cm) wide ring of well-rotted compost or manure around each plant the following spring, the first season of growth, when the ground has warmed up. On top of the compost, put down a layer of thick paper, heavy cardboard, old cloth grain bags, or worn-out nonsynthetic carpet and then a layer of old hay, straw, or if you live near the shore, eel grass if available, to hold down the paper. All these layers will eventually break down and form a nutritious mulch that will help conserve moisture and prevent weeds and grass from competing with the currants for nutrients. Renew these layers once a year, in spring or fall.
As the currant bushes grow, the bottom stems will spread out close to the ground. Therefore, keep the grass well mown all around the shrubs so that the fruit on the lower branches does not become entangled in grass.
In the early spring, prune out dead wood, as well as stems that look old. Until they are three years old, literature on the subject says red currant bushes should have no more than 10 to 12 stems. Vigorous bushes, however, can support more. It is hard to tell the age of each stem, but, in general, young stems are gray, and older ones are brown. The oldest stems are dark brown and knobby and have a smooth bark. Prune out the oldest stems and less vigorous older ones. Trim the lower branches of mature bushes as well if you wish to encourage an upright habit of growth.
You may prefer the sprawling habit, however. Such bushes are graceful as they sweep to the ground, and their bottom branches may "layer" themselves, or set down roots. In the spring, increase your supply of plants. Sever the rooted stem from the parent plant, and replant it in a prepared nursery bed that is protected from the wind and that has well-drained, fertile, friable soil. Set these plants fairly close together in rows and plant them rather deep. Water them well and transfer them to a permanent site the following spring.
In June, when the stems begin to form flower buds, watch out for the currant sawfly, whose larvae look like small green inchworms about three quarters of an inch long. These insects, if undeterred, can strip a plant of its leaves in no time. If they are evident, spray the bushes with a pyrethrin-based insecticide. We like Pyola (see Appendix for a source), an effective, organic spray that attacks insect adults, larvae, and eggs. Consult your local Cooperative Extension office for other sprays or dusts. Whatever you use, repeat the treatment a week later and until there is no more damage to the bush.
**Harvesting**
Red currants grow in clusters, turning from green to pale red to bright red. Not surprisingly, the fruit at the top of the bush usually ripens first; the berries at the bottom, last. In the northeast, red currants can be harvested in midsummer. For jelly, pick a mixture of ripe (top) and underripe (bottom) berries; for jam, a majority of ripe; for sauce, all ripe; for juice and wine, dead ripe or overripe. To pick fruit for jelly, juice, or wine, run your fingers down the main stems of the clusters, leaving the little stem on each berry intact. For jam and sass, pick each berry off its stem.
**Preserving, Freezing, And Cooking**
If you are going to grow red currants, you are obliged to make red currant jelly. In the following pages, there are two recipes to choose from. Throw some raspberries into the jelly bag, and then make Bar-le-Duc. After you have lined your pantry with jars of jelly, move on to red currant jam, sass, juice, and snub. But do not try drying red currants: they are too seedy. Red currants give a refreshingly tart quality to baked goods. Include them in any of the cooking recipes in the preceding chapter or try red currant pie topped with meringue. Review A Short Course in Fruit Preserving [p. 3–14] before preserving. Use white currants the same way as red currants. The variety Pink Champagne, a cross between red and white, is sweeter than either and can be eaten fresh.
**Red Currant Jelly I**
HERE IS ONE VERSION OF THE FABLED JELLY. EVEN IF YOU DO NOT USE IT TO SOOTHE A FEVERISH BROW OR A BURN, USE IT AS AN ACCOMPANIMENT TO MEAT DISHES. IT IS ALSO GREAT WITH CREAM CHEESE, ON HOT MUFFINS, BISCUITS, OR CORNBREAD. USE A MIXTURE OF RIPE AND UNDERRIPE BERRIES
Makes 1 1/2 to 2 pt (750 mL to 1 L)
2 1/2 qt red currants with stems (2.5 L)
1 cup water (250 mL)
Sugar
In a large preserving pot, combine red currants and water. Boil, covered, until currants are white, stirring occasionally to prevent sticking. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 3/4 cup (175 mL) sugar to each cup (250 mL) juice. Bring to a boil again. Skimming as necessary, boil, uncovered, for about 15 minutes or less or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Red Currant Jelly II**
THIS JELLY IS LESS INTENSELY FLAVORED THAN RED CURRANT JELLY I. USE A MIXTURE OF RIPE AND UNDERRIPE BERRIES.
Makes about 4 pt (2 L)
4 qt currants with stems (4 L)
1 qt water (1 L)
Sugar
In a large preserving pot, combine red currants and water. Simmer currants, covered, until white, stirring and mashing occasionally. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a boil again. Skimming as necessary, boil, uncovered, until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Variation:_ To make _red currant-raspberry jelly,_ substitute ripe and underripe raspberries for half the red currants. The proportion isn't crucial, however, so experiment as you please.
**Bar-le-Duc Jelly**
THIS IS NEITHER A TRUE JELLY NOR THE AUTHENTIC BAR-LE-DUC, NAMED AFTER A CITY IN FRANCE. THERE THE SEEDS OF EACH BERRY ARE REMOVED BEFORE BEING MADE INTO A CONCOCTION THAT IS A CROSS BETWEEN A JAM AND A JELLY. THE FOLLOWING IS WHAT I CALL BAR-LE-DUC. LIKE THE REAL THING, THE WHOLE CURRANT IS SUSPENDED IN THE JELLED JUICE. FOR THIS RECIPE, USE MAINLY RIPE RED CURRANTS, WITH SOME UNDERRIPE
Makes l 1/2 to 2 pt (750 mL to 1 L)
1 qt red currants without stems (1 L)
1/3 cup water (75 mL)
2 cups sugar (500 mL)
In a large stainless-steel pot, combine red currants and water. Cover and simmer until berries are just tender but still whole. Stir in sugar. Stirring gently, cook at a rolling boil, uncovered, until a small amount of mixture sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Red Currant Jam**
USE MAINLY RIPE RED CURRANTS, WITH SOME UNDERRIPE.
Makes 1 ½ to 2 pt (750 mL to 1 L)
4 cups red currants without stems (1 L)
2 2/3 cups sugar (650 mL)
In a large stainless-steel pot, mash red currants, cover, and heat until simmering. Stir in sugar and bring to a rolling boil. Stirring occasionally, boil, uncovered, for about 15 minutes or less or until mixture thickens and starts to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands and seal. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Red Currant Sass**
POUR A LITTLE RED CURRANT SASS ON BUTTERED PANCAKES, THEN A SPOONFUL OF BRANDIED FRUIT, THEN A DAB OF SOUR CREAM. USE RIPE RED CURRANTS.
Makes about 1 ½ qt (1.5 L)
2 qt red currants without stems (2 L)
3 cups sugar (750 mL)
Layer red currants and sugar consecutively in a large stainless-steel pot. Cover and let stand overnight. Bring mixture to a boil, uncover, and boil for 10 to 15 minutes or until mixture thickens.
Remove sass from heat and let subside. Stir. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Red Currant Juice**
Makes 2 to 3 qt (2 to 3 L)
6 qt red currants with stems (6 L)
2 cups water (500 mL)
Sugar
In a large preserving pot, mash dead-ripe red currants thoroughly. Add water to prevent scorching. Stirring frequently, simmer, covered, until juice runs freely. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice. Cover and bring to a boil. Stir in 1/2 to 1 cup (125 to 250 mL) sugar to each qt (L) juice. Bring to a boil, reduce heat, and simmer, uncovered, for 5 minutes. Remove juice from heat. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Freezing Red Currants**
You can easily freeze red currants if you're too busy to turn them into preserves during harvest season. Just pour cleaned, de-stemmed fruit into a freezer container or bag and seal. Or freeze with stems to use later for jelly and juice.
**Red Currant Snub**
THIS IS A VARIATION OF THE SHRUB DRINK OF NEW ENGLAND. IF YOU LIKE, USE A MIXTURE OF RASPBERRY AND RED CURRANT JUICE. WHEN SERVING SNUB, DILUTE IT WITH WATER TO TASTE AND POUR OVER ICE. OR LEAVE UNDILUTED, SPOON A LITTLE OVER PANCAKES AND TOP THAT WITH SOUR CREAM.
Makes 1 1/2 Pt (750 mL)
1 qt red currants with stems (1 L)
1 teacupful good brandy (175 mL)
2 cups sugar (500 mL)
In a preserving pot, mash red currants and simmer, covered, until white. Strain through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure 1 pt (500 mL) juice. Stir in sugar and boil mixture for 15 minutes. Remove from heat and let cool. Stir in brandy. Bottle and cork immediately. Store lying down in a cool cupboard.
**Thumbprint Cookies**
THE FIRST TIME I MADE THESE COOKIES IT WAS AT THE END OF THE FRUIT SEASON. I HAPPENED TO HAVE A LOT OF LITTLE JARS OF VARIOUS JAMS AND JELLIES IN THE REFRIGERATOR, SO I TRIED EACH AS A FILLING FOR MY THUMBPRINTS. THE CONSENSUS WAS THAT RED CURRANT JELLY WAS THE BEST, FOLLOWED BY RASPBERRY JAM AND ELDERBERRY JELLY. SEE WHAT YOU THINK.
Makes 2 1/2 dozen
1 full cup butter (250 mL)
2 tsp vanilla (10 mL)
1/2 cup sugar (125 mL)
2 scant cups flour (500 mL)
2 egg yolks
Red currant jelly [p. 79]
In a large bowl, cream butter; beat in sugar. Add egg yolks and beat all. Beat in vanilla and flour. Roll dough into balls the size of walnuts.
Place balls fairly close together on a lightly greased cookie sheet. Press a thumbprint into center of each and fill with red currant jelly (or your favorite jam or jelly). Bake in a 375°F (190°C) oven for 8 to 10 minutes. Note: You can flute sides of unbaked cookies by lightly pressing a sharp fork, held upright, against the edges of the dough, or roll out dough and cut with a fluted cookie cutter.
Thumbprint Cookies
**Red Currant Meat Sauce**
Makes 1 cup (250 mL)
1 cup red currant jelly [p. 79] (250 mL)
1 1/2 Tbsp chopped mint leaves (20 mL) Grated orange rind
In a bowl, gently break up jelly with a fork and stir in mint leaves and grated orange rind to taste. Let stand for at least 1 hour before using. Serve with lamb or veal.
**Red Currant-Raspberry Tart**
USE DEAD-RIPE, NOT OVERRIPE, BERRIES FOR THIS DELICIOUS TART.
3 cups red currants (750 ml)
1 cup raspberries (250 mL)
1 to 2 Tbsp cornstarch (15 to 25 mL)
Sugar to taste
Pastry [p. 46]
In a pot, cook together red currants and raspberries, covered, until soft and juicy. Remove from heat, stir in sugar to taste, and stir in cornstarch. Let stand for 10 minutes.
Pour filling into an unbaked 7-inch (18-cm) pie shell and make a lattice top with strips of dough. Bake in a 375°F (190°C) oven for 30 minutes or until pastry is browned and berries are bubbling.
**Red Currant Pie**
THIS SELDOM MADE OLD-FASHIONED PIE HAS GREAT FLAVOR AND A MELTING SMOOTHNESS WITH THE DELICATE MERINGUE TOPPING.
1 cup sugar (250 mL)
Unbaked pie shell [p. 46]
1/4 cup pastry flour (50 mL)
2 egg yolks
Meringue:
2 Tbsp water (25 mL)
2 egg whites
1 1/2 cups red currants (375 mL)
4 Tbsp sugar (75 mL)
In a large bowl, combine 1 cup (250 mL) sugar, flour, egg yolks, water, and red currants. Mix well. Pour into an unbaked 7-inch (18-cm) pie shell. Bake in a 350°F (180°C) oven for 35 to 40 minutes or until crust is browned and berries are bubbling.
Prepare meringue. Beat egg whites until stiff, gently folding in 4 Tbsp (75 mL) sugar. Pour meringue on top of baked pie and return to oven for 10 minutes or until meringue is lightly browned.
**_Gooseberries_**
North Americans have never been wild about gooseberries. True, gooseberries have been grown for hundreds of years in rural gardens, and there have been times, such as in the 18th century, when they have been prized for medicinal purposes, as well as for making jams, jellies, and pies. But many years have passed since fresh gooseberries have been available in the marketplace. Have you ever seen gooseberries on a grocery-store shelf? In an expensive specialty shop, perhaps. It is probably safe to say that few people under 40 have ever tasted gooseberries in any form.
On the other hand, the mere mention of the word _gooseberry_ sends transplanted Europeans into ecstasy. They begin to reminisce about their homeland—gooseberries, it seems, embody the lost charms of a half-remembered rural life. However small their gardens, many expatriate Europeans try to find room to plant at least one gooseberry bush, which in three years bears a fine crop of pale-green or red-striped fruit.
Gooseberries have much to recommend them: their bushes require only a small area to produce an abundant crop (5 to 10 qt/5 to 10 L per bush); their culinary uses are varied (though not so varied as strawberries or raspberries); and their high pectin content guarantees success even for the inexperienced jam or jelly maker. There are things to be said for the bush itself: whether in leaf, bud, bloom, or berry, it is attractive, suitable for planting around a house foundation or even in a corner of a cottage garden flower bed. Depending on the climate, a healthy bush will produce abundantly for 20 years and will stay ornamental for even longer than that. Finally, the shrubs are easy to grow in places where winters are freezing and summers are moderate.
Remember, however, gooseberries, which belong to the genus _Ribes,_ are alternate hosts to the disease white-pine blister rust. Check with the local Cooperative Extension office to see if there are any restrictions in your area (see Appendix to locate your local office). Plant gooseberries 1000 ft (300 m) away from any white-pine stands.
**Planting**
Although gooseberries are making a comeback among home gardeners, with few exceptions, there are still only a few varieties commonly available. In Canada, gooseberry plants may be fairly easy to come by, but in the United States, they are harder to find (see Appendix).
Gooseberry varieties, American or European, vary in color (from green to red), in size, and in time of ripening; in addition, some varieties have fewer thorns and therefore are easier to pick. Here, though, I must say something about what I regard as a cherished myth: red gooseberries. Invariably, whenever we have ordered a so-called red variety, it has turned out to be green. Not that we really mind, but I do wish plant nurseries would stop saying that every variety they offer is red; even the ubiquitous Pixwell, which everyone knows is green, is described in one catalog as pink. Poorman is offered as "brilliant" red; Clark (a European type), red; and Captivator, "dull" red and "almost" thornless. In any event, it probably won't matter, as Pixwell is, with few exceptions, the only one available. Recently, however, I saw a picture in a plant nursery catalogue where the gooseberries really _did_ look red. Maybe they really are.
Once you have tracked some down, choose a slope with a northern exposure for planting if possible. Gooseberries thrive where it is cool, though _not_ shady. The roots appreciate the good drainage and protection of a slope: they need to be shielded from excessive sun, which could scald them, and excessive shade, which could cause mildew. Soil that is a little too wet for other fruits suits gooseberries just fine as long as the roots are not standing in water (fertile clay loam is preferable). Few plants survive wet feet.
It is best to plant gooseberry shrubs in the spring, when they are dormant. To plant, trim rootstocks to 4 to 6 inches (10 to 15 cm) and put each one in a bucket of water while preparing holes. With a spade, make the holes 5 ft (1.5 m) apart and as deep as necessary to accommodate the roots. If you are planting several bushes, make rows about 6 to 10 ft (2 to 3 m) apart. Add a shovelful of well-rotted compost to each hole, water the holes, and plant the rootstocks in a little mound of earth, firmly tamping down the soil around them. In a dry spell, water the plants regularly until they show signs of new growth.
**Cultivating**
After planting, mulching is in order. This helps eliminate weeds, creates cool, moist conditions, and builds up the soil. In a 2-ft (60-cm) wide ring around each plant, set down a layer of well-rotted compost or manure, a layer of heavy paper, cardboard, or worn-out nonsynthetic carpet, and a layer of hay, straw, or, as we used in Cape Breton, eel grass. Renew this mulch each spring or fall.
Do not do much pruning the first few years after planting. In the early spring, remove dead or damaged branches with hand clippers. After the third year, when the gooseberry bushes get into high production, do a little more pruning if necessary but not much. The older the plant gets, though, the more you should prune every spring. Pay particular attention to the center of the plant, which should be kept airy because crowded branches encourage disease, powdery mildew in particular. A wet summer, together with a bush with too many stems, spells trouble. After removing the dead and damaged branches, remove any that are more than three years old so that the bush is left with a combination of one-, two-, and three-year-old branches. Fruit is produced on one-year-old wood and on the spurs of older wood. Also, many books recommend leaving only 5 to 7 shoots per plant, but more than a dozen can be left on vigorous bushes if they are not overcrowded.
Prune branches lying close to the ground if you wish to encourage an upright habit of growth—and if you do not like searching for berries tangled in grass. I prefer the sprawling habit because I like to find hidden treasures. There is always the possibility, too, that the low-lying branches will "layer" themselves, that is, send out roots into the ground.
Unfortunately, the hardier American gooseberry varieties do not layer themselves so readily as the European types, but a diligent search in the spring among low branches should yield a few plants. Roots can grow from more than one place on a single branch, so with judicious cutting, you can make several little plants. Set them fairly close together in weed-free, loose, and well-drained soil, and water and continue to weed thoroughly. Let the plants grow a year before transferring them to a permanent site. If the summer is wet, extra watering, of course, is not necessary.
There are two other, more difficult ways to propagate gooseberries. The first method is to make cuttings from new wood in the early spring. Choose medium-sized shoots that are 8 to 10 inches (20 to 25 cm) long and plant them 6 inches (15 cm) apart in cultivated garden soil. The secret to success with these cuttings is _deep_ planting: leave only 1 to 2 buds above ground. Alternatively, plant these cuttings in pots or beds in a cold frame or greenhouse as long as the soil is well worked and friable. Wherever you plant them, water them well during the growing season, protect them with straw in the late fall, and make sure they are not exposed to freezing temperatures. Following these rules, you should have healthy new plants the next spring.
The other method is "mound" layering. In August, after harvest, cut back the main branches severely to encourage new growth from the bottom of the plant. When new shoots begin to grow, heap dirt around them to encourage rooting. By autumn, or by the following spring, the new shoots should be well rooted. Detach them with clippers, plant them fairly deep in a well-prepared bed, and water and weed them. The next spring, move them to their permanent location. Once new plants are in their permanent site, mulch and prune as for other bushes.
One essential task for all bushes, new and old, is to watch out for an invasion of currant sawflies, little green worms, early in the growing season, before berries are formed. These insects, about three quarters of an inch long, can strip a plant entirely of its leaves overnight. If you see any sign of them, spray your bushes on a still day with a pyrethrin-based insecticide; we like the organic spray, Pyola, which works. Whatever you use, repeat the treatment a week later. Consult your local Cooperative Extension office for recommendations (see Appendix).
**Harvesting**
Like all fruits, gooseberries have three stages of ripening: underripe, ripe, and overripe. It is difficult for the novice to figure out exactly when to pick the fruit, so it is a good idea to keep the following pointers in mind. Underripe gooseberries are green in color, quite hard, and should be picked for jelly making. There is no need to rush out and pick these berries immediately, however, for they stand on the bush well, for a week, anyway. When gooseberries reach the ripe stage, they are still firm but not quite so hard, and their color has not changed noticeably. These berries are best for jams, canning, preserves, pies, and tarts, as well as for eating chilled, especially on a hot day. It is not advisable to wait for the third stage. By that time, many berries will have fallen to the ground and will have been quickly harvested by eager birds. Sometimes one is advised to wait until the fruit has turned yellowish (if it's a green variety) or dark red (if a red type) before harvesting, but experience proves that this is not practical. The fruit is supposed to be sweetest at this stage, but only the birds know for sure.
**Preserving, Canning, Freezing, And Cooking**
Someday I hope to be remembered with great affection and gratitude by new (and old) generations of gooseberry aficionados for having successfully made dozens of jams, jellies, and what not without removing either the stems (unless very evident) or the tails. No matter what you may read to the contrary, it is not necessary to top and tail gooseberries, an arduous task. I have processed gooseberries both ways, with absolutely no difference in the results, although in jam the tails are visible as dark specks.
Of course, try your hand at the old, reliable jams and jellies. But if you have an abundance of fruit, try chutney, marmalade (my favorite), and preserves. Gooseberries are also prized for pies and tarts; for these, use firm ripe berries. Though not so versatile as some other small fruits, gooseberries have special qualities that are irreplaceable. When underripe, they are delightfully sour; when ripe, they resemble green grapes—sweet with a little zing. Review Short Course [p. 3-14] before preserving.
**Gooseberry Jelly**
ALTHOUGH MOST GOOSEBERRIES ARE GREEN, THEIR JELLY IS A LOVELY SHADE OF AMBER-RED. THIS JELLY HAS A DISTINCT, DELICATE FLAVOR, AND IT IS ATTRACTIVE—PERFECT FOR GIFT GIVING. USE UNDERRIPE BERRIES.
Makes about 3 pt (1.5 L)
3 lb gooseberries (1.5 kg)
Sugar
4 1/2 cups cold water (1.125 L)
In a large preserving pot, cover gooseberries with water. Simmer, covered, until soft, mashing when berries begin to cook. Be careful not to scorch fruit. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Skimming as necessary, cook at a rolling boil, uncovered, for 10 to 15 minutes or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Note:_ 3 lb (1.5 kg) gooseberries equals about 6 heaping cups (1. 5 L).
**Gooseberry-Elderflower Jelly**
MAKING THIS JELLY IS ONE OF THE ANTICIPATED JOYS OF SUMMER. WHEN THE BANK OF ELDERBERRY PLANTS IS IN BLOSSOM, WE QUICKLY HARVEST SOME GOOSEBERRIES, THE ONES WE HAVE BEEN SAVING JUST FOR THIS TREAT. IT IS THE TENSION BETWEEN ANTICIPATING THE ELDERFLOWERS AND WATCHING OVER THE GOOSEBERRIES THAT GIVES THIS JELLY A SPECIAL PLACE. BESIDES, IT IS DELICIOUS—THE ELDERFLOWERS ADD A GRAPELIKE FLAVOR. USE SLIGHTLY UNDERRIPE GOOSEBERRIES.
Makes 4 to 6 Pt (1 to 1.5 L)
6 lb gooseberries (3 kg)
3 to 4 large bunches elder-flowers with short stems
1 1/2 pt water (750 mL)
Sugar
In a large preserving pot, cover gooseberries with water. Simmer, covered, until soft, stirring occasionally and mashing to form a pulp. Strain through a jelly bag [p. 28] and let drip for several hours or overnight.
Tie up elderflowers in a cheesecloth or muslin bag. Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover, bring to a boil, and stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a boil again and add elderflowers. Skimming as necessary, boil, uncovered, until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and retrieve elderflowers. Let subside and stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Gooseberry-Red Currant Bar-le-Duc**
THIS IS AN INTRIGUING VARIATION OF BAR-LE-DUC JELLY. USE SLIGHTLY RIPE GOOSEBERRIES AND RIPE, NOT TOO SOFT, RED CURRANTS. WHEN COOKING, TRY TO LEAVE THE BERRIES WHOLE.
Makes 1 1/2 to 2 pt (750 mL to 1 L)
1 qt gooseberries (1 L)
1/3 cup water (75 mL)
1/2 qt red currants without stems (500 mL)
3 cups sugar (750 mL)
In a large stainless-steel pot, combine gooseberries, red currants, and. water. Stirring gently, heat mixture, covered, until bubbling. Stir in sugar and boil, uncovered, for 10 to 15 minutes or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands.
Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Gooseberry Jam**
WHEN YOUR GOOSEBERRIES START TO SOFTEN, THE SECOND STAGE OF RIPENING, MAKE JAM. THIS JAM SETS QUICKLY, SO WATCH CAREFULLY.
Makes 1 1/2 pt (750 mL)
3 cups gooseberries (750 mL) Water or red currant juice
2 cups sugar (500 mL) Pinch salt
In a large stainless-steel pot, simmer gooseberries, covered, in a little water or red currant juice to prevent scorching. Mash if desired. When berries are bubbling, stir in sugar and salt. Bring to a boil. Stirring occasionally, boil, uncovered, for 10 to 15 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling water bath [p. 26] or steam canner for 15 minutes.
_Variation:_ To make _gooseberry-black currant jam,_ substitute 1/2 to 1 cup (125 to 250 mL) black currants for 1/2 to 1 cup (125 to 250 mL) gooseberries. We discovered this jam one summer when we were too busy making hay to pick the gooseberries, most of which fell to the ground. So we mixed black currants with the remaining gooseberries and discovered one of our favorite jams. The new fruit Josta apparently gives the same flavor, and if it does, it is a real bonus for the small-fruit gardener.
**Three-Fruit Jam**
FOR EVERY FRUIT, USE A MIXTURE OF RIPE AND SLIGHTLY UNDERRIPE BERRIES.
Makes about 5 pt (2.5 L)
1 qt gooseberries (1 L)
1 qt raspberries (1 L)
1 qt red currants (1 L) Sugar
Weigh fruit. In a large stainless-steel pot, simmer fruit, covered, for 20 minutes, mashing if desired and stirring occasionally to prevent scorching. Stir in a scant 2 cups (500 mL) sugar to each lb (500 g) fruit. Stirring as necessary, cook, uncovered, at a rolling boil for about 10 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling water bath [p. 26] or steam canner for 15 minutes.
**Gooseberry-Rhubarb Jam**
THIS JAM HAS A DELICIOUS TART FLAVOR AND LOVELY AMBER COLOR, ANOTHER REASON TO FREEZE TENDER RHUBARB STALKS. IN THIS CASE, STALKS OF THE RED TYPE MUST BE CUT EARLY IN THE SEASON BEFORE THEY GET STRINGY.
Makes about 3 pt (1.5L)
1 qt gooseberries (1 L)
1 qt cut-up tender red rhubarb stalks (1 L)
6 cups sugar (1.5 L)
1/2 cup water (125 mL)
Add water to fruit. Cover and bring to simmering. Remove cover, stir in sugar and bring to a rolling boil. Boil about 10 minutes. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling water bath [p. 26] or steam canner for 15 minutes.
**Gooseberry Marmalade**
THIS IS GUARANTEED TO BE A HIT AMONG THE MOST SOPHISTICATED GOURMETS. ONCE, SOME CUSTOMERS CAME LOOKING FOR CHUTNEY, BUT I DIDN'T HAVE ANY, SO I OFFERED THEM SOME OF MY GOOSEBERRY MARMALADE INSTEAD. THE HUSBAND OF ONE OF THE WOMEN LATER REMARKED, "THEY WENT GAGA OVER IT." USE FIRM RIPENING BERRIES.
Makes 3 to 3 /2 pt (1.5 to 1.75 L)
3 pt gooseberries (3.5 L) Sugar
2 lemons
1 cup water (250 mL)
4 oranges
Blanch gooseberries by pouring boiling water over them. Drain in a colander.
Peel lemons and oranges. Reserve peel. Slice pulp thinly; add to gooseberries. Turn fruit and sugar into a large stainless-steel pot, using 1 cup (250 mL) sugar to every cup (250 mL) fruit. Stir well. Cover and let stand overnight. Using scissors, cut reserved peel into thin shreds. Place in a bowl and add 1 cup (250 mL) water. Cover and let stand overnight as well.
Add peel-water mixture to fruit. Stirring frequently with a long-handled wooden spoon, simmer, uncovered, for about 1 hour or until mixture thickens.
Remove marmalade from heat and let subside. Stir. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Note:_ Do not increase the amounts because the marmalade may darken and lose its flavor.
**Gooseberry Chutney**
THE ORIGINAL DIRECTIONS CALLED FOR TOPPING AND TAILING AND HALVING THE GOOSEBERRIES. NEEDLESS TO SAY, THESE DIRECTIONS DON'T. SERVE CHUTNEY WITH COLD MEAT. USE UNDERRIPE GOOSEBERRIES.
Makes 2 to 3 Pt (1 to 1.5 L)
3 lb gooseberries (1.5 kg)
1 oz ginger (30 g)
2 oz salt (50 g)
1 cinnamon stick
1 pt malt vinegar (500 mL)
1 oz mustard seeds (30 g)
2 cups sugar (500 mL)
2 tsp cayenne pepper (10 mL)
Bruise gooseberries with a wooden stamper or mallet or chop them coarsely with a hand chopper. Place in a bowl and sprinkle with salt. Cover and let stand overnight. Drain and rinse.
In a large stainless-steel pot, combine 1/2 pt (250 mL) malt vinegar and sugar. Stirring to dissolve sugar, bring to a boil and simmer until mixture forms a thick syrup. Add remaining vinegar and gooseberries. Simmer, uncovered, for 10 minutes. Add ginger, cinnamon, mustard seeds, and cayenne pepper and, stirring occasionally, simmer for another 30 minutes or until thick. It should be a reddish color, not too dark.
Remove chutney from heat and retrieve cinnamon stick. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal immediately with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Note:_ 3 lb (1.5 kg) gooseberries equals about 6 _heaping_ cups (1.5 L).
**Gooseberry-Cream Snow**
Serves 2 to 3
2 cups ripe gooseberries (500 mL)
1 cup whipping cream [p. 58] (250 mL)
1/2 cup water (125 mL) Sugar
1 egg white
In a pot, simmer together gooseberries and water, covered, until berries are tender and pulpy. Remove lid halfway through cooking, stirring as necessary. With a wooden spoon, beat pulp until quite smooth. Add sugar to taste and chill completely.
In a bowl, beat whipping cream until very stiff. In a separate bowl, beat egg white until stiff, adding 1 Tbsp (15 mL) sugar; fold into cream. Beat gooseberry pulp into whipping-cream mixture a little at a time. Chill until ready to serve.
**Canned Gooseberries**
FOR VARIETY, ADD RED CURRANTS IN SOME FORM. USE FIRM RIPE BERRIES.
Makes 5 qt (5 L)
Gooseberries
Light syrup:
1 cup sugar (250 mL)
1 cup honey (250 mL)
1 qt water (1 L)
Prepare syrup. In a stainless-steel pot, combine sugar, honey, and water. Stirring to dissolve honey and sugar, bring to a boil. Pour 1/2 cup (125 mL) boiling syrup into each hot scalded jar [p. 28].
Fill each canning jar with gooseberries and shake jar gently so syrup is well distributed over fruit. If necessary, add more syrup to cover gooseberries, leaving 1/2 inch (1 cm) headroom. Adjust lids and process jars in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Note:_ A single batch of medium syrup yields about 5 cups (1.25 L). Allow about 1/2 to 1 cup (125 mL to 250 mL) syrup to each qt (L) jar. You can substitute another cup (250 mL) sugar for the honey.
**Freezing Gooseberries**
If you're too busy during the harvest to preserve gooseberries and you want to use them later in pie or any of the recipes below, just put them into a freezer bag or pack them into a freezer container without sugar and freeze.
**Gooseberry Pie or Tart**
THIS WAS A PRIZED NEW ENGLAND PIE BEFORE THE GOOSEBERRY BUSH FELL OUT OF FAVOR.
1 qt ripe gooseberries (1 L)
1 egg yolk
Unbaked pie shell [p. 46]
3 Tbsp heavy cream (50 mL)
1 cup sugar (250 mL)
Place gooseberries in an unbaked 9-inch (23-cm) pie shell and sprinkle with sugar. Bake in a 350°F (180°C) oven for about 20 minutes.
Meanwhile, in a bowl, lightly beat egg yolk and mix in cream. Drip mixture over baked pie. Return pie to oven for 15 minutes or until cream mixture forms a crust and berries are tender.
_Variation:_ Omit cream and egg. Precook gooseberries with sugar, adding a little water to prevent scorching. Pour into pie shell, and cover with a lattice top. Bake in a 350°F (180°C) oven for 35 minutes or until crust is browned and berries are tender.
**Fresh Stewed Gooseberries**
Serves 2 to 3
1/2 cup sugar or honey (125 mL)
1 cup water (250 mL)
1 lb ripe gooseberries (500 g)
In a stainless-steel pot, dissolve sugar or honey in water and bring to a boil. Add gooseberries, cover, and simmer for about 15 minutes or until tender. Uncover halfway through cooking. Drain gooseberries, reserving syrup. Reheat syrup and pour over fruit.
_Note:_ Do not overcook or berries will break down.
**Gooseberry Ice Cream**
WITH ITS PIQUANT FLAVOR, GOOSEBERRY ICE CREAM IS DELICIOUS.
Makes about 3 qt (3 L)
3 1/2 cups green gooseberries (875 mL)
1 1/2 cups sugar (375 mL)
1 qt cream (1 L)
In a pot, combine gooseberries and sugar. Cover and stew, removing lid halfway through cooking. Add more sugar to taste if desired (or use part honey). Put cooked gooseberries through a food mill and let cool.
Meanwhile, in the top of a double boiler, heat cream over a little boiling water. Let cool as well.
In a bowl, combine gooseberries and cream; stir well. Turn mixture into chilled container of hand ice-cream maker and churn until handle is hard to turn. Ripen for 1 to 2 hours in a cool spot, with the container packed in ice and covered with a burlap bag. Or make a smaller amount in an ice cream machine. Review "An Assortment of Basics" [p. 139–146].
**_Black Currants_**
I may as well tell you straight away that I am a black-currant fancier, one of a small but growing number of people in North America (the number is higher in Canada than in the United States) who think that the black currant has been unfairly neglected. The black currant shrub's hardiness, immunity to disease, and ability to produce great crops with little or no attention are hard to match. Given suitable growing conditions, the shrubs grow and produce fruit with more vigor than either gooseberry or red currant bushes. Black currant bushes are also beautiful, with their curved branches dipping down to the ground. In early summer, they are crammed with clusters of small greenish-white flowers and then with clusters of ripe black berries.
So why doesn't everyone have a black currant bush in his or her fruit garden? The charges are as follows: black currants are host to white-pine blister rust, black currants have an undesirable flavor, black currants are not worth growing.
Let's take a look at those accusations. Yes, black currants do carry white-pine blister rust, but so do other members of the _Ribes_ group (gooseberries and red currants). This fact has not prevented garden writers from promoting gooseberries and red currants, however, and as a result, those fruits are enjoying a modest comeback in the trade. On the other hand, the black currant suffers from bad publicity, often being singled out as the sole culprit of white-pine blister rust, without any further explanation. Therefore I think the following account is in order.
White-pine blister rust was inadvertently brought to North America around the turn of the century, the same time that the white pine was enjoying considerable commercial popularity. In the first stage of its development, the fungus lives on the _Ribes_ species, and it does little harm. But in the second stage, it lives inside the stem of the white pine, causing severe damage or death to the tree. By 1918, millions of acres of pine forests were infected by the fungus. At that time, the United States government took steps to prohibit the growth and sale of currants and gooseberries. Eventually, the white pine's commercial value declined, and in 1968 the government revoked the quarantine and eased the restrictions against _Ribes_ species. Each state then set up its own regulations.
By contrast, Canada, according to some horticulturists, has had a more enlightened policy toward _Ribes._ Though there is no shortage of white pines in Canada, there are no restrictions against currants and gooseberries. Instead, authorities have worked to combat the problem through breeding programs. Perhaps this attitude stems in part from the fact that there is a stronger British tradition there. Indeed, the British are avid fans of _Ribes,_ of black currants in particular.
When I was researching the original edition of this book, I decided to investigate state prohibitions against black currants. After all, if Americans were not allowed to grow black currants, what would be the use in telling them how wonderful they are? I was also interested in finding out if United States horticultural officials knew why black currants were pinpointed as the worst offenders against white pines. The more I looked into the matter, the more confused the situation appeared to be. My first inquiries, sent to plant associations and horticultural authorities, drew a blank. Nothing, it seemed, was known about black currants. No one knew if the plants were for sale anywhere in the United States; no one knew if the plants were allowed to be grown. In short, I got the definite feeling that black currants were regarded as pariahs of no horticultural interest.
Then I sent out inquiries to 16 state agricultural research stations and departments of conservation and to the United States Department of Agriculture. I asked about the status of black currants in their areas, whether or not they were thought to be more susceptible to white-pine blister rust than other _Ribes_ species and, finally, whether black currants had a future, and if so, what was it? In no time, I accumulated a rather bulky folder of information. The answers I received, although direct and to the point, were sometimes conflicting.
Maine and New Hampshire, where the white pine was still a viable commercial crop, outlaw black currants, with some restrictions on red currants and gooseberries. Rhode Island and Michigan prohibited the growth and sale of black currants but admittedly did not enforce the laws. (For example, a nursery in Ypsilanti, Michigan, received permission to sell black currant plants because there are not any white pines in that area.) It was generally felt, except in a few instances, that the question of enforcing prohibitions is academic, anyway: wild _Ribes_ species grow vigorously and cannot be controlled. Some officials in Alaska noted as well that even though there was no ban in their state, white-pine forests were unaffected. Some officials were adamant that black currants are more susceptible to white-pine blister rust than gooseberries and red currants, while others were just as adamant that all _Ribes_ are equally guilty. Many of my respondents, pointing to the inefficacy of past bans and the decline of the white-pine industry, said it was time to revise and review old, outdated laws. Some held out promise for a more lenient attitude toward black currants because of the availability of more effective controls, such as fungicides and disease-resistant varieties.
At the same time that these plant officials voiced differing views, scientists in Canada and individuals in the United States were continuing work to breed varieties resistant to white-pine blister rust, as well as types with bigger fruit and higher yields. The three varieties that still may be the most resistant to white-pine blister (Consort, Crusader, and Cornet) were all developed in Canada, at the Horticultural Research Institute of Ontario, at Vineland Station. Of these, Consort is considered the most resistant. It used to be the most popular and the most widely available in Canada, as well as the United States. In addition, two new varieties from Europe were then being tested in Canada; the Josta, also disease resistant, was developed in Germany. With the continued and growing interest in many "old-fashioned" fruits, there is no doubt that government-run experimental fruit stations in the United States will be moved to conduct similar research.
As for the charge that black currants have an undesirable flavor, I am convinced that most gardening writers in the United States have never tasted black currants in any form. Those few people who do go out of their way to try black currants are well rewarded. The late American gardening writer Lewis Hill once said that he and his wife were "converted to the taste when we bought a jar of 'confiture de cassis,' black-currant jam, in Canada." When I sent Hill a sample of our dried black currants, he was more than enthusiastic: "We are certainly going to expand our production. They are great!" _Update:_ The Hills went on to breed and select black currants at their Greensboro, Vermont plant nursery. The variety Hill's Kiev Select is one of Lewis Hill's selections.
What exactly is the flavor of black currants? In 1944, the great American horticulturist U. P. Hedrick said that the black currant has an "assertive flavor and aroma . . . it is most pleasant to eat out of hand or in culinary dishes." Others have called its taste "musky." It is certainly assertive, but I'm not sure about musky. The word _musky_ usually carries negative overtones, yet it also describes the appealing scent of many flavors.
To be honest, however, I think that black currants must be processed to be properly appreciated. Gardeners and commercial growers in Europe understand this well, although the British do enjoy a dish of fresh black currants and cream for breakfast. In any case, British children are as familiar with morning glasses of black currant juice as American children are with orange juice. In Europe, black currant is a highly prized flavoring for cordials, liqueurs, wines, desserts, and candies; rum with a shot of black currant syrup, I'm told, makes an incomparable drink.
Finally, are black currants worth growing? If you know nothing about a fruit except that it is a carrier of a dreaded disease (like Typhoid Mary) and tastes funny, you may well conclude that it is not. But if you have ever tasted black currant jam, jelly, wine, juice, or the dried berries, if you have ever seen a bush drooping under the weight of its fruit you will search from one end of the country to the other to find just one rootstock to plant in your garden. If you can find the _real_ thing, in whatever form, on the shelves of a specialty shop or in the pages of a British mail-order catalog, you can make up your own mind about the flavor of black currants. And from there, you can decide whether to grow them.
_Update_ : Since I wrote the above, some states have lifted their ban on growing black currants, while others have loosened restrictions in certain counties. Some states confine the planting of black currants to rust-resistant varieties, but the lists vary. The best advice is to consult your local Cooperative Extensive office (see Appendix) to find out about specific restrictions in your area. Although there are more black currant varieties available now than ever before, not all of them are noted for being rust-resistant; some bear larger, sweeter, juicier fruit, are earlier ripening, and may be mildew-resistant. Note that some of the plant nurseries which carry black currant shrubs list states to which they cannot ship plants. On the whole, however, the black currant picture is promising, especially since the fruits' health benefits have been publicized. Not only do black currants possess a high level of Vitamin C, the fruit has been found to be a rich source of potassium, phosphorous, iron, and other nutrients, and is high in antioxidants. It has, in fact, been described as the ultimate "superfruit" that may help fight cancer, heart disease, and Alzheimer's. With this kind of profile, surely it will not be long before black currants will be safer to grow throughout the country.
**Planting**
If there are white pines in your area or on your property, plant black currant rootstocks 1000 ft (300 m) from the stands of the trees.
In northern climates, plant black currant shrubs in the spring. One bush is more than enough for one person— the yield is enormous (2 gallons/8 L or more). We have six bushes, and we never calculate our yield. We are too busy picking quart after quart. The best site for black currants is a slope with a northern exposure to provide cool conditions. As long as the soil provides sufficient drainage for the roots, the particular type is not too important because it can be changed to suit the plant. As with other _Ribes,_ though, fertile clay loam is best.
To plant, trim plants to 4 to 6 inches (10 to 15 cm) and put them in a bucket of water while preparing holes. With a spade, make holes every 6 ft (2 m), 1 ft (30 cm) more than for red currants or gooseberries, in rows about 6 to 10 ft (2 to 3 m) apart. It's hard to believe, but in two or three years each small plant will mature into a candelabra of spreading branches. The tips of the branches may eventually touch, but that's okay. Just trim the offending branches whenever necessary.
If you have limited space, consider growing them, as well as gooseberries and red currants, in espalier form where branches are trained to grow on wires and supports, either against a wall or standing alone. But although production is great, maintenance is a lot of work, what with frequent tying, pinching, and pruning. But they are very attractive grown that way.
Add a shovelful of well-rotted compost to each hole and water it. Build a little mound of earth and set the roots over it, tamping the soil well around the plant. In a dry spell, water the plants regularly until they show signs of new growth.
**Cultivating**
Black currant bushes benefit from mulching as do other _Ribes_. Mulch keeps the plants cool, prevents weeds, and ultimately breaks down, adding organic matter to the soil. As already noted, we prefer a mulch that consists of three layers: manure or well-rotted compost; heavy paper, cardboard, or worn-out nonsynthetic carpet; hay, straw, or if you live close to a shore, eel grass. Whatever materials you use for mulch, it should be set down in a 2-ft (60-cm) wide ring around the plant, and it should be renewed once a year, in spring or fall.
When the plants are dormant, in early spring, remove dead and broken stems. Thin out vigorous shrubs that are more than three years old so that the center of the plant is not overcrowded. A dozen branches, a mix of one-, two-, and three-year-old ones, will provide more than enough berries if the growing conditions are satisfactory.
You will not need to worry about propagating black currant shrubs if their site provides them with cool conditions, adequate drainage, and fertile soil. They will reproduce themselves by self-layering—and much more vigorously than red currant plants. Make cuttings from the layered branches and plant them fairly close together in a nursery bed for one season before moving them to a permanent location. Remember, plant them in friable soil; weed and water them during the growing season as needed.
**Harvesting**
Harvesting black currants is an acquired art; it can start in late July and continue through August. Have a plan such as the following.
First, pick slightly underripe black currants for jelly. These berries, still a bit green, can be harvested quickly, as the stems can be left on. Second, pick the slightly riper, mostly black berries for jam, free of stems. You can wait several days between pickings, but do not wait too long—the birds may beat you to it. Third, pick the dead-ripe berries for juice. If you intend to make juice as a byproduct of making dried berries, pick those berries free of stems. If you want to make wine or mead or some other beverage, use dead-ripe currants as well, leaving on the stems.
**Preserving, Canning, Freezing, And Cooking**
There are few fruits so ill-served by modern cookbooks as black currants, but considering their history of neglect, this is not surprising. You may find a recipe for black currant jam, but more than likely the directions will be unnecessarily complicated, requiring a great deal of chopping and cooking and even the use of a food processor. Or the addition of commercial pectin, which seems excessive with a fruit so high in natural pectin and so quick setting.
The following recipes, our favorites over the years, have been gleaned from old cookbooks or adapted from other recipes. The procedure for jelly and jam making has been streamlined to reflect the general principles outlined in chapter one. It's hard to advise you what to do with your crop—the jam, jelly, juice, wine, and dried berries are all wonderful. But in cooking, dried black currants and black currant wine are particularly versatile. Dried currants are a nice addition to cookies and buns; black currant wine is suitable for chicken or beef dishes. Review A Short Course in Fruit Preserving [p. 3–14] before preserving.
**Black Currant Jelly**
AFTER YOU HAVE MADE THIS GREAT JELLY ONCE, YOU WILL WONDER WHY COMMERCIAL BLACK CURRANT JELLY OFTEN CONTAINS ARTIFICIAL COLORING. TRY THIS JELLY WITH CREAM CHEESE AND BREAD. USE A MIXTURE OF RIPE AND UNDERRIPE BLACK CURRANTS.
1 qt (1 L) juice makes about 2 pt (1 L)
Black currants with stems
Sugar
In a large preserving pot, cover at least 2 qt (2 L) black currants with water. Cover and boil until currants are soft, occasionally crushing fruit with a large wooden spoon, mallet, or paddle. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight. Reserve juice.
Bring pulp to a boil again and strain through a jelly bag, letting drip until dry.
Combine juice from both extractions. Measure juice and cook 4 cups at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil. Skimming as necessary, boil, uncovered, for about 15 minutes or less or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Black Currant Jam**
THIS IS A FAST-SETTING JAM. USE JUST-RIPE BLACK CURRANTS, WITH SOME SLIGHTLY UNDERRIPE.
Makes 1 1/2 to 2 pt (750 mL to 1 L)
4 cups black currants without stems (1 L)
3 cups sugar (750 mL)
1/2 cup water or grape juice (125 mL)
In a large stainless-steel pot, combine black currants and liquid. Mash if desired. Cover and bring to a boil. Stir in sugar and bring to a boil again. Reduce heat slightly. Stirring occasionally, cook, uncovered, for about 10 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Cottage Cheese**
TO MAKE CREAM CHEESE, YOU HAVE TO MAKE COTTAGE CHEESE FIRST. FIND A GALLON OR TWO OF UNPASTEURIZED MILK, PREFERABLY FROM A JERSEY COW. WITH COTTAGE CHEESE, PRACTICE MAKES PERFECT. DON'T GO THE RENNET ROUTE: YOU WILL BE DISAPPOINTED WITH THE BLAND PRODUCT.
Makes about 1 qt (1 L)
1 gallon unpasteurized milk (4 L) Cream
Salt
Pour unpasteurized milk into a large preserving pot. Cover and let stand in a warm place until milk forms a solid gelatinous mass similar to junket or custard, about 24 to 48 hours. To test, tip pot slightly to see if curd slides up side of pot. If so, it is ready for heating.
Set pot over low heat or in a large pan of hot water and gently break up curd by hand until it is uniform. Continue to heat slowly and stir until curds are fairly firm pieces and are separated from whey (curds are white and whey is yellowish). When curds are sufficiently firm, the mixture will probably be as hot as your hand can stand.
Remove curds from heat and drain through a large cheesecloth bag [see Jelly Bag, p. 28]. Let drip overnight.
Break up curds with a slotted spoon or a fork. Stir in cream and salt to taste. Place cottage cheese in a plastic container and chill thoroughly. If curds are grainy and hard, mixture was overcooked. If curds are soft, mixture was undercooked, and cottage cheese will sour quickly.
_Note:_ To hasten the initial thickening process, add 1/2 cup (125 mL) buttermilk.
**Cream Cheese**
TO MAKE CREAM CHEESE, FOLLOW THE RECIPE FOR COTTAGE CHEESE THROUGH TO DRAINING THE CURDS OVERNIGHT AND PROCEED.
Makes 1 qt (1 L)
1 qt dry curds (1 L)
1 cup heavy cream (250 mL)
1 /4 cup butter (50 mL)
1 1/2 scant tsp salt (7 mL)
1/2 tsp baking soda (2 mL)
Place dry curds in the top of a double boiler. With your hands, work butter into curds. Add baking soda. Place mixture over bottom of double boiler, making sure water does not touch top pot. Add some of cream to curds. Stirring occasionally and adding more cream if dry, slowly heat mixture for 10 minutes or until it turns gummy and sticky. As you stir, gummy strands will begin to form on bottom of pot.
Remove pot from heat and stir in remaining cream or enough to make a thick mixture the consistency of pudding. Stir in salt. Place cream cheese in a plastic container and refrigerate for several hours before using.
_Note:_ Do not worry if cream cheese is not completely smooth, as little lumps will eventually dissolve.
**Dried Black Currants**
THESE LOOK LIKE SMALL DARK RAISINS. DRIED CURRANTS SHOULD STILL HAVE A TENDER, SLIGHTLY MOIST QUALITY, BUT THEY SHOULD NOT BE STICKY. USE DEAD-RIPE BLACK CURRANTS.
Makes about 1 lb (500 g)
3 cups black currants without stems (750 mL)
1 cup sugar (250 mL)
Layer black currants and sugar consecutively in a medium-sized preserving pot. Cover and let stand overnight to draw out juices.
Bring mixture to a boil. Reduce heat and, stirring occasionally, simmer gently for 15 minutes. Strain black currants and reserve juice.
Place currants on trays lined with heavy brown paper and spread them out evenly. Set trays in direct sunlight. Stir currants daily to prevent sticking. After first day of drying, change paper. Drying takes about 3 to 4 days, depending on weather.
_Note:_ This recipe can be used to dry blueberries as well.
**Black Currant Juice**
VERY SIMILAR TO THE BRITISH PRODUCT, RIBENA, A BLACK CURRANT JUICE CONCENTRATE.
To make juice _,_ bring reserved juice from making dried currants to a boil and simmer for about 5 minutes. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes. To use, add 3 qt (3 L) water to 1 qt (1 L) syrup.
_Variation_ : To make Black Currant Juice without first making berries, add 1 cup sugar (250 mL) to 1 quart (1 L) fruit. Let sit to bring out juices. Stir, bring to a boil, simmer 10 minutes and pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes. To use, dilute with water to taste.
**Black Currant Wine**
THIS IS ONE OF THE BEST HOMEMADE WINES. USE IT AS A SUBSTITUTE IN ANY RECIPE CALLING FOR RED WINE, SUCH AS COQ AU VIN. OF COURSE, IT'S GRAND TO SERVE WITH SPAGHETTI OR A SPECIAL MEAL. USE CLEAN DEAD-RIPE OR OVERRIPE BLACK CURRANTS WITH OR WITHOUT STEMS.
Black currants
Sugar
Juice of 1 lemon or 2 Tbsp sugar
(25 mL) REAL Lemon
Wine or baking yeast
In a large preserving pot, cover black currants with water, using as much fruit as desired. Make about 5 to 10 extractions as for black currant jelly (p. 76), until pulp loses color.
Combine and measure liquid from extractions. Place in a large crock or plastic bucket. Stir in 2 lb (1 kg) sugar to each gallon (4 L) juice and mix in lemon juice.
Dissolve 1 Tbsp (15 mL) yeast in a little warm water and add to liquid. Stir mixture again and cover container. Let mixture ferment at 70° to 75°F (20° to 25°C) for 10 days.
Siphon contents into glass jugs, using cotton plugs as stoppers. Store wine at 60° to 65°F (15° to 18°C), away from direct sunlight, which could ruin the nice ruby color. When wine stops bubbling completely (in about 1 month), resiphon into permanent bottles. Seal and store in a cool, dark, dry place. If using corks, store bottles lying down.
_Note:_ If you do not make multiple extractions, the juice will be too strong and will make an unpalatable wine. Besides, you can make a lot of wine from a small amount of currants.
**Crème de Cassis**
FRIEND AND FRUIT MAVIN, LOIS ROSE, HAS BEEN MAKING THIS ANCIENT FRENCH LIQUEUR FOR ALMOST 20 YEARS, ADAPTED FROM JANE GRIGSON'S _JANE GRIGSON'S FRUIT BOOK_. THE CONSISTENCY, AS THE NAME IMPLIES, SHOULD BE CREAMY, ALTHOUGH THERE'S NO CREAM IN IT. THERE ARE MANY WAYS TO USE CRèME DE CASSIS BESIDES SIPPING IT SLOWLY OR MIXING INTO COCKTAILS. TRY OVER VANILLA ICE CREAM.
2 lb black currants (1 kg)
4 full cups red wine (1 L)
3 lb sugar (1 **½** kg)
1 1/2 pt brandy, gin, or vodka (700 mL)
Soak fruit in wine for 48 hours. Place several layers of fine cheesecloth over a pot. Process currants and wine in a blender until mush and pour into cheesecloth to drain. Pull ends together and twist hard to squeeze out all the liquid you can. Pour resulting juice into a pot, add sugar and stir at a very low heat for 2 hours until the juice is reduced a bit. Overheating can spoil the cassis, giving it a cooked taste (Lois uses a heat disperser under the cooking pot). The idea is to cook at a low enough temperature, above blood heat and lower than simmering, to reduce the liquid without evaporating all the alcohol. You really have to stick with it, stirring at intervals during the 2 hours. Cool, then add 3 parts juice to 1 part brandy, gin, or 151 vodka and bottle. Let sit a few months before opening.
**New Year's Punch for 20**
Drink straight or mixed with fruit juice and ice for a great summer drink.
5 gallons lukewarm water (20 L)
1 Tbsp cream of tartar (25 mL)
3 3/4 lb sugar (2.75 kg)
2 lemons
3/4 oz ginger (20 g)
1/2 tsp baking yeast (2 mL)
Place water in a large crock or pot, stir in sugar, ginger, and cream of tartar.
Cut lemons into thin slices and add to water mixture. Dissolve yeast in a little warm water; add to liquid. Stir, cover, and let stand overnight.
Siphon ginger ale into soda bottles and cap with a hand capper. Ready to use in 5 days. Store in a cool, dark, dry place.
**Black Mead**
MEAD IS AN ANCIENT DRINK MADE BY FERMENTING HONEY WITH OTHER INGREDIENTS. LET THIS DRINK AGE FOR AT LEAST A YEAR—IT'S WORTH THE WAIT. USE CLEAN DEAD-RIPE OR OVERRIPE BLACK CURRANTS WITH OR WITHOUT STEMS.
Makes 1 gallon (4 L)
4 lb black currants (2 kg) Juice of 2 lemons
2 lb honey (1 kg)
4 green apples, chopped
6 pt warm water (3 L)
1 Tbsp wine or baking yeast (15 mL)
1/4 tsp Marmite (1 mL)
1 Tbsp strong tea (15 mL) Cold water
In a large preserving pot, crush black currants with a wooden spoon or mallet. In a bowl, dissolve honey in warm water. Add Marmite, tea, lemon juice, and apples. Dissolve yeast in warm water. Add honey-apple mixture and yeast to crushed black currants. Stir, cover, and let stand for 3 days.
Strain, pressing pulp slightly. Add enough cold water to make 1 gallon (4 L) liquid. Cover and let ferment at 70° to 75°F (20° to 25°C) for 10 days. Siphon contents into glass jugs, using cotton plugs as stoppers. Store mead at 60° to 65°F (15° to 18°C), away from direct sunlight. When mead stops bubbling completely, resiphon into permanent bottles. Store in a cool, dark, dry place. If sealed with corks, store bottles lying down.
**Coq au Vin**
Serves 4
3 1/2 lb frying chicken, cut up (1.75 kg) Bacon drippings
Salt
Pepper
1/2 lb small onions, cut up (250 g)
1/2 lb fresh mushrooms or a good handful of dried (250 g)
2 Tbsp flour (25 ml)
2 cups black currant wine [p. 79] (500 ml)
1 large clove garlic, minced
2 to 3 shallots or ½ cup (125 mL) chopped onions or scallions
Several sprigs fresh parsley, chopped, or a few pinches of dried
1/4 tsp thyme (1 ml)
1/2 bay leaf
In a large frying pan, sauté chicken in bacon drippings, adding salt and pepper to taste. When chicken is browned on both sides, place in a roasting pan. Place in a 350°F (180°C) oven to await sauce.
Sauté onions and mushrooms in chicken-bacon fat. When onions begin to brown, remove pan from heat, stir in flour, return to heat, and gradually stir in wine. Add garlic, shallots, parsley, thyme, and bay leaf. Stirring constantly, simmer sauce for 1 to 2 minutes, adding more liquid (wine or water) as desired.
Remove chicken from oven. Pour sauce over chicken, cover, and continue baking for about 1 hour or until chicken is tender. While baking, baste occasionally, and before done, remove cover to brown. Serve with rice and more black currant wine.
Note: If using dried mushrooms, presoak them in boiling water for about 10 minutes and drain, reserving liquid to add to wine.
**Black Currant Sticky Buns**
Makes 3 dozen
1/4 tsp ginger (1 mL)
1 cup dried currants (p. 108) or a mixture of part raisins (250 mL)
1 tsp granulated sugar (5 mL)
1 Tbsp dry yeast (15 mL)
1/2 cup warm water (125 mL) Cinnamon (for mixing into dough) Melted butter
Sugar-cinnamon mix (to taste)
1 cup milk (250 mL)
1/2 cup shortening (125 mL)
1/3 cup honey (75 mL)
1 tsp salt (5 mL)
Syrup:
1 cup cold water (250 mL)
3 Tbsp butter (50 mL)
1 egg, well beaten
2 cups brown sugar (500 mL)
12 cups flour (part whole-wheat) (3 L)
2 Tbsp water (25 mL)
In a large bowl, sprinkle ginger, 1 tsp (5 mL) granulated sugar, and yeast on warm water. Let stand undisturbed until foaming.
In a saucepan, scald milk and melt in shortening. Stir in honey and salt. Remove from heat. Stir in cold water to cool mixture to warm. Add milk mixture to dissolved yeast; stir well. Add well-beaten egg and stir in a few cups of flour, beating well. Add currants. Mix in more flour until dough is stiff enough to knead.
Knead dough and when almost finished, sprinkle cinnamon on all sides and knead in. Cover bowl with a cloth and let dough rise in a warm place until doubled in size.
Meanwhile, prepare syrup. In an 11-inch (3.5-L) skillet, melt 3 Tbsp (50 mL) butter, add enough brown sugar to cover bottom of pan, and add 2 Tbsp (25 mL) water. (Add more brown sugar if you want stickier buns.) Bring mixture to a boil and cook for a few minutes. Pour onto 2 cookie sheets, tilting pans so they are evenly coated.
Divide dough into thirds. Roll out each piece into a rectangle about 1/4 inch (5 mm) thick. Spread each with melted butter and sprinkle with a little sugar-cinnamon mix. Roll up each piece longways and cut into buns 1 1/2 inches (4 cm) wide. Place buns, cut side down, on cookie sheets. Cover with a towel and let stand until dough begins to rise. Bake in a 350°F (180°C) oven for about 15 minutes. Remove to a platter immediately, flipping each bun so sticky side is up.
**Jellyroll with Black Currant Jam**
4 eggs, separated
1/2 tsp salt (2 mL)
3/4 cup sifted sugar (175 mL) Icing sugar
1 tsp vanilla (5 mL)
Black currant jam [p. 107] or jelly [p. 106]
3/4 cup cake flour (175 mL)
3/4 tsp baking powder (4 mL)
In a bowl, beat egg yolks until light. Gradually add sifted sugar and beat mixture until creamy. Add vanilla.
Sift together flour, baking powder, and salt. Gradually add to sugar-egg yolk mixture. Beat until smooth. In a separate bowl, beat egg whites until stiff; fold into batter.
Grease a shallow 7-inch x 11 -inch (2-L) baking pan and line it with wax paper. Pour in batter and bake in a 375° F (190°C) oven for 12 to 15 minutes.
Remove cake from oven and flop onto a dish towel well sprinkled with icing sugar. Trim off any hard edges, then roll up cake in towel. When ready to use, unroll and spread at least 1/2 cup (125 mL) black currant jam or desired amount over top of cake. Roll up cake, jam side in, wrap towel around it, and let sit for about 15 minutes before slicing and serving.
_Variation:_ Make your favorite sponge cake and cut into 2 layers. Thickly cover top of bottom layer with black currant jam. Add top layer and spread with whipped cream [p. 58]. Chill before serving.
**Black Currant-Leaf Cream**
I LEARNED ABOUT THIS DISH FROM A FRIEND WHO IS A FINE GARDENER AND COOK. IT IS WONDERFULLY REFRESHING IN THE HEAT OF SUMMER AND CAN BE USED TO GREAT EFFECT TO TOP PARFAITS, ICE CREAM, AND SORBETS. WHEN COOKING THE LEAVES YOU CAN TOSS IN A SMALL HANDFUL OF BERRIES FOR EXTRA BLACK CURRANT FLAVOR.
Serves 4
1 cup young black currant leaves (250 mL)
2 egg whites
1 cup whipped cream [p. 58] (250 mL)
2 cups sugar (500 mL)
1 cup water (250 mL) Touch lemon juice
In a pot, boil together black currant leaves, sugar, and water for 15 minutes, stirring to dissolve sugar. Strain, reserving syrup.
In a bowl, beat egg whites until stiff. _Gently_ pour hot syrup onto egg whites. Beat until mixture begins to thicken. Let cool.
Fold whipped cream and lemon juice into cooled mixture. Spoon into individual bowls and chill.
**Boodle's Fool**
Serves 6
3 cups black or red currants or gooseberries
(750 mL )
1 pt whipping cream [p. 58] (500 mL) Sugar or honey
1 stale cake, broken up, or equivalent amount of cookies
Prepare juice. In a large pot, combine about 3 cups (750 mL) ripe black currants with a little water. Cover and bring to a boil. Boil until currants are soft, occasionally crushing fruit with a large wooden spoon, mallet, or paddle. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight. Measure 1 cup (250 mL.) juice. Sweeten juice to taste with honey or sugar.
In a bowl, beat whipping cream halfway to stiff and add currant juice. Continue beating until mixture is frothy. Pour mixture over broken-up pieces of cake or over cookies. Chill for at least 4 hours.
**Black Currant Sorbet/Ice**
THIS DESSERT IS BOUND TO IMPRESS. IT IS NOT ONLY DELICIOUS AND COOLING, BUT A LOVELY ROSY PINK, ESPECIALLY BEAUTIFUL WHEN SERVED IN PARFAIT GLASSES, TOPPED WITH WHIPPED CREAM OR BLACK CURRANT LEAF CREAM AND EMBELLISHED WITH AN EDIBLE FLOWER. I LIKE TO USE ZEBRINA MALLOW (MALVA SYLVESTRIS ZEBRINA), A SMALL TO MEDIUM SIZED SOFT PINK TRUMPET WITH PURPLE STRIPES, LIKE A MINIATURE HOLLYHOCK. REVIEW "AN ASSORTMENT OF BASICS" [P.120] FOR USING AN ICE CREAM MACHINE.
Makes a little more than 1 qt (1 L)
2 cups unsweetened black currant juice from extraction 500 mL
1 cup water 250 mL
2 cups sugar 500 mL
Pre-mix juice and sugar in ice cream machine canister, cover with freezing cover and chill for faster freezing. When ready to begin freezing, turn on the machine, set canister in ice bucket, pour in water and layer ice and salt as directed. Remove freezing cover, insert paddle, and put mixing cover in place, adding more water according to directions. Turn on machine and add more salt and ice as directed when ice begins to melt. Continue freezing until sorbet is the consistency you want. This should take about 12-20 minutes. Remove canister, check sorbet and stir if needed, store in freezer for an hour or more, then you may remove sorbet to a plastic freezer container and return to freezer.
Variation: Substitute unsweetened red currant juice, or any unsweetened fruit juice, and proceed.
**_Elderberries_**
The elderberry often occupied a favored spot in the old-fashioned fruit garden—perhaps a corner where it could grow undisturbed, its branches spreading to 8 ft (2.5 m) when laden with fruit, the plant itself growing about 12 ft (4 m) under good conditions. In early July, the glorious elderblossoms, or elderblow, were picked for wine and jelly making. The buds were pickled, the florets were shaken into pancake and muffin batters to lighten and sweeten them. Later in the season, before the first frosts of autumn, the clusters of shiny purple-black fruit were picked mainly for wine making, although elderberry jelly and pie were also looked on with great favor. The experienced cook knew just where to find the hard sour green apples that, when combined with elderberries, made a firm, flavorful jelly. (Unlike many other small fruits, elderberries do not contain much natural pectin.) Large trays of ripe elderberries were set out in the sun to dry so that elderberry pie could be enjoyed throughout the winter.
When the creamy white umbels of elderflowers bloom, as giant saucers, they seem to embody the characteristics of summer—long sweet-scented sunny days. Then later, some time before the first frost, the ripening clusters of dark berries remind us that shorter days, cold nights, and the end of the growing season are close at hand. Even If you never harvest a single flower or berry, you may enjoy growing the elderberry bush just to look at it.
_Sambuccus canadensis,_ the elderberry of the old-fashioned fruit garden, grows wild in thickets all across Canada and the United States. Beware, though. If you gather elderberries in the wild, do not confuse the purple-black berry with the red variety, _Sambuccus pubens,_ which blossoms and fruits earlier. The red berries, reputedly, are inedible, though birds do enjoy them.
Luckily, cultivated elderberries, including European varieties ( _S. nigra_ ) and ornamental types that bear significantly, are widely available from plant nurseries. There are several ornamental varieties in my garden that are wonderful landscaping shrubs, including Black Lace (with pink flowers) and golden elderberry, a heavy bearer. Modern breeding has made elderberries even more attractive for preserving by enlarging the berries and shortening the ripening time of some varieties—Nova, for example. Elderberries are hardy, easy to grow, and adaptable to a wide range of soils and growing conditions. If you have the space, they certainly deserve to be grown in your fruit garden. High in Vitamin C, the berries' reputed health benefits are similar to black currants.
**Planting**
Elderberries thrive best in fertile, moist, and loamy soil. If your soil is heavy, try to plant rootstocks on a slope, preferably one with a southern exposure. If you can devote a whole southern-facing bank to the planting, the reward at blossom time will more than pay for the space. A slope will not only provide perfect drainage, but will help shelter the plants from early frost—important with later-ripening elderberries—and diseases such as mildew. Plant two varieties for best pollination. Nova, introduced in 1960 by the Agriculture Canada Experiment Station in Kentville, Nova Scotia, is often planted with York, a cultivar with extra large berries and also developed in Canada.
To plant, use a sharp spade and dig holes deep enough to receive the roots of the elderberry stock. Space the plants at least 6 ft (2 m) apart, with 8 to 10 ft (2.5 to 3 m) between rows. It is important not to let the roots dry out, so submerge them in a bucket of water until all the plants are set in the holes. Water each hole, place the plant in the center, spreading out the roots. Fill and cover the hole with soil and tamp the earth firmly into place. Water again, slowly. Next, lay down a 2-ft (60-cm) wide ring of mulch around the base of the plant: a thick layer of compost or manure; a layer of paper, cardboard, old nonsynthetic carpet, or cloth grain bags; and a thick top layer of straw or rotted sawdust. Once the elderberry plant is well established, in two or three years, the mulch need not be renewed if the soil is fairly fertile.
**Cultivating**
The only thing to do after mulching is to make sure the young plants receive enough water, particularly at fruiting time. If it is especially dry in late summer or early fall, water the bush to ensure good production.
Elderberry bushes will propagate themselves by sending up shoots outside the ring of the original plant. Allow the original plant to spread in a 2-ft (60-cm) circle, then remove all shoots that grow outside this area. Dig down with a sharp spade to remove the new stem and part of the old root and replant it in a nursery bed or in its permanent place, making sure the soil has adequate drainage. Early each spring, remove dead and winter-damaged stalks on all bushes.
**Harvesting**
If you want to harvest the blossoms for wine, jelly, or cooking, choose umbels in which most of the florets are newly opened; the flowers on mature umbels shatter easily. Cut the stalks with scissors, and store the flowering umbels with their stalks in a plastic bag in the refrigerator until you are ready to use them.
There is only one important thing to say about harvesting the berries: watch out for the birds! In September, gathering for their journey south, they will be eager to fill up on your elderberries. Netting is the most effective way to keep away birds—if used properly. Plastic or nylon netting is good because it is sturdy and can be reused. Cheesecloth is more effective, however, because the mesh is smaller. Follow the manufacturer's instructions when setting out nets, but do not try to cover too great an area with one net. It is better to concentrate on the best-producing shrubs. Let the birds have the berries from the lesser-producing ones. Several mature shrubs will provide an abundant crop, thousands and thousands of berries.
Check elderberry clusters every day. The berries on each cluster do not all ripen at once, so choose clusters in which most of the berries have ripened. Cut them off with scissors, leaving some stalk, as with harvesting the flowers; the stalks of really ripe fruit break off easily. Hint: it's easier to separate berries from stem if you freeze clusters and pull off the berries while the fruit is still very cold.
**Preserving, Canning, Freezing, And Cooking**
Use fresh flowers for wine, deep-fry fresh flowers dipped in batter, or shake the florets into pancake or bread batters to lighten and flavor them (and be sure to pick fresh flowers for adding to gooseberries for jelly [p. 72]. Dry flowers for tea. The first ripe berries should go into jelly because the earlier the apples are harvested for elderberry jelly, the better. Turn later berries into jam, preserves, wine, juice, or dried berries. As you will notice, elderberries are invariably coupled with another flavor. Alone, the elderberry, grapelike in flavor, tends to be insipid, so it needs a complement, usually something tart, such as apples, lemons, or vinegar.
Use dried elderberries and elderberry preserves in pies; elderberry juice makes an interesting custard pie. Don't forget to use elderberry wine in cooking wherever red wine is called for. Review A Short Course in Fruit Preserving [p. 4] before preserving.
**Elderflower Wine**
THIS IS MADE WITH THE BLOW, OR BLOSSOMS, AFTER MOST OF THE GREEN STALKS HAVE BEEN REMOVED. IT IS A CLEAR WHITE WINE, RATHER HEADY AND FRAGRANT.
Makes 1 gallon (4 L)
**2 qt elderflowers (2 L) Juice of 2 lemons**
**1 gallon boiling water (4 L)**
**2 1/2 lb sugar 1.25 kg)**
**1 oz wine or baking yeast (30 g)**
**1 yeast-nutrient tablet**
**or 1 Tbsp Marmite (15 mL)**
Place newly opened elderflowers in a large container and pour boiling water over them. Let mixture steep, covered, 2 to 3 days.
Dissolve yeast in a little warm water. Strain elderflower mixture and add lemon juice, sugar, yeast, and yeast nutrient or Marmite to liquid. Stir, cover, and let ferment at 70° to 75°F (20° to 25°C) for a week.
Siphon into jars, plug with cotton, and store away from light. Resiphon after 3 months and again when wine is done (when all bubbling ceases), at which time jars can be capped and stored in a cool, dark, dry place. If bottles are corked, store lying down.
**Dried Elderflowers**
You may want to try your hand at drying these glorious flowers to use for winter teas. The trick is quick drying to prevent the flowers from turning brown. You have done a good job if the florets have retained their creamy white color.
Gently shake newly opened elder florets loose from their clusters. Set them in a shallow pan immediately and dry them in a warm oven just after heat has been turned off. Check frequently, gently stirring flowers with a long-handled fork.
Remove dried florets from the oven as soon as they look and feel dry and crisp. Let cool completely in pan before storing in a jar—away from light.
**Eldeberry Jelly With Added Pectin**
ELDERBERRIES NEED ADDED PECTIN, WHETHER FROM APPLES, CRABAPPLES, OR FROM COMMERCIAL PECTIN. IF YOU CAN'T FIND TART APPLES OR CRABAPPLES, DON'T DESPAIR. THIS JELLY, MADE WITH TRADITIONAL PECTIN CRYSTALS, IS NOT OVERLY SWEET. I FOLLOWED THE AMOUNTS AND BOILING TIME FROM BILLY JOE TATUM, WHO DID NOT FAIL ME. READ ABOUT PECTINS [P. 28].
Makes about 3 pt (1.5L)
**3 3/4 cups elderberry juice (875.38 mL), extracted from 3-4 lb fruit (1.5-2 kg)**
**1/4 cup lemon juice (50 mL)**
**1 box pectin crystals**
**1/2 tsp butter (2 mL)**
**5 cups sugar (1.182 L)**
Crush fruit, heat and simmer for 15 minutes, strain. Stir pectin into juice until dissolved. Add lemon juice, butter (to prevent boiling over), and bring to a rolling boil that can't be stirred down. Stir in sugar, bring back to a boil and boil hard for 2 minutes. Let mixture subside, skim if necessary. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands, and seal. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Note:_ I have not tried commercial pectin with the recipes below. You may be able to adapt them, especially with the low or no sugar needed types.
**Elderberry-Apple Jelly**
IF SOME STEMS REMAIN ON THE ELDERBERRIES, USE HEAPING AMOUNTS.
Makes about 2 pt (1 L)
**3 lb tart apples or crabapples (1.5 kg)**
**1 qt water (1 L)**
**Sugar**
**2 qt elderberries (2 L)**
Cut up apples. In a large preserving pot, combine apples, early ripe elderberries, and water. Cover and simmer until fruit is very soft, stirring occasionally and mashing when mixture starts to bubble. Strain through a jelly bag [p. 28] and let drip overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil and, skimming as necessary, boil, uncovered, for 10 to 15 minutes or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Variations:_ To make _elderberry-orange jelly,_ put peel from 1 orange and half a stick of cinnamon in a small cheesecloth bag. Drop spice bag into elderberry-apple juice after sugar has been added. Remove it from pot just before jelly is poured into jars. Process as for Elderberry Jelly.
_Elderberry sass_ is a direct variant of elderberry-apple jelly and is our favorite pancake topping just after maple syrup. Simmer fruit as for jelly, but when cooking juice, add only 2 to 3 cups (500 to 750 mL) sugar to 1 qt (1 L) juice (you can cook more than 4 cups/1 L juice at a time). Boil mixture until it is reduced by one-third or one-quarter. Fill hot scalded jars [p. 28] to 1/2 inch (1 cm) from the top and seal at once with snap lids and screw bands. Process in a boiling water-bath [p. 26] or steam canner for 15 minutes.
**Spiced Elderberry Jelly**
Makes about 4 1/2 pt (2.25 L)
**6 lb tart fresh apples 3 kg**
**1 Tbsp ground cloves (or 1/4 cup/50 mL whole cloves) (25 mL)**
**4 qt elderberries with stems (4 L)**
**1 qt cider vinegar (1 L)**
**1 qt water (2 L)**
**Sugar**
**1 Tbsp ground cinnamon (or 1 /4 cup/50 mL cinnamon stick) (25 mL)**
Cut up apples. In a large preserving pot, combine early ripe elderberries, apples, cider vinegar, and water. Simmer, covered, until fruit is soft, stirring as needed. Mash fruit. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover, bring to a boil, and stir in cinnamon and cloves (if whole, put them in a cheesecloth bag). Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Boil, uncovered, for 15 minutes or until a small amount sheets off a metal spoon [p. 28-29]. Remove spice bag.
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Elderberry-Green Grape Jelly**
**Makes 2 to 3 pt (1 to 1.5 L)**
**2 qt elderberries (2 L)**
**4 cups water (1 L)**
**2 qt green grapes (2 L)**
**Sugar**
Place early ripe elderberries and ripe green grapes in separate preserving pots. Add 2 cups (500 mL) water to each pot, cover, and simmer until fruit is soft. Strain each mixture through its own jelly bag [p. 28] and let drip overnight.
Combine juice from each receiving pot and measure. Cook 4 cups (1 L) juice at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a boil again. Skimming as necessary, boil, uncovered, until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Elderberry Jam**
LEMON AND APPLE JUICE ARE ACIDIC AND HELP JELLING. YOU COULD ALSO USE SUMAC JUICE [P. 181]. USE LATER-RIPENING ELDERBERRIES.
Makes about 1 1/2 pt (750 mL)
**Juice of 2 lemons**
**1 lb elderberries without stems (500 g)**
**1/2 cup apple juice (125 mL)**
**2 cups sugar (500 mL)**
In a large stainless-steel pot, combine lemon and apple juice or apple pectin, and elderberries. Mash if desired. Bring to a boil, covered, and stir in sugar. Stirring frequently, boil, uncovered, 10 to 15 minutes or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Note:_ If you like, put cooked elderberries through a food mill before adding sugar.
**Elderberry Preserves**
BEFORE REFRIGERATION, OLD-TIMERS USED TO POUR THESE PRESERVES INTO A CROCK, THEN STORE, COVERED, IN A COOL CELLAR. USE LATER-RIPENING ELDERBERRIES.
Makes 2 1/2 pt (1.25 L)
**8 cups elderberries without stems (2 L)**
**1 pt vinegar (500 mL)**
**8 cups sugar (2 L)**
In a large stainless-steel pot, combine elderberries, vinegar, and sugar. Stir. Boil, uncovered, until thick, stirring as necessary. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] for 15 minutes.
**Elderberry Juice I**
ADD SOME RHUBARB JUICE [P. 44] TO THIS— A GREAT BREAKFAST DRINK. WHEN SERVING, DILUTE WITH WATER TO TASTE. USE DEAD-RIPE ELDERBERRIES.
**Elderberry clusters**
**Lemon or sumac juice [p. 181]**
**Sugar**
In a large preserving pot, cover elderberry clusters with water; remove most of the stalks which is easier after the berries are frozen. Simmer, covered, until soft and mushy. Pound fruit with a wooden mallet or large spoon to make sure all juice is cooked out of berries. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice. Heat, covered, to almost boiling. Add 1/2 cup (125 mL) sugar to each qt (L) juice, stirring until dissolved. Add lemon or sumac juice to taste. Bring mixture to a boil, reduce heat, and simmer, uncovered, for 5 to 10 minutes
Remove juice from heat. Pour into hot scalded jars [p. 28] leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Elderberry Juice II**
THIS IS A SIMPLE RECIPE: NO PRELIMINARY COOKING AND MASHING. USE DEAD-RIPE ELDERBERRIES, AND SERVE DILUTED WITH WATER TO TASTE.
Makes about 2 qt (2 L)
**4 qt elderberries without stems (4 L)**
**Vinegar**
**Sugar**
In a large preserving pot, cover elderberries with vinegar and let stand, covered, for 24 hours. Strain through a jelly bag [p. 28] and let drip until berries are dry.
Measure juice. Bring to a boil, covered, and stir in 2 cups (500 mL) sugar to each qt (L) juice. Simmer, covered, or with lid tilted, for 20 minutes, stirring as necessary.
Remove juice from heat. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Elderberry Cordial**
THIS MAKES A NICE AFTER-DINNER DRINK, AS WELL AS A TASTY BASE FOR A HOT TODDY IF YOU HAVE A COLD. ELDERBERRY JUICES AND DRINKS HAVE LONG BEEN ASSOCIATED WITH SOOTHING THE SYMPTOMS OF COLDS AND CONGESTION. USE DEAD-RIPE ELDERBERRIES.
Makes about 3 qt (3 L)
**8 qt elderberries without stems (3 L)**
**8 tsp whole allspice (40 mL)**
**1 cinnamon stick**
**2 qt water (2 L)**
**Sugar**
**2 tsp whole cloves (10 mL)**
**Brandy**
In a large preserving pot, combine elderberries, water, cloves, allspice, and cinnamon. Boil, covered, until berries are soft. Strain mixture. Measure juice and stir in 1 1/2 cups (375 mL) sugar to each qt (L) juice. Simmer, uncovered, until mixture begins to thicken, about 30 minutes.
Remove syrup from heat and let cool. Measure syrup and stir in 1 cup (250 mL) brandy to each qt (L) syrup. Pour into bottles and seal with screw caps. Store in a cool, dry cupboard.
_Note:_ To make _hot toddies,_ add hot water to desired amount of cordial.
**Elderberry Wine**
THE FOLLOWING PROPORTIONS MAKE A MEDIUM-DRY RED WINE; INCREASE OR DECREASE THE AMOUNT OF SUGAR FOR A SWEETER OR DRIER WINE. USE CLEAN DEAD-RIPE ELDERBERRIES.
**3 1/2 lb elderberries (1.75 kg)**
**1 Tbsp wine or baking yeast (15 mL)**
**1 lemon, sliced**
**7 pt water (3.5 L)**
**Sugar**
In a large preserving pot, combine elderberries, lemon, and water. Simmer, covered, until fruit is soft, mashing to expel juice. Strain mixture through a jelly bag [p. 28] and let drip overnight. Squeeze bag to extract trapped liquid. Simmer pulp, covered, and strain again.
Dissolve yeast in a little warm water. Combine and measure liquid from extractions. Place in a large crock or plastic bucket and stir in 2 lb (1 kg) sugar to each gallon (4 L) juice. Add yeast and stir mixture again. Cover container with a plastic bag and tie bag. Let mixture ferment at 70° to 75°F (20° to 25°C) until obvious bubbling ceases.
Siphon contents into glass jugs, using cotton plugs as stoppers. Store wine at 60° to 65°F (15° to 18°C), away from direct sunlight. When wine stops bubbling completely (in about 1 month), resiphon into permanent bottles. Seal and store in a cool, dark, dry place. If using corks, store bottles lying down.
**Dried Elderberries**
BEFORE USING DRIED ELDERBERRIES, STEEP THEM IN HOT WATER. SAVE THE JUICE AND USE IT IN COOKING OR BAKING. ADD SOME DRIED BERRIES TO WINTER APPLE PIES FOR MORE FLAVOR.
Harvest ripest clusters of elderberries and gently pull berries from their stems. Place on cookie sheets or large flat pans. Set them in sun or in a just-warm oven to dry. Stir them often and shake pan to prevent berries from sticking. Dry berries until free of liquid. To test, squeeze one.
**Freezing Elderberries**
Store elderberries, with or without stems, in a freezer bag or freezer container, for future use. No sugar is needed. Frozen elderberries are much easier to de-stem than fresh elderberries.
**Elderflower Pancakes**
ALL YOUR DELICIOUS SASSES WILL TASTE TERRIFIC ON THESE LIGHT ELDERFLOWER PANCAKES.
Serves 2
**1/2 cup elderflorets (125 mL)**
**1 Tbsp cornmeal (15 mL)**
**1/2 cup unbleached all-purpose flour (125 mL)**
**1 egg**
**2 Tbsp melted butter, shortening, or lard (25 mL)**
**1/2 tsp baking soda (2 mL)**
**2 Tbsp sugar (25 mL)**
**1/2 to 1 cup buttermilk**
**(125 to 250 mL)**
**1/2 tsp salt (2 mL)**
Cut clusters of elderflowers and shake off florets. Set florets aside. Sift together flour, baking soda, sugar, salt, and cornmeal.
In a bowl, beat egg well and stir in melted butter and 1/2 cup (125 mL) buttermilk, mixing only until flour is blended in; do not overstir. Stir in elderflorets and more buttermilk until batter reaches desired consistency.
Ladle pancakes onto a hot greased griddle and cook until nicely browned.
_Note:_ Lighten and flavor any bread, roll, or muffin recipe with elderflowers by substituting 1 cup (250 mL) florets for 1 cup (250 mL) flour.
_Variation:_ To make _buttermilk pancakes,_ omit florets and increase amount of flour to 1 cup (250 mL). Stir in additional buttermilk until batter reaches consistency of muffin dough or thinner, depending on how you like your pancakes. Great with all fruit toppings.
Elderflowers
**Elderflower-Peppermint Tea**
THIS IS A SOOTHING DRINK FOR RELIEF FROM COLD SYMPTOMS—OR IT'S JUST NICE TO HAVE ON A WINTER AFTERNOON. THE FOLLOWING PROPORTIONS ARE FOR DRIED FLOWERS AND MINT; INCREASE THE AMOUNTS IF USING FRESH BLOSSOMS AND LEAVES.
Makes 1 cup (250 mL)
**1 heaping tsp elderflower blossoms (5 mL)**
**1 cup boiling water (250 mL)**
**1 heaping tsp dried peppermint leaves (5 mL _)_**
Put elderflower blossoms and mint leaves in a tea egg. Place egg in a teacup. Add boiling water, cover, and let steep for a few minutes. Sweeten to taste with honey if desired.
**Elderflower Fritters**
SERVE THIS DESSERT ON A SUMMER EVENING.
Serves 2
**Elderflowers with stems**
**Batter:**
**Brandy**
**2 egg yolks**
**1 cinnamon stick (optional)**
**2/3 cup milk (150 mL)**
**1 cup flour (250 mL)**
**Orange juice**
**1 Tbsp sugar (15 mL)**
**Icing sugar**
**Dash salt**
On the morning of a sunny day, cut clusters of fully opened elderflowers, leaving about a 6-inch (15-cm) stem. Refrigerate in a plastic bag until 1 hour before cooking.
In a bowl, combine brandy and cinnamon stick. Add and submerge flower heads. Let stand for 1 hour.
Meanwhile, in a separate bowl, make batter by mixing together egg yolks, milk, flour, salt, and sugar.
Drain flower heads and shake dry. Holding each cluster by its stem, dip flowers in batter. Fry them immediately in 2 inches (5 cm) hot fat in a cast-iron frying pan. (Fry up to 3 clusters at once.) Remove when lightly browned, drain on paper, and sprinkle with a little icing sugar and a bit of orange juice. Serve at once.
**Elderberry Pie Deluxe**
**1 cup or less sugar (250 mL or less)**
**2 cups ripe elderberries (500 mL)**
**2 eggs, separated**
**2 Tbsp cornstarch (25 mL)**
**1 cup sour cream (250 mL)**
**Baked pie shell [p. 46]**
In a bowl, mix together sugar and egg yolks. Stir in sour cream until well blended. In a separate bowl, combine elderberries and cornstarch; fold into sugar mixture.
In a saucepan, cook berry mixture over low heat until thick, stirring often. Pour into a baked 9-inch (23-cm) pie shell.
In a bowl, beat egg whites until stiff and spread over pie filling. Bake pie in a 350°F (180°C) oven until meringue is lightly browned.
**Elderberry Pie**
**Pastry [p. 46]**
**1 Tbsp lemon juice or vinegar (15 mL)**
**3 cups ripe elderberries (750 mL)**
**1 cup or less sugar (250 mL or less)**
**Pinch salt**
**2 Tbsp flour (25 mL)**
Line a 9-inch (23-cm) pie plate with pastry and fill with elderberries. In a bowl, mix together remaining ingredients; pour over berries. Cover pie with a top crust and prick a few times with a fork. Bake in a 350°F (180°C) oven for about 40 minutes or until crust is browned and berries are bubbling.
_Note:_ To make with dried elderberries [p. 126], pour boiling water over 3 cups (750 mL) or more dried berries. Let steep until soft, drain, and proceed as above.
To make with elderberry preserves [p. 122], drain preserves and measure 2 1/2 to 3 cups (625 to 750 mL) elderberries. Omit lemon or vinegar and proceed. If using elderberry preserves that do not contain vinegar, do not omit lemon or vinegar.
**Elderberry-Custard Pie**
SERVE PIECES OF THIS ELEGANT PIE WITH WHIPPED CREAM.
**4 Tbsp flour (75 mL)**
**1 cup elderberry juice [p. 124] (250 mL)**
**1 cup or less sugar (250 mL or less)**
**1/4 tsp salt (2 mL)**
**1 egg, separated**
**1 cup milk (250 mL)**
**Unbaked pie shell [p. 46]**
In a bowl, combine flour, sugar, and salt. Stir in enough milk to make a paste.
In a saucepan, bring elderberry juice to a boil. Stir paste into hot juice. Simmer, stirring often to prevent lumps. Remove from heat and stir in remaining milk to cool mixture. Stir in egg yolk.
Beat egg white in a bowl until stiff; fold into juice mixture. Pour filling into an unbaked 9-inch (23-cm) pie shell and bake in a 350°F (180°C) oven for about 30 minutes.
**_Citron Melon_**
Until quite recently, I thought that the citron I grew in the garden was the same as the kind that was used to make candied citron. Those tiny pale-green translucent squares, sticky and sweet, are indispensable for fruitcake making. And every year, in the late fall, they turn up on food-store shelves in stacks of shallow plastic containers next to glacé cherries and candied pineapple.
I know now, thanks to _Wyman's Gardening Encyclopedia_ and my trusty _Ball Blue Book_ on preserving (1982), that there are two kinds of citron. One citron, the kind used for making those candied squares, grows on a tree and is called _Citrus medica._ The fruit is 6 to 10 inches (15 to 25 cm) long and 4 to 6 inches (10 to 15 cm) wide. The other citron, my citron, is a member of the watermelon family and is called _Citrullus lanatus_ var. _citroides,_ referred to as "the preserving melon" or "citron melon." Native to Africa, it was grown in America before 1863. Like other melons, it grows on a vine. When mature, these citrons are round, striped light and dark green, and weigh 5 to 8 lb (2.5 to 4 kg). Both tree and vine citron have thick rinds suitable for making the candied fruit, and in addition, the old-fashioned preserving melon has been used for hundreds of years as an easily grown, reliable source for making pickles, marmalade, and preserves, items that had a place on the well-stocked shelves of the winter larder years ago.
Although I might be able to grow _Citrus medica_ if I lived in southern Florida or in Texas, I know that with a little care I can always grow the preserving melon, even in my northern garden. Now regarded as an heirloom, seeds are hard to come by (see Appendix). There is little difference between the two varieties that may be available, green-or red-seeded. Either packet of seeds will bring a large crop of handsome striped fruit. If you enjoy the challenge of making things, you will get a kick out of making those little squares. If you can't stand fruitcake (try my recipe first before you make up your mind), make pickles, preserves, or marmalade. Friends will be glad to take some of the fruit off your hands, too, as citrons make marvelous centerpieces and fall decorations. With their tough outer rind, they keep for many months in a cool, sunny spot protected from freezing temperatures, and they are a handsome addition to the usual gourds, pumpkins, and Indian corn.
**Planting**
If you live in the northeast, there is a trick to growing citron—and all melons, for that matter. First, you must provide the plants with a long growing season, about 95 days. Second, you must provide them with more heat than is normally available during a typical summer. Both these conditions can be easily met by raising seedlings and then planting them out with a plastic mulch. Such a mulch absorbs and retains a great deal of heat even if there is little sun available.
To begin with, find suitable containers in which to grow the seedlings: old strawberry boxes, large-sized jiffy pots, or any recycled container of similar size that will break down when planted in soil. But line the strawberry boxes with moss or other material first, to keep the dirt from spilling out through the openings, and choose the sod from an area that has well-drained soil. The point is to raise seedlings in a container that is large enough to accommodate the growing roots and that can be planted intact without disturbing the roots.
About a month before your last-expected frost, in the spring, fill suitable containers—six will suffice unless you have a market for the fruit— with any rich well-drained friable soil that has been lightened with a little vermiculite. Tamp the soil, leaving watering space, water the containers, and set them in a cold frame, greenhouse, or on a sunny windowsill until the soil has warmed. Seventy-five to eighty degrees Fahrenheit (25° to 27°C) will give the best germination.
To plant, sow about 5 to 6 seeds in each container. Push the seeds gently into the soil until they are covered; if using sod, cover the seeds with soil. Keep the soil moist and warm, two imperatives for germinating citron seeds, as well as for growing seedlings, if you are using a cold frame, cover it with a heavy blanket on cool nights. When the seedlings are established, thin them to 2 to 3 per container. (Cut the extras with scissors.)
When transplanting the seedlings, it's best to give them their own patch so that they don't eventually interfere with other garden crops—citrons like to sprawl. Choose an area at one side of the garden, an area enriched, perhaps, with well-rotted compost. Citrons prefer light, sandy loam, but any garden soil will do if it is fairly rich and well drained. With a hoe, mark holes about 8 ft (2.5 m) apart, making little hills to provide good drainage. On a cloudy, windless, warm day, when all danger of frost has passed, set each well-watered container in its hill, firmly tamping the dirt around the stems of the seedlings. If you have used strawberry baskets, remove them by cutting down each side with scissors and slide the block of soil and roots gently into place.
As soon as the citrons are planted, they should be mulched. To mulch, take a piece of plastic (black is good because it absorbs heat, but any will suffice) about 18 inches x 30 inches (45 cm x 75 cm). Cut a slit in the middle and set the whole piece over the plant. Carefully lift the leaves through the slit and over the sides of the plastic so that the plant can grow on top while the roots remain underneath, heated by their little greenhouse. Heap dirt or rocks—rocks also retain heat—along the sides of the plastic to prevent blowing. In a dry spell, water the plants every day until they are well established.
**Cultivating**
During the early stages of growth, the area surrounding the young plants will probably need to be cultivated with a hoe. Weeds growing under the mulch, if there are any, will usually die from overheating. Aside from pulling large weeds that interfere with the spreading vines, there is little else to do. If you have raised healthy seedlings, the citron plants should be vigorous enough to shade out weeds, and the plastic mulch will help, too.
**Harvesting**
Citrons do not take kindly to cold, freezing temperatures, so harvest them all before the first frost. Leave about 2 inches (5 cm) of stem for a handle. One of the nicest things about harvesting citron, in addition to discovering more fruits under the canopy of leaves, is knowing that you needn't do anything with them for months, until all your other harvesting and processing work is done. Besides, you should leave citrons to ripen for a few months in a cool, sunny room. Be sure they are prominently displayed because everyone who sees them will admire them and want to know what they are.
**Preserving, Canning, And Cooking**
The citron is made up of an outer rind (not counting the thick skin) and an inner section of seeds. The outer rind is larger by far, and it is with this "meat" that you make candied citron, pickles, preserves, or marmalade. Citron is only edible when it is preserved, and even then it is rather bland and sweet. Nonetheless, its texture makes products interesting, and I include one recipe that calls for fresh rind. Review A Short Course in Fruit Preserving [p. 4] before preserving.
**Citron Cake**
THIS IS AN OLD RECIPE THAT USES FRESH CITRON.
1/4 lb butter (125 g)
1 1/2 tsp baking powder (7 mL)
1 cup sugar (250 mL)
1 Tbsp brandy (15 mL)
3 eggs, separated
1 cup citron rind, sliced thin and chopped (250 mL)
1/2 cup milk (125 mL)
1 cup flour (250 mL)
In a bowl, cream butter, add sugar, and beat well. Gradually add egg yolks and milk, mixing and beating thoroughly. Sift together flour and baking powder; add to sugar-egg mixture, beating well.
In a separate bowl, beat egg whites until stiff; fold into batter. Fold in brandy and citron. Pour batter into a greased 9-inch (22 cm) tube pan and bake in a 350°F (180°C) oven for about 1 hour or until knife or cake tester inserted near center comes out clean.
**Citron Marmalade**
THIS IS SIMILAR IN TASTE TO ORANGE OR CITRUS-FRUIT MARMALADE, BUT THE CITRON CUTS THE USUAL BITTERNESS.
Makes about 5 t (2.5 L)
3 lb citron (1.5 kg)
4 lb sugar (2 kg)
3 lb oranges and lemons, mixed (1.5 kg)
Cut citron into quarters, remove seeds, and peel outer skin. Slice rind very thin and cut into small pieces.
Thinly slice oranges and lemons and cut into small pieces, removing seeds. In a large stainless-steel pot, combine fruit and sugar. Cover and let stand overnight.
Bring to a boil, uncover, reduce heat, and simmer until thick, stirring as necessary.
Remove marmalade from heat and let subside. Stir. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Candied Citron**
ALTHOUGH MADE PRIMARILY FOR FRUITCAKE, CANDIED CITRON CAN BE INCLUDED IN MANY CAKE AND COOKIE RECIPES.
Makes about 1 lb (500 g)
2 medium-sized citrons
2 cups water (500 mL)
4 cups sugar (1 L)
2 Tbsp corn syrup (25 mL)
Cut citrons into quarters and remove seeds but do not peel. Place in a large preserving pot and add water. Cover and simmer until tender. Drain and cool.
Slice off outer skin and cut fruit into desired shapes, usually 1/4-inch (5-mm) squares. Drain overnight in a colander.
In a saucepan, combine sugar and 2 cups (500 mL) water and bring to a boil. Boil syrup until it reaches softball stage. Remove from heat and stir in citron squares. Let stand in syrup overnight.
Drain and roll squares in sugar. Place on a cookie sheet and dry slowly in a 250°F (120°C) oven. Turn off heat and let citron remain in oven until squares are quite firm.
_Variation:_ To make _candied orange peel_ or _candied lemon peel,_ substitute peel of 2 dozen medium-sized oranges or lemons for citron.
**Citron Pickles**
YOU CAN SUBSTITUTE PREPARED CITRON CHUNKS FOR OVERRIPE CUCUMBER IN ANY PICKLE RECIPE.
Makes about 4 pt (2 L)
4 lb citron rind (2 kg)
1/2 oz cinnamon sticks (15 g)
Water
1 Tbsp whole cloves (15 mL)
1 Tbsp whole allspice (15 mL)
1 pt cider vinegar (500 mL)
2 lemons, sliced thin
3 cups brown sugar (750 mL)
Salt
Cut citrons into quarters, scoop out seedy middle, and peel tough outer skin. Cut remaining fruit, or rind, into small chunks about 1 inch (2.5 cm) square or desired size. Weigh fruit.
In a large preserving pot, combine water and salt, using 3 Tbsp (50 mL) salt to each qt (L) water.
Use enough water to cover chopped rind. Place 4 lb (2 kg) rind in saltwater, cover, and let soak overnight.
Drain rind and cover with fresh boiling water. Simmer, covered, or with lid tilted, until citron is tender, checking occasionally to make sure water hasn't boiled away. Drain again.
In a separate pot, combine vinegar and sugar and bring to a boil. Tie up spices in a cheesecloth bag; add to vinegar-sugar mixture. Add citron and sliced lemons. Stirring occasionally, simmer whole mixture, uncovered, until fruit is clear.
Remove spice bag and ladle mixture into hot scalded jars [p. 28], leaving 1/2 inch (5 mm) headspace and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 27] or steam canner for 10 minutes.
**Citron Preserves**
SERVE THESE PRESERVES WITH CHOCOLATE ICE CREAM, OR DRAIN AND USE THEM IN FRUITCAKE AS A SUBSTITUTE FOR CANDIED CITRON.
Makes about 3 pt (1.5 L)
3 lb citron (1.5 kg)
Juice of 1 lemon
2 lb sugar (2 kg)
Cut citron into quarters, remove seeds, and peel outer skin. Slice rind very thin and cut into small pieces. Place in a large preserving pot. Add sugar, mix thoroughly, cover, and let stand overnight.
Bring mixture to a boil and simmer, uncovered, until citron is transparent. Add lemon juice and cook for a few more minutes.
Remove preserves from heat. Ladle into hot scalded jars [p. 28], leaving 1/2 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Nova Scotia Fruitcake**
Makes 2
4 oz candied citron, coarsely chopped [p. 135] (125 g)
2 oz candied lemon peel, coarsely chopped [p. 135] (50 g)
2 oz candied orange peel, coarsely chopped [p. 135] (50 g)
1/2 lb candied cherries, halved (250 g)
1 lb candied pineapple, shredded (500 g)
1 lb golden raisins (500 g)
1/2 lb seeded raisins (250 g)
4 oz dried black currants
[p. 108] (125 g)
1/2 cup dark rum, cognac,
or sherry (125 mL)
4 oz almonds, coarsely
chopped (125 g)
4 oz walnuts, coarsely
chopped (125 g)
4 oz pecans, coarsely
chopped (125 g)
2 cups unbleached all-purpose
flour, sifted (500 mL)
1/2 tsp mace (2 mL)
1/2 tsp cinnamon (2 mL)
1/2 tsp baking powder (2 mL)
1 tsp almond extract (5 mL)
1 Tbsp milk (15 mL)
1/2 cup butter (125 mL)
1 cup granulated sugar(250 mL)
1 packed cup brown
sugar (250 mL)
5 eggs
Rum, cognac, or sherry
In a large bowl, mix together candied citron, lemon peel, orange peel, cherries, pineapple, golden raisins, seeded raisins, and dried black currants. Stir in 1/2 cup (125 mL) rum to evenly coat ingredients. Cover and let stand overnight.
Combine fruits, almonds, walnuts, pecans, and 1/2 cup (125 mL) flour. Sift together remaining flour, mace, cinnamon, and baking powder. Set aside.
In a small bowl, stir almond extract into milk. In a large bowl, cream butter, add granulated and brown sugars and eggs, mixing well. Stir in milk mixture and then sifted dry ingredients. Beat well. Pour batter over fruit-nut mixture and mix well.
Grease two 9-inch (3-L) tube pans. Line each with greased wax paper, even tube in middle. Pack pans with batter, pressing batter firmly all around. Place a pan of hot water in bottom of oven. Bake cakes in a 275°F (140°C) oven for about 3 1/2 hours or less, until a sharp knife or cake tester inserted near center comes out clean. _Do not overbake._
Remove cakes from oven and let stand for 30 minutes. Turn out onto a cake rack, peel off paper, and let cakes cool. Wrap in rum-, cognac-, or sherry-soaked cheesecloth and place in deep tins lined with thick layers of wax paper, making sure each tin has a tight lid. As cloths dry out, dribble more liquor over them. This should probably be done twice over the space of 2 months. Let age for at least 4 months.
**_An Assortment of Basics_**
There are several more simple ways to preserve fruit or prepare it for the table. The small fruits are quite versatile and can be used interchangeably in any of the following basic recipes. Mix and match any way you like—all the recipes are guaranteed to please those lucky enough to taste the results.
Basic Butters, Cheeses, Pastes
Butters, cheeses, and pastes were ways to preserve a bountiful harvest of fruits, especially apples, peaches, plums, and quinces, but almost any fruit can be used. Basically, the fruit is reduced by cooking it up, skins, cores, or pits, with spices, if desired, tied up in cheesecloth. When the fruit is soft it is put through a food mill to remove skins and seeds (apples, quince) or pits (plums) to produce a smooth purée. Sugar is added and the mixture is cooked slowly, stirred often to prevent scorching, until the desired consistency: Butter is spreadable and should mound on a spoon; cheeses and pastes are drier and can be cut into slices, squares, or different shapes.
_For butters_ , add 1/2 to 1 cup (125 to 250 mL) sugar for 1 cup (250 mL) of purée, or according to taste. When the mixture is thick and mounds on a spoon without sliding off it is done; don't overcook. Pour or spoon into hot scalded canning jars or jelly jars [p. 6], leaving 1/4 inch (5 mm) headroom. Process jars in a boiling-water bath [p. 4] or steam canner for 10 minutes.
_For cheeses and pastes,_ add sugar by weight of the purée; for a less sweet product, add sugar by volume of the purée. Store in refrigerator for up to 3 months, or freeze.
**Basic Brandied Fruit**
BRANDIED FRUITS ARE, TO PUT IT MILDLY, A SENSATION WHEN SERVED OVER PANCAKES WITH A DAB OF SOUR CREAM AND A BIT OF RED CURRANT SASS. YOU NEEDN'T BRANDY MUCH FRUIT—A LITTLE GOES A LONG WAY, AND ONCE IT'S PRESERVED, THE MIXTURE CAN BE POURED INTO JARS, SEALED, PROCESSED, AND STORED UNTIL THE DAY YOU WANT TO DISH UP A DELUXE BREAKFAST. USE HIGH-QUALITY FIRM RIPE FRUIT, AND NEVER ADD MORE THAN 2 QT (2 L) AT A TIME. BESIDES STRAWBERRIES AND RASPBERRIES, CONSIDER CHERRIES, PEACHES (PEELED AND SLICED), APRICOTS, PINEAPPLE, BANANAS, ORANGE SLICES, AND BLUEBERRIES.
Brandy
Sugar
Fruit
Put 1 qt (1 L) brandy in a 2-gallon (8-L) crock or container that has a tight-fitting lid. Add fruit in season. Stir in 2 cups (500 mL) sugar to each qt (L) fruit. Stir well after each addition and keep container covered. After 2 months, ladle brandied fruit into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headspace, and seal at once. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Basic Fruit Shrub**
A SHRUB IS AN OLD, RELIABLE WAY TO PRESERVE FRESH-FRUIT FLAVOR. IT IS A CONCENTRATED LIQUID USED FOR MAKING SUMMER DRINKS, ALCOHOLIC OR NONALCOHOLIC. TURN ANY DEAD-RIPE OR OVERRIPE FRUIT INTO SHRUBS BY STEEPING IT IN CIDER VINEGAR AND HEATING THE RESULTING LIQUID WITH SUGAR. TO SERVE, ADD ABOUT 4 PARTS WATER TO 1 PART SHRUB, ADDING BRANDY TO TASTE IF DESIRED.
Fruit
Sugar
Cider vinegar
In a large preserving pot, cover fruit with cider vinegar. Cover and let stand for 24 hours.
Strain and measure juice. For each pt (500 mL) juice, add 1 pt (500 mL) sugar. Boil together, uncovered, for 10 minutes.
Skim mixture. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling water bath [p. 26] or steam canner for 10 minutes.
**Basic Fruit Soup**
THERE ARE MANY VERSIONS OF FRUIT SOUPS. SOME ARE SERVED HOT, SOME COLD. SOME ARE SERVED AS APPETIZERS, OTHERS AS DESSERTS. THIS RECIPE CAME FROM A MENNONITE FRIEND WHO LIVES IN CENTRAL AMERICA. THE SOUP IS SERVED COLD, AND THE FRUIT IS ONLY CUT UP, NOT PURÉED, AS IN MOST RECIPES. SOUP MADE FROM COOKED, PURÉED FRUIT IS USUALLY HEATED, THICKENED WITH CORNSTARCH, AND THEN THINNED WITH MILK OR CREAM.
Serves 5 to 6
2 cups cut-up fruit (500 mL)
1 heaping Tbsp flour or cornstarch (15 mL)
2 cups water (500 mL)
1 cup sugar (250 mL)
1 Tbsp sugar (15 mL)
1 qt creamy milk (1 L)
In a pot, combine fruit, water, and 1 cup (500 mL) sugar. Simmer, covered, for 10 minutes.
In a large bowl, make a paste out of flour or cornstarch, 1 Tbsp (15 mL) sugar, and a little milk. Gradually stir in all milk. Stir milk mixture into fruit. Chill thoroughly.
**Basic Berry Ice**
BERRY ICES ARE JUST FROZEN SYRUPS—CHILDREN LIKE THEM AS POPSICLES. USE SMALL FRUITS SUCH AS STRAWBERRIES, RASPBERRIES, AND RED CURRANTS.
Serves 5 to 6
4 to 6 cups berries (1 to 1.5 L)
4 cups water (1 L)
1 1/2 cups sugar (375 mL)
Prepare berry juice. Mash and strain ripe or overripe berries. Measure 2 cups (500 mL) juice. Set aside. (Or use juice left over from jelly making.)
In a large pot, combine sugar and water. Bring mixture to a boil, stirring to dissolve sugar. Remove from heat and stir in berry juice. Pour mixture into small molds or cups and freeze. To remove ice when serving, plunge molds into cold water.
_Note:_ You can substitute 3/4 to 1 cup (175 to 250 mL) honey for the sugar.
To make berry ice from presweetened canned juice or juice drained from preserves, omit sugar.
To make _red currant-raspberry ice,_ use 1 1/3 cups (325 mL) red currant juice and 2/3 cup (150 mL) raspberry juice.
To make popsicles, insert popsicle sticks in each mold when mixture begins to freeze. Then continue freezing.
**Basic Fruit Ice Cream**
WE USE A 6-QT (6-L) WHITE MOUNTAIN ICE CREAM MAKER FOR THIS. USE ANY RIPE FRUIT YOU WANT AND SWEET CREAM THAT IS FRESH AND NOT TOO THICK. TO INCREASE FRUIT FLAVOR, JUST DECREASE CREAM AND INCREASE FRUIT PURÉE.
Makes 4 qt (4 L)
3 qt sweet cream (3 L)
2 qt fruit (2 L)
3 cups sugar (750 mL)
Chill canister of ice-cream maker. In a pot, heat 1 qt (1 L) cream slowly. Stir in sugar and continue heating and stirring until it is dissolved in cream. Remove cream from heat and chill.
Meanwhile, purée fruit. Mash, crush, grind, or put it through a food mill or blender, depending on toughness and desired consistency. Press seedy fruits such as raspberries through a strainer if desired. Measure 1 qt (1 L) purée. Sprinkle purée with sugar to taste, stir, and let stand, covered, at room temperature until ready to use.
Pour chilled cream into chilled canister. Stir in remaining cream. Set canister in ice-cream-maker tub and pack rock salt and ice around it, using 1 part salt to 5 parts ice. Cover and churn until mixture is thick and handle starts to resist turning (about 700 turns).
Carefully remove lid of canister and pour in fruit, stirring it in carefully with a long-handled wooden spoon. Re-cover and churn until handle is difficult to turn (200 to 300 turns). Remove dasher and re-cover canister. Pack chipped ice all around canister where necessary, cover ice cream maker with burlap, and set in a cool place.
_Note:_ You can substitute 2 to 2 1/2 cups (500 to 625 mL) honey for the sugar.
Before making _black currant ice cream,_ stew black currants in a little water, adding sugar to taste. Put mixture through a food mill and add to nearly frozen ice cream, as above.
_Variation:_ To make _vanilla ice cream_ to go with fruit sauces _,_ omit fruit purée and use 4 qt (4 L) cream and 4 tsp (20 mL) vanilla. Heat 1 qt (1 L) cream, stir in sugar and vanilla, chill, and add to chilled canister with 3 qt (3 L) cream.
Several years ago, friends Sue and Henry Bass gave me a nifty electric ice cream maker so I could experiment with making small amounts of specialty ice cream, including fruit sherbert, sorbet/ices, and frozen fruit yogurt. Directions may vary according to the machine you use. The ones below are based on my 1990s Oster Ice Cream/Frozen Yogurt Maker which produces no more than 1 1/2 quarts (1.5 L) finished product and uses ice cubes and table salt for freezing. You will need approximately 2 cups table salt (500 mL) and 6 trays of small ice cubes, or 9 cups (2.12 L). Feel free to alter the recipes to achieve the taste you prefer (fruitier or sweeter), but be sure to confine yourself to the total liquid amount which is based on the capacity of the machine.
Strawberry Ice Cream
**Basic Fruit Sherbert**
SHERBERTS, AS DISTINCT FROM ICE CREAM, USE MILK AS THE BASE. INCREASE FRUIT JUICE AND DECREASE MILK FOR A FRUITIER FLAVOR. A LOT WILL DEPEND ON THE FRUIT JUICE YOU USE.
Makes about 1 1/2 quarts (1.5 L)
cups milk (750 mL)
2 cups unsweetened fruit juice (500 mL)
1 cup sugar (250 mL)
Combine ingredients in canister, stirring well. Cover with freezing cover, and chill for faster freezing. When ready to begin freezing, turn on the machine, set canister in ice bucket, pour in water and layer ice and salt as directed. Remove freezing cover, insert paddle, and put mixing cover in place, adding more water according to directions. Turn on machine and add more salt and ice as directed when ice begins to melt. Continue freezing until sherbert is the consistency you want. This should take about 20-25 minutes, or less in my experience. Remove canister, stir to blend if necessary, store in freezer for an hour or more, then you may remove sherbert to a plastic freezer container and return to freezer.
Strawberry Gelatin
**Basic Fruit Gelatin**
THERE ARE MANY WAYS TO GET THE JUICE FOR GELATIN: CANNED JUICES, JUICE DRAINED FROM PRESERVES, OR COOKED-AND-STRAINED FRESH FRUIT. ALL SORTS OF COMBINATIONS ARE POSSIBLE, TOO. WHEN SERVING GELATIN, WHIPPED CREAM WOULD NOT BE AMISS.
Serves 5 to 6
2 envelopes gelatin
3 cups presweetened juice (750 mL)
1 cup cold water (250 mL)
In a saucepan, sprinkle gelatin over cold water. Place over low heat and stir constantly until gelatin dissolves or no granules are visible.
Remove from heat and stir in juice. Pour gelatin into a Pyrex baking dish or bread pan and chill for several hours or until firm. If adding cut-up fresh fruit—sliced bananas, oranges, or strawberries—fold into gelatin when it begins to jell. Chill until firm.
_Note:_ To make with fresh fruit, cook desired fruit with a little water, mashing to extract as much juice as possible. When fruit is well heated and juicy, strain and measure 3 cups (750 mL) juice; stir in 1/2 cup (125 mL) sugar and proceed as above.
**Basic Frozen Fruit Yogurt**
Makes about 1 1/2 quarts (1.5 L)
1 cup fruit, fresh or frozen (250 mL)
1 cup sugar (250 mL)
1 quart plain yogurt (1 L)
Place fruit and sugar in a blender and process at medium speed until smooth. Pour into canister and combine with yogurt, stirring well. Cover with freezing cover, and chill for faster freezing. When ready to begin freezing, turn on the machine, set canister in ice bucket, pour in water and layer ice and salt as directed. Remove freezing cover, insert paddle, and put mixing cover in place, adding more water according to directions. Turn on machine and add more salt and ice as directed when ice begins to melt. Continue freezing until frozen yogurt is the consistency you want. This should take about 12-20 minutes. Remove canister, stir to blend if necessary, store in freezer for an hour or more, then you may remove frozen yogurt to a plastic freezer container and return to freezer.
**Basic Fruit Sorbet/Ice**
Makes about 1 quart (1 L)
1 cup water (250 mL)
2 cups unsweetened fruit juice (500 mL)
2 cups sugar (500 mL)
Sorbets and ices are synonymous; both use water, instead of cream or milk, as the base. Combine ingredients in canister, stirring well. Cover with freezing cover, and chill for faster freezing. When ready to begin freezing, turn on the machine, set canister in ice bucket, pour in water and layer ice and salt as directed. Remove freezing cover, insert paddle, and put mixing cover in place, adding more water according to directions. Turn on machine and add more salt and ice as directed when ice begins to melt. Continue freezing until sorbet/ice is the consistency you want. This should take about 12-20 minutes, or less in my experience. Remove canister, stir to blend if necessary, store in freezer for an hour or more, then you may remove sorbet/ice to a plastic freezer container and return to freezer.
_**Tree Fruits and Wild Fruits**_
There are, of course, many more fruits that you can grow and harvest if climate, soil, and space allow. Fruit trees that require a lot of sun, deep soil, careful pruning and, usually, cross-pollination add another dimension to the fruit garden. But even if you cannot grow them, some tree fruits, such as apples, are there for the picking, anyway, or are available from local orchards.
Fruits from the wild—blackberries, blueberries, cranberries—can be harvested and processed to complement the produce from a cultivated garden. It is exciting and satisfying to become acquainted with all the useful vegetation that survives from year to year without being tended.
Still other fruits, particularly pears and peaches, are often available in large quantities on the market. _The thing to remember is that no matter how you get your fruit, give some thought to its handling._ Decide how to take advantage fully of what a fruit has to offer. Turn high-pectin fruits into jams and jellies, overripe fruits into wines and juices, odds and ends of fruits into leathers. And, most important, always use the technology most suited to the task, the least damaging to the environment and, ultimately, the most satisfying. With a few simple hand tools, some ordinary kitchen equipment, and a basic knowledge of preserving, you can put the heart and soul back into the fruit harvest. Review Short Course [p. 4] before preserving.
**Tree Fruits**
If you decide to include tree fruits in your garden, study the varieties, and the methods of cultivation first. Take a look at dwarf and semi-dwarf trees, too—they take up less space than standard-sized varieties. Some dwarf types can even be grown in tubs.
**Apples**
The number of apple recipes here attests to the hardiness of apples in northern climates. In mid-June, the ground around our trees is thickly covered with white blossoms. On our farm, by fall, our porch was full of buckets of apples to turn into sundry products, to eat, to feed to livestock, or to give to friends and neighbors.
It has been said that the function of the apple tree is to "bottle sunshine." There are more kinds of apples to choose from now than there have been for some time, among both heirlooms and newer introductions. Standard types grow from 18 to 25 ft tall (5.5 to 7.6 m) and bear in 3 to 6 years. The semi-dwarf types may grow from 6 to 12 ft tall (1.8 to 3.7 m) and bear in their third or fourth year, but they don't last as long as standards. Bear in mind that you need two different varieties for proper pollination. Even if you don't grow your own, you can find many abandoned productive apple trees in the countryside. Many old varieties are still recognizable, and they are useful for more than cider making; for example, Wolf River for dried apples and Duchess for applesauce. Ultimately, you need not know their names, just their qualities. Whether in your backyard or in some deserted orchard, look for apples that lend themselves to the following recipes. And don't forget to ask for permission before you pick. Failing that, local orchards and farm stands are your best bet for buying better quality apples in quantity for preserving.
**Apple Butter III**
We can call this a jam to distinguish it from the other spiced-and-sugared apple butters. Sweet apples and fresh sweet cider are a must for this recipe. This butter is delightful on English muffins.
Apple cider
Apples
In a large preserving pot, boil fresh cider, uncovered, until reduced by half. Stir occasionally. Pare and quarter (no need to peel) enough sweet apples to fill up cider. Stirring frequently with a wooden spoon, boil slowly until mixture is consistency of marmalade. Remove butter from heat. Jar, seal, and process as for Apple Butter I.
**Dried Apple Rings**
THE DRIED APPLE INDUSTRY, DEVELOPED IN WESTERN NEW YORK, USED TO BE VERY IMPORTANT WHEN FRESH OR FROZEN APPLES WERE NOT WIDELY AVAILABLE. MANY EARLY APPLE PIE RECIPES, FOR INSTANCE, CALLED FOR USING DRIED APPLES. NOT ONLY WERE CONDITIONS FAVORABLE FOR GROWING APPLES IN WESTERN NEW YORK, BUT BY 1825, WITH THE ESTABLISHMENT OF THE ERIE CANAL, DRIED APPLES COULD BE SHIPPED ALL OVER THE WORLD. BY THE 1940S, HOWEVER, THE INDUSTRY DWINDLED, AS FRESH APPLES, NO LONGER A LUXURY, COULD BE SHIPPED ACROSS THE COUNTRY OR KEPT COLD IN THE HOME REFRIGERATOR. WE, LIKE THE PIONEERS OF OLD, SOUGHT ALTERNATIVE WAYS TO STORE OUR BOUNTIFUL APPLE HARVEST UNTIL THE NEXT GROWING SEASON. DRIED APPLE RINGS WERE THE ANSWER.
Peel and core fairly uniform large winter apples, reserving cores and peelings for jelly. Slice apple rings about 1/8 to 1/4 inch (3 to 5 mm) thick and hang them about 1/2 inch (1 cm) apart on clean sticks or dowels. Balance sticks across deep roasting pans and place pans wherever apple rings will be exposed to circulating dry air. Or suspend them above stove, where rings will receive some heat during cooking.
When rings feel dry and a little leathery or papery, not hard or crisp, and are pale amber in color, they are done. Store them in covered crocks or jars. They will keep almost indefinitely.
_Variation:_ Use the cores and peelings to make _apple-peelings jelly,_ the tastiest, fastest-setting jelly. Broken rings, of course, can be added to the pot. Just follow the recipe for apple jelly [p. 149], using 6 cups (1.5 L) cores and peelings and 4 cups (1 L) water.
**Apple Jelly**
Makes at least 2 pt (1 L)
5 lb apples (2.5 kg)
Sugar
5 cups cold water (1.25 L)
Cut up tart fresh apples and place in a large preserving pot. Add water, cover, and simmer until fruit is soft, stirring occasionally. Strain through a jelly bag [p. 28] and let drip for at least 4 or 5 hours.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 3/4 cup (175 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil. Skimming as necessary, boil, uncovered, for about 15 minutes or less, until a small amount sheets off a metal spoon [p. 28-29]. The time it takes to make jelly varies with apple types and the season, but do not boil more than 15 minutes.
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1 /4 inch (5 mm) from the top and seal at once with snap lids and screw bands, and seal. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Variations:_ To make _apple-mint jelly,_ add mint essence to each batch of juice. Prepare essence: In a cup, cover well-packed peppermint leaves with boiling water and let steep for several hours. Press down leaves with back of a spoon to extract all flavor. Strain and set aside 2 Tbsp (25 mL) mint juice, or essence. Add 2 Tbsp (25 mL) to each batch of apple juice after it has been brought to a boil a second time. Proceed as above.
To make _apple-rose geranium jelly,_ use reddish-green apples and proceed as above, adding 3 rose geranium leaves to each batch of juice. Remove leaves before jarring and sealing.
To make _apple-spice jelly,_ add 2 whole cloves, tied up in a muslin or cheesecloth bag, to every batch of juice. Bring to a boil and stir in 4 cups (1 L) sugar and 1/2 cup (125 mL) vinegar. Proceed as above, removing cloves before jarring and sealing.
**Apple Ginger**
I MAKE THIS EVERY YEAR TO SELL AT FALL CRAFT SALES AND IT ALWAYS SELLS OUT. IT IS AN OLD RECIPE THAT I DISCOVERED WHEN A FRIEND GAVE ME SOME PRESERVED GINGER AS A GIFT AND I LOOKED FOR WAYS TO USE IT IN PRESERVING. THE DIRECTIONS CALL FOR NOT ONLY USING THE FLESH OF THE APPLE, BUT THE PEELINGS AND CORES, TO PRODUCE A VERY HIGH PECTIN JUICE. APPLE GINGER IS A BEAUTIFUL AMBER COLOR AND THE CONSISTENCY OF A SPREAD, DELICIOUS WITH CREAM CHEESE ON TOAST, CRACKERS, OR WHATEVER. USE SWEET, RIPE EARLY-FALL APPLES. YOU CAN ALSO USE IT AS A GLAZE ON ROAST CHICKEN, PORK, OR HAM.
Makes 3 to 4 pt (1.5 to 2 L)
3 lb apples (1.5 kg)
1/2 lb preserved ginger, cut up in small
pieces (125 g)
Sugar
Pare and core apples; cut apples into small pieces. Place parings and cores in a large preserving pot. Cover with water and simmer, covered, until soft, stirring occasionally. Strain.
Measure juice and pour into a large stainless-steel pot. Stir in 1 cup (250 mL) sugar to every 1 1/2 cups (375 mL) juice. Stir in cut-up apples and 3 more cups (750 mL) sugar. Stir in preserved ginger; increase amount for stronger ginger flavor. Simmer, uncovered, until thick, stirring as necessary.
Remove apple ginger from heat. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Note:_ Preserved ginger can be found in supermarkets next to candied citron (it is already cut up in small pieces), or in specialty food stores, where the preserved ginger may be in larger pieces.
**Apple Sweetmeats**
THIS MAKES A CHUNKY APPLESAUCE THAT IS SWEET ENOUGH TO RETAIN ITS ORIGINAL NAME. USE LARGE, HARD, EARLY-FALL APPLES.
Makes about 3 qt (3 L)
3 cups sugar (750 mL)
4 to 5 lb unpeeled apple slices (2 to 2.5 kg)
1 qt water (1 L)
Cinnamon
In a large preserving pot, combine sugar and water, stirring to dissolve sugar. Bring to a boil.
Drop in apple slices. Simmer mixture, uncovered, until slices are soft.
Remove sweetmeats from heat and add cinnamon to taste. Ladle into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom. Adjust lids and process jars in a boiling-water bath [p. 26] or steam canner for 20 minutes.
**Old-Fashioned Applesauce**
TODAY, THERE IS A WHOLE GENERATION THAT DOES NOT KNOW THAT APPLESAUCE CAN BE MADE WITH JUST A HAND FOOD MILL. HERE IS THAT OLD-FASHIONED, TIME-HONORED WAY. USE LARGE AMOUNTS OF APPLES: SMALL ONES, BIG ONES, AND BLEMISHED ONES.
Makes about 1 qt (1 L)
3 to 4 lb apples (1.5 to 2 kg)
Cinnamon
Sugar
Cut up apples, without skinning or coring or trimming too fastidiously. Place in a large preserving pot, adding a little water to prevent scorching. Simmer, covered, until soft, adding more water if apples seem dry. Put mixture through a food mill and stir in sugar and cinnamon to taste. Reheat to boiling point.
Remove applesauce from heat. Pour into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom. Adjust lids and process jars in a boiling-water bath [p. 26] or steam canner for 20 minutes.
**Apple Butter I**
WHEN MAKING APPLE BUTTER, TRY TO FIND CIDER MADE FROM RUSSET APPLES. OTHERWISE, USE THE SWEETEST CIDER AVAILABLE. THE RESULT SHOULD BE A PLEASANT REDDISH-BROWN.
Makes about 8 pt (4 L)
8 qt apples (8 L)
3 lb sugar (1.5 kg)
2 qt cider (2 L)
3 tsp cinnamon (15 ml)
4 qt water (4 L)
1 1/2 tsp cloves (7 mL)
Cut apples into small pieces and place in a large preserving pot. Add cider and water. Stirring occasionally, simmer mixture, covered, until apples are soft. Put through a food mill.
Place pulp in a large preserving pot. Add sugar, cinnamon, and cloves. Stirring frequently with a wooden spoon, cook pulp slowly, uncovered, until very thick.
Remove butter from heat. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Apple Pectin**
MAKE THIS EARLY IN THE SEASON, WITH TART, JUICY APPLES. FOR MAKING JELLY FROM LOW PECTIN FRUITS, USE EQUAL AMOUNT OF APPLE PECTIN TO EXTRACTED JUICE, AND 1/2 CUP SUGAR (175 ML) FOR EVERY 1 CUP (250 ML) OF COMBINED JUICES.
Makes about 2 pt (1 L)
2 lb apples (1 kg)
2 Tbsp lemon juice (25 mL)
1 qt water (1 L)
Cut up apples without peeling or coring, and place in a large preserving pot. Add water, cover, add lemon juice, and simmer until apples are soft, about 40 minutes, stirring occasionally. Strain mixture through a jelly bag [p. 28] and let drip overnight.
In a large stainless-steel pot, bring juice to a rolling boil and cook, uncovered, for 15 minutes.
Remove pectin from heat. Pour into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process in a boiling-water bath [p. 26) or steam canner for 10 minutes.
**Apple Butter II**
THIS BUTTER IS DIFFERENT ENOUGH FROM APPLE BUTTER I THAT IT DESERVES A SEPARATE NAME. USE THE SWEETEST APPLES AVAILABLE.
Makes 2 1/2 pt (1.25 L)
2 qt peeled apple quarters (2 L)
1 1/2 tsp cinnamon (7 mL)
2 qt sweet cider (2 L)
1/2 tsp cloves (2 mL)
1 1/2 cups sugar (375 mL)
1/2 tsp allspice (2 mL)
In a large preserving pot, combine apple quarters and cider. Cook slowly for about 2 hours, stirring occasionally. Add sugar, cinnamon, cloves, and allspice. Continue cooking slowly until thick. Remove butter from heat. Jar, seal, and process as for Apple Butter I.
**Apple Crisp**
Serves 6
6 apples
2 Tbsp sugar (25 mL)
1/4 tsp cinnamon (1 mL)
1 Tbsp white wine, cider,
elderberry sass [p.128], or water (15 mL)
1/3 cup butter (80 mL)
1/3 cup brown sugar (80 mL)
3/4 cup flour (177 mL)
Generously butter a medium-sized bread pan. Core and thinly slice apples but do not peel. Fill bread pan almost to top. Sprinkle granulated sugar and cinnamon over apples; add white wine, cider, elderberry sass, or water if apples are dry.
In a bowl, make a crumb mixture by rubbing together butter, brown sugar, and flour. Sprinkle over apples and pat down to form a cover. Bake apple crisp in a 325°F (160°C) oven for about 1 hour or until apples are tender. Serve cold or hot with cream.
**Apple Cider**
This isn't really a recipe. But if you ever have bushels and bushels of apples around and you don't know what to do with them—have them turned into cider. We used to put down four barrels of cider every fall. One was for vinegar and the others for drinking throughout the year.
To make apple cider, you have to have a cider press and chopper. Or if you live in an area where there is a cider mill, you can pay to have your apples pressed. A bushel of apples will usually yield about 3 gallons (12 L) juice. Any kind of apples can be used— wildings or imperfect ones—but some types (Russet) are juicier and sweeter and make better cider. (To preserve sweet cider, freeze it in milk cartons or similar containers.)
If you want hard, or alcoholic, cider, plan on making a barrelful, which will require 15 to 20 bushels of apples. Put the juice in a charred oak barrel, the best kind for aging hard cider. Roll the barrel down cellar and place it on its side, up on blocks so that it's about 1 ft (30 cm) off the floor. Make sure the side with the bung hole is up. Drill a 1/3-inch (1-cm) hole straight through the bung and insert a 12-inch (30-cm) piece of plastic tubing. Place a glass of water beside the bung and run the other end of the tube into it, to form an air lock, allowing bubbles from fermentation to escape but preventing air from entering the barrel.
When you want to tap the barrel, drill a 1-inch (2.5-cm) hole through the top of the barrel about 2 1/2 inches (6 cm) from the bottom edge. Quickly drive in a spigot. Hard cider is best 9 or 10 months after it has been put down, when it is pale gold and tastes like a fine fruity dry wine.
**Apple Leather**
THIS IS A WONDERFUL WAY TO USE AN ABUNDANCE OF APPLES, PARTICULARLY LESS-PERFECT ONES.
Cut up any amount of apples. Trim them but not too meticulously. Place in a large preserving pot. Add a little water, only enough to prevent scorching. Cover, bring to a boil, stirring often. Uncover and simmer until apples are soft. Put mixture through a food mill. Add honey or corn syrup to taste if desired.
Spread pulp thinly on cookie sheets lined with one layer of heavy plastic wrap. Set pans in sun or in a just-warm oven or on top of a woodstove.
Turn fruit when it can be lifted off plastic without falling apart; it should be barely sticky to touch and still pliable. Remove plastic and dry other side of leather. Roll up leather in fresh plastic and store in jars or crocks. To use, unroll and cut off pieces with scissors.
_Note:_ Honey seems to make a softer, more pliable leather, but it does mask the apple taste. Sweetener is really unnecessary because the fruit is condensed, and, as a result, its natural sweet flavor is intensified.
**Curdie's Apple-Crumb Pie**
THIS IS A MENNONITE RECIPE THAT OUR DAUGHTER HAS MADE HER OWN. IT CALLS FOR TART APPLES AND A HEFTY AMOUNT OF SUGAR, SO IF YOU USE SWEET APPLES, YOU MAY WANT TO REDUCE THE SUGAR. WHATEVER YOU PREFER.
6 to 8 tart apples
1 tsp cinnamon (5 mL)
Pastry [p. 46]
3/4 cup flour (275 mL)
1 cup sugar (250 mL)
1/3 cup butter (75 mL)
Core and thinly slice apples but do not peel. Generously fill a 9-inch (1-L) pie plate with cut-up apples and then remove them. Place in a bowl and line pie plate with pastry.
In a separate bowl, mix together 1/2 cup (125 mL) sugar and cinnamon; mix into apples. Pour mixture into pie shell.
Combine remaining sugar and flour. Add butter and rub mixture together until it forms crumbs. Sprinkle over apples. Bake pie in a 350°F (180°C) oven for about 45 minutes.
_Note:_ If using dry apples, add some thinned elderberry sass [p. 121] to the apple mixture to make the pie juicier.
**Applesauce Cake**
THIS IS A WONDERFULLY MOIST CAKE THAT IMPROVES WITH AGE.
1/2 cup butter, shortening, or lard (125 mL)
1 cup nuts (250 mL)
1 cup chopped raisins and black currants
250 mL
3/4 cup sugar (175 ml,)
1 egg, well beaten
1/2 tsp cinnamon (2 mL)
1 tsp vanilla (5 mL)
1/4 tsp cloves (1 mL)
1 cup chopped dates (250 mL)
2 cups unbleached all-purpose flour (500 mL)
1 1/2 cups unsweetened applesauce (375 mL)
2 tsp baking soda (10 mL)
In a bowl, cream butter and add sugar; beat well. Add well-beaten egg, vanilla, and remaining ingredients. Blend. Pour batter into a greased 9-inch (2.5-L) springform pan. Bake in a 350°F (180°C) oven for about 1 hour or until a knife or cake tester inserted in center comes out clean.
_Note:_ To make with canned sweetened applesauce [p. 152], reduce amount of sugar to 1/4 to 1/2 cup (50 to 125 mL).
A bowl of applesauce
**Easy Apple Cake**
THIS CAKE IS MADE FOR THE JEWISH SABBATH IN THE RELIGIOUS COMMUNITY AISH HA TORAH IN JERUSALEM AND IT HAS BECOME OUR FAVORITE APPLE CAKE. IT'S KOSHER, WHICH MEANS THAT IT CAN BE SERVED WITH EITHER MEAT OR DAIRY MEALS IN AN OBSERVANT JEWISH HOME. USE LARGE SWEET, JUICY FIRM APPLES.
3 cups unpeeled apple slices
3 tsp baking powder (15 mL)
1 Tbsp cinnamon (15 mL)
4 eggs
1 3/4 cups sugar (425 mL)
1/4 cup orange juice (50 mL)
3 cups flour (750 mL)
1 Tbsp vanilla (15 mL)
1 tsp salt (5 mL)
1 cup salad oil (250 mL)
Place apple slices in a bowl. Stir in cinnamon and 4 Tbsp (75 mL) sugar.
In a large bowl, combine remaining sugar, flour, salt, and baking powder. Make a well in center and add eggs, orange juice, vanilla, and oil. Beat mixture until smooth.
Lightly grease a 9-inch (2.5-mL) springform pan. Spread one-third of dough on bottom of pan. Add half of apple slices and arrange evenly over top of dough. Cover apples with one-third of dough. Arrange rest of apple slices on top and cover with remaining dough.
Bake in a 350°F (180°C) oven for 75 minutes or until knife or cake tester inserted in center comes out clean. During baking, cover cake with tin foil for 20 minutes so it doesn't dry out.
**Applesauce Muffins**
VERY MOIST.
Makes 1 dozen
2 cups unbleached all-purpose flour (500 mL)
1 egg, lightly beaten
2 Tbsp melted butter (25 mL)
3/4 tsp salt (4 mL)
1 cup sweetened applesauce [p.152] (250 mL)
3 Tbsp sugar (50 mL)
3 tsp baking powder (15 mL)
1/2 cup milk (125 mL)
1/2 tsp cinnamon (2 mL)
Sift together flour, salt, sugar, baking powder, and cinnamon. Make a well in center. Add egg, butter, and applesauce; blend well. Stir in enough milk to make a smooth, creamy batter. Spoon into greased muffin tins and bake in a 400°F (200°C) oven for about 20 minutes.
**Crabapples**
If you plant crabapples, keep in mind that the prettiest-flowering varieties do not generally bear the best fruit for preserving. Wild crabapples are fine to use for jelly, marmalade, preserves, and sauce.
**Crabapple Jelly**
CRABAPPLES ARE HIGH IN PECTIN, SO THEY MAKE A FAST-SETTING JELLY. TAKE CARE NOT TO OVERCOOK! INCLUDE CRABAPPLE JUICE IN JELLIES USING FRUITS LOW IN PECTIN (ELDERBERRIES, FOR EXAMPLE.)
1 qt (1 L) juice makes about 2 pt (1 L)
Crabapples
Sugar
Halve larger crabapples. In a large preserving pot, cover at least 3 lb (1.5 kg) crabapples with cold water. Bring to a boil and reduce heat. Stirring occasionally, simmer, covered, until crabapples are soft. Strain mixture through a jelly bag [p. 28] and let drip for at least 5 hours.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil. Skimming as necessary, boil, uncovered, for about 10-15 minutes or until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 28] or steam canner for 10 minutes.
**Crabapple Marmalade**
THIS IS AN OLD RECIPE THAT YOU WILL NEVER FIND IN MODERN COOKBOOKS. THAT'S UNFORTUNATE BECAUSE THIS MARMALADE HAS CHARACTER AND DISTINCTION—TANGY IN TASTE AND ROSE VELVET IN COLOR.
Makes 6 pt (3 L)
6 lb crabapples (3 kg)
1 pt vinegar (500 mL)
2 oranges
2 tsp ground cloves (10 mL)
12 scant cups sugar (3 L)
2 tsp ground cinnamon (10 mL)
Halve larger crabapples. In a large preserving pot, combine crab-apples and a little water. Simmer, covered, until soft.
Meanwhile, cut oranges into quarters, removing and setting aside peel. Cut up orange quarters, removing seeds. In a small pot, simmer peel in water, covered, until soft. Drain, remove white skin, and cut into small pieces.
Put crabapple mixture through a food mill and place in a stainless-steel pot. Add sugar, vinegar, spices, orange pieces, and prepared peel. Stirring often, simmer mixture, uncovered, until thick.
Remove marmalade from heat and let subside. Stir. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Crabapple Preserves**
EAT THESE PRESERVES AS A DELICACY WITH ANY MEAL, EITHER AS A DESSERT OR AS A CONDIMENT, BY HOLDING ON TO THE STEMS AND DISCARDING THE SEEDS ONTO YOUR PLATE. DON'T FORGET TO EAT THE SYRUP.
Makes 5 TO 6 PT (2.5 TO 3 L)
Heavy Syrup:
6 lb crabapples (3 kg)
3 lb sugar (1.5 kg)
2 cups water (500 mL)
In a large preserving pot, dissolve sugar in water. Bring mixture to a boil, covered. Boil for a few minutes and add crabapples, carefully stirring with a long-handled fork to prevent damage. Bring to a boil and simmer, uncovered, for 5 minutes, stirring as necessary.
Remove preserves from heat. Ladle into hot scalded jars [p. 28], fill with syrup, leaving 1/2 inch headroom and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Variation:_ To make _crabapple sauce,_ bring preserves to a boil, covered. Put through a food mill, syrup and all. Serve with anything, adding cream if you like.
**Peaches**
No longer do northern gardeners have to pine for peaches. Reliance is dependably hardy to Zone 4, and although the fruit is small, flavor is intense. Many good varieties are available in warmer growing zones. Standard trees grow up to 15 feet (4.6 m) and have to be planted 20 ft apart (6.1 m). If space is a problem, look for semi-dwarf types that grow up to about 8 to 10 ft (2.4 to 3.0 m) and are planted 10 ft apart (3.0 m). Both are self-pollinating. If you can't grow your own peaches, superior tasting, good quality fruit is available at fruit stands in the summer. Better ask to taste one before you buy, though. In rainy years, peaches can be mealy and very poor for preserving. Sweet, firm, and juicy peaches are what you want. And freestone, otherwise you will have a miserable time separating fruit from pits. Be sure to add peaches to the brandied-fruit crock.
**Nellie's Peach Jam**
THIS SIMPLE RECIPE WAS DEVELOPED BY OUR BUSY DAUGHTER, NELL. SHE LIKES TO PRESERVE, BUT SHE DOESN'T HAVE A LOT OF TIME, SO WHEN SHE DOES MAKE JAM, SHE WANTS IT TO COME OUT WELL WITHOUT A LOT OF FUSS. SHE SAYS SHE NEVER THOUGHT ABOUT ADDING ADDITIONAL PECTIN, SINCE THE PEACHES JELL FAST ON THEIR OWN. SHE USES THE SAME PROPORTION OF FRUIT TO SUGAR AS FOR RASPBERRY JAM [P.67] AND ADDS LEMON JUICE TO KEEP THE CUT-UP PEACHES FROM DISCOLORING WHILE SHE'S PEELING THEM. USE FIRM, RIPE PEACHES.
Makes about 1 1/2 pt (700 mL)
1 quart cut-up and peeled peaches, about
10 good-sized (1 L)
3 cups sugar (700 mL) Lemon juice
To remove skins: immerse fruit in boiling water up to 1 minute, then plunge into ice water; slip off skins, halve each peach, remove pit, and cut up into medium chunks. Squeeze lemon juice over chunks so they don't discolor while you are preparing more fruit. Place cut-up peaches in a large stainless-steel pot, then mash the chunks to speed cooking (Nell says they break down easily). Cook fruit, covered, and when simmering, remove cover, stir in sugar. Bring to a rolling boil and cook, stirring often, about 10 minutes or until mixture thickens. Remove pot from heat. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Ginger-Peach Jam**
BOB MILLER AND HIS LATE WIFE DONNA BROOKS WERE EARLY FOLLOWERS OF THE ORIGINAL EDITION OF THIS BOOK AND INNOVATIVE FRUIT PRESERVERS, AS WELL AS FRUIT GROWERS. WHEN BOB GAVE US A JAR OF THIS JAM, WE WERE IMPRESSED WITH ITS FLAVOR AND TEXTURE. HE SAYS HE GOT IT FROM THE WEBSITE FOOD.COM WHERE IT IS CREDITED TO MIRJ. HE MADE A FEW CHANGES, WHICH I HAVE NOTED BELOW. I AM EVER IMPRESSED WITH STUDENTS WHO LEAD THE TEACHER.
Makes about 2 1/2 pt (1.25 L)
2 lb peaches (about 6 cups) (0.9 kg)
3 cups sugar (700 mL)
3 1/2 Tbsp grated ginger (50 mL)
2 Tbsp lemon juice (30 mL)
1/4 tsp allspice (1 mL)
Place peaches in boiling water for one minute. Plunge them into ice water. Slip off skins, halve fruit and remove pits. Cut peaches into small pieces. In a large stainless-steel pot mix all ingredients and simmer about 30 minutes or until mixture thickens and mounds on a spoon. Remove pot from heat. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Peach-Orange Marmalade**
I AMENDED THIS RECIPE FROM A BAG OF SUGAR, AND CONTRARY TO MY EXPECTATIONS, IT'S NOT TOO SWEET. THE ADDITION OF ORANGES SPEEDS JELLING. USE FIRM, RIPE PEACHES.
Makes about 2 1/2 pt (1.25 L)
3 large juice oranges
1 lemon
3 lb peaches, about 12 medium (1.5 kg)
6 cups sugar (1.5 L)
Grate rinds of oranges and the lemon. Squeeze juices and remove seeds (don't strain). Peel peaches (optional) as for peach jam and cut up into small pieces. Put cut-up peaches in large stainless-steel pot, add citrus juices and grated rind. Stir in sugar, and bring mixture bring to a boil, uncovered, stirring to prevent sticking. Reduce heat and let simmer slowly until thickened, about 35-45 minutes. Remove pot from heat. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Peach Butter**
USE DEAD-RIPE OR OVERRIPE JUICY PEACHES FOR THIS DELICIOUS BUTTER.
Makes about 5 pt (2.5 L)
4 lb peaches (2 kg)
Cloves
Sugar
Cinnamon
Allspice
Cut up peaches but do not peel. Place in a large preserving pot and add enough water to prevent scorching. Simmer, covered, until fruit is tender. Put peaches through a food mill. Add 1/2 cup (125 mL) sugar to each cup (250 mL) pulp. Add spices to taste if desired.
In a large stainless-steel pot, cook mixture slowly, uncovered, until thick, stirring often to prevent sticking.
Remove butter from heat. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Note:_ You can turn any excess fruit—grapes, pears, plums, cherries—into butters by following the above recipe. Less-juicy fruits require more water, up to 1 cup (250 mL) to each cup (250 ml) pulp.
**Canned Peaches**
SWEET, RIPE PEACHES, CANNED IN A LIGHT SYRUP, ARE THE GLORY OF THE WINTER PANTRY, DELICIOUS BY THEMSELVES OR MIXED WITH A VARIETY OF PRESERVED FRUITS, ESPECIALLY STRAWBERRIES. I NO LONGER REMOVE SKINS BEFORE ADDING TO THE SYRUP, AN INNOVATION I HOPE EARNS FOLLOWERS, SINCE I DETECT NO DIFFERENCE IN THE FINISHED PRODUCT AND A LOT OF TIME AND MESS SAVED TO DEVOTE TO SOMETHING MORE ENJOYABLE. OF COURSE, BE SURE TO WASH FRUIT WELL.
About 6 qt (6 L)
10 medium-large peaches (should make 1 qt/1 L of pieces)
Sugar
Quarter peaches and cut up in bite-size pieces. As you work, drop pieces into 1 gallon (3 L) water with 2 Tbsp (15 mL) salt to prevent discoloring. Add 1/2 cup (125 mL) sugar for each quart (1 L) of cut up fruit. Let sit to draw out juices, then heat fruit to boiling point in a large preserving pot, covered. Fill hot scalded jars [p. 28] to 1/2 inch (1 cm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 25 minutes.
**Peach Leather**
Use ripe fruit in any condition. Trim blemished parts. Cut peaches in half, remove pits, and chop peach halves into small pieces. Place in a large preserving pot, adding water if necessary. Simmer until fruit is soft, about 10 minutes, stirring in sugar or honey to taste if desired. Put mixture through a food mill. Spread pulp thinly on cookie sheets lined with one layer of heavy plastic wrap. Set pans in sun or in a just-warm oven or on top of a woodstove.
Turn fruit when it can be lifted off plastic without falling apart; it should be barely sticky to touch and still pliable. Remove plastic and dry other side of leather. Roll up leather in fresh plastic and store in jars or crocks. To use, unroll and cut off pieces with scissors.
**Pears**
The world of pears now includes Asian pears, round, juicy, and crisp, and best for eating fresh. The best variety for preserving is Bartlett, the one commonly available in grocery stores. Max-Red, a sport, is reported to be sweeter. Note that pears require two different types for proper pollination. Standard types grow from 18 to 20 ft tall (5.5 to 6.1 m) and should be spaced 20 to 25 ft apart (6.1 to 7.6 m). As with all the dwarf types, dwarf pears are a blessing for those with less space to devote to fruit trees: they reach 8 to 10 ft tall (2.4 to 3.0 m) and should be planted 10 ft (3.0 m) apart. Wild or neglected pear trees often bear small hard fruit. It depends on the type, but pears usually need to be stored in a cool place before they are usable. Sometimes it's for as little as a few days; sometimes it's for as long as a couple of months.
Canned peaches in an upscale European canning jar
**Plain Pear Preserves**
CANNED PEARS ARE NOT OVERLY SWEET IF THEY ARE PRESERVED IN A THIN OR MODERATE SYRUP. WHEN I MAKE PEAR PRESERVES, I DON'T PEEL THE FRUIT, ALTHOUGH MOST PEOPLE DO. HERE IS THE PREFERRED METHOD, BUT TRY MINE SOMETIME, TOO. USE BARTLETT PEARS OR ANOTHER HARD VARIETY.
Pears
Light syrup:
1 qt water (1 L)
2 cups sugar (500 mL)
Peel, halve, and core pears. Prepare syrup. In a large preserving pot, boil together sugar and water, stirring to dissolve sugar. Add pears and simmer mixture for 5 minutes.
Pack pears into hot scalded canning jars [p. 28] and pour 1/2 to 1 cup (125 to 250 mL) syrup over pears, leaving 1/2 inch (1 cm) headroom. Adjust lids and process jars in a boiling-water bath [p. 26] or steam canner. Process 1-pt (50-mL) jars for 20 minutes, l-qt (l-L) jars for 25 minutes.
**Pears-in-Wine**
Makes 1 1/2 pt (750 mL)
6 pears
1 piece cinnamon stick
1 cup sugar (250 mL)
1 slice lemon
1/2 cup elderberry wine [p. 125] or other red wine (225 mL)
Peel and core sweet, firm ripe pears; halve if large. In a large stainless-steel pot, combine sugar, wine, cinnamon, and lemon. Bring to a boil. Reduce heat and drop in pears. Cook over medium heat until fruit is tender. _Do not overcook._
Discard cinnamon. Fill hot scalded jars [p. 28] with pears and pour 1/2 to 1 cup (125 to 250 mL) syrup over them, leaving 1/2 inch (1 cm) headroom. Make sure pears are covered with syrup. Seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Dried Pears**
Pears are one of the easiest fruits to dry, and the result is very sweet.
Slice pears vertically, leaving skin and seeds to retain flavor. Lay thin slabs on a tray or cookie sheet lined with heavy plastic wrap. Place tray near cooking area, where it will receive extra heat, or in direct sunlight. Turn fruit daily.
To hasten drying, put tray in oven after heat has been turned off or in a just-warm oven. When they feel dry and pliable, pears are done. Store in covered crocks or jars.
**Pear Leather**
Chop overripe pears into small pieces, peel and all. Place in a large preserving pot, cover, and simmer until soft. Add water if necessary, to prevent scorching. Add sugar to taste if desired. Put mixture through a food mill.
Spread pulp thinly on cookie sheets lined with one layer of heavy plastic wrap. Set pans in sun or in a just-warm oven or on top of a woodstove.
Turn fruit when it can be lifted off plastic without falling apart; it should be barely sticky to touch and still pliable. Remove plastic and dry other side of leather. Roll up leather in fresh plastic and store in jars or crocks. To use, unroll and cut off pieces with scissors.
**Plums**
Of all tree fruits, plums are the easiest for the home gardener to grow and they are beautiful in bloom. Plum trees like heavy soil, and they produce abundant crops sooner than other fruit trees. Some varieties, though, are hardier and more disease resistant. Japanese plums require two different varieties for proper pollination. You could plant two different types anyway, one for preserving and one for eating fresh. As with other fruit trees, dwarf types are available if you are cramped for growing space: Dwarf plum trees grow from 8 to 10 ft tall (2.4 to 3.0 m) and should be planted 10 ft apart (3.0 m); standard types reach 15 ft (4.6 m) and should be planted 20 ft apart (6.1 m). Plum varieties are constantly subject to change, so if you are interested in growing them, consult your local Cooperative Extension for recommended types (see Appendix).
**Plum Preserves**
THIS IS AN EASY WAY TO PRESERVE A LOT OF PLUMS IF YOU ARE LUCKY ENOUGH TO BE OVERWHELMED BY THEM, AS WE ARE ON OCCASION. THE PITS ARE LEFT IN, SO BEWARE WHEN YOU EAT THEM. USE FIRM, NOT OVERRIPE, PLUMS, PREFERABLY A PRESERVING VARIETY, SUCH AS DAMSON.
Makes about 1 qt (1 L)
Heavy syrup:
1 1/2 to 2 lb plums (750 g to 1 kg)
3/4 cup sugar (175 mL)
1/2 cup water (125 mL)
Prepare syrup. In a large stainless-steel pot, stir sugar into water. Cover and bring to a boil, stirring as necessary. Add plums and simmer, uncovered, for 5 minutes, stirring carefully with a long-handled fork, only as necessary.
Remove preserves from heat. Ladle into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Note:_ Always do plums in small batches to keep them from breaking down. You can also use the above recipe for making cherry preserves.
**Plum Jam**
THIS IS A KNOCKOUT ON VANILLA ICE CREAM.
Makes 1 pt (500 mL)
Plums
Sugar
If plums are freestone, remove pits before cooking. Otherwise, in a large stainless-steel pot, combine plums and a small amount of water. Simmer, covered, until soft. Cool mixture and remove pits.
Simmer plums, uncovered, until they form a pulp that is fairly thick and measures 2 cups (500 mL). Add 1 1/2 cups (375 mL) sugar, stir often, and simmer until mixture thickens.
Remove jam from heat and let subside. Stir. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Quince**
Quince is a small, graceful tree with grayish bark, crooked stems, goblet-like white flowers in late spring, and if you're lucky, perfumed golden fruit that hangs at the tips of its branches by fall. Even if I never got a single fruit, I would want to grow fruiting quince ( _Cydonia oblonga_ ) for its old-fashioned aura and ornamental qualities (not as showy as flowering quince, _Chaenomeles_ spp., but elegant in its way). Fruiting quince grows to 10 to 12 ft (3.0 to 3.7 m) so it can easily be tucked into a shrub border as long as it receives a half day of full sun. It grows best, and is less prone to disease, in a site with good air circulation. It is self-pollinating, will grow in most soils, and is hardy to -25° F (-32° C). Quinces are high in Vitamin C and pectin-rich, so their extracted juice helps low-pectin fruit to jell. They are known for their affinity for apples in any form. Add a few small, cut-up pieces to applesauce and apple pies to impart their distinct flavor. See Appendix for these hard-to-find fruit trees.
**Wild Fruits**
Before we owned a farm, we made seasonal forays to pick wild berries because we wanted to add more fruit to our diet and our income was very limited. Like a little army of squirrels, we packed into the old Dodge truck and came home with buckets of fruit. Before the first frost, we always remembered to get some sumac stalks, too, to flavor winter drinks.
**Quince Jelly**
THIS IS A BEAUTIFUL PINK COLOR AND SLIGHTLY TART.
Quince
Sugar
Wipe fuzz from fruits. Cut up in small pieces, nearly cover with water, bring to a boil and simmer, with the cover tilted, for about 2 hours until quite tender. Drain through a jelly bag overnight [p. 28]. Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover and bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil. Skimming as necessary, boil, uncovered, for about 10-15 minutes or until a small amount sheets off a metal spoon [p. 28-29]. Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Quince Paste**
Pastes are a more condensed form of fruit butters. You should be able to cut a fruit paste into little squares. There are many old recipes for making quince paste. This one comes from Jeanne Leblanc, who learned about quince paste on a visit to Spain, and likes to serve it on toast or rice crackers with a thin slice of goat gouda.
Peel, core, and quarter at least 6 large quinces. (Jeanne says that the core seeds tend to gum up the food mill, and little flecks of skin remain in the purée, which is ok if you like it that way.) Put the pieces in a large preserving pot and add water to cover. Bring to a boil and simmer until quinces are soft, which may take up to 45 minutes. Drain, cool, and run through a food mill to make a purée.
Put the purée in a large pot, place over medium heat and add an equal amount of sugar by volume. Bring the mixture to a boil, reduce to a simmer and cook, stirring regularly, until it darkens slightly and thickens. The purée is ready when it quickly clots on a frozen plate, when a wooden spoon stands up in the pot, or when the purée drips in sheets from the spoon. This usually takes about 30 minutes.
Pour the hot, sweetened purée into a bowl or mold and leave to set for at least 24 hours. Unmold when ready to eat. Slice and serve.
Refrigerate the remaining quince paste for up to 3 months. Jeanne preserves hers like any jam and processes it for 5 minutes in a boiling-water bath [p. 26]. She sometimes freezes some purée so she can thaw it, add sugar, and make more quince paste during the year.
Later, we lived on a remote back-country farm without any vehicle at all. We got almost all our fruit from what we planted, except for an annual expedition to the local shore for cranberries. Occasionally, we enjoyed a harvest of chokecherries from bushes the birds so thoughtfully planted along the 1/2 mile lane into our farm. Wherever we live, though, we continue to explore the wild for edible fruits.
**Barberries**
Early settlers first brought the tart barberry ( _Berberis vulgaris_ ) to New England. Barberries are still found in New England, growing along banks or wherever escapees find conditions favorable. The attractive, but thorny, bushes bear long clusters of juicy, oblong, brilliant orange-red berries, usually harvested in the early fall. A domesticated ornamental variety, _Berberis thunbergii,_ bears inedible fruit and in some areas is considered invasive.
**Blackberries**
Blackberries grow wild over much of North America, but the farther south you go, the bigger and juicier the berries get. In warm climates, they grow in large thickets, and you can spend days picking them by the bucket, for jam or for syrup or to eat fresh with other fruits (particularly peaches) and lots of thick cream. Breeders appear to have been busy with blackberries. If you want to grow them, look for new varieties that bear on first year canes (primocane bearing). These are Prime Jim, Prime Jan, and Prime Ark, hardy to Zone 4, because they can freeze to the ground over winter and still produce a crop the following summer. _Note:_ Dewberries and closely related boysenberries, youngberries, and loganberries, can be used interchangeably in the recipes below.
**Barberry Sauce**
JIGS, EVER ON THE LOOK OUT FOR THE POSSIBILITIES OF EDIBLE WILD FRUITS, GOT THE IDEA FOR PRESERVING BARBERRIES FROM OUR NOW TATTERED FERNALD'S _E DIBLE WILD PLANTS OF EASTERN NORTH AMERICA_. THIS SAUCE IS MORE LIKE A CONFECTION—POPULAR WITH CHILDREN.
Barberries
Molasses
In a large stainless-steel pot, bring equal amounts of barberries and molasses to a boil, covered. Skim out barberries and boil syrup, uncovered, until reduced, by half. Make sure it does not burn. Add barberries and bring to a rolling boil.
Remove sauce from heat. Stir. Ladle into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands, and process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Blackberry Jam**
USE A MIXTURE OF RIPE AND SLIGHTLY UNDERRIPE BERRIES. THE LARGER WILD BERRIES FROM THE MID-ATLANTIC STATES AND FROM THE SOUTH MAKE A LESS SEEDY JAM.
Makes about 1 1/2 pt (750 mL)
1 qt blackberries (1 L)
3 cups sugar (750 mL)
2 Tbsp lemon juice (25 mL)
In a large stainless-steel pot, mash blackberries to expel juice. Bring to simmering point, covered. Stir in sugar and lemon juice. Stirring frequently, boil, uncovered, for about 15 minutes or less, or until mixture thickens and begins to cling to bottom of pot.
Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
_Note:_ For less-seedy jam, put part of cooked berries through a food mill before adding sugar.
**Blackberry Syrup**
YOU CAN MAKE JUICE FROM THIS SYRUP AS WELL—JUST DILUTE IT WITH WATER TO TASTE. USE DEAD-RIPE BLACKBERRIES. USE OVER ICE CREAM, ADD TO PIES, PRESERVES, OR FRUIT DISHES OF ANY KIND.
Makes 6 to 8 pt (3 to 4 L)
6 qt blackberries (6 L _)_
Sugar
2 1/2 qt water (2.5 L)
In a large preserving pot, mash blackberries. Add water, cover, and bring to a boil. Stirring occasionally, simmer for 10 minutes or until berries are soft and juice runs freely. Strain through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 6 cups (1.5 L) at a time in a large stainless-steel pot. Bring to a boil, covered. Stir in 1/2 cup (125 mL) sugar to each cup (250 mL) juice. Boil, uncovered, for about l0 to 15 minutes or until mixture thickens.
Remove syrup from heat and let subside. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Curdie's Blackberry Pie**
INGENIOUS AND SIMPLE, THIS PIE IS A GREAT DESSERT FOR HOT MIDSUMMER DAYS. OUR DAUGHTER REMEMBERS IT WELL FROM VISITS TO A VERMONT FARM.
2 cups sugar (500 mL)
Baked pie shell [p. 46]
1 cup water (250 mL)
Whipped cream [p. 58]
1 qt blackberries (1 L)
In a pot, boil together sugar and water, uncovered, to form a syrup, about 5 minutes. Stir until sugar is dissolved. Pour over blackberries and place mixture in a baked 9-inch (23-cm) pie shell. Top with whipped cream and refrigerate until ready to serve.
**Blueberries**
Wild blueberries thrive in northern New England and eastern Canada, and when we have been unable to grow them, we always look for them in the wild. Sometimes, if we can't pick them, a friend picks them for us and we barter for some of Jigs' great smoked fish. The cultivated high-bush blueberry grows into a large shrub if given acid, well-drained soil and moderate moisture. There are a growing number of varieties available to the home gardener; some are self-pollinating, others need another variety for proper pollination. High-bush blueberries grow to 5 ft (1.5 m); plant them 4 ft (1.2 m) apart. On the farm we raised a long hedge of these and the plants were so heavily laden that we could not keep up with them and so invited friends to come and pick. They are beautiful at every stage, from flowering to fruiting, and in the fall, the foliage turns deep crimson. No wonder blueberries are popular plants. Breeders have been busy developing them for earlier ripening, hardiness, larger or tastier fruit. There are dwarf types for growing in tubs if you're limited for space. Because the berries don't all ripen at once, you will need several bushes to have enough to preserve. Watch out for birds, and net if you see damage. Blueberries, like black currants and elderberries, are high in Vitamin C and other health benefits.
Note: Huckleberries may be substituted for blueberries in any of these recipes.
**Blueberry Preserves**
BLUEBERRIES ARE ONE OF THE EASIEST FRUITS TO CAN, AND COOKING THEM SEEMS TO BRING OUT THEIR FLAVOR. USE FIRM RIPE BERRIES.
Blueberries
Sugar
In a large preserving pot, combine blueberries and sugar, using 1/4 to 1/2 cup (50 to 125 mL) sugar to each qt (L) blueberries. Bring to a boil, stirring frequently to prevent sticking.
Remove preserves from heat. Ladle into hot scalded canning jars [p. 28], leaving 1/2 inch (1 cm) headroom, and adjust lids. Process jars for 10 minutes in a boiling-water bath [p. 26] or steam canner.
**Blueberry Jam**
THIS IS FAST-SETTING WITH HOMEMADE APPLE PECTIN [P. 149].
Makes about 2 pt (1 L)
1 qt blueberries, some a little under-ripe (1 L)
1 cup apple pectin (250 mL)
2 Tbsp lemon juice (25 mL)
3 cups sugar (750 mL)
Heat berries, pectin and lemon juice to simmering, stir in sugar, and bring to a rolling boil. Boil hard for 10 minutes or until thickened. Remove jam from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Blueberry Cobbler**
1 qt blueberries (1 L)
1 Tbsp baking powder (15 mL)
1 cup sugar (250 mL)
1/4 tsp salt (1 mL)
1/4 cup melted butter (50 mL)
3 Tbsp shortening or lard (50 mL)
Juice of 1/2 a lemon
1 egg
1 cup cake flour 250 mL
1/4 cup milk 50 mL
Pinch nutmeg
Place blueberries, sugar, butter, and lemon juice in a greased baking dish. Mix.
Mix and sift together flour, nutmeg, baking powder, and salt. Cut in shortening with a pastry cutter until evenly distributed.
In a bowl, beat together egg and milk; stir into flour-shortening mixture. Drop spoonfuls of dough over blueberries to cover. Bake cobbler in a 350°F (180°C) oven for about 40 minutes or until dough is lightly browned and berries are bubbling.
_Note:_ To make with blueberry preserves (above), reduce sugar to taste. Stir 1 heaping Tbsp (15 mL) flour into blueberry mixture and proceed.
**Blueberry Cake**
1/2 cup butter (125 mL)
1 tsp baking powder (5 mL)
3/4 cup sugar (175 mL)
1 1/2 cups blueberries (375 mL)
2 eggs, well beaten
1/3 cup milk (75 mL)
1 1/2 cups flour (375 mL)
1 tsp vanilla (5 mL)
Pinch salt
In a bowl, cream butter with sugar; add well-beaten eggs. Mix and sift together flour, salt, and baking powder; stir into butter-sugar mixture. Stir in blueberries. Add milk and stir mixture well but do not overheat. Add vanilla. Pour batter into a shallow buttered pan and bake in a 350° F (180°C) oven for 35 minutes or until cake is lightly browned and knife inserted in center comes out clean.
**Vera's Blueberry Kolachky**
A LONG TIME CZECH FRIEND ONCE MADE THIS WONDERFUL CONFECTION FOR US.
1/4 cup warm water (50 mL)
1 large egg, well beaten
1 tsp granulated sugar (5 mL)
3 cups flour (750 mL)
1/4 tsp ginger (1 mL)
Softened shortening or salad oil
Blueberries
1 Tbsp dry yeast (15 mL)
1/2 cup milk (125 mL)
1/4 cup butter (50 mL)
Topping:
1/3 cup sugar (75 mL)
1 cup brown sugar (250 mL)
1/2 tsp salt (2 mL)
1/2 cup cold butter (125 mL)
1/2 cup cold water (125 mL)
1 Tbsp cinnamon (15 mL)
In a large bowl, combine warm water, 1 tsp (5 mL) granulated sugar, ginger, and yeast. Let stand in a warm place until mixture is dissolved and yeast begins to foam.
In a saucepan, scald milk, stirring in butter until melted. Dissolve 1/3 cup (75 mL) sugar and salt in hot milk. Remove from heat. Stir in cold water to cool milk mixture to warm; add to yeast mixture. Add well-beaten egg and 1 cup (250 mL) flour. Beat well. Mix in remaining flour or enough to get dough to clear bowl.
Place dough on a board sprinkled with flour and cover with bowl, turned upside down. Let dough rest at least 5 minutes. Then knead until smooth and elastic, adding more flour if necessary to make a smooth, satiny dough. Return dough to bowl and brush top of it with softened shortening or a little salad oil. Cover bowl with a towel and leave dough to double in bulk.
Punch down dough. Place in a 10-inch x 14-inch (4-L) pan, flattening dough to fit bottom of pan and pulling it up along sides to form a rim. Fill depressed dough with fresh blueberries, right up to rim.
Prepare topping. In a bowl, mix together brown sugar, cold butter, and cinnamon. Work the mixture until it forms coarse crumbs. Sprinkle over blueberries.
Let kolachky rise until light and bake in a 350°F (180°C) oven for about 20 minutes or until dough is firm and blueberries are bubbling.
**Cape Breton Blueberry Muffins**
I AM INDEBTED TO THE _CAPE BRETON BICENTENNIAL DANCERS_ COOKBOOK OF FAVORITE RECIPES FOR THESE BEST-EVER MUFFINS.
Makes about 15
1 cup sugar (250 mL)
2 eggs
1/2 cup butter or margarine (125 mL 2)
cups flour 500 mL
2 tsp baking powder (10 mL)
1/2 tsp salt (2 mL)
1/2 cup milk (125 mL)
2 1/2 cups blueberries, fresh or frozen
(625 mL)
Cream sugar, butter, and eggs. Mix flour, baking powder, and salt. Add alternately with milk. Mix until just smooth. Add blueberries, lightly dusted with a little of the flour mixture so they don't sink to the bottom of the batter. Spoon into greased muffin tins, or paper cups. Sprinkle with a little sugar (it makes a frosted glaze). Bake at 375° F (190° C) for about 25 minutes or until golden brown.
**Freezing Blueberries**
Blueberries freeze well without sugar but retain better flavor if lightly coated with sugar: about 1/2 cup (125 mL) for every qt (L) of fruit. Freeze in freezer containers. These are excellent in blueberry muffins.
Cape Breton blueberry muffins
**Chokecherries**
Chokecherries grow from coast to coast in North America along the banks of streams. Its dark purple fruit is small and grows in grapelike clusters. The berries, which ripen in early fall, are inedible raw, but they make fine jellies and sasses. _Prunus, or virginiana_ Midnight Schubert, with deep wine-red leaves, is an attractive non-suckering variety worth looking for.
This recipe originated when Jigs was the Jelly King. We knew nothing about whether fruits were high or low in pectin. Jigs found this recipe, it used apples, and it worked. We wouldn't have had a spare penny to buy commercial pectin, in any case.
**Chokecherry-Apple Jelly**
Makes about 2 pt (1 L)
2 qt chokecherries (2 L)
2 cups water (500 mL)
3 1/4 lb chopped apples (1.5 kg)
Sugar
In a large preserving pot, combine ripe chokecherries and tart fresh apples. Simmer, covered, until soft, stirring as necessary to prevent sticking. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Bring to a boil, covered, and stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a rolling boil and, skimming as necessary, boil, uncovered, until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Variation:_ To make _chokecherry-apple sass,_ follow directions for chokecherry-apple jelly. After juice is brought to a boil, stir in 112 cup (125 mL) sugar to each cup (250 mL) juice. Boil, uncovered, until thick. Let subside. Jar and seal with snap lids and screw bands. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Cranberries**
There are two kinds of cranberries to harvest from the wild, _Vaccinium macrocarpon,_ the bog cranberry, and _Viburnum trilobum,_ high-bush cranberry. They are unrelated, but they share characteristics. They both bear acid fruit high in natural pectin. Bog cranberries are the kind served on Thanksgiving. We well remember picking bog cranberries on the shores of Cape Breton on sunny, cold days, with the wind at our backs. Bog cranberries grow on creeping small-leaf plants that thrive in bogs and wet places, as well as in barren sandy areas. The fruit is best picked in late fall after a frost. High-bush cranberries grow mainly by walls and fences, in woods and low places. The large shrubs bear berries that are brighter in color than bog cranberries, but similar in size, and are best picked before frost. Both types grow in northern New England and eastern Canada, but high-bush cranberries are more common in the northeastern states. They are ornamental, with showy white flowers in the spring and crimson leaves in the fall, as well as clusters of gleaming bright red berries. We now grow a dwarf version to 6 ft (1.8 m).
**Cranberry Jelly**
Makes about 2 pt (1 L)
1 qt bog or high-bush cranberries (1 L)
2 cups water (500 mL)
2 cups sugar (500 mL)
In a large stainless-steel pot, combine cranberries and water. Simmer, covered, until skins of cranberries crack or pop. Put hot fruit through a food mill. Add sugar and bring mixture to a boil. Skimming as necessary, boil, uncovered, about for 5 minutes.
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Cranberry Preserves**
THESE ARE EASIER TO MAKE THAN THE TRADITIONAL CRANBERRY JELLY. IF YOU PREFER JAM TO JELLY, CHANCES ARE YOU WILL PREFER THESE PRESERVES.
Makes about 1 qt (1 l)
1 qt bog cranberries (1 L)
2 cups water (500 mL)
1 1/2 cups sugar (375 mL)
In a large stainless-steel pot, combine cranberries, sugar, and water. Cover and bring to a boil. Uncover and, stirring occasionally, cook slowly for about 10 minutes or until cranberry skins break.
Remove preserves from heat. Ladle into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Cranberry Relish**
THIS RELISH WILL KEEP FOR SEVERAL MONTHS IN THE REFRIGERATOR. LIME GIVES IT A PIQUANT TOUCH.
Makes about 1 qt (1 L)
1 qt bog cranberries (1 L)
1 lime (optional)
1 orange
1 cup sugar (250 mL)
1 lemon
Cut lime, orange, and lemon into quarters. Remove seeds but do not peel. Grind all fruit twice with a vegetable grinder or food processor, saving juice. Place in a large bowl and stir in juice and sugar thoroughly. Pack relish into tightly covered containers and chill. Freeze extra.
_Note:_ If you include lime, make sure you use a heaping cup (250 mL) sugar.
**Cranapple Juice**
Apples
Sugar
Bog cranberries
Cut up desired amount of apples and place in a large preserving pot with desired amount of cranberries. Simmer, covered, until soft. Strain through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice. Add 1 cup (250 mL) sugar to each qt (L) juice. Boil, uncovered, for 5 minutes.
Remove juice from heat. Pour into hot scalded jars [p. 28], leaving 1/2 inch (1 cm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 15 minutes.
**Cranapple Jelly**
1 qt (1 L) juice makes about 2 pt (1 L)
Bog cranberries
Tart apples
Use equal amount of cranberries to apples. Cut up apples. In a large preserving pot, combine fruit with cold water. Simmer, covered, until soft, stirring as necessary. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Bring juice to a boil, covered, and stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Bring to a boil again. Skimming as necessary, boil, uncovered, until a small amount sheets off a metal spoon [p.28-29].
Remove from heat and let jelly subside, skimming off froth if desired. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
_Variation:_ To make _cranapple sass,_ cook any proportion of apples and cranberries as above. Stir in 1/2 cup (125 mL) sugar to each cup (250 mL) juice and boil, uncovered, until mixture thickens. Pour into hot scalded jars [p. 28], leaving 1/4 inch (5 mm) headroom, and seal at once with snap lids and screw bands. Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Wild Grapes**
Wild grapes grow as far north as New Brunswick, but they are more commonly found in the mid-Atlantic and southern states. If you are lucky enough to find them, pick them before someone else does— in early fall when some of the berries are still underripe. Make a jelly that you will never forget.
**Original Wild Grape or Venison Jelly**
THIS IS JIGS' SPECIALTY. ONCE FROSTED FRUIT IS SAID TO SPEED JELLING.
Makes 2 to 3 pt (1 to 1.5 L)
5 lb wild grapes (2.5 kg)
1/2 cinnamon stick
1 cup vinegar (250 mL) Sugar
1 Tbsp whole cloves (15 mL)
In a large preserving pot, combine grapes, vinegar, cloves, and cinnamon stick. Stirring occasionally, simmer, covered, until grapes are soft. Mash. Strain through a jelly bag [p. 28] and let drip for several hours or overnight.
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Boil for 20 minutes, uncovered, and stir in 1 cup (250 ml) sugar to each cup (250 mL) juice. Skimming as necessary, boil rapidly until a small amount sheets off a metal spoon [p. 28-29].
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands, Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Wild Grape or Venison Jelly II**
THE ORIGINAL RECIPE DID NOT CALL FOR APPLES BUT THEY HELP JELLING.
Makes about 4 1/2 pt (2.25 L)
4 qt wild grapes with stems (4 L)
6 lb tart fresh apples (3 kg)
1/4 cup whole cloves (50 mL )
1 qt cider vinegar (1 L)
1 qt water (1 L) Sugar
1 /4 cup stick cinnamon (50 mL)
Cut up apples. In a large preserving pot, combine grapes, apples, cider vinegar, and water and spices. Simmer, covered, until fruit is soft, stirring as needed. Mash fruit. Strain mixture through a jelly bag [p. 28] and let drip for several hours or overnight
Measure juice and cook 4 cups (1 L) at a time in a large stainless-steel pot. Cover, bring to a boil. Stir in 1 cup (250 mL) sugar to each cup (250 mL) juice. Boil, uncovered, for 15 minutes or until a small amount sheets off a metal spoon [p.28-29]. Remove spice bag.
Remove jelly from heat and let subside. Stir, skimming if desired. Fill hot scalded jars [p. 28] to 1/4 inch (5 mm) from the top and seal at once with snap lids and screw bands, Process in a boiling-water bath [p. 26] or steam canner for 10 minutes.
**Sumac**
Several varieties of staghorn sumac, _Rhus typhina,_ grow across the United States and Canada. They can be easily distinguished from poison sumac by their fuzzy red fruit—the poisonous variety carries white fruit. The red fruit is extremely acid and therefore is useful as a lemon substitute. Pick the fruiting branches in the early fall, about the same time as chokecherries and elder berries and before the expected rains damage the fruit. Even though it's considered a weed tree, staghorn sumac is a nice addition to the back of a large flower border if you have the space and the leaves turn a brilliant scarlet in fall. Dig up young shoots and plant them where they will have room to spread their attractive limbs. Some people call staghorn sumac "the velvet tree," because it has long velvety hairs on its branches.
**Sumac Juice**
SUBSTITUTE UNSWEETENED SUMAC JUICE FOR THE WATER IN ANY ELDERBERRY JAM, JELLY, JUICE, OR SASS RECIPE. FRUITING SUMAC STALKS WILL KEEP OVER WINTER IN A COOL, DRY PLACE, AWAY FROM DIRECT LIGHT.
Sumac
Sugar
Place desired amount of sumac heads in a large pot. Cover with water and pound and stir for about 10 minutes, until an extraction of juice is apparent. It has a nice pink color.
Strain through a jelly bag [p. 28] made of several layers of cheesecloth. Add sugar to taste, stirring well.
**_Appendix_**
**Local Cooperative Extension Offices**
http://www.csrees.usda.gov/ Extension/
All you have to do is click on the map to find your local office. There you should be able to find answers to questions about growing fruit in your area and the best varieties (more correctly, cultivars) to meet your particular needs.
In Canada, contact your Provincial Department of Agriculture.
**The Nearest U-Pick**
Http://www.pickyourown.org/
This site can help you find "pick-your-own farms" in every state and in Canada. It is a wealth of information, including where to find canning equipment and ice cream machines.
**HARD-TO-FIND & DWARF & SEMI-DWARF FRUIT STOCK**
These plant nurseries sell, among other fruits, black and red currant, gooseberry, elderberry, and fruiting quince:
**Miller Nurseries**
<http://millernurseries.com/>
Ph: 1-800-836-9630
Miller's in New York State also offers Pink Champagne Currant (a cross between red and white currants); a good selection of semi-dwarf and dwarf fruit trees; Heirloom Apples, including Wolf River, the large ones we used to make into apple rings; Dwarf Blueberries to grow in tubs; Reliance Peach; High-bush Cranberry; and Jostaberry.
**Raintree Nursery**
<http://www.raintreenursery.com/>
Ph: 1-800-391-8892
Raintree in Washington State offers "The finest Old & New Cultivars From Around the World," including an unusually wide selection of Black Currants; ornamental Elderberries; Huckleberries; many kinds of Blueberries (including wild), and those best for the South and Pacific Northwest.
**One Green World**
<http://onegreenworld.com/>
Ph: 1-877-353-4028
**One Green World in Oregon,**
although specializing in fruits for the Pacific Northwest climate, also offers hardy fruits, including Day Neutral Strawberries; and Honeyberry (hardy to -44 ° F), something to try if you have difficulty growing Blueberries.
**Gurney's Seed & Nursery Co.**
<http://gurneys.com/>
Ph: 513-354-1491
Gurney's in Indiana is a source for traditional fruits as well as Elderberry, Red Currant, Jostaberry, and Gooseberry, also a range of Blueberries, including a dwarf type; Strawberries and a Pyramidal Strawberry Bed to grow them in a small space; traditional and semi-dwarf and dwarf fruit trees including Apples, Peaches, and Pears.
**Saskatoon Farm**
<http://www.saskatoonfarm.com/>
Ph: 1-800-463-2113
This farm in Alberta, Canada, offers Gooseberry and Black Currant plants, as well as High-bush Cranberry, Honeyberry, the desirable Midnight Schubert Chokecherry, and even sells Chokecherry Syrup. There is a U-Pick operation on the farm.
**Northern Alberta Permaculture Institute**
<http://napi.ca/>
Here you will find an extensive listing of seed and plant sources in Canada.
**Organic Insect Spray**
**Gardens Alive!**
<http://www.gardensalive.com/>
Ph: 513-354-1482
Source for the fast-acting Pyola insect spray for dealing with the currant sawfly. It is available from Gurney's, too.
**Hard-To-Find seeds of Citron Melon (Citrullus lanatus var. citroides):**
**Comstock Garden Seeds**
<http://comstockferre.com/>
Ph: 860-571-6570
**Seed Savers Exchange**
<http://www.seedsavers.org/>
Ph: 563-382-5990
**Ozark Seed Bank**
<http://onegarden.org/>
**Seeds of Diversity (Canada)**
<http://www.seeds.ca/>
**Canning Equipment & Related Items**
Canning is more popular than ever before with more people putting up their own food. Check local sources first.
**Lehman's**
<http://www.lehmans.com/>
Ph: 1-888-438-5346
Located in Ohio in the heart of Amish country, this is a one-stop shop for the traditional water bath canner, canning jars (including the elegant European type), lids and rings, a hand-crank ice cream maker, hand food mill, and the hard-to-find steam canner and steam juicer.
**Ace Hardware**
<http://www.acehardware.com/home/index.jsp>
At this site you can find the Ace Hardware store nearest you. There you should be able to find canning equipment as well as jelly jars and all types of pectin.
**Harvest Essentials**
<http://www.harvestessentials.com/capi.html>
Offers a range of canning and preserving essentials as well as steam canners and a nifty hand food mill, La Petite Sauce Maker and Food Strainer.
**Bernadin Home Canning (Canada)**
<http://www.bernardin.ca/>
Ph: 905-731-3384
Everything you need for canning: water bath canner, jars, lids and rings, pectins.
**Cheesemaking Supplies**
**New England Cheesemaking Supply Company**
<http://www.cheesemaking.com/>
Ph: 413-397-2012
You will find what you need to make all the dairy recipes in this book, including the non-electric Yogotherm for making yogurt.
In Canada, contact **Glengarry Cheesemaking Dairy Supply** for similar items.
<http://glengarrycheesemaking.on.ca/>
Ph: 1-888-816-0903 or 613-347-1141
**Manual Food Processor**
**Starfrit**
<http://www.starfrit.com/>
I bought my Starfrit manual food processor in Canada at a local hardware store. You can get the same thing in the U.S. from <http://www.amazon.com/>
**Small Electric Ice Cream Machines**
Amazon also carries the Oster Ice Cream/Frozen Yogurt Maker, and other brands, for making small batches of sorbets and the like.
**_Bibliography_**
Ahern, Nell Giles, ed. _The Boston Globe's Chocolate Cook Book._ Boston: The Globe Newspaper Co., 1955.
Aish Ha Torah Women's Organization. _The Taste of Shabbos._ Spring Valley, N.Y.: Philipp Feldheim, 1987.
_Ball Blue Book._ rev. ed. Muncie: Ball Corp., 1974; 1982..
_Ball Blue Book_. Altrista Consumer Products. Daleville, Indiana: Jarden Home Brands, 2011.
Beeton, Isabella. _Mrs Beeton's All About Cookery._ London: Pan Books, 1963.
_Bernadin Home Canning Guide._ Toronto: Bernadin of Canada, 1975; 1995.
Blasberg, C. H. _Growing Strawberries in Vermont._ Vermont Agricultural Experiment Station. Pamphlet 15. Burlington, 1948.
Bowles, Ella Shannon, and Dorothy S. Towle. _Secrets of New England Cooking._ New York: M. Barrows, 1947.
Brattleboro, Vermont, Woman's Club. _My Ladye's Coke Book._ Brattleboro, 1924.
Brown, Marion. _Pickles & Preserves._ New York: Wilfred Funk, 1955.
Canada. Department of Agriculture. _Elderberry Cultivation in Eastern Canada._ Publication 1280. Ottawa, 1966.
Department of Agriculture. _Growing Red Raspberries in Eastern Canada._ Publication 1196. Ottawa, 1964.
Department of Agriculture. _Growing Strawberries in Eastern Canada._ Publication 1170. Ottawa.
Department of Agriculture. _Planting and Growing Rhubarb._ Publication 1369. Ottawa, 1968.
Claiborne, Craig. _An Herb & Spice Book._ New York: Bantam, 1963.
Downes, Muriel, and Rosemary Hume. _Jams, Preserves & Pickles._ New York: Weathervane Books.
Farmer, Fannie. _The Boston Cooking School Cook Book._ Boston, 1896.
Fernald, M. L., et al. _Edible Wild Plants of Eastern North America._ New-York: Harper & Bros., 1958.
Fraser, S. _American Fruits._ Orange Judd Co., 1924.
Fuller, A. S., _The Small Fruit Culturist._ 1867.
Gibbons, E, _Stalking the Wild Asparagus._ New York: David McKay, 1962.
Grigson, Jane, _Jane Grigson's Fruit Book_. New York: Atheneum, 1982.
"Handbook on Pruning." _Brooklyn Botanical Garden Record, Plants and Gardens,_ 37, 2 (1981).
Hedrick, U. P. _Fruits for the Home Garden,_ repr. New York: Dover, 1973.
Hill, Lewis. _Fruits and Berries for the Home Garden._ Garden Way, Vt., 1980.
The Home Institute of the New York Herald Tribune. _America's Cookbook._ New York: Charles Scribner's Sons, 1943.
Mosser, Marjorie. _Foods of Old New England._ Garden City: Doubleday, 1957.
_The New Settlement Cookbook._ New York: Simon & Schuster, 1954.
Petrides, G. A. _A Field Guide to Trees and Shrubs._ Cambridge, Mass.: Houghton Mifflin, 1958.
Ricketson, C. L. _Currants and Gooseberries._ Vineland Station, Ont.: Horticultural Research Institute of Ontario, 1966.
Sanford, S. N. F. _New England Herbs._ Boston: New England Museum of Natural History, 1937.
Shoemaker, J. S., _Small Fruit Culture._ New York: McGraw-Hill, 1955.
_Vegetable Growing._ New York: Wiley & Sons, 1953.
Showalter, Mary Emma. _Mennonite Community Cookbook._ Scottdale, Pa.: Herald Press, 1974.
Tompkins, J. P., and D. K. Ourecky. _Raspberry Growing in New York State._ Cornell University Extension Publication Information. Bulletin 155. Ithaca.
Tatum, Billy Joe. _Wild Foods Field Guide and Cookbook_. New York: Workman, 1976.
_Two-Hundred Favorite Recipes of The Cape Breton Bicentennial Dancers_. Winnipeg, Manitoba: Gateway Publishing Co. Ltd.
United States. Department of Agriculture. _Strawberry Varieties in the U.S._ Farmers' Bulletin 1043. Washington, D.C., 1958.
Wyman, D. _Wyman's Gardening Encyclopedia._ New York: Macmillan, 1972.
**_Index_**
Apples, _See also_
Chokecherries, 151,
159, 160, 166, 168
Cranberries, 128,
151, 161-64
Elderberries, xi, xx,
6, 12, 97-112, 155,
166, 168
Apple Butter I, 134
Apple Butter II, 135
Apple Butter III, 130
Apple Cider, 130, 135
Apple Crisp, 135
Apple Ginger, 132
Apple Jelly, 130, 131
Apple Leather, 14, 136
Apple-Mint Jelly, 131
Apple Pectin, 104,
134, 156
Apple-Peelings Jelly,
130
Apple-Rose
Geranium Jelly, 131
Applesauce Cake,
137
Applesauce Muffins,
139
Apple-Spice Jelly, 131
Apple
Sweetmeats, 133
Curdie's Apple-
Crumb Pie, 137
Dried Apple Rings,
130
Easy Apple Cake,
139
Old-Fashioned
Applesauce, 133
Varieties, 129
Bailey, Liberty Hyde,
xix
Barberries, 151, 152
Barberry Sauce, 152
_Berberis_
_thunbergii,_ 152
_Berberis vulgaris,_ 151
Basic Berry Ice, 123,
Basic Brandied Fruit,
121
Basic Butters,
Cheeses, Pastes, 120
Basic Frozen Fruit
Yogurt, 127
Basic Fruit Gelatin, 126
Basic Fruit Ice Cream,
124
Basic Fruit Sherbert,
125
Basic Fruit Shrub, 122
Basic Fruit Sorbet/Ice,
125
Basic Fruit Soup, 122
Bass, Sue and Henry,
124
Beverages, _See also_
_specific fruits,_ 91, 107
Ginger Ale, 91
Hot Toddies, 107
New Year's Punch for
20, 91
Blackberries, xi, 128,
152-54
Blackberry Jam, 153
Blackberry Syrup,
154
Curdie's Blackberry
Pie, 154
Varieties, 152
Black Currants, _See_
_also_ Gooseberries
Black Currant Ice
Cream, 124
Black Currant Jam,
73, 83, 85, 86, 95
Black Currant Jelly,
xxii, 86, 90
Black Currant Juice,
83, 89, 96
Black Currant-Leaf
Cream, 95
Black Currant
Sorbet/Ice, 96
Black Currant
StickyBuns, 94
Black Currant Wine,
85, 90, 93
Black Mead, 92
Boodle's Fool, 96
Crème de Cassis, 91
Cultivating, 84, 85
Disease-Resistant
Varieties, 81
Dried Black
Currants, 82, 85,
88, 119
Freezing, 85, 96
Harvesting, 85
Jellyroll with Black
Currant Jam, 95
Planting, 83, 84
Preserving, Canning,
Freezing, and
Cooking, 85
Reputation, 79-83
Sponge Cake with
Black Currant Jam,
95
White-Pine Blister
Rust, xxi, 55, 79, 81
Blueberries, xi, 12, 88,
128, 155-60, 168,
Blueberry Cake, 157
Blueberry Cobbler,
157
Blueberry Jam, 156
Blueberry Preserves,
156, 157
Cape Breton
Blueberry Muffins,
159
Freezing
Blueberries, 159
Varieties, 155
Vera's Blueberry
Kolachky, 158
Boiling-Water Bath
Term and Method,
4-6, 10
Boysenberry, 152
Bread, _See also_ 24,
86, 100
Rhubarb, 24 English
Muffins, 21, 130
Brooks, Donna, 143
Cakes, _See also_
Apples, Black
Currants,
Blueberries,
Citron, Rhubarb
Chocolate Cake
Supreme, 52
Nova Scotia
Fruitcake, 119
Super Chocolate
Roll, 53
Two-Egg Cake, 37
Viennese Chocolate
Torte, 39
Candied Fruit, _See_
_also_ Citron, 116-19
Candied Lemon
Peel, 116, 119
Candied Orange
Peel, 117, 119
Cereal
Cracked-Wheat
Cereal, 31
Cheese
Cottage Cheese, 87,
88
Cream Cheese, xxii,
35, 58, 86-8, 132
Chicken, 85, 132
Coq au vin, 90, 93
Chokecherries, 159,
160, 166
Chokecherry-Apple
Jelly, 160
Chokecherry-Apple
Sass, 160
_Prunus virginiana_
Midnight
Schubert, 159, 168
Citron, 113-19, 132, 168
Candied Citron, 113,
116-19, 132
Citron Cake, 116
Citron Marmalade,
117
Citron Pickles, 116
Citron Preserves, 118
_Citrullus lanatus_ var _._
_citroides,_ 168
_Citrus medica,_ 113,
114
Cultivating, 115
Harvesting, 115, 116
Planting, 114, 115
Preserving, Canning,
and Cooking, 116
Cold Pack, 5
Term, 5
Cookies, 20, 62
Rollies, 20
Thumbprint Cookies,
62
Crabapples 102, 140,
141
Crabapple Jelly, 140
Crabapple
Marmalade, 141
Crabapple
Preserves, 141
Crabapple Sauce,
141
Cranberries, 128,
161-64
Cranapple Jelly, 164
Cranapple Juice, 163
Cranapple Sass, 164
Cranberry Jelly, 161
Cranberry
Preserves, 161
Cranberry Relish,
162
_Vaccinium_
_macrocarpon,_ 161
_Viburnum trilobum,_
161
Cream
Crème Fraiche, 34
Devonshire Cream,
35, 52
Sour Cream, 24, 34,
61, 62, 111, 121
Sweet Cream, 18, 24,
34, 35, 40, 52, 124
Whipping Cream,
34, 38, 76, 96
Whipped Cream, 37,
39, 50, 52, 53, 95,
96, 112, 126, 154
Dewberries, 152
Dried Berries, 12, 13
_See also_ Black
Currants, 83,
Elderberries
Steps, 12, 13
Elderberries, xi, 6,
97-112, 155
Cultivating, 99
Dried Elderberries,
100, 108, 112
Elderberry-Apple
Jelly, 102
Elderberry Cordial,
107
Elderberry-Custard
Pie, 112
Elderberry‑Green
Grape Jelly, 103
Elderberry Jam, 104
Elderberry Jelly With
Added Pectin, 101
Elderberry Juice I,
105
Elderberry Juice II,
105
Elderberry‑Orange
Jelly, 102
Elderberry Pie, 112
Elderberry Pie
Deluxe, 111
Elderberry
Preserves, 104
Elderberry Sass,
102, 135
Elderberry Wine,
108
Freezing, 100, 109
Harvesting, 99, 100
Planting, 98, 99
Preserving, Canning,
Freezing, and
Cooking, 100
Sambuccus
canadensis, 98
Sambuccus pubens,
98
Spiced Elderberry
Jelly, 103
Varieties, 98
Elderflowers, 98, 100,
101, 111 See also
Gooseberries, 100
Dried Elderflowers,
101
Elderflower
Fritters, 111
Elderflower
Pancakes, 110
Elderflower‑
Peppermint Tea,
110
Elderflower Wine,
100
Equipment, xix, 3, 4,
167-69
Extraction, Term and
Method, 5
Fruit, See also specific
fruits
Three‑Fruit Jam, 74
Gooseberries, 6, 55,
65-78, 80
Canned
Gooseberries, 77
Cultivating, 67-9
Currant Sawfly, 57,
168
Freezing, 71
Fresh Stewed
Gooseberries, 78
Gooseberry‑Black
Currant Jam, 73
Gooseberry
Chutney, 76
Gooseberry‑Cream
Snow, 76
Gooseberry‑
Elderflower Jelly,
72
Gooseberry Ice
Cream, 78
Gooseberry Jam, 73
Gooseberry Jelly, 71
Gooseberry
Marmalade, 75
Gooseberry Pie or
Tart, 77
Gooseberry‑Red
Currant
Bar‑le‑Duc, 73
Gooseberry
Rhubarb Jam, 74
Harvesting, 69-71
Planting, 66, 67
Preserving, Canning,
Freezing, and
Cooking, 71
Pyola spray, 69 _See_
_also_ Red Currant,
54-64
Varieties, 66, 68
White‑Pine Blister
Rust, 66
Grapes, 164, 165 See
also Elderberries, xi,
6, 97-112, 155
Original Wild Grape
or Venison Jelly,
165
Wild Grape or
Venison Jelly II, 165
Grigson, Jane, 91
Headroom
Term, 5
Hedrick, U. P., 82
Hill, Lewis, 82
Hot Pack
Term, 5
Huckleberries, 155,
168
Ice Cream, See also
Gooseberries
Basic Fruit Ice,
Cream, 124
Vanilla Ice Cream,
124
Jam, See also specific
fruits
Steps, 9, 10
Jelly, See also specific
fruits
Steps, 8, 9
Jelly Bag,
Term and Method,
6
Josta, 73, 81, 167
Juice, _See also_ Pectin
_specific fruits_
Steps, 13
Leather, 13, 14, 34,
136
_See also_ Apples,
Peaches, Pears,
Strawberries, 34
Steps, 13, 14
LeBlanc, Jeanne, xiv,
151
Loganberry, 152
Malinich, Tim, xi, xiv,
Malva sylvestris
Zebrina, xii, 96,
Miller, Bob, xiv, 143
Pancakes, 110 _See also_
Elderflowers, 110
Buttermilk Pancakes,
110
Pastry
Swisher Family
Never-Fail Pie
Crust, 20
Peaches, 142-45
Canned Peaches, 145
Ginger-Peach Jam,
143
Nellie's Peach Jam,
142
Peach Butter, 144
Peach Leather, 145
Peach-Orange
Marmalade, 144
Varieties, 142
Pears, See also Apples,
129-40
Dried Pears, 147
Pear Leather, 147
Pears-in-Wine, 147
Raspberry Jam, 45
Plain Pear Preserves,
146
Varieties, 145
Pectin
Term, 6
Pies, 77, 112, 137 See
also specific fruits
Plums
Plum Jam, 149
Plum Preserves, 148
Varieties, 148
Preserves, See also
specific fruits
Steps, 11, 12
Quince, 150, 151
Chaenomeles spp.,
150
Cydonia oblonga,150
Quince Jelly, 150
Quince Paste, 151
Raspberries, 40-53
See also
Red Currants, 54-64
Cultivating, 42, 43
Fall‑bearing, 41
Freezing, 47
Harvesting, 43
Plain Canned
Raspberry
Preserves I, 46
Plain Canned
Raspberry
Preserves II, 46
Planting, 41
Preserving, Canning,
Freezing and
Cooking, 43, 44
Raspberries Canned
in Red Currant
Juice, 45
Raspberry‑Flavored
White Wine, 49
Raspberry Jam, 45
Raspberry‑Jam
Squares, 49
Raspberry Juice, 48
Raspberry Pudding,
or Rothe Gruetze, 50
Raspberry‑Red
Currant Juice, 48
Raspberry‑Red
Currant Sass, 48
Raspberry Sass, 47
Raspberry Slump, 52
Raspberry Spritz, 50
Varieties, 41
Red Currants, See also
Gooseberries,
Raspberries
Bar‑le‑Duc Jelly, 60
Cultivating, 55-57
Currant Sawfly, 57
Freezing, 58
Harvesting, 57, 58
Planting, 55
Preserving, Canning,
Freezing, and
Cooking, 58
Pyola spray, 57
Red Currant Jam, 60
Red Currant Jelly I,
58
Red Currant Jelly II,
59
Red Currant Juice, 61
Red Currant Meat
Sauce, 63
Red Currant Pie, 64
Red Currant‑
Raspberry Ice,
123
Red Currant‑
Raspberry Jelly, 59
Red Currant‑
Raspberry Tart, 64
Red Currant Sass, 61
Red Currant Sorbet/
Ice
Red Currant Snub,
62
Varieties, 55
White-Pine Blister
Rust, 55
Rhubarb, 15-25
Canned Rhubarb, 18
Canned Strawberry
Cultivating, 17
Freezing, 19
Harvesting, 17
Planting, 16
Preserving, Canning,
Freezing, and
Cooking, 18
Rhubarb Bread, 24
Rhubarb Fool, 24
Rhubarb Juice, 22
Rhubarb Marmalade
I, 19
Rhubarb Marmalade
II, 23
Rhubarb Pie, 25
Rhubarb Sherbert,
22
Rhubarb-Strawberry
Jam, 20
Rhubarb Upside-
Down Cake, 25
Rhubarb Wine, 23
Stewed Rhubarb, 24
Varieties, 16
Rose, Lois, 91
Sass, _See also specific_
_fruits_
Steps, 10, 11
Scalded Jars
Term and Method, 6
Sheets Off a Spoon
Term and Method, 6,
7
Steam Canner
Term and Method, 4,
5
Steam Juicer
Term and Method, 4,
5
Strawberries, 26-39
See also Rhubarb
Canned Strawberry
Preserves, 32
Chunky Strawberry
Sauce, 32
Cultivating, 29, 30
Fragaria chiloensis,
26
Fragaria virginiana,
26
Freezing, 31
Frozen Whole
Strawberries, 32
Harvesting, 30
Planting, 27, 28
Preserving, Canning,
Freezing, and
Cooking, 30-2
Strawberries and
Cream, 34
Strawberry Bavarian
Cream, 38
Strawberry Frosting,
36
Strawberry Juice, 33
Strawberry Leather,
34
Strawberry‑Rhubarb
Parfait, 39
Strawberry Sass, 33
Strawberry
Shortcake, 37
Strawberry Tart, 37
Swisher Family
Never-Fail Pie
Crust, 20
Varieties, 26, 27, 29
World's Best
Strawberry Jam,
31
Sumac, 166
_Rhus typhina,_ 166
Sumac Juice, 166
Whipped Cream, 37,
39, 50, 52, 53, 95, 96,
112, 126, 154
White-Pine Blister
Rust, _See also_
Black Currants,
Gooseberries,
Red Currants, 54-
64
Genus _Ribes,_ xxi, 55,
66
Yogurt, 32, 35, 36, 127
Greek Yogurt and
Yogurt Cheese
Spread, 36
| {
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} | 3,944 |
This hip bag/bumbag can be personalised with embroidery of your choice, making it perfect for the festival season or a great small travel pouch to keep your valuables with you at all times!
One size- adjustable strap with a plastic buckle.
Front antique brass zipper closure.
This is a pigment dyed good so should always be washed separately before wearing to avoid bleeding dye issues. Pigment dyed refers to washed down colours that continue to soften and age with washing. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,100 |
We don't need a national Latino museum
The northwest corner of the National Museum of African American History and Culture in Washington. (Jahi Chikwendiu/The Washington Post)
By Mike Gonzalez
Mike Gonzalez is a senior fellow at the Heritage Foundation.
Should we build a Latino Smithsonian museum? Some Hispanic politicians think so. Piggybacking on the attention garnered by the opening this weekend of the National Museum of African American History and Culture, they have renewed a push for the creation of a National Museum of the American Latino.
It's an idea that sounds good — until you think about it for about three seconds.
This is not just because museums are for dead things ("The Louvre is a morgue; you go there to identify your friends," the French artist Jean Cocteau famously complained), but because it would breathe life into concepts from which we need to move away.
The Latino museum is being championed by Rep. Xavier Becerra (D-Calif.), who doesn't even bother to hide the "me-tooism." Just a couple of weeks before the opening of the African American Museum, Becerra introduced a bill calling for the Latino museum to be placed in the Arts and Industries Building on the Mall. Sen. Robert Menendez (D-N.J.) moved parallel legislation in the Senate.
"It provides inspiration, and it really does give you locomotion to try to move this forward," Becerra told The Post. "So many [of the African American Museum's supporters] have come to me and said, 'You're next.' It pumps you up."
And that's just it. Of all the reasons this is a bad idea, we can start with the fact that the experiences of African Americans cannot be compared to those of any other group — especially immigrants and their descendants.
That would include the vast majority of the 56 million people the Census Bureau instructs to identify themselves as "Hispanic" — who can't all be descended from the estimated 100,000 people who chose to remain in the Southwest at the conclusion of the Mexican War in 1848.
The notion that they constitute an ethno-racial pentagon along with African Americans, Asian Americans, Native Americans and non-Latino whites is a dubious social construct of very recent pedigree. That a museum would help perpetuate this division — literally cement it — is a second reason to oppose it.
Dividing the country along these cleavages — an official policy that began only in the late 1970s and quickly migrated to the academy, the labor market and the culture — has contributed to a degree of social fragmentation that is only now becoming apparent.
What started as a perhaps well-meaning concept stands behind much of today's palpable societal angst. Even liberals are starting to worry about what national fracturing is doing to social solidarity.
The multicultural dispensation that resulted depends on indoctrinating members of four of the groups into believing that they are historical victims of the fifth. This is on its face a nonsensical proposition for those who willingly came here, and for their descendants, and has led to misallocations of priorities and funds.
Many non-Latino whites are disadvantaged socially, as this year's runaway bestseller "Hillbilly Elegy" by J.D. Vance makes abundantly clear. As a very good review in last week's New Yorker explained, poor whites also face economic and cultural barriers to upward mobility.
At the same time, many members of the designated minorities also are very socially advantaged and do not need set-asides to get a government contract or be accepted into Harvard University.
Which is the third important reason the Smithsonian should not open a Latino national museum: Such an institution could only perpetuate the notion of victimhood.
This is a corrosive idea because it tells individuals, especially the young, that they lack agency, that their problems were created by others. We don't have to imagine what politicians would do with this — we see it every day.
There are small museums here and there for German Americans, Italian Americans and Jewish Americans, which is fine. There could be a repository for the definitive story of Cuban Americans, most of whom are here as a result of the traumatic dislocation caused by the Cuban Revolution; for Mexican Americans, whose incredible cultural imprint in the Southwest is at least as important as that of Vance's Scots-Irish in Appalachia; for Puerto Ricans, etc.
But, please, no Smithsonian museum for an ethnicity created by 1970s federal bureaucrats. Defenders of immigration make the case that today's immigrants will assimilate as members of previous surges did — which is what undoubtedly will happen, but only if they are treated as those earlier arrivals were.
That is, as immigrants on their way to being Americans, not as members of a permanent national minority.
Robert L. Wilkins: Magnificent, awful, profound: The stories the new African American museum will tell
Jonathan Capehart: Wow, Hispanics loathe Donald Trump. Like, a lot. But enough to vote? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,683 |
newswire article reporting united states 25.Mar.2006 17:44
alternative media | government | katrina aftermath
Anti-American Barbara Bush Orders Her Katrina Donation to Be Invested in Her Company
author: joe broadhurst e-mail: joejoehead@resist.ca
Un-f'n-believable! Barbara Bush ordered her Katrina Donation to be specifically spent at lil' Neil's educational software company Ignite!. She is an investor in Ignite! Barbara Bush is such a psycho that she ordered her donation to be invested in her own company. The "Oh, these people are much better off" is nothin' but psycho....
Here is the list of investors in Neil's (and Barbara's) company Ignite!:
Neil Bush's Investors
By Paul Kiel - March 24, 2006, 6:31 PM
Yesterday, Josh pointed out that the business model for Neil Bush's education company Ignite! seems to be that "Neil goes around the world finding international statesmen, bigwigs and criminals who want to 'invest' in Ignite! as a way to curry favor with the brother in the White House."
But just who are those statesmen, bigwigs and criminals?
The company declines to name private investors, but documents filed with the SEC show that it raised $7.1 million from 53 investors. There are a number of unnamed investors from the UAE, Saudi Arabia, and the British Virgin Islands.
Here are the ones we do know about. It's quite an international grab bag:
-- Hamza El Khouli, an associate of Egyptian President Hosni Mubarak and chairman of First Arabian Development and Investment Company (Lawrence Journal World)
-- Les and Anne Csorba, Bush contributors who worked in the first Bush White House in the White House Personnel Office and the Office of Personnel Management (Houston Chronicle).
-- Former Iranian Ambassador Hushang Ansary, a Houston businessman (Houston Chronicle).
-- Sofidiv Inc., a division of the Moet Hennessy Louis Vuitton luxury goods company (Houston Chronicle)
-- Mohammed Al Saddah of the Ultra Horizon Co. in Kuwait (Houston Chronicle).
-- Winston Wong (alternately, Winston Wang), a Taiwan businessman who started the Grace Semiconductor Manufacturing Corp. with the eldest son of China's president, Jiang Zemin (Washington Post)
-- Joseph Peacock, company secretary at Crest Investment, where Neil Bush was co-chairman; they specialize in the energy sector. Peacock is also involved with many companies owned by Jamal Daniel - Daniel is a Syrian-American businessman with business ties to the Bush family (St. Petersburg Times)
-- Boris Berezovky, a Russian billionaire living in London with ties to many Russian politicians, including Boris Yeltsin. He is sought by Russian prosecutors and Interpol for fraud (Moscow Times, Washington Post)
-- Knowledge Universe, a company chaired by former junk bond king Michael Milken (Philadelphia Inquirer, AP)
-- The Nagase Brothers education company in Japan (AP, NY Times)
-- Timothy Bridgewater, the chief executive of Interlink Management, the venture capital firm he and Neil started, also one of the Republican "Pioneers" who helped raise $100,000 for George W. Bush during the 2000 campaign (Philadelphia Inquirer)
-- Bush's parents
Now check this out:
Former first lady's donation aids son
Katrina funds earmarked to pay for Neil Bush's software program
By CYNTHIA LEONOR GARZA
Copyright 2006 Houston Chronicle
Former first lady Barbara Bush donated an undisclosed amount of money to the Bush-Clinton Katrina Fund with specific instructions that the money be spent with an educational software company owned by her son Neil.
Since then, the Ignite Learning program has been given to eight area schools that took in substantial numbers of Hurricane Katrina evacuees.
"Mrs. Bush wanted to do something specifically for education and specifically for the thousands of students flooding into the Houston schools," said Jean Becker, former President Bush's chief of staff. "She knew that HISD was using this software program, and she's very excited about this program, so she wanted to make it possible for them to expand the use of this program."
The former first lady plans to visit a Houston Independent School District campus using the Ignite program today to call on local business leaders to support schools and education.
The trip to Fleming Middle School is intended to showcase Bush's commitment to education for both Houston-area and New Orleans evacuee students, according to a press release issued Wednesday by Ignite.
Fleming, which has more than 170 New Orleans students, was one of eight area schools chosen by the Harris County Department of Education to receive a donated COW, or Curriculum on Wheels, multimedia program after Hurricane Katrina.
Neil Bush founded Austin-based Ignite Learning, which produces the COW program, in 1999.
Becker said she wasn't at liberty to divulge how much money the Bush family gave to the hurricane funds, but said the "rest of their donation was not earmarked for anything."
Nationally, some other donors also specified how they wanted their donations spent, Becker said.
For example, one man wanted his money to go to Habitat for Humanity but via the former presidents' fund. Nearly $1 million has been raised for the local fund and more than $120 million for the national.
Regarding the fact that Bush's earmarked donation also benefited her son's company, Becker said, "Mrs. Bush is obviously an enthusiastic supporter of her son. She is genuinely supportive of his program," and has received many letters from educators who support it. Bush "honestly felt this would be a great way to help the (evacuee) students."
Barbara and Neil Bush presented the donated programs to Houston-area schools this winter.
Districts that received the free curriculum include Houston, Alvin, Katy, Pearland and Spring and the New Orleans West charter school.
There are 40 Ignite programs being used in the Houston area, and 15 in the Houston school district, said Ken Leonard, president of Ignite.
Information about the effectiveness of the program, through district-generated reports, was not readily available Wednesday, according to an HISD spokeswoman.
Two years ago, the school district raised eyebrows when it expanded the program by relying heavily on private donations.
In February 2004, the Houston school board unanimously agreed to accept $115,000 in charitable donations from businesses and individuals who insisted the money be spent on Ignite. The money covered half the bill for the software, which cost $10,000 per school.
The deal raised conflict of interest concerns because Neil Bush and company officials helped solicit the donations for the HISD Foundation, a philanthropic group that raises money for the district.
HISD school principals decide for themselves whether to spend their budgeted money on Ignite.
Leonard said that in the past six to eight months, the company has hired national sales representatives across the country — in Florida, New York, Pennsylvania, Georgia and Nevada — in hopes of expanding beyond Texas. Currently, about 80 percent of the company's customers are from Texas.
Last year, Neil Bush reportedly toured former Soviet Union countries promoting Ignite with Russian tycoon Boris Berezovsky.
According to the Times of London, Berezovsky, a former Kremlin insider now living in Britain, is wanted on criminal charges in Moscow accusing him of seeking to stage a coup against President Vladimir Putin.
The purpose of today's event is to showcase everyone's efforts in helping the hurricane evacuee students who ended up in Houston, Leonard said.
"We have a role, but we're not the leader in this," Leonard said. He also acknowledged that his company will benefit from the former first lady's visit.
Barbara Bush is expected to observe both teachers and students using the Ignite Learning program while touring classrooms, according to the Ignite press release.
During a short reception, teachers and students will give testimonials about the program and Bush will "encourage community business leaders to have a stronger presence in supporting schools and education," the press release said.
The free-standing instructional tools that are not dependent on the Internet. They include a built-in computer, projector and speakers and come pre-loaded with science and social studies courses.
homepage: http://intl-news.blogspot.com/2006/03/anti-american-barbara-bush-orders-her.html | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 582 |
Investigation of the role of G1/S cell cycle mediators in cellular senescence
Experimental Cell Research
Cellular senescence is a state of irreversible cell cycle arrest in which normal cells at the end of their lifespan fail to enter into DNA synthesis upon serum or growth factor stimulation. We examined whether proteins required for G1/S cell cycle progression were irreversibly down-regulated in senescent human fibroblasts. Both the 44- and 42-kDa forms of the MAP-kinase protein were expressed at similar levels in young and senescent cells.
CDC2-CDC28 Kinases
Cell Aging
Cyclin-Dependent Kinase 2
Cyclins
Protein Kinases
Protein-Serine-Threonine Kinases
Protein-Tyrosine Kinases
Tumor Suppressor Protein p53
Afshari, C. A.
Vojta, P. J.
Annab, L. A.
Futreal, P. A.
Willard, T. B.
Barrett, J. C.
Sequential extension of proliferative lifespan in human fibroblasts induced by over-expression of CDK4 or 6 and loss of p53 function
Replicative senescence is thought to be a significant barrier to human tumorigenesis, which in human fibroblasts, and many other cell types, can be overcome experimentally by combined loss of function of p53 and Rb 'pathways'. To avoid the confounding pleiotropic effects of HPVE7 frequently used in such studies, here we have employed retroviral vectors over-expressing CDK4 or CDK6 as a more representative model of naturally-occurring mutations targeting the Rb pathway.
Cell Transformation, Viral
Clone Cells
Cyclin-Dependent Kinase Inhibitor p16
Genes, p16
Genes, Retinoblastoma
Genetic Vectors
Oncogene Proteins, Viral
Papillomavirus E7 Proteins
Recombinant Fusion Proteins
Retinoblastoma Protein
Morris, Mark
Hepburn, Peter
Wynford-Thomas, David
Cyclin-dependent kinases 7 and 9 specifically regulate neutrophil transcription and their inhibition drives apoptosis to promote resolution of inflammation
Cell Death and Differentiation
Terminally differentiated neutrophils are short-lived but the key effector cells of the innate immune response, and have a prominent role in the pathogenesis and propagation of many inflammatory diseases. Delayed apoptosis, which is responsible for their extended longevity, is critically dependent on a balance of intracellular survival versus pro-apoptotic proteins. Here, we elucidate the mechanism by which the cyclin-dependent kinase (CDK) inhibitor drugs such as R-roscovitine and DRB (5,6-dichloro-1-beta-D-ribofuranosylbenzimidazole) mediate neutrophil apoptosis.
Dichlororibofuranosylbenzimidazole
Hep G2 Cells
HL-60 Cells
Protein Kinase Inhibitors
Purines
RNA Polymerase II
Leitch, A. E.
Lucas, C. D.
Marwick, J. A.
Duffin, R.
Haslett, C.
Rossi, A. G.
Cyclin-dependent kinase inhibition by flavoalkaloids
Mini Reviews in Medicinal Chemistry
Chromone alkaloids and flavoalkaloids are an important group of natural products possessing promising medicinal properties. A chromone alkaloid rohitukine is a major bioactive chemical constituent of plant Dysoxylum binectariferum (Meliaceae) Hook. which is phylogenetically related to the Ayurvedic plant, D. malabaricum Bedd. used for treatment of rheumatoid arthritis. This chromone alkaloid led to discovery of two synthetic flavoalkaloids: flavopiridol (Sanofi) and P-276-00 (Piramal) which have reached to advanced stages of clinical development for cancer treatment.
Clinical Trials as Topic
Drug Evaluation, Preclinical
Jain, S. K.
Bharate, S. B.
Vishwakarma, R. A.
Cell cycle control as a basis for cancer chemoprevention through dietary agents
Frontiers in Bioscience: A Journal and Virtual Library
The development of cancer is associated with disorders in the regulation of the cell cycle. The purpose of this review is to briefly summarize the known sequence of events that regulate cell cycle progression with an emphasis on the checkpoints and the mechanisms cell employ to insure DNA stability in the face of genotoxic stress. Key transitions in the cell cycle are regulated by the activities of various protein kinase complexes composed of cyclin and cyclin-dependent kinases (CDK) molecules.
Anticarcinogenic Agents
Genistein
Stilbenes
Meeran, Syed Musthapa
Katiyar, Santosh Kumar
Piceatannol, a natural analog of resveratrol, inhibits progression through the S phase of the cell cycle in colorectal cancer cell lines
The Journal of Nutrition
Piceatannol, a naturally occurring analog of resveratrol, was previously identified as the active ingredient in herbal preparations in folk medicine and as an inhibitor of p72(Syk). We studied the effects of piceatannol on growth, proliferation, differentiation and cell cycle distribution profile of the human colon carcinoma cell line Caco-2. Growth of Caco-2 and HCT-116 cells was analyzed by crystal violet assay, which demonstrated dose- and time-dependent decreases in cell numbers. Treatment of Caco-2 cells with piceatannol reduced proliferation rate.
Alkaline Phosphatase
Caco-2 Cells
Dose-Response Relationship, Drug
S Phase
Tumor Cells, Cultured
Wolter, Freya
Clausnitzer, Antje
Akoglu, Bora
Stein, Jürgen
Indirubins inhibit glycogen synthase kinase-3 beta and CDK5/p25, two protein kinases involved in abnormal tau phosphorylation in Alzheimer's disease. A property common to most cyclin-dependent kinase inhibitors?
The bis-indole indirubin is an active ingredient of Danggui Longhui Wan, a traditional Chinese medicine recipe used in the treatment of chronic diseases such as leukemias. The antitumoral properties of indirubin appear to correlate with their antimitotic effects. Indirubins were recently described as potent (IC(50): 50-100 nm) inhibitors of cyclin-dependent kinases (CDKs). We report here that indirubins are also powerful inhibitors (IC(50): 5-50 nm) of an evolutionarily related kinase, glycogen synthase kinase-3beta (GSK-3 beta).
Calcium-Calmodulin-Dependent Protein Kinases
CDC2 Protein Kinase
Cyclin B
Dopamine and cAMP-Regulated Phosphoprotein 32
Glycogen Synthase Kinase 3
Glycogen Synthase Kinases
Inhibitory Concentration 50
Molecular Structure
Neostriatum
Phosphoproteins
Phosphothreonine
Piperidines
Staurosporine
tau Proteins
Leclerc, S.
Garnier, M.
Hoessel, R.
Marko, D.
Bibb, J. A.
Snyder, G. L.
Greengard, P.
Biernat, J.
Wu, Y. Z.
Mandelkow, E. M.
Eisenbrand, G.
Meijer, L.
Bufalin inhibits platelet-derived growth factor-BB-induced mesangial cell proliferation through mediating cell cycle progression
Biological & Pharmaceutical Bulletin
Bufalin, a traditional Chinese medicine, has been reported as a protective factor in many tumors. We therefore investigated the effect of bufalin on platelet-derived growth factor (PDGF)-BB-induced proliferation of cultured rat mesangial cells. The effect of bufalin on cell proliferation and its underlying mechanisms were investigated in cultured rat mesangial cells (MCs) by the methylthiazoletetrazolium (MTT) assay, flow cytometry, reverse transcription-polymerase chain reaction (RT-PCR), Western blotting, and cyclin-dependent kinases (CDK)2 and CDK4 kinase assays.
Bufanolides
Glomerular Mesangium
Platelet-Derived Growth Factor
Proto-Oncogene Proteins c-sis
Zhang, Aiqing
Zheng, Jun
Gong, Jing
Li, Shanwen
Zeng, Zhifeng
Gan, Weihua
Natural product triptolide mediates cancer cell death by triggering CDK7-dependent degradation of RNA polymerase II
Triptolide is a bioactive ingredient in traditional Chinese medicine that exhibits diverse biologic properties, including anticancer properties. Among its many putative targets, this compound has been reported to bind to XPB, the largest subunit of general transcription factor TFIIH, and to cause degradation of the largest subunit Rpb1 of RNA polymerase II (RNAPII). In this study, we clarify multiple important questions concerning the significance and basis for triptolide action at this core target.
Diterpenes
Epoxy Compounds
Phenanthrenes
Manzo, Stefano Giustino
Zhou, Zhao-Li
Wang, Ying-Qing
Marinello, Jessica
He, Jin-Xue
Li, Yuan-Chao
Ding, Jian
Capranico, Giovanni
Miao, Ze-Hong | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 795 |
\section{Introduction}
The Bohmian interpretation of nonrelativistic quantum mechanics (QM)
\cite{bohm1,bohm2,bohmPR1,holbook}
is the best known and most successfull attempt to explain quantum phenomena
in terms of ``hidden variables'', that is, objective properties of the system
that are well defined even in the absence of measurements.
According to this interpretation, particles allways have continuous and deterministic trajectories
in spacetime, while all quantum uncertainties are an artefact of the ignorance of the
initial particle positions. The wave function plays an auxiliary role, by acting as a
pilot wave that determines motions of particles for given initial positions.
Yet, in its current form, the Bohmian interpretation
is not without difficulties. An important
nontrivial issue is to make the Bohmian interpretation
of many-particle systems compatible with special relativity.
Manifestly relativistic-covariant
Bohmian equations of motion of many-particle systems have been proposed
in \cite{durr96} and further studied in \cite{nikrel05,nikrel06},
but for a long time it has not been known how to associate probabilistic predictions
with such relativistic-covariant equations of motion.
Recently, a progress has been achieved by realizing that the {\it a priori} probability density
$\rho({\bf x},t) \propto |\psi({\bf x},t)|^2$ of a single particle at the space-position ${\bf x}$
at time $t$ should {\em not} be interpreted as a probability density in space satisfying
$\int d^3x \, \rho({\bf x},t)=1$, but as a probability density in {\em spacetime}
satisfying $\int d^3x \, dt\, \rho({\bf x},t)=1$ \cite{nikprobrel07,nikprobrel09}.
The usual probabilistic interpretation in space is then recovered as a conditional probability,
corresponding to the case in which the time of detection has been observed.
For an $n$-particle wave function it generalizes to
$\int d^4x_1 \cdots \int d^4x_n \, \rho(x_1,\cdots , x_n)=1$, where
$\rho(x_1,\cdots , x_n) \propto |\psi(x_1,\cdots , x_n)|^2$ and
$x_a \equiv ({\bf x}_a, t_a)$. As shown in \cite{nikprobrel07,nikprobrel09},
such a manifestly relativistic-invariant probabilistic interpretation is compatible with
the manifestly relativistic-covariant Bohmian equations of motion.
In this paper we confirm this compatibility, through a more
careful discussion in Appendix \ref{APPB}.
The main remaining problem is how to make the Bohmian interpretation
compatible with quantum field theory (QFT), which predicts that particles can
be created and destructed. How to make a theory that describes deterministic continuous
trajectories compatible with the idea that a trajectory may have a singular point
at which the trajectory begins or ends?
One possibility is to explicitly break the rule of continuous deterministic evolution,
by adding an additional equation that specifies stochastic breaking of the trajectories
\cite{durrcr1,durrcr2}. Another possibility is to introduce an additional
continuously and deterministically evolving hidden variable that specifies
effectivity of each particle trajectory \cite{nikcr1,nikcr2}. However,
both possibilities seem rather artificial and contrived. A more elegant possibility
is to replace pointlike particles by extended strings, in which case the Bohmian
equation of motion automatically contains
a continuous deterministic description of particle creation and destruction
as string splitting \cite{nikprobrel07}. However, string theory is not yet an
experimentally confirmed theory, so it would be much more appealing if
Bohmian mechanics could be made consistent without strings.
The purpose of this paper is to generalize the relativistic-covariant Bohmian interpretation
of relativistic QM \cite{nikprobrel09} describing
a fixed number of particles, to relativistic QFT that describes systems
in which the number of particles may change. It turns out that the
natural Bohmian equations of motion for particles described by QFT automatically
describe their creation and destruction, without need to add any additional structure
to the theory. In particular, the new artificial structures that has been added
in \cite{durrcr1,durrcr2} or \cite{nikcr1,nikcr2} turn out to be completely
unnecessary.
The main new ideas of this paper are introduced in Sec.~\ref{SEC2}
in a non-technical and intuitive way. This section serves as a motivation
for styding the technical details developed in the subsequent sections, but
a reader not interested in technical details may be
satisfied to read this section only.
Sec.~\ref{SEC3} presents in detail several new
and many not widely known conceptual and technical results
in standard QFT that do not depend on the interpretation of quantum theory.
As such, this section
may be
of interest even for readers not interested in interpretations of quantum theory.
The main purpose of this section, however, is to prepare the theoretical framework needed
for the physical interpretation studied in the next section.
Sec.~\ref{SEC4} finally deals with the physical interpretation.
The general probabilistic interpretation and its relation to the usual probabilistic rules
in practical applications of QFT is discussed first, while
a detailed discussion of the interpretation in terms
of deterministic particle trajectories (already indicated in Sec.~\ref{SEC2})
is delegated to the final part of this section.
Finally, the conclusions are drawn in Sec.~\ref{SEC5}.
In the paper we use units $\hbar=c=1$ and the metric signature $(+,-,-,-)$.
\section{Main ideas}
\label{SEC2}
In this section we formulate our main ideas in a casual and
mathematically non-rigorous way, with the intention to develop an
intuitive understanding of our results, and to motivate the formal developments that will
be presented in the subsequent sections.
As a simple example, consider a QFT state of the form
\begin{equation}\label{e2.1}
|\Psi\rangle = |1\rangle + |2\rangle ,
\end{equation}
which is a superposition of a 1-particle state $|1\rangle$ and a 2-particle state
$|2\rangle$. For example, it may represent an unstable particle for which
we do not know if it has already decayed into 2 new particles (in which case it is
described by $|2\rangle$) or has not decayed yet (in which case it is
described by $|1\rangle$). However, it is known that
one allways observes either one unstable particle
(the state $|1\rangle$) or two decay products (the state $|2\rangle$). One never
observes the superposition (\ref{e2.1}). Why?
To answer this question, let us try with a Bohmian approach.
One can associate a 1-particle wave function
$\Psi_1(x_1)$ with the state $|1\rangle$ and a 2-particle wave function
$\Psi_2(x_2,x_3)$ with the state $|2\rangle$, where $x_A$ is the spacetime
position $x^{\mu}_A$, $\mu=0,1,2,3$, of the particle labeled by $A=1,2,3$.
Then the state (\ref{e2.1}) is represented by a superposition
\begin{equation}\label{e2.2}
\Psi(x_1,x_2,x_3) = \Psi_1(x_1) + \Psi_2(x_2,x_3) .
\end{equation}
However, the Bohmian interpretation of such a superposition will describe
{\em three} particle trajectories. On the other hand, we should observe
either one or two particles, not three particles. How to explain that?
To understand it intuitively, we find it instructive to first understand an
analogous but much simpler problem in Bohmian mechanics. Consider
a single non-relativistic particle moving in 3 dimensions. Its wave function
is $\psi({\bf x})$, where ${\bf x}=\{ x^1,x^2,x^3 \}$ and the
time-dependence is suppressed. Now let us assume that the particle
can be observed to move either along the $x^1$ direction (in which case it is described
by $\psi_1(x^1)$) or on the $x^2$-$x^3$ plane (in which case it is described
by $\psi_2(x^2,x^3)$). In other words, the particle is observed to move
either in one dimension or two dimensions, but never in all three dimensions.
But if we do not know which of these two possibilities will be realized, we describe
the system by a superposition
\begin{equation}\label{e2.3}
\psi(x^1,x^2,x^3) = \psi_1(x^1) + \psi_2(x^2,x^3) .
\end{equation}
However, the Bohmian interpretation of the superposition (\ref{e2.3})
will lead to a particle that moves in all three dimensions. On the other hand,
we should observe that the particle moves either in one dimension or two dimensions.
The formal analogy with the many-particle problem above is obvious.
Fortunately, it is very well known how to solve this analogous problem involving one
particle that should move either in one or two dimensions. The key is to take into account
the properties of the {\em measuring apparatus}. If it is true that one allways observes
that the particle moves either in one or two dimensions, then the total wave function
describing the entanglement between the measured particle and the measuring apparatus
is not (\ref{e2.3}) but
\begin{equation}\label{e2.4}
\psi({\bf x},y) = \psi_1(x^1)E_1(y) + \psi_2(x^2,x^3)E_2(y) ,
\end{equation}
where $y$ is a position-variable that describes the configuration of the measuring
apparatus. The wave functions $E_1(y)$ and $E_2(y)$ do not overlap.
Hence, if $y$ takes a value $Y$ in the support of $E_2$, then this value is not
in the support of $E_1$, i.e., $E_1(Y)=0$. Consequently, the motion of the
measured particle is described by the conditional wave function \cite{durrequil1}
$\psi_2(x^2,x^3)E_2(Y)$.
The effect is the same as if (\ref{e2.3}) collapsed to $\psi_2(x^2,x^3)$.
Now essentially the same reasoning can be applied to the superposition (\ref{e2.2}).
If the number of particles is measured, then instead of (\ref{e2.2}) we actually have
a wave function of the form
\begin{equation}\label{e2.5}
\Psi(x_1,x_2,x_3,y) = \Psi_1(x_1)E_1(y) + \Psi_2(x_2,x_3)E_2(y) .
\end{equation}
The detector wave functions $E_1(y)$ and $E_2(y)$ do not overlap.
Hence, if the particle counter is found in the state $E_2$, then the measured
system originally described by (\ref{e2.2}) is effectively described by $\Psi_2(x_2,x_3)$.
\begin{figure*}[t]
\includegraphics[width=13cm]{QFTtraj1.eps}
\caption{\label{fig1}
{\it Left:} The destruction of particle 1 and survival of particle 2, as seen in spacetime.
The dot on trajectory 1 denotes a singular point of destruction at $x^{0}=T$.
{\it Right:} The same process as seen on the $x^{0}_1$-$x^{0}_2$ plane
of the configuration space. Instead of a singular point, we have a continuous
curve that asymptotically approaches the dashed line $x^{0}_1=T$.}
\end{figure*}
Now, what happens with the particle having the spacetime position $x_1$?
In general, its motion in spacetime may be expected to be
described by the relativistic Bohmian equation
of motion \cite{durr96,nikrel05,nikrel06}
\begin{equation}\label{e2.6}
\frac{dX^{\mu}_1(s)}{ds} = \frac{
\frac{i}{2} \Psi^*\!\stackrel{\leftrightarrow\;}{\partial^{\mu}_1}\! \Psi }{\Psi^*\Psi} ,
\end{equation}
where $s$ is an auxiliary scalar parameter along the trajectory. However, in our case
the effective wave function does not depend on $x_1$, i.e., the derivatives in
(\ref{e2.6}) vanish. Consequently, all 4 components of the 4-velocity (\ref{e2.6})
are zero. The particle does not change its spacetime position $X^{\mu}_1$.
It is an object without an extension not only in space, but also in time.
It can be thought of as a pointlike particle that exists only at one instant of time
$X^{0}_1$. It lives too short to be detected. Effectively, this particle
behaves as if it did not exist at all.
Now consider a more realistic variation of the measuring procedure, taking into account
the fact that the measured particles become entangled with the measuring apparatus
at some finite time $T$.
Before that, the wave function of the measured particles is really well described by
(\ref{e2.2}). Thus, before the interaction with the measuring apparatus, all 3
particles described by (\ref{e2.2}) have continuous trajectories in spacetime.
All 3 particles exist. But at time $T$, the total wave function significantly changes.
Either (i) $y$ takes a value from the support of $E_2$ in which case
$dX_1^{\mu}/ds$ becomes zero, or (ii) $y$ takes a value from the support of $E_1$
in which case $dX_2^{\mu}/ds$ and $dX_3^{\mu}/ds$ become zero.
After time $T$, either the particle 1 does not longer change its spacetime position,
or the particles 2 and 3 do not longer change their spacetime positions.
The effect is the same as if the particle 1 or the particles 2 and 3 do not exist
for times $t>T$. In essence, this is how relativistic Bohmian interpretation
describes the particle destruction. In order for this mechanism to work, we see
that it is essential that each particle possesses not only its own space coordinate ${\bf x}_A$,
but also its own time coordinate $x^0_A$.
The corresponding particle trajectories are illustrated by Fig.~\ref{fig1}.
The picture on the left shows the trajectories of particles 1 and 2 in spacetime,
for the case in which the particle 1 is destructed at time $T$.
The trajectory of the destructed particle looks discontinuous. However,
the trajectories of all particles are described by continuous functions
$X^{\mu}_A(s)$ with a common parameter $s$,
so the set of all 3 trajectories (because $A=1,2,3$) in the 4-dimensional
spacetime can be viewed as a single continuous
trajectory in the $3\cdot 4 =12$ dimensional configuration space.
The picture on the right of Fig.~\ref{fig1}
demonstrates the continuity on the $x^{0}_1$-$x^{0}_2$
plane of the configuration space.
One may object that the mechanism above works only in a very special case
in which the absence of the overlap between $E_1(y)$ and $E_2(y)$ is {\em exact}.
In a more realistic situation this overlap is negligibly small, but not exactly zero.
In such a situation neither of the particles will have exactly zero 4-velocity.
Consequently, neither of the particles will be really destroyed.
Nevertheless, the measuring apparatus will still behave as if some particles
have been destroyed. For example, if $y$ takes value $Y$
for which $E_1(Y)\ll E_2(Y)$, then for all practical purposes the measuring apparatus
behaves as if the wave function collapsed to the second term in (\ref{e2.5}).
The particles with positions $X_2$ and $X_3$ also behave in that way.
Therefore, even though the particle with the position $X_1$ is not really destroyed,
an effective wave-function collapse still takes place. The influence of the particle
with the position $X_1$ on the measuring apparatus described by $Y$
is negligible, which is effectively the same
as if this particle has been destroyed.
\begin{figure}[t]
\includegraphics[width=6cm]{QFTtraj2.eps}
\caption{\label{fig2}
A ``real'' particle (having non-zero 4-velocity) surrounded by a sea of ``vacuum'' particles
(having zero 4-velocities).}
\end{figure}
Of course, the interaction with the measuring apparatus is not the only mechanism
that may induce destruction of particles. Any interaction with the environment
may do that. (That is why we use the letter $E$ to denote the states of the
measuring apparatus.) Or more generally, any interactions among particles
may induce not only particle destruction, but also particle creation.
Whenever the wave function $\Psi(x_1,x_2,x_3,x_4, \ldots )$ does not really vary
(or when this variation is negligible)
with some of $x_A$ for some range of values of $x_A$,
then at the edge of this range a trajectory of the particle $A$ may exhibit
true (or apparent) creation or destruction.
In general, a QFT state may be a superposition of $n$-particle states
with $n$ ranging from $0$ to $\infty$. Thus, $\Psi(x_1,x_2,x_3,x_4, \ldots )$
should be viewed as a function that lives in the space of infinitely many coordinates
$x_A$, $A=1,2,3,4,\ldots, \infty$. In particular, the 1-particle wave function
$\Psi_1(x_1)$ should be viewed as a function $\Psi_1(x_1,x_2, \ldots)$
with the property $\partial^{\mu}_A \Psi_1 =0$ for $A=2,3,\ldots, \infty$.
It means that any wave function in QFT describes an infinite number of particles,
even if most of them have zero 4-velocity. As we have already explained, particles
with zero 4-velocity are dots in spacetime. The initial spacetime position of any particle
may take any value, with the probability proportional to
$|\Psi(x_1,x_2, \ldots)|^2$. Thus, the Bohmian particle trajectories associated
with the 1-particle wave function $\Psi_1(x_1,x_2, \ldots)$ take a form as in
Fig.~\ref{fig2}. In addition to one continuous particle trajectory, there is also an infinite
number of ``vacuum'' particles which live for an infinitesimally short time.
It is intuitively clear that a particle that lives for an infinitesimally short time
is not observable. However, we have an infinite number of such particles,
so could their overall effect be comparable, or even overwhelming,
with respect to a finite number of ``real'' particles that live for a finite
time? There is a simple intuitive argument that such an effect should
not be expected. The number of ``vacuum'' particles is equal
to the cardinal number of the set on natural numbers, denoted by $\aleph_0$.
This set has a measure zero with respect to a continuous trajectory,
because a continuous trajectory corresponds to a set of real numbers, the cardinal number of
which is $2^{\aleph_0}$ \cite{penrose}. Intuitively,
the number of points on a single continuous trajectory is infinitely times larger
than the number of points describing the ``vacuum'' particles.
Consequently, the contribution of the ``vacuum'' particles to any
measurable effect is expected to be negligible.
The purpose of the rest of the paper is to further elaborate
the ideas presented in this section
and to put them into a more precise framework.
\section{Interpretation-independent aspects of QFT}
\label{SEC3}
\subsection{Measurement in QFT as entanglement with the environment}
\label{SEC3.1}
Let $\{|i\rangle \}$ be some orthonormal basis of 1-particle states.
A general normalized 1-particle state is
\begin{equation}\label{e3.1}
|\Psi_1\rangle = \sum_i c_i |i\rangle ,
\end{equation}
where the normalization condition implies $ \sum_i |c_i|^2=1$.
From the basis $\{|i\rangle \}$ one can construct the $n$-particle basis
$\{|i_1,\ldots, i_n\rangle \}$, where
\begin{equation}\label{e3.2}
|i_1,\ldots, i_n\rangle = S_{\{i_1,\ldots, i_n\}}|i_1\rangle \cdots |i_n\rangle .
\end{equation}
Here $S_{\{i_1,\ldots, i_n\}}$ denotes the symmetrization over all $\{i_1,\ldots, i_n\}$
for bosons, or antisymmetrization for fermions. The most general state in QFT
describing these particles can be written as
\begin{equation}\label{e3.3}
|\Psi\rangle = c_0 |0\rangle +
\sum_{n=1}^{\infty} \sum_{i_1,\ldots, i_n} c_{n;i_1,\ldots,i_n}
|i_1,\ldots, i_n\rangle ,
\end{equation}
where the vacuum $|0\rangle$ is also introduced.
Now the normalization condition implies
$|c_0|^2+\sum_{n=1}^{\infty} \sum_{i_1,\ldots, i_n} |c_{n;i_1,\ldots,i_n}|^2 =1$.
Now let as assume that the number of particles is measured. It implies that the particles
become entangled with the environment, such that the total state describing both
the measured particles and the environment takes the form
\begin{eqnarray}\label{e3.4}
|\Psi\rangle_{\rm total} & = & c_0 |0\rangle |E_0\rangle
\\
& & + \sum_{n=1}^{\infty} \sum_{i_1,\ldots, i_n} c_{n;i_1,\ldots,i_n}
|i_1,\ldots, i_n\rangle |E_{n;i_1,\ldots,i_n} \rangle .
\nonumber
\end{eqnarray}
The environment states $|E_0\rangle$, $|E_{n;i_1,\ldots,i_n} \rangle$ are
macroscopically distinct. They describe what the observers really observe.
When an observer observes that the environment is in the state
$|E_0\rangle$ or $|E_{n;i_1,\ldots,i_n} \rangle$, then one says that the
original measured QFT state is in the state $|0\rangle$ or
$|i_1,\ldots, i_n\rangle$, respectively. In particular, this is how the number of
particles is measured in a state (\ref{e3.3}) with an uncertain number of particles.
The probability that the environment will be found in the state
$|E_0\rangle$ or $|E_{n;i_1,\ldots,i_n} \rangle$ is
equal to $|c_0|^2$ or $|c_{n;i_1,\ldots,i_n}|^2$, respectively.
Of course, (\ref{e3.3}) is not the only way the state $|\Psi\rangle$ can be expanded.
In general, it can be expanded as
\begin{equation}\label{e3.5}
|\Psi\rangle = \sum_{\xi} c_{\xi} |\xi\rangle ,
\end{equation}
where $|\xi\rangle$ are some normalized (not necessarily orthogonal)
states that do not need to have a definite number of particles.
A particularly important example are coherent states (see, e.g., \cite{bal}),
which minimize the products of uncertainties of fields and their
canonical momenta. Each coherent state is a superposition
of states with all possible numbers of particles, including zero.
The coherent states are overcomplete and not orthogonal. Yet, the
expansion (\ref{e3.5}) may be an expansion in terms of coherent states $|\xi\rangle$
as well.
Furthermore, the entanglement with the environment does not
necessarily need to take the form (\ref{e3.4}). Instead, it may take a
more general form
\begin{equation}\label{e3.6}
|\Psi\rangle_{\rm total} = \sum_{\xi} c_{\xi} |\xi\rangle |E_\xi\rangle ,
\end{equation}
where $|E_\xi\rangle$ are macroscopically distinct.
In principle, the interaction with the environment may create the entanglement
(\ref{e3.6}) with respect to any set of states $\{ |\xi\rangle \}$.
In practice, however, some types of expansions are preferred.
This fact can be explained by the theory of decoherence \cite{schloss}, which
explains why states of the form of (\ref{e3.6}) are stable only for some particular
sets $\{ |\xi\rangle \}$. In fact, depending on details of the interactions
with the environment, in most real situations the entanglement takes either the form
(\ref{e3.4}) or the form (\ref{e3.6}) with coherent states $|\xi\rangle$.
Since coherent states minimize the uncertainties of fields and their canonical momenta,
they behave very much like classical fields. This explains why experiments in quantum
optics can often be better described in terms of fields rather than particles
(see, e.g., \cite{bal}). In fact, the theory of decoherence can explain
under what conditions the coherent-state basis becomes preferred over
basis with definite numbers of particles \cite{zeh,zurek}.
There is one additional physically interesting class of sets $\{ |\xi\rangle \}$.
They may be eigenstates of the particle number operator defined with respect to
{\em Bogoliubov transformed} (see, e.g., \cite{bd}) creation and destruction operators.
Thus, even the vacuum may have a nontrivial expansion of the form
\begin{equation}\label{u3.3}
|0\rangle = c_0 |0'\rangle +
\sum_{n=1}^{\infty} \sum_{i'_1,\ldots, i'_n} c_{n;i'_1,\ldots,i'_n}
|i'_1,\ldots, i'_n\rangle ,
\end{equation}
where the prime denotes the $n$-particle states with respect to the Bogoliubov
transformed number operator. In fact, whenever the two definitions of particles
are related by a Bogoliubov transformation, the vacuum for one definition of particles is a
squeezed state when expressed in terms of particles of the other definition of particles
\cite{grish}. Thus, if the entanglement with the environment takes the form
\begin{eqnarray}\label{u3.4}
|\Psi\rangle_{\rm total} & = & c_0 |0'\rangle |E_0\rangle
\\
& & + \sum_{n=1}^{\infty} \sum_{i'_1,\ldots, i'_n} c_{n;i'_1,\ldots,i'_n}
|i'_1,\ldots, i'_n\rangle |E_{n;i'_1,\ldots,i'_n} \rangle ,
\nonumber
\end{eqnarray}
then the vacuum (\ref{u3.3}) may appear as a state with many particles.
Indeed, this is expected to occur when the particle detector is accelerated
or when a gravitational field is present \cite{bd}. The theory of decoherence
can explain why the interaction with the environment leads to an entanglement of the
form of (\ref{u3.4}) \cite{decohunruh1,decohunruh2,decohunruh3}.
Thus, decoherence induced by interaction with the environment
can explain why do we observe either a definite number
of particles or coherent states that behave very much like classical fields.
However, decoherence alone cannot explain why do we observe
some particular state of definite number of particles and not some other,
or why do we observe some particular coherent state and not some other.
\subsection{Free scalar QFT in the particle-position picture}
\label{SEC3.2}
Consider a free scalar hermitian field operator $\hat{\phi}(x)$ satisfying the
Klein-Gordon equation
\begin{equation}\label{e3.7}
\partial^{\mu}\partial_{\mu}\hat{\phi}(x)+ m^2\hat{\phi}(x) =0.
\end{equation}
The field can be decomposed as
\begin{equation}\label{e3.8}
\hat{\phi}(x)=\hat{\psi}(x)+\hat{\psi}^{\dagger}(x) ,
\end{equation}
where $\hat{\psi}$ and $\hat{\psi}^{\dagger}$ can be expanded as
\begin{eqnarray}\label{e3.9}
& \hat{\psi}(x)=\displaystyle\int d^3k \, f({\bf k}) \, \hat{a}({\bf k})
e^{-i[\omega({\bf k})x^0-{\bf k}{\bf x}]} , &
\nonumber \\
& \hat{\psi}^{\dagger}(x)=\displaystyle\int d^3k \, f({\bf k}) \, \hat{a}^{\dagger}({\bf k})
e^{i[\omega({\bf k})x^0-{\bf k}{\bf x}]} . &
\end{eqnarray}
Here
\begin{equation}\label{e3.9'}
\omega({\bf k})=\sqrt{{\bf k}^2+m^2}
\end{equation}
is the $k^0$ component
of the 4-vector $k=\{ k^{\mu} \}$, and
$\hat{a}^{\dagger}({\bf k})$ and $\hat{a}({\bf k})$
are the usual creation and destruction operators, respectively. The function
$f({\bf k})$ is a real positive function which we do not specify explicitly
because several different choices appear in the literature, corresponding to
several different choices of normalization. All subsequent equations will
be written in forms that do not depend on this choice.
We define the operator
\begin{equation}\label{e3.10}
\hat{\psi}_n(x_{n,1}, \ldots , x_{n,n}) = d_n
S_{ \{x_{n,1}, \ldots , x_{n,n} \} }
\hat{\psi}(x_{n,1}) \cdots \hat{\psi}(x_{n,n}) .
\end{equation}
The symbol $ S_{ \{x_{n,1}, \ldots , x_{n,n} \} }$ denotes the symmetrization,
reminding us that the expression
is symmetric under the exchange of coordinates $\{x_{n,1}, \ldots , x_{n,n} \}$.
(Note, however, that the product of operators on the right hand side of (\ref{e3.10})
is in fact automatically symmetric because
the operators $\hat{\psi}(x)$ commute, i.e., $[\hat{\psi}(x),\hat{\psi}(x')]=0$.)
The parameter $d_n$ is a normalization constant determined by the normalization condition
that will be specified below.
The operator (\ref{e3.10}) allows us to define $n$-particle states
in the basis of particle spacetime positions, as
\begin{equation}\label{e3.11}
|x_{n,1}, \ldots , x_{n,n}\rangle = \hat{\psi}^{\dagger}_n(x_{n,1}, \ldots , x_{n,n})
|0\rangle .
\end{equation}
All states of the form (\ref{e3.11}), together with the vacuum $|0\rangle$, form
a complete and orthogonal basis in the Hilbert space of physical states.
If $|\Psi_n\rangle$ is an arbitrary (but normalized) $n$-particle state,
then this state can be represented by the $n$-particle wave function
\begin{equation}\label{e3.12}
\psi_n(x_{n,1}, \ldots , x_{n,n}) = \langle x_{n,1}, \ldots , x_{n,n} |\Psi_n\rangle .
\end{equation}
We also have
\begin{equation}\label{e3.12.1}
\langle x_{n,1}, \ldots , x_{n,n} |\Psi_{n'}\rangle =0 \;\;{\rm for}\;\; n\neq n' .
\end{equation}
We choose the normalization constant $d_n$ in (\ref{e3.10}) such that the following
normalization condition is satisfied
\begin{equation}\label{e3.13}
\int d^4x_{n,1}\cdots \int d^4x_{n,n} \, | \psi_n(x_{n,1}, \ldots , x_{n,n})|^2 =1 .
\end{equation}
However, this implies that the wave functions
$\psi_n(x_{n,1}, \ldots , x_{n,n})$ and $\psi_{n'}(x_{n',1}, \ldots , x_{n',n'})$,
with different values of $n$ and $n'$,
are normalized in different spaces. On the other hand, we want these wave functions
to live in the same space, such that we can form superpositions of wave functions
describing different numbers of particles. To accomplish this, we define
\begin{equation}\label{e3.14}
\Psi_n(x_{n,1}, \ldots , x_{n,n})=\sqrt{ \frac{{\cal V}^{(n)}}{{\cal V}} } \,
\psi_n(x_{n,1}, \ldots , x_{n,n}) ,
\end{equation}
where
\begin{equation}\label{e3.15}
{\cal V}^{(n)}=\int d^4x_{n,1}\cdots \int d^4x_{n,n} ,
\end{equation}
\begin{equation}\label{e3.16}
{\cal V}=\prod_{n=1}^{\infty} {\cal V}^{(n)} ,
\end{equation}
are volumes of the corresponding configuration spaces.
In particular, the wave function of the vacuum is
\begin{equation}\label{e3.17}
\Psi_0=\frac{1}{\sqrt{{\cal V}}} .
\end{equation}
This provides that all wave functions are normalized in the same configuration space
as
\begin{equation}\label{e3.18}
\int {\cal D}\vec{x} \, | \Psi_n(x_{n,1}, \ldots , x_{n,n})|^2 =1 ,
\end{equation}
where we use the notation
\begin{equation}\label{e3.19}
\vec{x}=(x_{1,1},x_{2,1},x_{2,2},\ldots ),
\end{equation}
\begin{equation}\label{e3.20}
{\cal D}\vec{x} = \prod_{n=1}^{\infty} \, \prod_{a_{n}=1}^{n} d^4x_{n,a_{n}} .
\end{equation}
Note that the physical Hilbert space does not contain non-symmetrized
states, such as a 3-particle state $|x_{1,1}\rangle |x_{2,1},x_{2,2}\rangle$.
It also does not contain states that do not satisfy (\ref{e3.9'}).
Nevertheless, the notation can be further simplified by introducing an extended
kinematic Hilbert space that contains such unphysical states as well.
Every physical state can be viewed as a state in such an extended Hilbert space,
although most of the states in the extended Hilbert space are not physical.
In this extended space it is convenient to denote the pair of
labels $(n,a_n)$ by a single label $A$. Hence, (\ref{e3.19}) and (\ref{e3.20})
are now written as
\begin{equation}\label{e3.21}
\vec{x}=(x_1,x_2,x_3,\ldots ),
\end{equation}
\begin{equation}\label{e3.22}
{\cal D}\vec{x} = \prod_{A=1}^{\infty} d^4x_A .
\end{equation}
Similarly, (\ref{e3.16}) with (\ref{e3.15}) is now written as
\begin{equation}\label{e3.23}
{\cal V}=\int \prod_{A=1}^{\infty} d^4x_A .
\end{equation}
The particle-position basis of this extended space is denoted by $|\vec{x})$ (which should be
distinguished from $|\vec{x}\rangle$ which would denote a symmetrized state
of an infinite number of physical particles).
Such a basis allows us to write
the physical wave function (\ref{e3.14}) as a wave function
on the extended space
\begin{equation}\label{e3.25}
\Psi_n(\vec{x})=(\vec{x}|\Psi_n\rangle .
\end{equation}
Now (\ref{e3.18}) takes a simpler form
\begin{equation}\label{e3.18'}
\int {\cal D}\vec{x} \, | \Psi_n(\vec{x})|^2 =1 .
\end{equation}
The normalization (\ref{e3.18'}) corresponds to the normalization
in which the unit operator on the extended space is
\begin{equation}\label{e3.24}
1=\int {\cal D}\vec{x} \, |\vec{x}) (\vec{x}| ,
\end{equation}
while the scalar product is
\begin{equation}\label{e3.24'}
(\vec{x}|\vec{x}')=\delta(\vec{x}-\vec{x}') ,
\end{equation}
with $\delta(\vec{x}-\vec{x}') \equiv \prod_{A=1}^{\infty}\delta^4(x_A-x'_A)$.
A general physical state can be written as
\begin{equation}\label{e3.26}
\Psi(\vec{x})=(\vec{x}|\Psi\rangle =
\sum_{n=0}^{\infty}c_n \Psi_n(\vec{x}) .
\end{equation}
It is also convenient to write this as
\begin{equation}\label{e3.27}
\Psi(\vec{x})=\sum_{n=0}^{\infty} \tilde{\Psi}_n(\vec{x}) ,
\end{equation}
where the tilde denotes a wave function that is not necessarily normalized.
The total wave function is normalized, in the sense that
\begin{equation}\label{e3.26.1}
\int {\cal D}\vec{x} \, |\Psi(\vec{x})|^2=1 ,
\end{equation}
implying
\begin{equation}\label{e3.26.2}
\sum_{n=0}^{\infty}|c_n|^2=1 .
\end{equation}
Next, we introduce the operator
\begin{equation}\label{e3.28}
\Box = \sum_{A=1}^{\infty} \partial_A^{\mu} \partial_{A\mu} .
\end{equation}
From the equations above (see, in particular, (\ref{e3.7})-(\ref{e3.12})),
it is easy to show that $\Psi_n(\vec{x})$ satisfies
\begin{equation}\label{e3.29}
\Box\Psi_n(\vec{x}) +nm^2 \Psi_n(\vec{x}) =0 .
\end{equation}
Introducing a hermitian number-operator $\hat{N}$ with the property
\begin{equation}\label{e3.30}
\hat{N}\Psi_n(\vec{x}) = n \Psi_n(\vec{x}) ,
\end{equation}
one finds that a general physical state (\ref{e3.26}) satisfies the generalized
Klein-Gordon equation
\begin{equation}\label{e3.31}
\Box\Psi(\vec{x})+m^2\hat{N}\Psi(\vec{x})=0.
\end{equation}
We also introduce the generalized Klein-Gordon current
\begin{equation}\label{e3.32}
J^{\mu}_A(\vec{x})=
\frac{i}{2} \Psi^*(\vec{x})\!\stackrel{\leftrightarrow\;}{\partial^{\mu}_A}\! \Psi (\vec{x}) .
\end{equation}
From (\ref{e3.31}) one finds that, in general, this current is not conserved
\begin{equation}\label{e3.33}
\sum_{A=1}^{\infty}\partial_{A\mu}J^{\mu}_A(\vec{x}) = J(\vec{x}) ,
\end{equation}
where
\begin{equation}\label{e3.34}
J(\vec{x})=-\frac{i}{2}m^2
\Psi^*(\vec{x})\!\stackrel{\;\leftrightarrow}{\hat{N}}\! \Psi (\vec{x}) ,
\end{equation}
and $\Psi' \!\stackrel{\;\leftrightarrow}{\hat{N}}\! \Psi \equiv
\Psi' (\hat{N}\Psi) - ( \hat{N} \Psi') \Psi$.
From (\ref{e3.34}) we see that the current is conserved in two special cases:
(i) when $\Psi=\Psi_n$ (a state with a definite number of physical
particles), or (ii) when $m^2=0$ (any physical state of massless particles).
In the extended Hilbert space it is also useful to introduce the momentum picture
through the Fourier transforms. We define
\begin{equation}\label{e3.35}
\Psi_{\vec{k}}(\vec{x})= \sqrt{ \frac{(2\pi)^{4\aleph_0}}{{\cal V}} }
(\vec{x}|\vec{k})
=\frac{e^{-i\vec{k}\vec{x}}}{\sqrt{{\cal V}}} ,
\end{equation}
where $\vec{k}\vec{x}\equiv \sum_{A=1}^{\infty} k_{A\mu}x_A^{\mu}$
and $\aleph_0=\infty$ corresponds to the number of different values of the
label $A$.
In the basis of momentum eigenstates $|\vec{k})$ we have
\begin{equation}\label{e3.36}
1= \int {\cal D}\vec{k}\, |\vec{k}) (\vec{k}| ,
\end{equation}
\begin{equation}\label{e3.37}
(\vec{k}|\vec{k}')=\delta(\vec{k}-\vec{k}') .
\end{equation}
It is easy to check that the normalizations as above make the Fourier transform
\begin{equation}\label{e3.38}
\Psi(\vec{k})=(\vec{k}|\Psi\rangle=\int{\cal D}\vec{x}\,
(\vec{k}|\vec{x})(\vec{x}|\Psi\rangle
\end{equation}
and its inverse
\begin{equation}\label{e3.39}
\Psi(\vec{x})=(\vec{x}|\Psi\rangle=\int {\cal D}\vec{k}\,
(\vec{x}|\vec{k})(\vec{k}|\Psi\rangle
\end{equation}
consistent. We can also introduce the momentum operator
\begin{equation}\label{e3.40}
\hat{p}_{A\mu}=i\partial_{A\mu}.
\end{equation}
The wave function (\ref{e3.35}) is the momentum eigenstate
\begin{equation}\label{e3.41}
\hat{p}_{A\mu}\Psi_{\vec{k}}(\vec{x}) =
k_{A\mu}\Psi_{\vec{k}}(\vec{x}) .
\end{equation}
In particular, the wave function of the physical vacuum is given by (\ref{e3.17}), so
\begin{equation}\label{e3.42}
\hat{p}_{A\mu}\Psi_0(\vec{x}) = 0 .
\end{equation}
We see that (\ref{e3.17}) can also be written as
\begin{equation}\label{e3.35'}
\Psi_0(\vec{x})=\frac{e^{-i\vec{0}\vec{x}}}
{\sqrt{{\cal V}}} ,
\end{equation}
showing that the physical vacuum can also be represented as
\begin{equation}\label{e3.43}
|0\rangle = |\vec{k}=\vec{0}) .
\end{equation}
Intuitively, it means that the vacuum can be thought of
as a state with an infinite number of particles,
all of which have vanishing 4-momentum.
Similarly, an $n$-particle state can be thought of as a state with an infinite number of particles,
where only $n$ of them have a non-vanishing 4-momentum.
Finally, let us rewrite some of the
main results of this (somewhat lengthy) subsection in a form
that will be suitable for a generalization in the next subsection.
A general physical state can be written in the form
\begin{equation}\label{s1}
|\Psi\rangle = \sum_{n=0}^{\infty} c_n |\Psi_n\rangle =
\sum_{n=0}^{\infty} |\tilde{\Psi}_n\rangle .
\end{equation}
The corresponding unnormalized $n$-particle wave functions are
\begin{equation}\label{s2}
\tilde{\psi}_n(x_{n,1},\ldots,x_{n,n}) =
\langle 0|\hat{\psi}_n(x_{n,1},\ldots,x_{n,n})|\Psi\rangle .
\end{equation}
There is a well-defined transformation
\begin{equation}\label{s3}
\tilde{\psi}_n(x_{n,1},\ldots,x_{n,n}) \rightarrow \tilde{\Psi}_n(\vec{x})
\end{equation}
from the physical Hilbert space to the extended Hilbert space, so that
the general state (\ref{s1}) can be represented by a single wave function
\begin{equation}\label{s4}
\Psi(\vec{x})=\sum_{n=0}^{\infty} c_n \Psi_n(\vec{x})
=\sum_{n=0}^{\infty} \tilde{\Psi}_n(\vec{x}).
\end{equation}
\subsection{Generalization to the interacting QFT}
\label{SEC3.3}
In this subsection we discuss the generalization of the results of the preceding subsection
to the case in which the field operator $\hat{\phi}$ does not satisfy the free
Klein-Gordon equation (\ref{e3.7}). For example,
if the classical action is
\begin{equation}\label{e3.44}
S=\int d^4x \left[ \frac{1}{2}(\partial^{\mu}\phi) (\partial_{\mu}\phi)
-\frac{m^2}{2}\phi^2 - \frac{\lambda}{4}\phi^4 \right] ,
\end{equation}
then (\ref{e3.7}) generalizes to
\begin{equation}\label{e3.45}
\partial^{\mu}\partial_{\mu}\hat{\phi}_H(x)+
m^2\hat{\phi}_H(x) +\lambda \hat{\phi}_H^3(x)=0,
\end{equation}
where $\hat{\phi}_H(x)$ is the field operator in the Heisenberg picture.
(From this point of view, the operator $\hat{\phi}(x)$
defined by (\ref{e3.8}) and (\ref{e3.9}) and satisfying the free Klein-Gordon
equation (\ref{e3.7}) is the field operator in the interaction (Dirac) picture.)
Thus, instead of (\ref{s2}) now we have
\begin{equation}\label{s2'}
\tilde{\psi}_n(x_{n,1},\ldots,x_{n,n}) =
\langle 0|\hat{\psi}_{nH}(x_{n,1},\ldots,x_{n,n})|\Psi\rangle ,
\end{equation}
where $|\Psi\rangle$ and $|0\rangle$ are states in the Heisenberg picture.
Assuming that (\ref{s2'}) has been calculated (we shall see below how
in practice it can be done), the rest of the job is straightforward.
One needs to make the transformation (\ref{s3}) in the same way
as in the free case, which leads to an interacting variant of (\ref{s4})
\begin{equation}\label{s4'}
\Psi(\vec{x})=\sum_{n=0}^{\infty} \tilde{\Psi}_n(\vec{x}) .
\end{equation}
The wave function (\ref{s4'}) encodes the complete information about the
properties of the interacting system.
Now let us see how (\ref{s2'}) can be calculated in practice. Any operator
$\hat{O}_H(t)$ in the Heisenberg picture depending on a single time-variable $t$
can be written in terms of operators in the interaction picture as
\begin{equation}\label{e3.46}
\hat{O}_H(t)=\hat{U}^{\dagger}(t)\hat{O}(t)\hat{U}(t) ,
\end{equation}
where
\begin{equation}\label{e3.47}
\hat{U}(t)=Te^{-i\int_{t_0}^t dt' \hat{H}_{\rm int}(t')} ,
\end{equation}
$t_0$ is some appropriately chosen ``initial'' time, $T$ denotes the time ordering,
and $\hat{H}_{\rm int}$
is the interaction part of the Hamiltonian expressed as a functional of field operators
in the interaction picture (see, e.g., \cite{chengli}).
For example, for the action (\ref{e3.44}) we have
\begin{equation}\label{e3.48}
\hat{H}_{\rm int}(t)=\frac{\lambda}{4} \int d^3x \, :\!\hat{\phi}^4({\bf x},t)\!: ,
\end{equation}
where $:\;:$ denotes the normal ordering.
The relation (\ref{e3.46}) can also be inverted, leading to
\begin{equation}\label{e3.49}
\hat{O}(t)=\hat{U}(t)\hat{O}_H(t)\hat{U}^{\dagger}(t) .
\end{equation}
Thus, the relation (\ref{e3.10}), which is now valid in the interaction picture,
allows us to write an analogous relation in the Heisenberg picture
\begin{eqnarray}\label{e3.10'}
\hat{\psi}_{nH}(x_{n,1}, \ldots , x_{n,n}) & = & d_n
S_{ \{x_{n,1}, \ldots , x_{n,n} \} }
\nonumber \\
& & \hat{\psi}_H(x_{n,1}) \cdots \hat{\psi}_H(x_{n,n}) ,
\end{eqnarray}
where
\begin{equation}\label{e3.50}
\hat{\psi}_H(x_{n,a_n})=\hat{U}^{\dagger}(x^0_{n,a_n})
\hat{\psi}(x_{n,a_n})\hat{U}(x^0_{n,a_n}) .
\end{equation}
By expanding (\ref{e3.47}) in powers of $\int_{t_0}^t dt' \hat{H}_{\rm int}$,
this allows us to calculate (\ref{e3.10'}) and (\ref{s2'}) perturbatively.
In (\ref{s2'}), the states in the Heisenberg picture $|\Psi\rangle$ and $|0\rangle$
are identified with the states in the interaction picture at the initial time
$|\Psi(t_0)\rangle$ and $|0(t_0)\rangle$, respectively.
To demonstrate that such a procedure leads to a physically sensible result,
let us see how it works in the special (and more familiar) case of the equal-time
wave function. It is given by $\tilde{\psi}_n(x_{n,1},\ldots,x_{n,n})$
calculated at $x^0_{n,1}=\cdots=x^0_{n,n}\equiv t$.
Thus, (\ref{s2'}) reduces to
\begin{eqnarray}\label{e3.51}
& \tilde{\psi}_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t) = d_n
\langle 0(t_0)| \hat{U}^\dagger(t) \hat{\psi}({\bf x}_{n,1},t) \hat{U}(t) &
\nonumber \\
& \cdots
\hat{U}^\dagger(t) \hat{\psi}({\bf x}_{n,n},t) \hat{U}(t) |\Psi(t_0)\rangle . &
\end{eqnarray}
Using $\hat{U}(t)\hat{U}^\dagger(t)=1$ and
\begin{equation}\label{e3.52}
\hat{U}(t) |\Psi(t_0)\rangle = |\Psi(t)\rangle , \;\;\;\;
\hat{U}(t) |0(t_0)\rangle = |0(t)\rangle ,
\end{equation}
the expression further simplifies
\begin{eqnarray}\label{e3.53}
& \tilde{\psi}_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t) = &
\nonumber \\
& d_n
\langle 0(t)| \hat{\psi}({\bf x}_{n,1},t) \cdots
\hat{\psi}({\bf x}_{n,n},t) |\Psi(t)\rangle . &
\end{eqnarray}
In practical applications of QFT in particle physics, one usually calculates the
$S$-matrix, corresponding to the limit $t_0\rightarrow -\infty$,
$t\rightarrow\infty$. For Hamiltonians that conserve energy (such as (\ref{e3.48}))
this limit provides the stability of the vacuum, i.e., obeys
\begin{equation}\label{e3.54}
\lim_{t_0\rightarrow -\infty, \; t\rightarrow\infty} \hat{U}(t) |0(t_0)\rangle =
e^{-i\varphi_0} |0(t_0)\rangle ,
\end{equation}
where $\varphi_0$ is some physically irrelevant phase \cite{bd2}.
Essentially, this is because the integrals of the type
$\int_{-\infty}^{\infty} dt' \cdots$ produce $\delta$-functions
that correspond to energy conservation, so the vacuum remains stable
because particle creation from the vacuum would violate energy conservation.
Thus we have
\begin{equation}\label{e3.55}
|0(\infty)\rangle =e^{-i\varphi_0}|0(-\infty)\rangle \equiv e^{-i\varphi_0}|0\rangle .
\end{equation}
The state
\begin{equation}\label{e3.55.1}
|\Psi(\infty)\rangle=\hat{U}(\infty)|\Psi(-\infty)\rangle
\end{equation}
is not trivial, but whatever it is, it has some expansion of the form
\begin{equation}\label{e3.56}
|\Psi(\infty)\rangle = \sum_{n=0}^{\infty} c_{n}(\infty)|\Psi_{n}\rangle ,
\end{equation}
where $c_{n}(\infty)$ are some coefficients.
Plugging (\ref{e3.55}) and (\ref{e3.56})
into (\ref{e3.53}) and recalling (\ref{e3.10})-(\ref{e3.12.1}),
we finally obtain
\begin{equation}\label{e3.57}
\tilde{\psi}_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};\infty)=
e^{i\varphi_0} c_n(\infty) \psi_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};\infty) .
\end{equation}
This demonstrates the consistency of (\ref{s2'}), because (\ref{e3.55.1})
should be recognized as the standard description of evolution from
$t_0\rightarrow -\infty$ to $t\rightarrow \infty$ (see, e.g., \cite{chengli,bd2}),
showing that the coefficients $c_n(\infty)$ are the same as those described
by standard $S$-matrix theory in QFT.
In other words, (\ref{s2'}) is a natural many-time generalization of the concept of
single-time evolution in interacting QFT.
\subsection{Generalization to other types of particles}
\label{SEC3.4}
In Secs.~\ref{SEC3.2} and \ref{SEC3.3} we have discussed in detail
scalar hermitian fields, corresponding to spinless uncharged particles.
In this subsection we briefly discuss how these results can be
generalized to any type of fields and the corresponding particles.
In general, fields $\phi$ carry some additional labels which we
collectively denote by $l$, so we deal with fields $\phi_l$.
For example, spin-1 field carries a
vector index, fermionic spin-$\frac{1}{2}$ field carries a spinor index,
non-Abelian gauge fields carry internal indices of the gauge group, etc.
Thus Eq.~(\ref{e3.10}) generalizes to
\begin{eqnarray}\label{e3.10gen}
& \hat{\psi}_{n,L_n}(x_{n,1}, \ldots , x_{n,n}) = &
\nonumber \\
& d_n
S_{ \{x_{n,1}, \ldots , x_{n,n} \} }
\hat{\psi}_{l_{n,1}}(x_{n,1}) \cdots \hat{\psi}_{l_{n,n}} (x_{n,n}) , &
\end{eqnarray}
where $L_n$ is a collective label $L_n=(l_{n,1}, \ldots , l_{n,n} )$.
The symbol $S_{ \{x_{n,1}, \ldots , x_{n,n} \} }$ denotes symmetrization (antisymmetrization) over bosonic (fermionic) fields describing the same type of particles.
Hence, it is straightforward to make the appropriate generalizations of all results
of Secs.~\ref{SEC3.2} and \ref{SEC3.3}. For example, (\ref{e3.27})
generalizes to
\begin{equation}\label{e3.27gen}
\Psi_{\vec{L}} (\vec{x}) = \sum_{n=0}^{\infty} \sum_{L_n} \tilde{\Psi}_{n,L_n}
(\vec{x}) ,
\end{equation}
with self-explaining notation.
To further simplify the notation, we introduce the column
$\Psi\equiv \{\Psi_{\vec{L}} \}$ and the row
$\Psi^{\dagger}\equiv \{\Psi^*_{\vec{L}} \}$.
With this notation, the appropriate generalization of (\ref{e3.26.1}) can be written as
\begin{equation}\label{e3.26.1gen}
\int {\cal D}\vec{x} \, \sum_{{\vec{L}}} \Psi^*_{\vec{L}}(\vec{x})
\Psi_{\vec{L}}(\vec{x}) \equiv \int {\cal D}\vec{x} \,
\Psi^{\dagger}(\vec{x}) \Psi(\vec{x})
=1 .
\end{equation}
For the case of states that contain fermionic particles, Eq.~(\ref{e3.26.1gen})
requires further discussion. As a simple example, consider a
1-particle state describing one electron.
In this case, (\ref{e3.26.1gen}) can be reduced to
\begin{equation}\label{e3.58}
\int d^4x \, \psi^{\dagger}(x)\psi(x) =1 ,
\end{equation}
where $\psi$ is a Dirac spinor. In this expression, the quantity $\psi^{\dagger}\psi$
must transform as a Lorentz scalar. At first sight, it may seem to be in contradiction
with the well-known fact that $\psi^{\dagger}\psi=\bar{\psi}\gamma^0\psi$
transforms as a time-component of a Lorentz vector. However, there is no
true contradiction. Let us explain.
The standard derivation that $\bar{\psi}\gamma^{\mu}\psi$ transforms as a vector \cite{bd1}
starts from the assumption that the matrices $\gamma^{\mu}$ do not transform
under Lorentz transformations, despite of carrying the index ${\mu}$.
However, such an assumption
is not necessary. Moreover, in curved spacetime such an assumption is inconsistent \cite{bd}.
In fact, one is allowed to define the transformations of $\psi$ and $\gamma^{\mu}$
in an arbitrary way, as long as such transformations do not affect the transformations
of measurable quantities, or quantities like $\bar{\psi}\gamma^{\mu}\psi$
that are closely related to measurable ones.
Thus, it is much more natural to deal with a differently defined transformations
of $\gamma^{\mu}$ and $\psi$, such that $\gamma^{\mu}$ transforms as a vector
and $\psi$ transforms as a scalar under Lorentz transformations of spacetime coordinates \cite{bd}.
The spinor indices of $\gamma^{\mu}$ and $\psi$ are then reinterpreted as indices
in an internal space. With such redefined transformations, (\ref {e3.58}) is fully consistent.
The details of our transformation conventions are presented in Appendix \ref{APPA}.
\section{The physical interpretation}
\label{SEC4}
\subsection{Probabilistic interpretation}
In this subsection we adopt and further develop the probabilistic interpretation
introduced in \cite{nikprobrel09} (and partially inspired by earlier results
\cite{stuc1,stuc2,horw,kypr,fanchi}).
The quantity
\begin{equation}\label{e4.1}
{\cal D}P=\Psi^{\dagger}(\vec{x}) \Psi(\vec{x}) \, {\cal D}\vec{x}
\end{equation}
is naturally interpreted as the probability of finding the system in the
(infinitesimal) configuration-space volume ${\cal D}\vec{x}$ around a
point $\vec{x}$ in the configuration space. Indeed, such an interpretation
is consistent with our normalization conditions such as
(\ref{e3.26.1}) and (\ref{e3.26.1gen}). In more physical terms, it gives the
joint probability that the particle $1$ is found at the spacetime position
$x_1$, particle $2$ at the spacetime position $x_2$, etc.
Similarly, the Fourier-transformed
wave function $\Psi(\vec{k})$ defines the probability
$\Psi^{\dagger}(\vec{k}) \Psi(\vec{k}) {\cal D}\vec{k}$, which is the
joint probability that the particle $1$ has the 4-momentum
$k_1$, particle $2$ the 4-momentum $k_2$, etc.
As a special case, consider an $n$-particle state
$\Psi(\vec{x})=\Psi_n(\vec{x})$. It really depends only on
$n$ spacetime positions $x_{n,1},\ldots x_{n,n}$. With respect to
all other positions $x_B$, $\Psi$ is a constant. Thus, the probability
of various positions $x_B$ does not depend on $x_B$; such a particle can be found
anywhere and anytime with equal probabilities. There is an infinite number of such
particles. Nevertheless, the Fourier transform of such a wave function reveals
that the 4-momentum $k_B$ of these particles is necessarily zero; they have neither
3-momentum nor energy. For that reason, such particles can be thought of as ``vacuum'' particles. In this picture, an $n$-particle state $\Psi_n$ is thought of as a state describing
$n$ ``real'' particles and an infinite number of ``vacuum'' particles.
To avoid a possible confusion with the usual notions of vacuum
and real particles in QFT, in the rest of the paper
we refer to ``vacuum'' particles as {\it dead} particles
and ``real'' particles as {\it live} particles. Or let us be
slightly more precise: We say that the
particle $A$ is dead if the wave function in the momentum space
$\Psi(\vec{k})$ vanishes for all values of $k_A$ except $k_A=0$.
Similarly, we say that the particle $A$ is live if it is not dead.
The properties of live particles associated with the state $\Psi_n(\vec{x})$ can also be
represented by the wave function $\psi_n(x_{n,1},\ldots,x_{n,n} )$. By averaging over
physically uninteresting dead particles, (\ref{e4.1}) reduces to
\begin{eqnarray}\label{e4.2}
dP & = & \psi_n^{\dagger}(x_{n,1},\ldots,x_{n,n}) \psi_n(x_{n,1},\ldots,x_{n,n})
\nonumber \\
& & \times \, d^4x_{n,1}\cdots d^4x_{n,n},
\end{eqnarray}
which involves only live particles. This describes the probability when neither the space
positions of detected particles nor times of their detections are known.
To relate it with a more familiar probabilistic interpretation of QM, let us consider
the special case; let us assume that the first particle
is detected at time $x^0_{n,1}$, second particle at time $x^0_{n,2}$, etc.
In this case, the detection times are known, so (\ref{e4.2}) is no longer
the best description of our knowledge about the system. Instead, the relevant
probability derived from (\ref{e4.2})
is the {\em conditional} probability
\begin{eqnarray}\label{e4.3}
dP_{(3n)} & = &
\frac{ \psi_n^{\dagger}(x_{n,1},\ldots,x_{n,n}) \psi_n(x_{n,1},\ldots,x_{n,n}) }
{N_{x^0_{n,1},\cdots,x^0_{n,n}}}
\nonumber \\
& & \times \, d^3x_{n,1}\cdots d^3x_{n,n},
\end{eqnarray}
where
\begin{eqnarray}\label{e4.4}
N_{x^0_{n,1},\cdots,x^0_{n,n}} & = & \int \psi_n^{\dagger}(x_{n,1},\ldots,x_{n,n})
\psi_n(x_{n,1},\ldots,x_{n,n})
\nonumber \\
& & \times \, d^3x_{n,1}\cdots d^3x_{n,n}
\end{eqnarray}
is the appropriate normalization factor.
(For more details regarding the meaning and limitations of (\ref{e4.3}) in the
1-particle case see Appendix \ref{APPB}.)
The probability (\ref{e4.3}) is sometimes also postulated as a fundamental axiom
of many-time formulation of QM \cite{tomonaga}, but here
(\ref{e4.3}) is derived from a more fundamental and more general expression (\ref{e4.2}) (which, in turn, is derived from an even more general axiom (\ref{e4.1})).
An even more familiar expression is obtained by studding a special case
of (\ref{e4.3}) in which $x^0_{n,1}=\cdots =x^0_{n,n}\equiv t$, so that
(\ref{e4.3}) reduces to
\begin{eqnarray}\label{e4.5}
dP_{(3n)} & = &
\frac{ \psi_n^{\dagger}({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t)
\psi_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t) }
{N_{t}}
\nonumber \\
& & \times \, d^3x_{n,1}\cdots d^3x_{n,n},
\end{eqnarray}
where $N_{t}$ is given by (\ref{e4.4}) evaluated at $x^0_{n,1}=\cdots =x^0_{n,n}\equiv t$.
Now let us see how the wave functions representing the states in interacting QFT are interpreted
probabilistically. Consider the wave function $\tilde{\psi}_n(x_{n,1},\ldots,x_{n,n})$
given by (\ref{s2'}). For example, it may vanish for small values of
$x^0_{n,1}, \dots, x^0_{n,n}$, but it may not vanish for their large values. Physically, it means
that these particles cannot be detected in the far past (the probability is zero), but that
they can be detected in the far future. This is nothing but a probabilistic description of
the creation of $n$ particles that have not existed in the far past. Indeed,
the results obtained in Sec.~\ref{SEC3.3} (see, in particular, (\ref{e3.57}))
show that such probabilities are consistent with the probabilities of particle creation obtained
by the standard $S$-matrix methods in QFT.
Having developed the probabilistic interpretation, we can also calculate the average values
of various quantities. We are particularly interested in average values of the 4-momentum
$p^{\mu}_A$. In general, its average value is
\begin{equation}\label{e4.6}
\langle p^{\mu}_A \rangle= \int {\cal D}\vec{x} \, \Psi^{\dagger}(\vec{x}) \hat{p}^{\mu}_A
\Psi(\vec{x}) ,
\end{equation}
where $\hat{p}^{\mu}_A$ is given by (\ref{e3.40}).
If $\Psi(\vec{x})=\Psi_n(\vec{x})$, then (\ref{e4.6}) can be reduced to
\begin{eqnarray}\label{e4.7}
\langle p^{\mu}_{n,a_n} \rangle & = & \int d^4x_{n,1}\cdots d^4x_{n,n}
\\
& & \psi_n^{\dagger}(x_{n,1},\ldots,x_{n,n}) \hat{p}^{\mu}_{n,a_n}
\psi_n(x_{n,1},\ldots,x_{n,n}) .
\nonumber
\end{eqnarray}
Similarly, if the times of detections are known and are all equal to $t$, then the average
space-components of momenta are given by a more familiar expression
\begin{eqnarray}\label{e4.8}
\langle {\bf p}_{n,a_n} \rangle & = & N_t^{-1} \int d^3x_{n,1}\cdots d^3x_{n,n}
\\
& & \psi_n^{\dagger}({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t) \hat{{\bf p}}_{n,a_n}
\psi_n({\bf x}_{n,1},\ldots,{\bf x}_{n,n};t) .
\nonumber
\end{eqnarray}
Finally, note that (\ref{e4.6}) can also be written in an alternative form
\begin{equation}\label{e4.9}
\langle p^{\mu}_A \rangle= \int {\cal D}\vec{x}\, \rho(\vec{x}) U^{\mu}_A(\vec{x}) ,
\end{equation}
where
\begin{equation}\label{e4.10}
\rho(\vec{x})=\Psi^{\dagger}(\vec{x})\Psi(\vec{x})
\end{equation}
is the probability density and
\begin{equation}\label{e4.11}
U^{\mu}_A(\vec{x})=\frac{J^{\mu}_A(\vec{x})}{\Psi^{\dagger}(\vec{x})\Psi(\vec{x})} .
\end{equation}
Here $J^{\mu}_A$ is given by an obvious generalization of (\ref{e3.32})
\begin{equation}\label{e4.12}
J^{\mu}_A(\vec{x})=
\frac{i}{2} \Psi^{\dagger}(\vec{x})\!\stackrel{\leftrightarrow\;}{\partial^{\mu}_A}\!
\Psi (\vec{x}) .
\end{equation}
The expression (\ref{e4.9}) will play an important role in the next subsection.
\subsection{Particle-trajectory interpretation}
The idea of the particle-trajectory interpretation is that each particle has some trajectory
$X^{\mu}_A(s)$, where $s$ is an auxiliary scalar parameter that parameterizes the trajectories.
Such trajectories must be consistent with the probabilistic interpretation (\ref{e4.1}).
Thus, we need a velocity function $V^{\mu}_A(\vec{x})$, so that the trajectories
satisfy
\begin{equation}\label{e4.13}
\frac{dX^{\mu}_A(s)}{ds}=V^{\mu}_A(\vec{X}(s)) ,
\end{equation}
where the velocity function must be such that the following conservation equation is obeyed
\begin{equation}\label{e4.14}
\frac{\partial \rho(\vec{x})}{\partial s} +
\sum_{A=1}^{\infty}\partial_{A\mu}[\rho(\vec{x}) V^{\mu}_A(\vec{x}) ] =0.
\end{equation}
Namely, if a statistical ensemble of particle positions in spacetime has the distribution
(\ref{e4.10}) for some initial $s$, then (\ref{e4.13}) and (\ref{e4.14}) will
provide that this statistical ensemble will also have the distribution
(\ref{e4.10}) for {\em any} $s$, making the trajectories consistent with (\ref{e4.1}).
The first term in (\ref{e4.14}) trivially vanishes: $ \partial \rho(\vec{x})/\partial s =0$.
Thus, the condition (\ref{e4.14}) reduces to the requirement
\begin{equation}\label{e4.15}
\sum_{A=1}^{\infty}\partial_{A\mu}[\rho(\vec{x}) V^{\mu}_A(\vec{x}) ] =0.
\end{equation}
In addition, we require that the average velocity should be proportional to the average momentum
(\ref{e4.9}), i.e.,
\begin{equation}\label{e4.16}
\int {\cal D}\vec{x}\, \rho(\vec{x}) V^{\mu}_A(\vec{x})
= {\rm const} \times \int {\cal D}\vec{x}\, \rho(\vec{x}) U^{\mu}_A(\vec{x}) .
\end{equation}
In fact, the constant in (\ref{e4.16}) is physically irrelevant, because it can allways be
absorbed into a rescaling of the parameter $s$ in (\ref{e4.13}). The physical 3-velocity
$dX^{i}_A/dX^{0}_A$, $i=1,2,3$, is not affected by such a rescaling. Thus, in the rest
of the analysis we fix
\begin{equation}\label{e4.17}
{\rm const}=1 .
\end{equation}
As a first guess, Eq.~(\ref{e4.16}) with (\ref{e4.17}) suggests that one could take
$V^{\mu}_A=U^{\mu}_A$. However, it does not work in general. Namely,
from (\ref{e4.10}) and (\ref{e4.11}) we see that
$\rho U^{\mu}_A =J^{\mu}_A$, and we have seen in (\ref{e3.33})
that $ J^{\mu}_A$ does not need to be conserved. Instead, we have
\begin{equation}\label{e4.18}
\sum_{A=1}^{\infty}\partial_{A\mu}[\rho(\vec{x}) U^{\mu}_A(\vec{x})] = J(\vec{x}) ,
\end{equation}
where $J(\vec{x})$ is some function that can be calculated explicitly whenever
$\Psi(\vec{x})$ is known. So, how to find the appropriate function $V^{\mu}_A(\vec{x})$?
The problem of finding $V^{\mu}_A$ is solved in \cite{nikcr2} for a very general case
(see also \cite{struy}). Since the detailed derivation is presented in \cite{nikcr2},
here we only present the final results. Applying the general method developed in \cite{nikcr2},
one obtains
\begin{equation}\label{e4.19}
V^{\mu}_A(\vec{x})=U^{\mu}_A(\vec{x}) +
\rho^{-1}(\vec{x}) [e^{\mu}_A + E^{\mu}_A(\vec{x})] ,
\end{equation}
where
\begin{equation}\label{e4.20}
e^{\mu}_A=-{\cal V}^{-1} \int {\cal D}\vec{x}\, E^{\mu}_A(\vec{x}) ,
\end{equation}
\begin{equation}\label{e4.21}
E^{\mu}_A(\vec{x})=\partial^{\mu}_A
\int {\cal D}\vec{x}'\, G(\vec{x},\vec{x}') J(\vec{x}') ,
\end{equation}
\begin{equation}\label{e4.22}
G(\vec{x},\vec{x}') = \int \frac{{\cal D}\vec{k}}{(2\pi)^{4\aleph_0}}
\frac{e^{i\vec{k}(\vec{x}-\vec{x}')}}{\vec{k}^2} .
\end{equation}
Eqs.~(\ref{e4.21})-(\ref{e4.22}) provide that (\ref{e4.19}) obeys (\ref{e4.15}),
while (\ref{e4.20}) provides that (\ref{e4.19}) obeys (\ref{e4.16})-(\ref{e4.17}).
We note two important properties of (\ref{e4.19}). First, if $J=0$ in (\ref{e4.18}),
then $V^{\mu}_A=U^{\mu}_A$. In particular, since $J=0$ for free fields in states
with a definite number of particles (it can be derived for any type of particles
analogously to the derivation of (\ref{e3.34}) for spinless uncharged particles),
it follows that $V^{\mu}_A=U^{\mu}_A$ for such states.
Second, if $\Psi(\vec{x})$ does not depend
on some coordinate $x^{\mu}_B$, then both $U^{\mu}_B=0$ and
$V^{\mu}_B=0$. [To show that $V^{\mu}_B=0$, note first that $J(\vec{x})$ defined
by (\ref{e4.18}) does not depend on $x^{\mu}_B$ when $\Psi(\vec{x})$ does not depend
on $x^{\mu}_B$. Then the integration over $dx'^{\mu}_B$ in (\ref{e4.21}) produces
$\delta(k^{\mu}_B)$, which kills the dependence on $x^{\mu}_B$ carried by
(\ref{e4.22})].
This implies that dead particles have zero 4-velocity.
The results above show that the relativistic Bohmian trajectories are compatible with the
spacetime probabilistic interpretation (\ref{e4.1}). But what about the more
conventional space probabilistic interpretations (\ref{e4.3}) and (\ref{e4.5})?
{\it A priori}, these Bohmian trajectories are not compatible with (\ref{e4.3}) and (\ref{e4.5}).
Nevertheless, as discussed in more detail in Appendix \ref{APPB} for the 1-particle case,
the compatibility between measurable predictions of the Bohmian interpretation and that of the
``standard'' purely probabilistic interpretation restores when the appropriate theory
of quantum measurements is also taken into account.
Having established the general theory of particle trajectories by the results above,
now we can discuss particular consequences.
The trajectories are determined uniquely if the initial spacetime positions
$X^{\mu}_A(0)$ in (\ref{e4.13}), for all $\mu=0,1,2,3$, $A=1,\ldots, \infty$,
are specified. In particular,
since dead particles have zero 4-velocity, such particles do not really have trajectories
in spacetime. Instead, they are represented by dots in spacetime, as in
Fig.~\ref{fig2} (Sec.~\ref{SEC2}). The spacetime positions of these dots are
specified by their initial spacetime positions.
Since $\rho(\vec{x})$ describes probabilities for particle creation and destruction,
and since (\ref{e4.14}) provides that particle trajectories are such that
spacetime positions of particles
are distributed according to $\rho(\vec{x})$, it implies that particle trajectories are
also consistent with particle creation and destruction. In particular, the trajectories
in spacetime may have beginning and ending points, which correspond to points
at which their 4-velocities vanish (for an example, see Fig.~\ref{fig1}). For example,
the 4-velocity of the particle A vanishes if the conditional wave function
$\Psi(x_A,\vec{X}')$ does not depend on $x_A$ (where $\vec{X}'$ denotes
the actual spacetime positions of all particles except the particle $A$).
One very efficient mechanism of destroying particles is through the interaction
with the environment, such that the total quantum state takes the form
(\ref{e3.4}). The environment wave functions
$(\vec{x}|E_0\rangle$, $(\vec{x}|E_{n;i_1,\ldots,i_n} \rangle$ do not overlap,
so the particles describing the environment can be in the support of only one of these
environment wave functions. Consequently, the conditional wave function
is described by only one of the terms in the sum (\ref{e3.4}), which effectively
collapses the wave function to only one of the terms in (\ref{e3.3}). For example,
if the latter wave function is $(\vec{x}|i_1,\ldots,i_n \rangle$, then it depends on only
$n$ coordinates among all $x_A$. All other live particles from sectors with
$n'\neq n$ become dead, i.e., their 4-velocities become zero which appears as
their destruction in spacetime.
More generally, if the overlap between the environment wave functions is negligible
but not exactly zero, then particles from sectors with
$n'\neq n$ will not become dead, but their influence on the environment will still be
negligible, which still provides an effective collapse to $(\vec{x}|i_1,\ldots,i_n \rangle$.
Since decoherence is practically irreversible (due to many degrees of freedom involved),
such an effective collapse is irreversible as well.
Another physically interesting situation is when the entanglement with the environment
takes the form (\ref{e3.6}), where $|\xi\rangle$ are coherent states.
In this case, the behavior of the environment can very well be described in terms
of an environment that responds to a presence of classical fields.
This explains how classical fields may appear at the
macroscopic level, even when the microscopic ontology is described
in terms of particles.
Since $|\xi\rangle$ is a superposition of states with all possible numbers of particles,
trajectories of particles from sectors with different numbers of particles coexist;
there is an infinite number of live particle trajectories in that case.
Similarly, an entanglement of the form of (\ref{u3.4}) explains how accelerated detectors
and detectors in a gravitational field may detect particles in the vacuum.
For example, let us consider the case of a uniformly accelerated detector. In this case,
$|0'\rangle$ corresponds to the Rindler vacuum, while $|0\rangle$ is referred to as
the Minkowski vacuum \cite{bd}. The particle trajectories described by (\ref{e4.13})
are those of the Minkowski particles.
The interaction between the Minkowski vacuum and the accelerated detector creates
new Minkowski particles.
For instance, if the detector is found in the state $|E_0\rangle$, then the Minkowski
particles are in the state $|0'\rangle$, which is a squeezed state
describing an infinite number of live particle trajectories.
Such a view seems particularly appealing from the point of view of
recently discovered renormalizable Horava-Lifshitz gravity \cite{horava}
that contains an absolute time and thus a preferred definition of particles
in a classical gravitational background \cite{nikhorava}.
Let us also give a few remarks on measurements of 4-momenta and 4-velocities.
If $|i\rangle$ in (\ref{e3.1}) denote the 4-momentum eigenstates, then
(\ref{e3.4}) describes a measurement of the particle 4-momenta. Since the 4-momentum
eigenstates are also the 4-velocity eigenstates, (\ref{e3.4}) also describes a measurement
of the particle 4-velocities. Thus, as discussed also in more detail in \cite{nikcr1}, even though
the Bohmian particle velocities may exceed the velocity of light, they cannot exceed
the velocity of light when their velocities are measured. Instead, the effective wave function associated
with such a measurement is a momentum eigenstate of the form of (\ref{e3.35}),
where $k_A^2=m^2$ for live particles and $k_A=0$ for dead particles.
This also explains why dead particles are not seen in experiments: their 3-momenta
and energies are equal to zero.
Finally, we want to end this subsection with some conceptual remarks concerning
the physical meaning of the parameter $s$.
This parameter can be thought of as an evolution parameter, playing
a role similar to that of the absolute Newton time $t$ in the usual formulation
of nonrelativistic Bohmian interpretation \cite{bohm1,bohm1}. To make the similarity
with such a usual formulation more explicit, it may be useful to think of $s$
as a coordinate parameterizing a ``fifth dimension'' that exists independently
of other 4 dimensions with coordinates $x^{\mu}$. However, such a
5-dimensional view should not be taken too literally.
In particular, while the time $t$ is
measurable, the parameter $s$ is not measurable.
Given the fact that $s$ is not measurable, what is the physical meaning
of the claim that particles have the distribution $\rho({\vec x})$ at some $s$?
We can think of it in the following way: We allways measure spacetime positions
of particles at some values of $s$, but we do not know what these values are.
Consequently, the probability density that describes our knowledge is described
by $\rho({\vec x})$ averaged over all possible values of $s$. However, since
$\rho({\vec x})$ does not depend on $s$, the result of such an averaging
procedure is trivial, giving $\rho({\vec x})$ itself.
\section{Conclusion}
\label{SEC5}
In this paper we have extended the Bohmian interpretation of QM, such that
it also incorporates a description of particle creation and destruction described
by QFT. Unlike the previous attempts
\cite{durrcr1,durrcr2} and \cite{nikcr1,nikcr2} to describe the creation and destruction
of pointlike particles within the Bohmian interpretation, the approach
of the present paper incorporates the creation and destruction
of pointlike particles {\em automatically}, without adding any additional
structure not already present in the equations that describe the continuous
particle trajectories.
One reason why it works is the fact that we work with a many-time
wave function, so that the 4-velocity of each particle may vanish separately.
Even though the many-time wave function plays a central role,
we emphasize that the many-time wave function,
first introduced in \cite{tomonaga}, is a natural concept
when one wants to treat time on an equal footing with space, even if one does
not have an ambition to describe particle creation and destruction
\cite{nikprobrel09}.
Another, even more important reason why it works is the entanglement with the
environment, which explains an effective wave function collapse into
particle-number eigenstates (or some other eigenstates)
even when particles are not really created or destroyed.
As a byproduct, in this paper we have also obtained many technical results that allow to represent
QFT states with uncertain number of particles
in terms of many-time wave functions. These results may be
useful by they own, even without the Bohmian interpretation (see, e.g., \cite{nikpure}).
\section*{Acknowledgements}
This work was supported by the Ministry of Science of the
Republic of Croatia under Contract No.~098-0982930-2864.
| {
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} | 9,035 |
Q: KnockoutJS not updating view I am having an issue with Knockout which seems a bit puzzling.
I have a list of contacts, which I want to bind to the UI (observableArray).
However I do not need the items themselves to each be observable, since they are only updated through a dialog box and I don't need each field tracked separately.
I have the following jsFiddle to demonstrate my issue:
http://jsfiddle.net/EsgGg/12/
var c = contacts()[0];
c.name="James";
contacts.splice( 0, 1, c );
// the contacts observableArray is now correct but the UI is unchanged
For some reason the splice method does not update the View??
Thanks in advance.
I really think that Knockout should allow a trigger('change') or some other method on observables to make this type of thing easier.
A: Knockout has a change trigger on observables. It's called valueHasMutated()
contacts.valueHasMutated();
But in fact you are better off if every value that shows on the screen (and is subject to occasional change) is an actual observable in your view model.
contacts()[0].name("James"); // done
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,035 |
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Yang, Victoria. "Is Multiculturalism Really Alive in Canada? A Qualitative Case Study of Chinsese Workers in Montreal's Chinatown." Canadian Content: The McGill Undergraduate Journal of Canadian Studies Vol. 5 (Spring 2013): 100–118.
Dreisziger, Nándor. "'Our Unfortunate Hungarians': Early Hungarian Settlement in Montreal. A Speech by Mária Bagossy Fehér." Hungarian Studies Review Vol. 40, no. 1 (Spring 2013): 69–74. http://www.hungarianstudies.org/HSR2013.pdf.
O'Donnell, Brendan. "Defining a Minority: A Bibliographic Sketch of English Quebec History." Québec Studies Vol. 56 (Fall/Winter 2013): 113–136.
Nigam, Sunita. "Not Just for Laughs: Sugar Sammy, Stand-up Comedy, and National Performance." Québec Studies (Winter , Special Issue 2013): 117–133.
Linteau, Paul-André. The History of Montréal: The Story of a Great North American City. Translated by Peter McCambridge. Montreal: Baraka Books, 2013.
Fiore, Anna-Maria. "Le capital social collectif des Sud-Asiatiques de Montréal : De l'entre soi au mainstream." Canadian Ethnic Studies / Études ethniques au Canada Vol. 45, no. 1–2 (2013): 237–260.
Abukhattala, Ibrahim. "What Arab Students Say about Their Linguistic and Educational Experiences in Canadian Universities." International Education Studies Vol. 6, no. 8 (2013): 31–37. https://files.eric.ed.gov/fulltext/EJ1068646.pdf.
Gubbay Helfer, Sharon. "Rome Among the Bishops: A Reflection on David Rome and His Contributions to Dialogue." Last modified August 27, 2012. http://cjs.concordia.ca/publications/working-papers-in-canadian-jewish-studies/documents/Workingpapers1GubbayHelfer.pdf.
Anctil, Pierre. "H.-M. Caiserman et l'École littéraire de Montréal. Vers une exploration en yiddish du Canada français." Revue d'histoire de l'Amérique française Vol. 66, no. 1 (t 2012): 65–83. http://www.erudit.org/revue/haf/2012/v66/n1/1021082ar.pdf.
Chapman, Mary. "The 'Thrill' of Not Belonging: Edith Eaton (Sui Sin Far) and Flexible Citizenship." Canadian Literature/Littérature canadienne Vol. 212 (Spring 2012): 191–195. http://canlit.ca/article/the-thrill-of-not-belonging/.
Simon, Sherry. Cities in Translation: Intersections of Language and Memory. Milton Park, Abingdon, Oxon, UK & New York, NY: Routledge, 2012.
Martel, Marcel. "'Riot' at Sir George Williams: Giving Meaning to Student Dissent." In Debating Dissent: Canada and the Sixties, edited by Lara Campbell, Dominique Clément, and Gregory S. Kealey, 97–114. Toronto, ON: University of Toronto Press, 2012.
Kirkland, Elizabeth. "Mothering Citizens: Elite Women in Montreal, 1890-1914." PhD dissertation, McGill University, 2012. http://www.collectionscanada.gc.ca/obj/thesescanada/vol2/QMM/TC-QMM-106277.pdf.
Gubbay Helfer, Sharon. "Rome Among the Bishops: An Immigrant Jew Explores The Unknown Worlds of French Canada." Canadian Jewish Studies / Études juives canadiennes Vol. 20 (2012): 15–55. http://cjs.journals.yorku.ca/index.php/cjs/article/view/36099.
Desforges, Josée. "Entre création et destruction : les comportements des types du juif et du canadien français dans les caricatures antisémites publiées par Adrien Arcand à Montréal entre 1929 et 1939." Master's Thesis, Université du Québec à Montréal, 2012. http://www.collectionscanada.gc.ca/obj/thesescanada/vol2/QMUQ/TC-QMUQ-5248.pdf.
Boulard, Danièle. "Un regard sur les pratiques mentorales comme voie d'intégration pour les immigrants : deux solitudes montréalaises." PhD dissertation, Université du Québec à Montréal, 2012. http://www.collectionscanada.gc.ca/obj/thesescanada/vol2/QMUQ/TC-QMUQ-4504.pdf.
Anctil, Pierre. "A Community in Transition: The Jews of Montréal." Contemporary Jewry Vol. 31, no. 3 (October 2011): 225–245.
Robinson, Ira. "'The Other Side of the Coin': The Anatomy of a Public Controversy in the Montreal Jewish Community, 1931." Studies in Religion/Sciences religieuses Vol. 40, no. 3 (September 2011): 271–282.
Croteau, Jean-Philippe. "Les commissions scolaires et les immigrants à Toronto et à Montréal (1900-1945) : quatre modèles d'intégration en milieu urbain." Francophonies d'Amérique No. 31 (Printemps 2011): 49–85. https://www.erudit.org/fr/revues/fa/2011-n31-fa055/1008547ar.pdf.
Wong, Alan. "The Disquieting Revolution: A Genealogy of Reason and Racism in the Québec Press." Global Media Journal -- Canadian Edition Vol. 4, no. 1 (2011): 145–162. http://www.gmj.uottawa.ca/1101/v4i1_wong.pdf.
Robinson, Ira. "The Bouchard-Taylor Commission and the Jewish Community of Quebec in Historical Perspective." In Religion, Culture and the State: Reflections on the Bouchard-Taylor Report, edited by Howard Adelman and Pierre Anctil, 58–68. Toronto, ON: University of Toronto Press, 2011.
Pichette, Amanda. "Representations of Muslim Women in the Quebec News Print Media." Master's Thesis, Concordia University, 2011. http://www.collectionscanada.gc.ca/obj/thesescanada/vol2/QMG/TC-QMG-974063.pdf.
McClure, Helen R. "The Crystallization of a Moral Panic: A Content Analysis of Anglophone Canadian Print Media Discourse on Arabs and Muslims Pre- and Post-9/11." PhD dissertation, American University, 2011. http://aladinrc.wrlc.org/bitstream/handle/1961/11076/McClure_american_0008E_10138display.pdf?sequence=1.
Alton, Caitlin J. "Cultural Diversity in Mile End: Everyday Interactions Between Hasidim and Non-Hasidim." Master's Thesis, Concordia University, 2011. http://spectrum.library.concordia.ca/7399/1/Alton_MA_S2011.pdf.
Riches, Caroline, and Xiao Lan Curdt-Christiansen. "A Tale of Two Montréal Communities: Parents' Perspectives on Their Children's Language and Literacy Development in a Multilingual Context." Canadian Modern Language Review/La revue canadienne des langues vivantes Vol. 66, no. 4 (June 2010): 525–555.
Rosenberg, Michael. "Montreal's Anglophone Community and Minority Ethnocultural NGOs." Canadian Diversity/Diversité canadienne Vol. 8, no. 2 (Spring 2010): 18–23. http://static1.1.sqspcdn.com/static/f/633158/9324644/1289241710847/Spring+2010.pdf?token=%2FZ5n70Dp8KP74kwP5lcLm8gNLkI%3D.
Dere, Jessica, Andrew G. Ryder, and Laurence J. Kirmayer. "Bidimensional Measurement of Acculturation in a Multiethnic Community Sample of First-Generation Immigrants." Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement Vol. 42, no. 2 (April 2010): 134–138.
Troper, Harold. The Defining Decade. ; The Canadian Jewish Community During The 1960's. Toronto, ON: University of Toronto Press, 2010.
Mansah, Joseph, and David Firang. "The African Diaspora in Montreal and Halifax: A Comparative Overview of 'the Entangled Burdens of Race, Class and Space.'" In The African Diaspora in the U.S. and Canada at the Dawn of the 21st Century, edited by John W. Frazier, Joe T. Darden, and Norah F. Henry, 35–48. Albany, NY: State University of New York Press, 2010.
Gingras, Philip. "Invisible Migrants: The Case of Russians in Montreal." Master's Thesis, Concordia University, 2010. http://www.collectionscanada.gc.ca/obj/thesescanada/vol2/002/MR71005.PDF.
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Romuald Pasławski (ur. 1 stycznia 1895 w Olibowie, zm. 17 października 1979 w Warszawie) – kapitan taborów Wojska Polskiego, kawaler Orderu Virtuti Militari, uczestnik I wojny światowej, wojny polsko-bolszewickiej i kampanii wrześniowej.
Życiorys
Urodził się 1 stycznia 1895 we wsi Olibów, w ówczesnym powiecie dubieńskim guberni wołyńskiej, w rodzinie Mariana i Michaliny z Buderawskich. Od 11 maja 1915 w armii rosyjskiej, w której walczył na froncie francuskim. Na własną prośbę przeniósł się do II Korpusu Polskiego w Rosji, gdzie służył jako żołnierz w 16 pułku strzelców. 11 maja 1918 dowodził plutonem w bitwie pod Kaniowem, po której dostał się do niewoli niemieckiej.
Za udział w bitwie w której: "szybką i samodzielną akcją zapobiegł okrążeniu Legii Oficerskiej oraz 16 p. strz. przez siły nieprzyjaciela" został odznaczony Orderem Virtuti Militari.
Po odzyskaniu przez Polskę niepodległości od listopada 1919 w Wojsku Polskim. Walczył w wojnie polsko-bolszewickiej w szeregach 34 pułku piechoty. 5 maja 1920 został mianowany podporucznikiem. Do rezerwy przeniesiony 10 kwietnia 1922. 8 stycznia 1924 został zatwierdzony w stopniu porucznika ze starszeństwem z 1 czerwca 1919 i 2163. lokatą w korpusie oficerów rezerwy piechoty. Posiadał wówczas przydział w rezerwie do 81 pułku piechoty w Grodnie. Na stopień kapitana rezerwy został awansowany ze starszeństwem z 19 marca 1939 i 3. lokatą w korpusie oficerów taborowych.
Po wojnie prowadził Zakłady Mechaniczne w Warszawie. Mieszkał w Milanówku, w willi Józefin.
Po kampanii wrześniowej dostał się do niewoli niemieckiej. Przebywał w Oflagu VI E Dorsten.
Od 1945 pracował w warszawskim Zjednoczeniu Budowy Elektrowni. Zmarł 17 października 1979 w Warszawie, pochowany na warszawskim Cmentarzu Komunalnym Północnym.
Żonaty dwukrotnie. Od 1922 z Magdaleną Butrymowicz. Od 1948 z Józefą Rudzińską. Córka Eliza (1923).
Ordery i odznaczenia
Krzyż Srebrny Orderu Wojskowego Virtuti Militari nr 6733
Krzyż Walecznych
Krzyż Niepodległości – 23 grudnia 1933 "za pracę w dziele odzyskania niepodległości"
Przypisy
Bibliografia
Linki zewnętrzne
Jeńcy Oflagu VI E Dorsten
Kapitanowie taborów II Rzeczypospolitej
Ludzie związani z Milanówkiem
Odznaczeni Krzyżem Srebrnym Orderu Virtuti Militari (II Rzeczpospolita)
Odznaczeni Krzyżem Walecznych (II Rzeczpospolita)
Odznaczeni Medalem Niepodległości
Oficerowie 81 Pułku Strzelców Grodzieńskich
Pochowani na cmentarzu komunalnym Północnym w Warszawie
Uczestnicy wojny polsko-bolszewickiej (strona polska)
Urodzeni w 1895
Zmarli w 1979
Żołnierze II Korpusu Polskiego w Rosji 1917–1918 | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/nearly-lorentz-coordinate-systems.930549\/","text":"# I Nearly Lorentz Coordinate Systems\n\n1. Nov 4, 2017\n\n### Silviu\n\nHello! I am reading Schutz A first course in GR and he introduces the Nearly Lorentz coordinate systems as ones having a metric such that $g_{\\alpha\\beta} = \\eta_{\\alpha\\beta} + h_{\\alpha\\beta}$, with h a small deviation from the normal Minkowski metric. Then he introduces the Background Lorentz transformations (this is section 8.3 in the second edition) in which all the points are transformed as $x^{\\bar\\alpha}=\\Lambda_\\beta^{\\bar\\alpha}x^\\beta$. Applying this transformation to g he gets in the end that h transforms as $h_{\\bar\\alpha\\bar\\beta}=\\Lambda_\\mu^{\\bar\\alpha}\\Lambda_\\nu^{\\bar\\beta} h_{\\mu\\nu}$ and from here he says that we can treat h as if it was a tensor in SR and this simplify the calculations a lot. Can someone explain to me why ones need all these calculations for this? I am sure I am missing something but h is the difference between g and $\\eta$ so isn't it a tensor, just because it is the difference between 2 tensors? Why do you need a proof for it? Moreover Schutz says that h \"it is, of course, not a tensor, but just a piece of $g_{\\alpha\\beta}$\". So can someone explain to me why isn't h a tensor and why my logic is flawed? Thank you!\n\n2. Nov 4, 2017\n\n### martinbn\n\nWhy is $\\eta_{\\alpha\\beta}$ a tensor?\n\n3. Nov 4, 2017\n\n### Silviu\n\nWell a tensor (in this case a $(0,2)$ tensor) is a function that turns 2 vectors into a real number. $\\eta$ is a 4x4 matrix so it behaves like a $(0,2)$ tensor, when applied to a 4D vector (which is our case).","date":"2017-12-16 19:18:44","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\": 0.9051706790924072, \"perplexity\": 297.2180601255402}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948588420.68\/warc\/CC-MAIN-20171216181940-20171216203940-00771.warc.gz\"}"} | null | null |
Home > Christian Living > Last Place Fantasy Football Team Invokes Matthew 19:30 Loophole, Declared ...
Christian Living Scripture
Last Place Fantasy Football Team Invokes Matthew 19:30 Loophole, Declared To Be First
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PORTLAND, OR—Derrick Martinez' fantasy football team is dead last in his church's annual league, having gone 0-13 so far this year. The rest of the men at church had counted him out as a playoff contender, pointing out that it's mathematically impossible for him to make a comeback.
But Martinez had one final trick up his sleeve: after his thirteenth loss this weekend, he cited Jesus' words in Matthew 19:30 to the rest of the league: "But many that are first shall be last; and the last shall be first." The league commissioner was forced to rearrange the rankings, putting Martinez's team in the top slot and kicking the first-place team to last.
"It's a great loophole," he told reporters. "I use it every year. I intentionally tank the first dozen weeks or so and then pull ahead at the last second with the ol' 'the last shall be first' trick. It's a fantastic reversal move that most people don't know about."
At publishing time, Martinez was horrified to discover the now-last-place team had also cited the rule, pushing him back to last place once again.
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Archaeologists Discover Pair Of Crocs Worn By Judas Iscariot | {
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The set design here is based on what I imagine spelling bees are like in Geography City. That's why there's no audience apart from the contestants. York probably didn't even send out a press release.
First and foremost: the San Diego Comic-Con ham scans are up. Infinite thanks to Kris Straub of Starslip Crisis for his assistance and bandwidth. And also to all the artists who participated.
Welcome to anyone who has been linked here via Starslip or Fleen, or anywhere else for that matter. If you want to sample our wares, I recommend starting here. Terror Island is a gamepiece photocomic, telling the story of two roommates trying to decide whose turn it is to go get the groceries, while their friends variously help or hinder their grocery shopping related schemes.
I am very very excited about this spelling bee storyline we just started, as well as the storylines coming up after it. Then again, I'm pretty excited about everything having to do with Terror Island, so, that's not really news.
White Town's 'Your Woman' is exceptionally good, and a while after it came out, I completely forgot about it. I rediscovered it from my friend Dennis's music collection, and then again when the video was linked from some forums I hang out on. So I figured I'd share it with you all.
Also, you should all ask Ben about the book I got him for his birthday. It's about fonts and morality. | {
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\section{\label{sect:intro}introduction}
The ultimate performance of a ring-based accelerator is determined not
only by certain critical global parameters, such as beam emittance, but
also by local properties of the beam at particular points of interest
(POI). The capability of diagnosing and controlling local beam
parameters at POIs, such as beam size and divergence, is crucial for a
machine to achieve its design performance. Examples of POIs in a
dedicated synchrotron light source ring include the undulator locations,
from where high brightness X-rays are generated. In a collider, POIs
are reserved for detectors in which the beam-beam luminosity is
observed. However, beam diagnostics elements, such as beam position
monitors (BPM) are generally placed outside of the POIs as the POIs are
already occupied. An intuitive, but quantitatively unproven belief, is
that the desired beam properties at the POIs can be achieved once the
beam properties are well-controlled at the location of the BPMs.
Using observational data at BPMs to indirectly predict the beam
properties at POIs can be viewed as a regression problem and can be
treated as a supervised learning process: BPM readings at given
locations are used as a training dataset. Then a ring optics model with
a set of quadrupole excitations as its arguments is selected as the
hypothesis. From the dataset, an optics model needs to be generalized
first. Based on the model, the unknown beam properties at POIs can be
predicted. However, there exists some systematic error and random
uncertainty in the BPMs' readings, and the quantity of BPMs (the
dimension of the training dataset) is limited. Therefore, the
parameters in the reconstructed optics model have inherent
uncertainties, as do the final beam property predictions at the POIs.
The precision and accuracy of the predictions at the POIs depend on the
quantity of BPMs, their physical distribution pattern around the ring,
and their calibration, resolution, etc. When a BPM system is designed
for a storage ring, however, it is more important to consider the
inverse problem: i.e. How are the BPM system technical requirements
determined in order to observe whether the ring achieves its desired
performance? In this paper, we developed an approach to address this
question with Bayesian Gaussian regression.
In statistics, a Bayesian Gaussian
regression~\cite{Rasmussen,Bishop:2006} is a Bayesian approach to
multivariate regression, i.e. regression where the predicted outcome is
a vector of correlated random variables rather than a single scalar
random variable. Every finite collection of the data has a normal
distribution. The distribution of generalized arguments of the
hypothesis is the joint distribution of all those random variables.
Based on the hypothesis, a prediction can be made for any unknown
dataset within a continuous domain. In our case, multiple BPMs'
readings are normally distributed around their real values. The
standard deviations of the Gaussian distributions are BPM's resolutions.
A vector composed of quadrupoles' mis-settings is the argument to be
generalized. The prediction at the POIs is the function of this vector.
The continuous domain is the longitudinal coordinate $s$ along a storage
ring.
To further explain this approach, the remaining sections are outlined as
follows: Sect.~\ref{sect:performance} introduces the relation between
machine performance and beam diagnostics system capabilities.
Sect.~\ref{sect:GPmodel} explains the procedure of applying the Bayesian
Gaussian regression in the ring optics model reconstruction, and the
prediction of local optics properties at POIs. In
Sect.~\ref{sect:nsls2}, the National Synchrotron Light Source II
(NSLS-II) storage ring and its BPM system are used to illustrate the
application of this approach. Some discussions and a brief summary is
given in Sect.~\ref{sect:summary}.
\section{machine performance and beam diagnostics capability\label{sect:performance}}
As mentioned previously, ultimate performance of a ring-based
accelerator relies heavily on local beam properties at particular POIs.
Consider a dedicated light source ring. Its ultimate performance is
measured by the brightness of the X-rays generated by undulators. The
brightness of undulator emission is determined by the transverse size of
both the electron and photon beam and their angular divergence at their
source points~\cite{Lindberg,Walker,Chubar,Hidas}. Therefore, the
undulator brightness performance $\mathcal{B}$ depends on the ring's
global emittance and the local transverse optics parameters,
\begin{eqnarray}
\mathcal{B} & \propto &
\frac{1}{\Sigma_x\Sigma_x^{\prime}\Sigma_y\Sigma_y^{\prime}}
\nonumber\\ \Sigma_{x,y} & = &
\sqrt{\epsilon_{x,y}\beta_{x,y}+\eta_{x,y}^2\sigma_{\delta}^2+\sigma^2_{ph}}
\nonumber\\ \Sigma_{x,y}^{\prime} & = &
\sqrt{\epsilon_{x,y}\gamma_{x,y}+\eta_{x,y}^{\prime2}
\sigma_{\delta}^2+\sigma^{\prime2}_{ph}}. \label{eq:brightness}
\end{eqnarray}
Here $\epsilon_{x,y}$ are the electron beam emittances, which represent
the equilibrium between the quantum excitation and the radiation damping
around the whole ring. $\beta,\gamma$ are the Twiss
parameters~\cite{Courant:1997rq}, $\eta,\eta^{\prime}$ are the
dispersion and its derivative at the undulators' locations,
$\sigma_{\delta}$ is the electron beam energy spread
$\sigma_{ph}=\frac{\sqrt{\lambda L_u}}{2\pi}$ and
$\sigma^{\prime}_{ph}=\frac{1}{2}\sqrt{\frac{\lambda}{L_u}}$ are the
X-ray beam diffraction ``waist size'' and its natural angular
divergence, respectively. The X-ray wavelength $\lambda$, is determined
based on the requirements of the beam-line experiments, and $L_u$ is the
undulator periodic length. The emittance was found to be nearly
constant with small $\beta$-beat (see Sect.~\ref{sect:nsls2}).
Therefore, monitoring and controlling the local POI's Twiss parameters
is crucial.
The final goal of beam diagnostics is to provide sufficient, accurate
observations to reconstruct an online accelerator model. Modern BPM
electronics can provide the beam turn-by-turn (TbT) data, which is
widely used for the beam optics characterization and the model
reconstruction. Based on the model, we can predict the beam properties
not only at the locations of monitors themselves, but more importantly
at the POIs. The capability of indirect prediction of the Twiss
parameters at POIs eventually defines the BPM system requirements on TbT
data acquisition. Based on Eq.~\eqref{eq:brightness}, how precisely one
can predict the bias and the uncertainty of Twiss parameters $\beta$ and
$\eta$ at locations of undulators is the key problem in designing a BPM
system. Therefore, to specify the technical requirements of a BPM
system, the following questions need to be addressed: in order to make
an accurate and precise prediction of beam properties at POIs, how many
BPMs are needed? How should the BPMs be allocated throughout the
accelerator ring, and how precise should the BPM TbT reading be?
In the following section a method of reconstructing the linear optics
model, and determining the brightness performance for a ring-based light
source will be discussed. For a collider ring, its luminosity is
determined only by the beam sizes at the interaction
points~\cite{Herr:2003em}. Gaussian regression analysis can therefore
be applied to predict its $\beta^*$ and luminosity as well.
\section{\label{sect:GPmodel}Gaussian regression for model reconstruction and prediction}
When circulating beam in a storage ring is disturbed, a BPM system can
provide its TbT data at multiple longitudinal locations. TbT data of
the BPMs can be represented as an optics model plus some random reading
errors,
\begin{equation}\label{eq:tbt}
x(s)_i =
A(i)\sqrt{\beta(s)}\cos\left[i\cdot2\pi\nu+\phi(s)\right]+\varepsilon(s)_i,
\end{equation}
here $i$ is the index of turns, $A(i)$ is a variable dependent on turn
number, $\beta(s)$ is the envelope function of Twiss parameters at $s$
location, $\nu$ is the betatron tune, $\phi$ is the betatron phase, and
$\varepsilon(s)_i$ is the BPM reading noise~\cite{Calaga,Langner,Cohen},
which generally has a normal distribution. Based on the accelerator
optics model defined in Eq.~\eqref{eq:tbt}, we can extract a set of
optics Twiss parameters at all BPM
locations~\cite{Castro:1996,Irwin:1999,Huang:2005,Tomas:2017}.
Recently, Ref.~\cite{Hao:2019lmn} proposed using a Bayesian approach to
infer the mean (aka expectation) and uncertainty of Twiss parameters at
BPMs simultaneously. The mean values of $\beta$ represent the most
likely optics pattern. The random BPM reading error and the
simplification of the optics model can result in some uncertainties,
$\varepsilon_{\beta}$, in the inference process,
\begin{equation}\label{eq:beta}
\beta = \beta(s,\bs{K})+\varepsilon_{\beta}(s),
\end{equation}
here $\bs{K}$ is a vector composed of all normalized quadrupole focusing
strengths, and $\varepsilon_{\beta}$ is the inference uncertainty.
Unless otherwise stated, bold symbols, such as ``$\bs{X}$'', are used to
denote vectors and matrices throughout this paper. In accelerator
physics, the deviation from the design model $\beta_0$ is often referred
to as the $\beta$-beat. From the point of view of model reconstruction,
the $\beta$-beat is due to quadrupole excitation errors and can be
determined by
\begin{equation}\label{eq:beta-beat}
\Delta\beta=\beta(s,\bs{K}_0+\Delta\bs{K})-\beta_0(s,\bs{K}_0)
\approx\bs{M}\Delta\bs{K},
\end{equation}
where $\bs{K}_0$ represents the quadrupoles' nominal setting and
$\beta_0$ is the nominal envelope function along $s$. $\bs{M}$ is the
response matrix composed of elements
$M_{i,j}=\frac{\partial\beta_{s_i}}{\partial K_j}$ observed by the BPMs.
The dependency of $\beta$ on $\bs{K}$ is not linear in a complete optics
model. However, when quadrupole errors are small enough, the dependence
can be approximated as a linear relation as illustrated in
Fig.~\ref{fig:linear}. The approximation holds for most operational
storage rings, and other diffraction limited light sources under design
or construction. A linear approximation allows us to use the linear
regression approach for this process. Eq.~\eqref{eq:beta} or
\eqref{eq:beta-beat} is a hypothesis with the unknown arguments $\bs{K}$
or $\Delta\bs{K}$, which need to be generalized from BPM measurement
data.
\begin{figure}[!ht]
\centering \includegraphics[width=1.\columnwidth]{linear.png}
\caption{\label{fig:linear} $\beta_x$ dependency on the excitation error
of a quadrupole observed by a BPM at the NSLS-II ring. The dependency
is nonlinear. However, when the quadrupole error is confined to a
small range $[-0.25\%,0.25\%]$, it can be approximated as a linear
dependence as shown in the zoomed-in window. At modern storage rings,
such as NSLS-II, quadrupole excitation errors due to a power supply's
mis-calibration and/or magnetic hysteresis are much less than 0.25\%.}
\end{figure}
Given a set of measured optics parameters $\beta$s at multiple locations
$s$ from BPM TbT data, the posterior probability of the quadrupole error
distribution $p(\Delta\bs{K}|\bs{\beta})$ can be given according to
Bayes theorem~\cite{Li:2019kch},
\begin{eqnarray}
p(\Delta\bs{K}|\bs{\beta}) & = &
\frac{p(\bs{\beta}|\Delta\bs{K})
p(\Delta\bs{K})}{p(\bs{\beta})} \nonumber \\ &
\propto &
p(\bs{\beta}|\Delta\bs{K})p(\Delta\bs{K}). \label{eq:bayes}
\end{eqnarray}
Here $p(\bs{\beta}|\Delta\bs{K})$ is referred to as the likelihood function,
\begin{eqnarray}
\mathcal{N}(\bs{\beta}|\bar{\bs{\beta}},\sigma_{\beta}^2) & = &
\frac{1}{\sqrt{2\pi}\sigma_{\beta}}
\exp\left[-\frac{(\bs{\beta}-\bar{\bs{\beta}})^2}{2\sigma_{\beta}^2}\right]
\nonumber \\
& \approx & \frac{1}{\sqrt{2\pi}\sigma_{\beta}}
\exp\left[-\frac{(\Delta\bs{\beta}-\bs{M}\Delta
\bs{K})^2}{2\sigma_{\beta}^2}\right]. \label{eq:likelihood}
\end{eqnarray}
Here $\bar{\beta}=\mathbb{E}(\beta)$ and $\sigma^2_{\beta}$ are the
expectation value and the variance of the normal distribution of
measured $\beta$s. Once the expectation value of the optics measurement
is extracted from the TbT data, a prior quadrupole excitation error
distribution $p(\Delta\bs{K})$ can be determined by comparing them
against the design optics model,
\begin{eqnarray}\label{eq:prior}
p(\Delta\bs{K}) &=&\mathcal{N}(\Delta\bs{K}|0,\sigma_{\Delta
K}^2) \nonumber \\ & = & \frac{1}{\sqrt{2\pi}\sigma_{\Delta K}}
\exp\left[-\frac{\Delta\bs{K}^2}{2\sigma^2_{\Delta K}}\right],
\end{eqnarray}
in which the variance $\sigma^2_{\Delta K}$ of the prior distribution
$p(\Delta\bs{K})$ is linearly proportional to the mean value of the
measured $\beta$-beat,
\begin{equation}\label{eq:prior2}
\sigma_{\Delta K} \sim \kappa|\Delta\beta| =
\kappa|\bar{\beta}-\beta_0|.
\end{equation}
Here ``$\sim$'' in Eq.~\eqref{eq:prior2} describes a statistically
proportional relationship between $\beta$-beats (in the unit of ``m'')
and quadrupole strength error $\Delta K$ (in units of $m^{-2}$). The
coefficient $\kappa$ can be computed based on the optics model either
analytically or numerically before carrying out any measurements. In the
NSLS-II ring, $\kappa \approx 1.6\times10^{-3} m^{-3}$, i.e. a $0.25 m$
$\beta$-beat ($\frac{\Delta\beta}{\beta}\approx1\%$) corresponds to a
distribution of quadrupole errors with the standard deviation
$4\times10^{-4} m^{-2}$ ($\frac{\Delta K}{K}\approx0.12\%$) as shown in
Fig. 1 in Ref. ~\cite{Li:2019kch}. Qualitatively, the relative
$\beta$-beat and quadrupole error, i.e. $\frac{\Delta\beta}{\beta}$ and
$\frac{\Delta K}{K}$ are often used in accelerator literature.
Here the absolute $\Delta\beta$ and $\Delta K$ are used simply because
they were adapted to our quantitative implementation.
Both the likelihood function and the prior distribution are generally
normally distributed. Therefore, the posterior distribution is a normal
distribution by summing over the arguments of the exponentials in
Eq.~\eqref{eq:likelihood} and \eqref{eq:prior},
\begin{equation}\label{eq:posterior1}
(\Delta\bs{\beta}-\bs{M}\Delta\bs{K})^T\bs{S}_{\beta}^{-1}
(\Delta\bs{\beta}-\bs{M}\Delta\bs{K})+
\Delta\bs{K}^T\bs{S}_{K}^{-1}\Delta\bs{K}.
\end{equation}
Here
\begin{equation}\label{eq:S}
\bs{S}_{\beta}^{-1} = \frac{1}{\sigma_{\beta}^2}\bs{I}, \;\;
\bs{S}_{K}^{-1} = \frac{1}{\sigma_{\Delta K}^2}\bs{I}.
\end{equation}
The identity matrix $\bs{I}$ is used in Eq.~\eqref{eq:S} because all
BPMs' resolutions are assumed to have the same values $\sigma_{\beta}$.
In reality, however, $\bs{S}_{\beta}^{-1}$ needs to be replaced with a
diagonal matrix with different elements if the BPMs' resolutions are
different. The quadrupoles' error distribution matrix $\bs{S}_{K}^{-1}$
needs to be processed in the same way if necessary. The mean value of
the posterior, corresponding to the most likely quadrupole error
distribution, can be used to implement the linear optics correction as
explained in Ref.~\cite{Li:2019kch},
\begin{equation}\label{eq:posteriorMean}
\bs{m} =
\sigma^{-2}_{\beta}\bs{A}^{-1}
\bs{M}^T\Delta\bs{\bar{\beta}},
\end{equation}
where
$\bs{A}=\left[\sigma^{-2}_{\beta}\bs{M}^T\bs{M}+\sigma^{-2}_{\Delta
K}\bs{I}\right]$. Adding an extra term $\sigma^{-2}_{\Delta
K}\bs{I}$ to prevent overfitting is known as the regularization
technique. The posterior variance represents the uncertainty of
quadrupole errors.
\begin{equation}\label{eq:posteriorVariance}
\bs{\Sigma_K^2}=\bs{A}^{-1}.
\end{equation}
Given $\beta$-beats observed at $s$, the posterior generalizes an optics
model, in which the quadrupoles errors are normally distributed,
\begin{equation}\label{eq:posterior2}
p(\Delta\bs{K}|\Delta\bs{\beta},s)=\mathcal{N}(\Delta\bs{K}|\bs{m},\bs{\Sigma_K^2}),
\end{equation}
with the mean value and the variance given by
Eq.~\eqref{eq:posteriorMean} and \eqref{eq:posteriorVariance}
respectively.
Thus far, the optics are measured at the locations of the BPMs, and the
corresponding quadrupole error distributions are generalized based on
the measurements. To confirm the machine brightness performance, we
need to predict the beam properties at POIs. To do so, the output of
all possible posterior quadrupole error distributions must be averaged,
\begin{eqnarray}
p(\Delta\bs{\beta}_*|\bs{s}_*,\Delta\bs{\beta},\bs{s}) & = & \int
p(\Delta\bs{\beta}_*|\bs{s}_*,\Delta\bs{K})
p(\Delta\bs{K}|\Delta\bs{\beta},\bs{s}) \mathrm{d}\bs{K} \nonumber \\ & =
& \mathcal{N}(\bs{m}_*,\bs{\Sigma}_*^2). \label{eq:prediction}
\end{eqnarray}
Here $\Delta\bs{\beta}_*$ is the predicted result at POIs' locations
$\bs{s}_*$ given the measured $\Delta\bs{\beta}$ at $\bs{s}$. The mean
values and the variances of the predicted distributions at POIs are
\begin{eqnarray}
\bs{m}_* & = & \sigma^{-2}_{\beta}\bs{M}_*\bs{A}^{-1}\bs{M}^T\Delta\bs{\bar{\beta}} \nonumber \\
\bs{\Sigma}_*^2 & = &\bs{M}_*\bs{A}^{-1}\bs{M}^T_*, \label{eq:predictionMeanVar}
\end{eqnarray}
$\bs{M}_*$ is the Jacobian matrix of the optics response to quadrupole
errors observed at POIs. The difference between the mean value
$\bs{m}_x$ and the real $\beta$ at a POI is referred to as the
predicted bias. By substituting the bias and uncertainty back into
Eq.~\eqref{eq:brightness}, we can estimate how accurate the brightness
could be measured for given BPMs' resolutions. Based on the desired
brightness resolution, we can determine the needed quantity and
resolution of BPMs.
\section{\label{sect:nsls2} Application to NSLS-II ring}
In this section, we use the NSLS-II ring and its BPM system TbT data
acquisition functionality to demonstrate the application of this
approach. NSLS-II is a $3^{rd}$ generation dedicated light source. All
undulator source points (POIs) are located at non-dispersive straights.
Typical photon energy from undulators is around 10 $keV$, with
corresponding wavelengths around 0.124 $nm$. The undulators' period
length is 20 $mm$. The horizontal beam emittance is 0.9 $nm\cdot rad$
including the contribution from 3 damping wigglers. The emittance
coupling ratio can be controlled to less than 1\%. At its 15 short
straight centers, the Twiss parameters are designed to be as low as
$\beta_{x,y}=1.80,\,1.20\,m$, and $\alpha_{x,y}=0$ to generate the
desired high brightness x-ray beam from the undulators.
The horizontal emittance growth with an optics distortion was studied by
carrying out a lattice simulation. With $\beta$-beat at a few percent,
the corresponding $\alpha-$ and $\gamma$-distortions were generated by
adding some normally distributed quadrupole errors based on
Eq.~\eqref{eq:prior} and \eqref{eq:prior2}. The horizontal emittance
was found to grow slightly with the average $\beta$-beats as
illustrated in Fig.~\ref{fig:emit_vs_bb}. When there is about a 1\%
horizontal $\beta$-beat ($\sim 0.14 m$), the emittance increases by only
about 0.1\%, which is negligible. Therefore, in the following
calculation, the emittance was represented as a constant.
\begin{figure}[!ht]
\centering \includegraphics[width=1.\columnwidth]{emit_vs_bb.png}
\caption{\label{fig:emit_vs_bb} Beam horizontal emittance growth with
the average $\beta_x$-beat for the NSLS-II ring. If the global
$\beta_x$-beat can be controlled within 1\%($\sim 0.15m$), the
emittance growth is negligible.}
\end{figure}
Degradation of an undulatorís brightness is determined by its local
optics distortion which can be evaluated with Eq.~\eqref{eq:brightness}.
Multi-pairs of simulated $\beta-\alpha$ were incorporated into the
previously specified undulator parameters to observe the dependence of
the X-ray brightness on the $\beta$-beat (see Fig.~\ref{fig:brg_vs_bb}).
A change of approximately 1\% of the $\beta_{x,y}$ in the transverse
plane can degrade the brightness by about 1\%. In other words, in order
to resolve a 1\% brightness degradation, the predictive errors of the
ring optics (including the bias and uncertainty) at the locations of
undulators should be less than 1\%. Because multiple undulators are
installed around the ring, the predicted performance needs to be
evaluated at all POIs simultaneously.
\begin{figure}[!ht]
\centering \includegraphics[width=1.\columnwidth]{brg_vs_bb.png}
\caption{\label{fig:brg_vs_bb} Brightness degradation of an undulator at
a low-$\beta$ straight due to the average $\beta$-beats in the
horizontal and vertical planes. Each dot represents a set of simulated
optics distortions. The brightness degradation has an approximate
linear dependence on $\beta$-beat.}
\end{figure}
There exist two types of errors in Eq.~\eqref{eq:tbt} which can
introduce uncertainties in characterizing the optics parameters at BPMs.
First, due to radiation damping, chromatic decoherence and nonlinearity,
a disturbed bunched-beam trajectory is not a pure linear undamping
betatron oscillation~\cite{Meller:1987ug}. A reduced model (for
example, assuming $A$ is a constant), will introduce systematic
errors~\cite{Malina,Carla,Franchi,Langner}. The second error source is
the BPM TbT resolution limit, which results in random noise. At
NSLS-II, the BPM TbT resolution at low beam current ($<2\; mA$) is
inferred as $\sim10-15\;\mu m$. When a $2^{nd}$ order polynomial
function is used to represent the turn-dependent amplitude $A(i)$, the
inferred $\beta$ function resolution at BPMs can be reached as low as
0.5\%~\cite{Hao:2019lmn}.
First we studied the dependence of predictive errors on the quantity of
BPMs. A comprehensive simulation was set up to compare the Gaussian
regression predictive errors with the real errors. A linear optics
simulation code was used to simulate the distorted optics due to a set
of quadrupole errors. The $\beta$-beats observed at the BPMs were marked
as the ``real'' values. On top of the real values, 0.5\% random errors
were added to simulate one-time measurement uncertainty seen by the
BPMs. A posterior distribution Eq.~\eqref{eq:posteriorMean} and
\eqref{eq:posteriorVariance} of the quadrupole errors was obtained by
reconstructing the optics model with the likelihood function
Eq.~\eqref{eq:likelihood}, and the prior distribution \eqref{eq:prior}
and \eqref{eq:prior2}. The predicted optics parameters with their
uncertainties were then calculated based on another likelihood function
between quadrupoles and the locations of undulators with
Eq.~\eqref{eq:prediction}.
The results of comparison are illustrated in Fig.\ref{fig:IDVsBPM}. As
with any regression problem, the training data distribution (i.e. the
BPM locations) should be as uniform as possible within the continuous $s$
domain. There are 30 cells in the NSLS-II ring, and each cell has 6
BPMs. Equal numbers of BPMs were selected from each cell to make the
training data uniformly distributed. The goal was to predict all
straight section optics simultaneously. The predicted performance was
therefore evaluated by averaging at multiple straight centers.
Initially, one BPM was selected per cell. The number of selected BPMs
was then gradually increased to observe the evolution of predictive
errors. It was found that utilizing more BPMs improved the predicted
performance, as expected. Both the bias and uncertainty were reduced
with the quantity of BPMs. However, the improvement became less and
less apparent once more than 4 BPMs per cell were used.
Since there are 6 BPMs per cell at the NSLS-II ring, we chose different
BPM combinations. We found that some patterns/combinations of BPMs were
better used to capture/measure these types of optics distortions. For
example, each end of the straight sections needs one BPM to observe the
ID, and at least one BPM needs to be located inside the achromat arc in
order to observe the dipoles. The distribution of the BPMs does not
need to be uniform in the longitudinal $s$ direction, instead, they
should be uniform along the betatron phase propagation. Collider rings
would see this effect more clearly due to the existence of interaction
points. However, for most light source rings, including the NSLS-II
ring, the phase propagation along the longitudinal direction is mostly
quite linear in the longitudinal direction.
\begin{figure*}
\centering \includegraphics[width=1.\textwidth]{BPM_number.png}
\caption{\label{fig:IDVsBPM} A zoomed-in view of predicted means and
variances of $\beta$ observed at both BPMs (the training set) and
undulators (POIs) for a section (spanning 3 cells, 4 POIs) of the
NSLS-II ring. Black and red dots represent the real $\beta$ values at
BPMs and POIs. Black crosses are the data observed by the BPMs. The
light blue lines with a shadow are the predictions at the BPMs, and
the green error bars are the final prediction at POIs. From subplot 1
to 6, the quantity of BPMs used increases gradually. A large set of
training data (i.e. using more BPMs) for the regression does improve
the accuracy and precision of the predicted results at POIs. However,
the improvement becomes less apparent after using more than 120 BPMs.}
\end{figure*}
Next, we studied the effect of $\beta$ measurement resolution on the
predictive errors. A similar analysis was carried out but with
different $\beta$-resolution as illustrated in
Fig.~\ref{fig:BPMprecision}. By observing Fig.~\ref{fig:BPMprecision},
several conclusions can be drawn: (1) The degradation of the $\beta$
resolution reduced the accuracy of the generalized optics model.
However, this can be improved by applying a more complicated optics
model~\cite{Hao:2019lmn}. Thus, the BPM TbT resolution is the final
limit on the resolution of $\beta$ parameters. In order to accurately
and precisely predict the beam properties at POIs, improving the
resolution of BPMs is crucial. (2) After a certain point, the predicted
performance is not improved significantly with the quantity of BPMs as
seen in both Fig~\ref{fig:IDVsBPM} and \ref{fig:BPMprecision}. The
advantage of reduction of predictive errors will gradually level out
once enough BPMs are used. Meaning that quantitatively, the improvement
in error reduction will eventually become negligible compared to the
cost of adding more BPMs. The higher the resolution each individual BPM
has, the less number of BPMs are needed. There should be a compromise
between the required quality and quantity of BPMs to achieve an expected
predictive accuracy. (3) The quality (resolution) is much more
important than the quantity of BPMs from the point of view of optics
characterization. For example, at NSLS-II, in order to resolve 1\%
brightness degradation, at least 120 BPMs with a $\beta$ resolution
better than 1\% are needed, or 90 BPMs with a 0.75\% resolution, etc.
Having more BPMs than is needed creates no obvious, significant
improvement. Having 60 high precision (0.5\% $\beta$-resolution) BPMs
yields a better performance than having 180 low precision (1\%) BPMs in
this example.
\begin{figure}[!ht]
\centering \includegraphics[width=1.\columnwidth]{BPM_resolutionVsVar.png}
\caption{\label{fig:BPMprecision} Predictive $\beta$-beat errors
(including bias and uncertainties) at the locations of undulator
(POIs). $\beta$s are observed with different number of BPMs and
different resolutions. The resolution of $\beta$ is the final limit on
predictive errors. The higher the resolution each individual BPM has,
the less number of BPMs are needed.}
\end{figure}
\section{\label{sect:summary} Discussion and summary}
A systematic approach has been proposed to analyze a BPM systemís
technical requirements in this manuscript. The approach is based on the
resolution requirements for monitoring a machine's ultimate performance.
The Bayesian Gaussian regression is useful in statistical data
modelling, such as reconstructing a ring's optics model from beam TbT
data. The optics properties of the ring are contained in a collection
of data having a normal distribution. From past experience in designing
and commissioning various accelerators, many will intuitively realize
that having more BPMs does not always significantly improve diagnostics
performance and is therefore not necessarily cost-effective for an
accelerator design. Using the Gaussian regression method, however,
confirmed that quantitatively. More importantly, a reasonable compromise
can be reached between the quality (resolution) and the quantity of BPMs
using this method.
It is worth noting that our approach is simplified as a linear
regression by assuming a known linear dependence of optics distortion on
quadrupole errors. If a ring's optics are significantly different from
the design model, this assumption is not valid. In our case, we needed
to iteratively calculate the likelihood function $\bs{M}$ by
incorporating the posterior mean of quadrupole errors
Eq.~\eqref{eq:posteriorMean} and compare it to the optics model until
the best convergence was reached. This was not discussed in this paper,
however, because our analysis applies best to machines whose optics are
quite close to their design model. Other important effects on X-ray
brightness, such as quadrupolar errors from sextupole feed down, skew
quadrupoles, longitudinal misalignments of quadrupoles and BPMs,
systematic gain errors in BPMs, magnet fringe field etc. are not
addressed in detail here. These effects are neglected at the NSLS-II
ring because either they are small compared with the quadrupole
excitation errors and hysteresis, or their effects have been integrated
into our optics model. The Gaussian regression method outlined here,
however, can be expanded to take them into account if necessary.
In a ring-based accelerator, BPMs are used for multiple other purposes,
such as orbit monitoring and optics characterization, etc. In this
paper we only concentrated on a particular use case of TbT data to
characterize the linear optics, and then to predict X-ray beam
brightness performance. A similar analysis can be applied to the orbit
stability, and dynamic aperture reduction due to $\beta$-beat as well.
An accelerator's BPM system needs to satisfy several objectives
simultaneously. Therefore the Gaussian regression approach could/should
be extended to a higher dimension parameter space to achieve an optimal
compromise among these objectives.
\begin{acknowledgments}
We would like to thank Dr. O. Chubar, Dr. A. He, Dr. D. Hidas and
Dr. T. Shaftan (BNL) for discussing the undulator brightness evaluation,
and Dr. X. Huang (SLAC) for other fruitful discussions. This research
used resources of the National Synchrotron Light Source II, a
U.S. Department of Energy (DOE) Office of Science User Facility operated
for the DOE Office of Science by Brookhaven National Laboratory under
Contract No. DE-SC0012704. This work is also supported by the National
Science Foundation under Cooperative Agreement PHY-1102511, the State of
Michigan and Michigan State University.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,873 |
Upgrades version of bash found in $PATH with 4.3.27(1)-release which is the lastest official major/minor/patch level.
* Upgrade shell script: https://gist.github.com/sgviking/7bb38938187e36308175
## Requirements/Assumptions
### Host machine
* python2
* python fabric library
### Server being updated
* wget
* build tools
* gpg
* tar
* sudo access
### SELinux Warning
I hacked in quick support for SELinux systems into the above upgrade shell script. Being far from an SELinux expert you should double check my work if you are running systems with SELinux enabled. Previously the upgrade script, when ran on a SELinux enabled system, would lock you out because of the change to bash. The new script turns off SELinux, installs bash, reapplies the proper context label and turns on SELinux.
## Using fabric script
The fabric script (fabfile.py) includes two functions: version and bashup
### Setting up python virtual environment
git clone https://github.com/SocialGeeks/bashup43.git
cd bashup43/
virtualenv -p python2 .
source bin/activate
pip install -r requirements.txt
### Updating multiple hosts (roles)
You can edit fabfile.py and your hosts into env.roledefs. The roledefs can be named anything you want and you can create as many or few hosts and roledefs as needed.
fab version -R rolename | tee rolename_version.log
fab bashup -R rolename | tee rolename_bashup.log
# verify script worked
fab version -R rolename
### Updating single host
fab version -H [IP_ADDRESS]
fab bashup -H [IP_ADDRESS]
# verify script worked
fab version -H [IP_ADDRESS]
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,055 |
Trump Blasts Intel Report, Calls CNN 'Fake News' at First Presser as President-Elect
10:04 AM PST, January 11, 2017 - Inside Edition
Playing Why President-Elect Trump Refused To Answer Questions From CNN
At his first press conference in nearly six months, Donald Trump answered questions in the wake of the release of a shocking report that alleges Russia has compromising intelligence on him.
In the lead-up to his first meeting with the press as president-elect, press secretary Sean Spicer and Vice President-elect Mike Pence both denounced BuzzFeed's decision to publish the full text of a dossier that accuses Trump of engaging in "perverted" sexual acts and alleges his campaign surrogates had contact with Putin allies throughout the election.
Read: Barack Obama's Tearful Final Address as President: 'Yes We Did' [FULL TEXT]
Reading from prepared remarks, Spicer called BuzzFeed's move a "sad and pathetic attempt to get clicks." The future VP said the reports can "only attributed to media bias."
"And the American people are sick and tired of it," Pence added, before introducing Trump.
Trump immediately blasted the "nonsense" report, which he said "maybe" came from "intelligence agencies."
"It should never have been written, it should never have been had, it should never have been released," said Trump, who called it "a tremendous blot" on news organizations, namely BuzzFeed and CNN, who the Trump team says published the memos.
Following the press conference, CNN clarified that they published the briefings Trump received, not the memos or the full text dossier. BuzzFeed published the entirety of the unverified claims.
"I'm also very much of a germophobe, believe me," Trump said in an apparent reference to the sexual acts described in the report.
After answering a couple of questions regarding the Russia accusations and hacks, Trump declined to answer a reporter's question regarding his refusal to release his tax returns.
"Only ones who care about my tax returns are the reporters," Trump said, then turned the mic over to an attorney who outlined efforts to sever ties between Trump and his company.
After retaking the stage, Trump would later refer to BuzzFeed as "a failing pile of garbage."
The soon-to-be president also refused to take questions from CNN's Jim Acosta, dismissing his organization as "fake news."
Read: President Obama Serves as Groomsman-in-Chief at Aide's Wedding
Other notable moments from the press conference:
*While claiming the RNC was not successfully hacked, Trump said for the first time that Russia is responsible for the DNC hacks.
*Trump plans to hand over full control of his company to his sons, while Ivanka Trump intends to sever ties with her family's business.
*Trump reaffirmed his belief that Obamacare is a "disaster" and pushed for its full repeal and replacement.
*Also reaffirmed his commitment to building a wall along the Mexican border, for which the U.S. will be "reimbursed."
*Trump says he will donate profits from foreign government payments to his businesses to the U.S. Treasury.
*Trump's attorney said the president-elect will not completely divest from his organization because that would be "impossible."
Read: Russia Denies Having Compromising Intel on Trump, Calls Report 'Pulp Fiction'
Russia Denies Having 'Perverted' Information on Donald Trump
Serial Killer Says True Crime Writer Made Up Interview With Him | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,968 |
package org.semanticweb.cogExp.OWLFormulas;
/** Class represents integer values to be part of cardinality restrictions
*
* @author marvin
*
*/
public class OWLInteger implements OWLAtom{
private int value;
public OWLInteger(int value){
this.value = value;
}
public int getValue(){
return value;
}
@Override
public boolean equals(Object other){
if (other == null) return false;
if (other == this) return true;
if (!(other instanceof OWLInteger))return false;
boolean valsSame = value==(((OWLInteger) other).getValue());
if (!valsSame){return false;}
// System.out.println("class name equal? " + classname + " / " + ((OWLClassName) other).getName() + " AND " + ontologyname + " / " + ((OWLClassName) other).ontologyname);
return valsSame;
}
@Override
public int hashCode() {
int hash = 1;
hash = hash * 17 + this.value;
// for (OWLFormula form : this.tail){
// hash = hash * 31 + form.hashCode();
// }
return hash;
}
@Override
public String toString(){
return "|" + value + "|";
}
@Override
public OWLInteger clone(){
return new OWLInteger(value);
}
@Override
public boolean isSymb() {
return false;
}
@Override
public boolean isVar() {
return false;
}
@Override
public boolean isClassName() {
return false;
}
@Override
public boolean isIndividual() {
return false;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,598 |
TSMC hints at securing Apple SoC production deal including 2 nm chips for the next 5 years
Apple could become TSMC's largest customer in a few years. (Image Source: GSMDome)
The Taiwanese foundries are already building a new R&D center for 2 nm technologies and beyond in Hsinchu, which should be ready some time in 2021. The exact timeline for the 2 nm volume production is not yet known, but TSMC reports that it has been collaborating with some of its customers on the development of this advanced node. Market analysts claim that Apple will be first to launch 2 nm SoCs thanks to a five-year production deal with TSMC.
Bogdan Solca, Published 08/26/2020
Apple Business iPad Pro iPhone Laptop MacBook Tablet
Apart from its plans for the 5 nm, 4 nm and 3 nm nodes, TSMC also revealed new information regarding the future of its 2 nm nodes during the online technology symposium streamed on August 25.
TSMC reports that it has just begun constructing the research and development center, which will be focusing on technologies for the 2 nm nodes and beyond in Hsinchu Baoshan. This new center should be ready in 2021 and will become the company's main production site for all technologies beyond 2 nm. TSMC already invested more than NT$600 billion into this project and the new facilities are expected to accommodate at least 10,000 new employees.
Additionally, TSMC mentioned that it has been working closely with certain customers on the development of the 2 nm node, and market observers believe that one of the main customers involved in this process is Apple. AMD could also be involved, but market analysts point out that Apple will eventually become more important than AMD, as the Cupertino company is planning to design its own ARM-based laptop processors and advanced integrated / discrete GPUs.
Since Apple is fabless, it usually collaborates with TSMC's and Samsung's fabs in order to produce the SoCs needed for all the iPhone and iPad products. However, the new plans to ditch Intel's CPUs on all of its laptop and desktop products, together with all the hints at future GPU lineups suggest that Apple has chosen TSMC to be the main chip supplier, and, according to market analysts, it looks like Samsung is out of the picture for at least 5 years.
DigiTimes 1 2
MediaTek and Intel Foundry Services partner to make next-gen smart device chips 07/26/2022
TSMC reportedly planning on building six new manufacturing facilities in the U.S. 03/02/2021
Intel has reportedly outsourced some of its CPU production to TSMC 01/28/2021
Apple secures 80 percent of TSMC's 5 nm production capacity for the coming year 12/21/2020
TSMC holds an opening ceremony for its 3nm production facility 11/25/2020
Report suggests Nvidia could launch improved RTX 3000 GPUs produced on TSMC's 7 nm node in 2021 10/09/2020
Most consumers from Germany or Spain surveyed prefer to buy phones online now, whereas those from the UK do not 10/03/2020
Apple announces September 15 event as Apple Watch Series 6 and new iPad Air launch proves a no-show; iPhone 12 series due in October 09/08/2020
Apple rejected update for Facebook app that informed users of Apple's 30% cut of in-app purchases 08/28/2020
Apple certifies seven iPad model numbers with the EEC ahead of possible iPad or iPad Air refresh as alleged manual leaks 08/27/2020
120 Hz refresh rate, adaptive refresh and LiDAR camera options showcased in Apple iPhone 12 Pro Max leaks 08/26/2020
Apple may face another class-action lawsuit for display cable flaw in 2016 and 2017 MacBook Pros 08/20/2020
Apple threatens to terminate Epic's developer account 08/18/2020
An iPad that folds into an iPhone? Apple's apparently working on an eye-popping foldable for 2023, with a 3nm A16X and voice calling 08/12/2020
Intel's upcoming 10 nm and 7 nm CPUs/GPUs rumored to be produced on TSMC's 6 nm and 5 nm nodes, respectively 07/28/2020
Xperia 5 II leaks: Sony is bringing...
NVIDIA GeForce RTX 3090: 24 GB of G...
Bogdan Solca - Senior Tech Writer - 1919 articles published on Notebookcheck since 2017
I first stepped into the wondrous IT&C world when I was around seven years old. I was instantly fascinated by computerized graphics, whether they were from games or 3D applications like 3D Max. I'm also an avid reader of science fiction, an astrophysics aficionado, and a crypto geek. I started writing PC-related articles for Softpedia and a few blogs back in 2006. I joined the Notebookcheck team in the summer of 2017 and am currently a senior tech writer mostly covering processor, GPU, and laptop news.
contact me via: Facebook
> Notebook / Laptop Reviews and News > News > News Archive > Newsarchive 2020 08 > TSMC hints at securing Apple SoC production deal including 2 nm chips for the next 5 years
Bogdan Solca, 2020-08-26 (Update: 2020-08-26) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,574 |
{"url":"https:\/\/ncertmcq.com\/tag\/maths-class-8-rd-sharma-solutions\/page\/8\/","text":"## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.7\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.7\n\nOther Exercises\n\nFind the square root of the following numbers in decimal form :\n\nQuestion 1.\n84.8241\nSolution:\n\nQuestion 2.\n0.7225\nSolution:\n\nQuestion 3.\n0.813604\nSolution:\n\nQuestion 4.\n0.00002025\nSolution:\n\nQuestion 5.\n150.0625\nSolution:\n\nQuestion 6.\n225.6004\nSolution:\n\nQuestion 7.\n3600.720036\nSolution:\n\nQuestion 8.\n236.144689\nSolution:\n\nQuestion 9.\n0.00059049\nSolution:\n\nQuestion 10.\n176.252176\nSolution:\n\nQuestion 11.\n9998.0001\nSolution:\n\nQuestion 12.\n0.00038809\nSolution:\n\nQuestion 13.\nWhat is that fraction which when multiplied by itself gives 227.798649 ?\nSolution:\n\nQuestion 14.\nsquare playground is 256.6404 square metres. Find the length of one side of the playground.\nSolution:\nArea of square playground = 256.6404 sq. m\n\nQuestion 15.\nWhat is the fraction which when multiplied by itself gives 0.00053361 ?\nSolution:\n\nQuestion 16.\nSimplify :\n\nSolution:\n\nQuestion 17.\nEvaluate $$\\sqrt { 50625 }$$ and hence find the\u00a0value of $$\\sqrt { 506.25 } +\\sqrt { 5.0625 }$$.\nSolution:\n\nQuestion 18.\nFind the value of $$\\sqrt { 103.0225 }$$ and hence And the value of\n\nSolution:\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.7 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.6\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.6\n\nOther Exercises\n\nQuestion 1.\nFind the square root of:\n\nSolution:\n\nQuestion 2.\nFind the value of :\n\nSolution:\n\nQuestion 3.\nThe area of a square field is 80\u00a0$$\\frac {244 }{ 729 }$$ square metres. Find the length of each side of the field.\nSolution:\n\nQuestion 4.\nThe area of a square field is 30 $$\\frac {1 }{ 4}$$ m2. Calculate the length of the side of the squre.\nSolution:\n\nQuestion 5.\nFind the length of a side of a square playground whose area is equal to the area of a rectangular field of dimensions 72 m and 338 m.\nSolution:\nLength of rectangular field (l) = 338 m\nand breadth (b) = 72 m\n\u2234 Area = 1 x 6= 338 x 72 m2\n\u2234 Area of square = 338 x 72 m2 = 24336 m2\nand length of the side of the square\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.6 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.5\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.5\n\nOther Exercises\n\nQuestion 1.\nFind the square root of each of the\u00a0long division method.\n(I) 12544\n(ii) 97344\n(iii) 286225\n(iv) 390625\n(v) 363609\n(vi) 974169\n(vii) 120409\n(viii) 1471369\n(ix) 291600\n(x) 9653449\n(xi) 1745041\n(xii) 4008004\n(xiii) 20657025\n(xiv) 152547201\n(jcv) 20421361\n(xvi) 62504836\n(xvii) 82264900\n(xviii) 3226694416\n(xix)6407522209\n(xx) 3915380329\nSolution:\n<\n\nQuestion 2.\nFind the least number which must be subtracted from the following numbers to make them a perfect square :\n(i) 2361\n(ii) 194491\n(iii) 26535\n(iv) 16160\n(v) 4401624\nSolution:\n(i) 2361\nFinding the square root of 2361\n\nWe get 48 as quotient and remainder = 57\n\u2234 To make it a perfect square, we have to subtract 57 from 2361\n\u2234 Least number to be subtracted = 57\n(ii) 194491\nFinding the square root of 194491\n\nWe get 441 as quotient and remainder = 10\n\u2234 To make it a perfect square, we have to subtract 10 from 194491\n\u2234 Least number to be subtracted = 10\n(iii) 26535\nFinding the square root of 26535\n\nWe get 162 as quotient and 291 as remainder\n\u2234 To make it a perfect square, we have to subtract 291 from 26535\n\u2234 Least number to be subtracted = 291\n(iv)16160\nFinding the square root of 16160\n\nWe get 127 as quotient and 31 as remainder\n\u2234 To make it a perfect square, we have to subtract 31 from 16160\n\u2234 Least number to be subtracted = 31\n(v) 4401624\nFind the square root of 4401624\n\nWe get 2098 as quotient and 20 as remainder\n\u2234 To make it a perfect square, we have to subtract 20 from 4401624\n\u2234 Least number to be subtracted = 20\n\nQuestion 3.\nFind the least number which must be added to the following numbers to make them a perfect square :\n(i) 5607\n(ii) 4931\n(iii) 4515600\n(iv) 37460\n(v) 506900\nSolution:\n(i) 5607\n\nFinding the square root of 5607, we see that 742 = 5607- 131 =5476 and 752 = 5625\n\u2234 5476 < 5607 < 5625\n\u2234 5625 \u2013 5607 = 18 is to be added to get a perfect square\n\u2234 Least number to be added = 18\n(ii) 4931\n\nFinding the square root of 4931, we see that 702= 4900\n\u2234 712 = 5041 4900 <4931 <5041\n\u2234 5041 \u2013 4931 = 110 is to be added to get a perfect square.\n\u2234 Least number to be added =110\n(iii) 4515600\n\nFinding the square root of 4515600, we see\nthat 21242 = 4511376\nand 2 1 252 = 45 1 56 25\n\u2234 4511376 <4515600 <4515625\n\u2234 4515625 \u2013 4515600 = 25 is to be added to get a perfect square.\n\u2234 Least number to be added = 25\n(iv) 37460\n\nFinding the square root of 37460\nthat 1932 = 37249, 1942 = =37636\n\u2234 37249 < 37460 < 37636\n\u2234 37636 \u2013 37460 = 176 is to be added to get a perfect square.\n\u2234 Least number to be added =176\n(v) 506900\n\nFinding the square root of 506900, we see that\n7112 = 505521, 7122 = 506944\n\u2234 505521 < 506900 < 506944\n\u2234 506944 \u2013 506900 = 44 is to be added to get a perfect square.\n\u2234 Least number to be added = 44\n\nQuestion 4.\nFind the greatest number of 5 digits which is a perfect square.\nSolution:\nGreatest number of 5-digits = 99999 Finding square root, we see that 143 is left as remainder\n\n\u2234 Perfect square = 99999 \u2013 143 = 99856 If we add 1 to 99999, it will because a number of 6 digits\n\u2234 Greatest square 5-digits perfect square = 99856\n\nQuestion 5.\nFind the least number of four digits which is a perfect square.\nSolution:\nLeast number of 4-digits = 10000\nFinding square root of 1000\nWe see that if we subtract 39\n\nFrom 1000, we get three digit number\n\u2234 We shall add 124 \u2013 100 = 24 to 1000 to get a\nperfect square of 4-digit number\n\u2234 1000 + 24 = 1024\n\u2234 Least number of 4-digits which is a perfect square = 1024\n\nQuestion 6.\nFind the least number of six-digits which is a perfect square.\nSolution:\nLeast number of 6-digits = 100000\n\nFinding the square root of 100000, we see that if we subtract 544, we get a perfect square of 5-digits.\n4389 \u2013 3900 = 489\nto 100000 to get a perfect square\nPast perfect square of six digits= 100000 + 489 =100489\n\nQuestion 7.\nFind the greatest number of 4-digits which is a perfect square.\nSolution:\nGreatest number of 4-digits = 9999\n\nFinding the square root, we see that 198 has been left as remainder\n\u2234 4-digit greatest perfect square = 9999 \u2013 198 = 9801\n\nQuestion 8.\nA General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row.\nSolution:\nTotal number of soldiers = 8160 Soldiers left after arranging them in a square = 60\n\u2234 Number of soldiers which are standing in a square = 8160 \u2013 60 = 8100\n\nQuestion 9.\nThe area of a square field is 60025 m2. A man cycle along its boundry at 18 km\/hr. In how much time will be return at the starting point.\nSolution:\nArea of a square field = 60025 m2\n\nQuestion 10.\nThe cost of levelling and turfing a square lawn at Rs. 250 per m2 is Rs. 13322.50. Find the cost of fencing it at Rs. 5 per metre ?\nSolution:\nCost of levelling a square field = Rs. 13322.50\nRate of levelling = Rs. 2.50 per m2\n\nand perimeter = 4a = 4 x 73 = 292 m Rate of fencing the field = Rs. 5 per m\n\u2234 Total cost of fencing = Rs. 5 x 292 = Rs. 1460\n\nQuestion 11.\nFind the greatest number of three digits which is a perfect square.\nSolution:\n3-digits greatest number = 999\n\nFinding the square root, we see that 38 has been left\n\u2234 Perfect square = 999 \u2013 38 = 961\n\u2234 Greatest 3-digit perfect square = 961\n\nQuestion 12.\nFind the smallest number which must be added to 2300 so that it becomes a perfect square.\nSolution:\nFinding the square root of 2300\n\nWe see that we have to add 704 \u2013 700 = 4 to 2300 in order to get a perfect square\n\u2234 Smallest number to be added = 4\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.5 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.4\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.4\n\nOther Exercises\n\nQuestion 1.\nWrite the possible unit\u2019s digits of the square root of the following numbers. Which of these numbers are odd square roots ?\n(i) 9801\n(ii) 99856\n(iii) 998001\n(iv) 657666025\nSolution:\n(i)\u00a0 In $$\\sqrt { 9801 }$$ \u2234 the units digits is 1, therefore, the units digit of the square root can be 1 or 9\n(ii) In $$\\sqrt { 799356 }$$ \u2234 the units digit is 6\n\u2234 The units digit of the square root can be 4 or 6\n(iii) In $$\\sqrt { 7998001 }$$ \u2234 the units digit is 1\n\u2234 The units digit of the square root can be 1 or 9\n(iv) In 657666025\n\u2234 The unit digit is 5\n\u2234 The units digit of the square root can be 5\n\nQuestion 2.\nFind the square root of each of the following by prime factorization.\n(i) 441\n(ii) 196\n(iii) 529\n(iv) 1764\n(v) 1156\n(vi) 4096\n(vii) 7056\n(viii) 8281\n(ix) 11664\n(x) 47089\n(xi) 24336\n(xii) 190969\n(xiii) 586756\n(xiv) 27225\n(xv) 3013696\nSolution:\n\nQuestion 3.\nFind the smallest number by which 180 must be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square so obtained.\nSolution:\nFactorising 180,\n\n180 = 2 x 2 x 3 x 3 x 5\nGrouping the factors in pairs we see that factor 5 is left unpaired.\n\u2234 Multiply 180 by 5, we get the product 180 x 5 = 900\nWhich is a perfect square\nand square root of 900 = 2 x 3 x 5 = 30\n\nQuestion 4.\nFind the smallest number by which 147 must be multiplied so that it becomes a perfect square. Also, find the square root of the number so obtained.\nSolution:\nFactorising 147,\n\n147 = 3 x 7\u00d77\nGrouping the factors in pairs of the equal factors, we see that one factor 3 is left unpaired\n\u2234 Multiplying 147 by 3, we get the product 147 x 3 = 441\nWhich is a perfect square\nand its square root = 3\u00d77 = 21\n\nQuestion 5.\nFind the smallest number by which 3645 must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.\nSolution:\nFactorising 3645\n\n3645 = 3 x 3 3 x 3 x 3 x 3 x 5\nGrouping the factors in pair of the equal factors, we see t at one factor 5 is left unpaired\n\u2234 Dividing 3645 by 5, the quotient 729 will be the perfect square and square root of 729 = 27\n\nQuestion 6.\nFind the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained.\nSolution:\nFactorsing 1152,\n\n1152 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3\nGrouping the factors in pairs of the equal factors, we see that factor 2 is left unpaired.\n\u2234 Dividing by 2, the quotient 576 is a perfect square .\n\u2234 Square root of 576, it is 24\n\nQuestion 7.\nThe product of two numbers is 1296. If one number is 16 times the others find the numbers.\nSolution:\nProduct of two numbers = 1296\nLet one number = x\nSecond number = 16x\n\n\u2234 First number = 9\nand second number = 16 x 9 = 144\n\nQuestion 8.\nA welfare association collected Rs. 202500 as donation from the residents. If each paid as many rupees as there were residents find the number of residents.\nSolution:\nTotal donation collected = Rs. 202500\nLet number of residents = x\nThen donation given by each resident = Rs. x\n\u2234 Total collection = Rs. x x x\n\nQuestion 9.\nA society collected Rs. 92.16. Each member collected as many paise as there\u00a0were members. How many members were there and how much did each contribute?\nSolution:\nTotal amount collected = Rs. 92.16 = 9216 paise\nLet the number of members = x\nThen amount collected by each member = x\npaise\n\n\u2234 Number of members = 96\nand each member collected = 96 paise\n\nQuestion 10.\nA school collected Rs. 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school ?\nSolution:\nTotal fee collected = Rs. 2304\nLet number of students = x\nThen fee paid by each student = Rs. x\n\u2234 x x x = 2304 => x2 = 2304\n\u2234 x = $$\\sqrt { 2304 }$$\n\nQuestion 11.\nThe area of a square field is 5184 m2. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field.\nSolution:\nThe area of a square field = 5184 m2\nLet side of the square = x\n\n\u2234 side of square= 72 m\n\u2234 Perimeter, of square field = 72 x 4 m = 288 m\nPerimeter of rectangle = 288 m\nLet breadth of rectangular field (b) = x\nThen length (l) = 2x\n\u2234 Perimeter = 2 (l + b)\n= 2 (2x + x) = 2 x 3x = 6x\n= 2 (2x + x) = 2 x 3x = 6x\n\n\u2234 Length of rectangular field = 2x = 2 x 48 = 96 m\nand area = l x b = 96 x 48 m2\n= 4608 m2\n\nQuestion 12.\nFind the least square number, exactly divisible by each one of the numbers :\n(i) 6, 9,15 and 20\n(ii) 8,12,15 and 20\n\nSolution:\n\nLCM of 6, 9, 15, 20 = 2 x 3 x 5 x 3 x 2 = 180\n=2 x 2 x 3 x 3 x 5\nWe see that after grouping the factors in pairs, 5 is left unpaired\n\u2234 Least perfect square = 180 x 5 = 900\n\nWe see that after grouping the factors,\nfactors 2, 3, 5 are left unpaired\n\u2234 Perfect square =120 x 2 x 3 x 5 = 120 x 30 = 3600\n\nQuestion 13.\nFind the square roots of 121 and 169 by the method of repeated subtraction.\nSolution:\n\nQuestion 14.\nWrite the prime factorization of the following numbers and hence find their square roots. ^\n(i) 7744\n(ii) 9604\n(iii) 5929\n(iv) 7056\nSolution:\nFactorization, we get:\n(i) 7744 = 2 x 2 x 2 x 2 x 2 x 2 x 11 x 11\n\nGrouping the factors in pairs of equal factors,\n\nQuestion 15.\nThe students of class VIII of a school donated Rs. 2401 for PM\u2019s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.\nSolution:\nTotal amount of donation = 2401\nLet number of students in VIII = x\n\u2234 Amount donoted by each student = Rs. x\n\nQuestion 16.\nA PT teacher wants to arrange maximum possible number of 6000 students in a Held such that the number of rows is equal to the number of columns. Find the number of rows if 71 were left out after arrangement.\nSolution:\nNumber of students = 6000\nStudents left out = 71\n\u2234 Students arranged in a field = 6000 \u2013 71=5929\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.4 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.3\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.3\n\nOther Exercises\n\nQuestion 1.\nFind the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication :\n(i) 25\n(ii) 37\n(iii) 54\n(iv) 71\n(v) 96\nSolution:\n(i) (25)2\n\nQuestion 2.\nFind the squares of the following numbers using diagonal method :\n(i) 98\n(ii) 273\n(iii) 348\n(iv) 295\n(v) 171\nSolution:\n\nQuestion 3.\nFind the squares of the following numbers :\n(i) 127\n(ii) 503\n(iii) 451\n(iv) 862\n(v) 265\nSolution:\n(i) (127)2 = (120 + 7)2\n{(a + b)2 = a2 + lab + b2}\n= (120)2 + 2 x 120 x 7 + (7)2\n= 14400+ 1680 + 49 = 16129\n\n(ii) (503)2 = (500 + 3)2\n{(a + b)2 = a2 + lab + b1}\n= (500)2 + 2 x 500 x 3 + (3)2\n= 250000 + 3000 + 9 = 353009\n\n(iii) (451)2 = (400 + 51)2\n{(a + b)2 = a2 + lab + b2}\n= (400)2 + 2 x 400 x 51 + (5l)2\n= 160000 + 40800 + 2601 = 203401\n\n(iv) (451)2 = (800 + 62)2\n{(a + b)2 = a2 + lab + b2}\n= (800)2 + 2 x 800 x 62 + (62)2\n= 640000 + 99200 + 3844 = 743044\n\n(v) (265)2\n{(a + b)2 = a2 + 2ab + b2}\n(200 + 65)2 = (200)2 + 2 x 200 x 65 + (65)2\n= 40000 + 26000 + 4225 = 70225\n\nQuestion 4.\nFind the squares of the following numbers\n(i) 425\n(ii) 575\n(iii) 405\n(iv) 205\n(v) 95\n(vi) 745\n(vii) 512\n(viii) 995\nSolution:\n(i) (425)2\nHere n = 42\n\u2234 n (n + 1) = 42 (42 + 1) = 42 x 43 = 1806\n\u2234 (425)2 = 180625\n\n(ii) (575)2\nHere n = 57\n\u2234 n (n + 1) = 57 (57 + 1) = 57 x 58 = 3306\n\u2234 (575)2 = 330625\n\n(iii) (405)2\nHere n = 40\n\u2234 n (n + 1) = 40 (40 + 1) -40 x 41 = 1640\n\u2234 (405)2 = 164025\n\n(iv) (205)2\nHere n = 20\n\u2234 n (n + 1) = 20 (20 + 1) = 20 x 21 = 420\n\u2234 (205)2 = 42025\n\n(v) (95)2\nHere n = 9\n\u2234 n (n + 1) = 9 (9 + 1) = 9 x 10 = 90\n\u2234 (95)2 = 9025\n\n(vi) (745)2\nHere n = 74\n\u2234 n (n + 1) = 74 (74 + 1) = 74 x 75 = 5550\n\u2234 (745)2 = 555025\n\n(vii) (512)2\nHere a = 1, b = 2\n\u2234 (5ab)2 = (250 + ab) x 1000 + (ab)2\n\u2234 (512)2 = (250 + 12) x 1000 + (12)2\n= 262 x 1000 + 144\n= 262000 + 144 = 262144\n\n(viii) (995)2\nHere n = 99\n\u2234 n (n + 1) = 99 (99 + 1) = 99 x 100 = 9900\n\u2234 (995)2 = 990025\n\nQuestion 5.\nFind the squares of the following numbers using the identity (a + b)1 = a2 + lab + b1\n(i) 405\n(ii) 510\n(iii) 1001\n(iv) 209\n(v) 605\nSolution:\na + b)2 = a2 + lab + b2\n\n(i) (405)2 = (400 + 5)2\n= (400)2 + 2 x 400 x 5 + (5)2\n= 160000 + 4000 + 25 = 164025\n\n(ii) (510)2 = (500 + 10)2\n= (500)2 + 2 x 500 x 10 x (10)2\n= 250000 + 10000 + 100\n= 260100\n\n(iii) (1001)2 = (1000+1)2\n= (1000)2 + 2 X 1000 x 1 + (1)\n= 1000000 + 2000 + 1\n=1002001\n\n(iv) (209)2 = (200 + 9)2\n= (200)2 + 2 x 200 x 9 x (9)2\n= 40000 + 3600 +81\n= 43681\n\n(v) (605)2 = (600 + 5)2\n= (600)2 + 2 x 600 x 5 +(5)2\n= 360000 + 6000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 25\n=366025\n\nQuestion 6.\nFind the squares of the following numbers using the identity (a \u2013 b)2 = a2 \u2013 2ab + b2 :\n(i) 395\n(ii) 995\n(iii) 495\n(iv) 498\n(v) 99\n(vi) 999\n(vii) 599\nSolution:\na \u2013 b)2 = a2 \u2013 lab + b2\n\n(i) (395)2 = (400 \u2013 5)2\n= (400)2 \u2013 2 x 400 x 5 + (5)2\n= 160000-4000 + 25\n= 160025-4000\n= 156025\n\n(ii) (995)2 = (1000 \u2013 5)2\n= (1000)2 \u2013 2 x 1000 x 5 + (5)2\n= 1000000- 10000 + 25\n= 1000025- 10000\n= 990025\n\n(iii) (495)2 = (500 \u2013 5)2\n= (500)2 \u2013 2 x 500 x 5 + (5)2\n= 250000 \u2013 5000 + 25\n= 250025 \u2013 5000\n= 245025\n\n(iv) (498)2 = (500 \u2013 2)2\n= (500)2 \u2013 2 x 500 x 2 + (2)2\n= 250000 \u2013 2000 + 4\n= 250004 \u2013 2000\n= 248004\n\n(v) (99)2 = (100 \u2013 l)2\n= (100)2 \u2013 2 x 100 x 1 + (1)2\n= 10000 \u2013 200 + 1\n= 10001 \u2013 200\n= 9801\n\n(vi) (999)2 = (1000- l)2\n= (1000)2 \u2013 2 x 1000 x 1+ (1)2\n= 1000000-2000+1\n= 10000001-2000=998001\n\n(vii)\u00a0(599)2 = (600 \u2013 1)2\n= (600)-2 x 600 X 1+ (1)2\n= 360000 -1200+1\n= 360001 \u2013 1200 = 358801\n\nQuestion 7.\nFind the squares of the following numbers by visual method :\n(i) 52\n(ii) 95\n(iii) 505\n(iv) 702\n(v) 99\nSolution:\n(a + b)2 = a2 \u2013 ab + ab + b2\n(i) (52)2 = (50 + 2)2\n= 2500 + 100 + 100 + 4\n= 2704\n\n(ii) (95)2 = (90 + 5)2\n= 8100 + 450 + 450 + 25\n= 9025\n\n(iii) (505)2 = (500 + 5)2\n= 250000 + 2500 + 2500 + 25\n= 255025\n\n(iv) (702)2 = (700 + 2)2\n= 490000 + 1400+ 1400 + 4\n= 492804\n\n(v) (99)2 = (90 + 9)2\n= 8100 + 810 + 810 + 81\n= 9801\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.3 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.2\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.2\n\nOther Exercises\n\nQuestion 1.\nThe following numbers are not perfect squares. Give reason :\n(i) 1547\n(ii) 45743\n(iii) 8948\n(iv) 333333\n\nSolution:\nWe know that if the units digit is 2, 3, 7 or 8 of a number, then the number is not a perfect square.\n\n(i) \u2234 1547 has 7 as units digit.\n\u2234 It is not a perfect square.\n\n(ii) 45743 has 3 as units digit\n\u2234 It is not a perfect square.\n\n(iii) \u00a0\u2234 8948 has 8 as units digit\n\u2234 It is not a perfect square.\n\n(iv) \u00a0\u2234 333333 has 3 as units digits\n\u2234 It is not a perfect square.\n\nQuestion 2.\nShow that the following numbers are not perfect squares :\n(i) 9327\n(ii) 4058\n(iii) 22453\n(iv) 743522\n\nSolution:\n(i) 9327\n\u2234 The units digit of 9327 is 7\n\u2234 This number can\u2019t be a perfect square.\n\n(ii) 4058\n\u2234 The units digit of 4058 is 8\n\u2234 This number can\u2019t be a perfect square.\n\n(iii) 22453\n\u2234 The units digit of 22453 is 3\n.\u2234 This number can\u2019t be a perfect square.\n\n(iv) 743522\n\u2234 The units digit of 743522 is 2\n\u2234 This number can\u2019t be a perfect square.\n\nQuestion 3.\nThe square of which of the following numbers would be an odd number ?\n(i) 731\n(ii) 3456\n(iii) 5559\n(iv) 42008\nSolution:\nWe know that the square of an odd number is odd and of even number is even. Therefore\n(i) Square of 731 would be odd as it is an odd number.\n(ii) Square of 3456 should be even as it is an even number.\n(iii) Square of 5559 would be odd as it is an odd number.\n(iv) The square of 42008 would be an even number as it is an even number.\nTherefore suqares of (i) 731 and (ii) 5559 will be odd numbers.\n\nQuestion 4.\nWhat will be the units digit of the squares of the following numbers ?\n(i) 52\n(ii) 977\n(iii) 4583\n(iv) 78367\n(v) 52698\n(vi) 99880\n(vii) 12796\n(viii) 55555\n(ix) 53924\n\nSolution:\n\n(i) Square of 52 will be 2704 or (2)2 = 4\n\u2234 Its units digit is 4.\n\n(ii) Square of 977 will be 954529 or (7)2 = 49 .\n\u2234 Its units digit is 9\n\n(iii) Square of 4583 will be 21003889 or (3)2 = 9\n\u2234 Its units digit is 9\n\n(iv) IS 78367, square of 7 = 72 = 49\n\u2234 Its units digit is 9\n\n(v) In 52698, square of 8 = (8)2 = 64\n\u2234 Its units digit is 4\n\n(vi) In 99880, square of 0 = 02 = 0\n\u2234 Its units digit is 0\n\n(vii) In 12796, square of 6 = 62 = 36\n\u2234 Its units digit is 6\n\n(viii) In In 55555, square of 5 = 52 = 25\n\u2234 Its units digit is 5\n\n(ix) In 53924, square pf 4 = 42 = 16\n\u2234 Its units digit is 6\n\nQuestion 5.\nObserve the following pattern\n1 + 3 = 22\n1 + 3 + 5 = 32\n1+34-5 + 7 = 42\nand write the value of 1 + 3 + 5 + 7 + 9 +\u2026\u2026\u2026\u2026upto n terms.\nSolution:\nThe given pattern is\n1 + 3 = 22\n1 + 3 + 5 = 32\n1+3 + 5 + 7 = 42\n1+3 + 5 + 7 + 9 +\u2026\u2026\u2026\u2026\u2026\u2026 upto n terms (number of terms)2 = n2\n\nQuestion 6.\nObserve the following pattern :\n22 \u2013 12 = 2 + 1\n32 \u2013 22 = 3 + 2\n42\u00a0\u2013 32 = 4 + 3\n52\u00a0\u2013 42 = 5 + 4\nFind the value of\n(i) 1002 \u2013 992\n(ii) 1112 \u2013 1092\n(iii) 992 \u2013 962\nSolution:\nFrom the given pattern,\n22 \u2013 12 = 2 + 1\n32 \u2013 22 = 3 + 2\n42 \u2013 32 = 4 + 3\n52 \u2013 42 = 5 + 4\nTherefore\n(i) 1002-99\u00b0 = 100 + 99\n\n(ii) 1112 \u2013 1092 = 1112 \u2013 1102\u2013 1092\n= (1112 \u2013 1102) + (1102 \u2013 1092)\n= (111 + 110) + (110+ 109)\n= 221 + 219 = 440\n\n(iii) 992 \u2013 962 = 992 \u2013 982 + 982 \u2013 972 + 972 \u2013 962\n= (992 \u2013 982) + (982 \u2013 972) + (972 \u2013 962)\n= (99 + 98) + (98 + 97) + (97 + 96)\n= 197 + 195 + 193 = 585\n\nQuestion 7.\nWhich of the following triplets are Pythagorean ?\n(i) (8, 15, 17)\n(ii) (18, 80, 82)\n(iii) (14, 48, 51)\n(iv) (10, 24, 26)\n(vi) (16, 63, 65)\n(vii) (12, 35, 38)\nSolution:\nA pythagorean triplet is possible if (greatest number)2 = (sum of the two smaller numbers)\n\n(i) 8, 15, 17\nHere, greatest number =17\n\u2234 (17)2 = 289\nand (8)2 + (15)2 = 64 + 225 = 289\n\u2234 82 + 152 = 172\n\u2234 8, 15, 17 is a pythagorean triplet\n\n(ii) 18, 80, 82\nGreatest number = 82\n\u2234 (82)2 = 6724\nand 182 + 802 = 324 + 6400 = 6724\n\u2234 182 + 802 = 822\n\u2234 18, 80, 82 is a pythagorean triplet\n\n(iii) 14, 48, 51\nGreatest number = 51\n\u2234 (51)2 = 2601\nand 142 + 482 = 196 + 2304 = 25 00\n\u2234 512\u2260\u00a0142 + 482\n\u2234 14, 48, 51 is not a pythagorean triplet\n\n(iv) 10, 24, 26\nGreatest number is 26\n\u2234 262 = 676\nand 102 + 242 = 100 + 576 = 676\n\u2234 262 = 102 + 242\n\u2234 10, 24, 26 is a pythagorean triplet\n\n(vi) 16, 63, 65\nGreatest number = 65\n\u2234 652 = 4225\nand 162 + 632 = 256 + 3969 = 4225\n\u2234 652 = 162 + 632\n\u2234 16, 63, 65 is a pythagorean triplet\n\n(vii) 12, 35, 38\nGreatest number = 38\n\u2234 382 = 1444\nand 122 + 352 = 144 + 1225 = 1369\n\u2234 382 \u2260122 + 352\n\u2234 12, 35, 38 is not a pythagorean triplet.\n\nQuestion 8.\nObserve the following pattern\n\nSolution:\nFrom the given pattern\n\nQuestion 9.\nObserve the following pattern\n\nand find the values of each of the following :\n(i) 1 + 2 + 3 + 4 + 5 +\u2026.. + 50\n(ii) 31 + 32 +\u2026 + 50\nSolution:\nFrom the given pattern,\n\nQuestion 10.\nObserve the following pattern\n\nand find the values of each of the following :\n(i) 12 + 22 + 32 + 42 +\u2026\u2026\u2026\u2026\u2026 + 102\n(ii) 52 + 62 + 72 + 82 + 92 + 102 + 12\n2\nSolution:\nFrom the given pattern,\n\nQuestion 11.\nWhich of the following numbers are squares of even numbers ?\n121,225,256,324,1296,6561,5476,4489, 373758\nSolution:\nWe know that squares of even numbers is also are even number. Therefore numbers 256, 324,1296, 5476 and 373758 have their units digit an even number.\n\u2234 These are the squares of even numbers.\n\nQuestion 12.\nBy just examining the units digits, can you tell which of the following cannot be whole squares ?\n\n1. 1026\n2. 1028\n3. 1024\n4. 1022\n5. 1023\n6. 1027\n\nSolution:\nWe know that a perfect square cam at ends with the digit 2, 3, 7, or 8\n\u2234 By examining the given number, we can say that 1028, 1022, 1023, 1027 can not be perfect squares.\n\nQuestion 13.\nWrite five numbers for which you cannot decide whether they are squares.\nSolution:\nA number which ends with 1,4, 5, 6, 9 or 0\ncan\u2019t be a perfect square\n2036, 4225, 4881, 5764, 3349, 6400\n\nQuestion 14.\nWrite five numbers which you cannot decide whether they are square just by looking at the unit\u2019s digit.\nSolution:\nA number which does not end with 2, 3, 7 or 8 can be a perfect square\n\u2234 The five numbers can be 2024, 3036, 4069, 3021, 4900\n\nQuestion 15.\nWrite true (T) or false (F) for the following statements.\n(i) The number of digits in a square number is even.\n(ii) The square of a prime number is prime.\n(iii) The sum of two square numbers is a square number.\n(iv) The difference of two square numbers is a square number.\n(vi) The product of two square numbers is a square number.\n(vii) No square number is negative.\n(viii) There is not square number between 50 and 60.\n(ix) There are fourteen square number upto 200.\n\nSolution:\n(i) False : In a square number, there is no condition of even or odd digits.\n(ii) False : A square of a prime is not a prime.\n(iii) False : It is not necessarily.\n(iv) False : It is not necessarily.\n(vi) True.\n(vii) True : A square is always positive.\n(viii) True : As 72 = 49, and 82 = 64.\n(ix) True : As squares upto 200 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 which are fourteen in numbers.\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.2 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1\n\nOther Exercises\n\nQuestion 1.\nWhich of the following numbers are perfect squares ?\n(i)484\n(ii) 625\n(iii) 576\n(iv) 941\n(v) 961\n(vi) 2500\nSolution:\n\nGrouping the factors in pairs, we have left no factor unpaired\n\u2234 484 is a perfect square of 22\n\n\u2234 Grouping the factors in pairs, we have left no factor unpaired\n\u2234 625 is a perfect square of 25.\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 576 is a perfect square of 24\n(iv) 941 has no prime factors\n\n\u2234 941 is not a perfect square.\n(v) 961 =31 x 31\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 961 is a perfect square of 31\n\nGrouping the factors in pairs, we see that no factor is left impaired\n\u2234 2500 is a perfect square of 50 .\n\nQuestion 2.\nShow that each of the following* numbers is a perfect square. Also find the number whose square is the given number in each case :\n(i) 1156\n(ii) 2025\n(iii) 14641\n(iv) 4761\nSolution:\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 1156 is a perfect square of 2 x 17 = 34\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n2025 is a perfect square of 3 x 3 x 5 =45\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 14641 is a perfect square of 11\u00d711 = 121\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 4761 is a perfect square of 3 x 23 = 69\n\nQuestion 3.\nFind the smallest number by which the given number must be multiplied so that the product is a perfect square.\n(i) 23805\n(ii) 12150\n(iii) 7688\nSolution:\n\nGrouping the factors in pairs of equal factors, we see that 5 is left unpaird\n\u2234 In order to complete the pairs, we have to multiply 23805 by 5, then the product will be the perfect square.\nRequid smallest number = 5\n(ii) 12150 = 2 x 3 x 3\u00d73 x 3\u00d73 x 5\u00d75\n\nGrouping the factors in pairs of equal factors, we see that factors 2 and 3 are left unpaired\n\u2234 In order to complete the pairs, we have to multiply 12150 by 2 x 3 =6 i.e., then the product will be the complete square.\n\u2234 Required smallest number = 6\n\nGrouping the factors in pairs of equal factors, we see that factor 2 is left unpaired\n\u2234 In order to complete the pairs we have to multiply 7688 by 2, then the product will be the complete square\n\u2234 Required smallest number = 2\n\nQuestion 4.\nFind the smallest number by which the given number must be divided so that the resulting number is a perfect square.\n(i) 14283\n(ii) 1800\n(iii) 2904\nSolution:\n\nGrouping the factors in pairs of equal factors, we see that factors we see that 3 is left unpaired\nDeviding by 3, the quotient will the perfect square.\n\nGrouping the factors in pair of equal factors, we see that 2 is left unpaired.\n\u2234 Dividing by 2, the quotient will be the perfect square.\n\nGrouping the factors in pairs of equal factors, we see that 2 x 3 we left unpaired\n\u2234 Dividing by 2 x 3 = 6, the quotient will be the perfect square.\n\nQuestion 5.\nWhich of the following numbers are perfect squares ?\n11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121\nSolution:\n11 is not a perfect square as 11 = 1 x 11\n12 is not a perfect square as 12 = 2\u00d72\u00a0x 3\n16 is a perfect square as 16 = 2\u00d72 x 2\u00d72\n32 is not a perfect square as 32 = 2\u00d72 x 2\u00d72 x 2\n36 is a perfect square as 36 =\u00a02\u00d72\u00a0x\u00a03\u00d73\n50 is not a perfect square as 50 = 2 x 5\u00d75\n64 is a perfect square as 64 = 2\u00d72 x 2\u00d72 x 2\u00d72\n79 is not a perfect square as 79 = 1 x 79\n81 is a perfect square as 81 = 3\u00d73 x 3\u00d73\n111 is not a perfect square as 111 = 3 x 37\n121 is a perfect square as 121 = 11 x 11\nHence 16, 36, 64, 81 and 121 are perfect squares.\n\nQuestion 6.\nUsing prime factorization method, find which of the following numbers are perfect squares ?\n\u2234 189,225,2048,343,441,2916,11025,3549\nSolution:\n\nGrouping the factors in pairs, we see that are 3 and 7 are left unpaired\n\u2234 189 is not a perfect square\n\nGrouping the factors in pairs, we see no factor left unpaired\n\u2234 225 is a perfect square\n\nGrouping the factors in pairs, we see no factor left unpaired\n\u2234 2048 is a perfect square\n\nGrouping the factors in pairs, we see that one 7 is left unpaired\n\u2234 343 is not a perfect square.\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 441 is a perfect square.\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 2916 is a perfect square.\n\nGrouping the factors in pairs, we see that no factor is left unpaired\n\u2234 11025 is a perfect square.\n\nGrouping the factors in pairs, we see that 3, no factor 7 are left unpaired\n\u2234 3549 is a perfect square.\n\nQuestion 7.\nBy what number should each of the following numbers be multiplied to get a perfect square in each case ? Also, find the number whose square is the new number.\n(i) 8820\n(ii) 3675\n(iii) 605\n(iv) 2880\n(v) 4056\n(vi) 3468\nSolution:\n\nGrouping the factors in pairs, we see that 5 is left unpaired\n\u2234 By multiplying 8820 by 5, we get the perfect square and square root of product will be\n= 2 x 3 x 5 x 7 = 210\n\nGrouping the factors in pairs, we see that 3 is left unpaired\n\u2234 Multiplying 3675 by 3, we get a perfect square and square of the product will be\n= 3 x 5 x 7 = 105\n\nGrouping the factors in pairs, we see that 5 is left unpaired\n\u2234 Multiplying 605 by 5, we get a perfect square and square root of the product will be\n= 5 x 11 =55\n\nGrouping the factors in pairs, we see that 5 is left unpaired\n\u2234 Multiplying 2880 by 5, we get the perfect square.\nSquare rooi of product will be = 2 x 2 * 2 \u2013 3 x 5 = 120\n\nGrouping the factors in pairs, we see that 2 and 3 are left unpaired\n\u2234 Multiplying 4056 by 2 x 3 i.e., 6, we get the perfect square.\nand square root of the product will be\n= 2 x 2 x 3 x 13 = 156\n\nGrouping the factors in pairs, we see that 3 is left unpaired\n\u2234 Multiplying 3468 by 3 we get a perfect square, and square root of the product will be 2 x 3 x 17 = 102\n\nGrouping the factors in pairs, we see that 2 and 3 are left unpaired\n\u2234 Multiplying 7776 by 2 x 3 or 6 We get a perfect square and square root of the product will be\n= 2 x 2 x 2 x 3 x 3 x 3 = 216\n\nQuestion 8.\nBy what numbers should each of the following be .divided to get a perfect square in each case ? Also find the number whose square is the new number.\n(i) 16562\n(ii) 3698\n(iii) 5103\n(iv) 3174\n(v) 1575\nSolution:\n\nGrouping the factors in pairs, we see that 2 is left unpaired\n\u2234 Dividing by 2, we get the perfect square and square root of the quotient will be 7 x 13 = 91\n\nGrouping the factors in pairs, we see that 2 is left unpaired,\n\u2234 Dividing 3698 by 2, the quotient is a perfect square\nand square of quotient will be = 43\n\nGrouping the factors in pairs, we see that 7 is left unpaired\n\u2234 Dividing 5103 by 7, we get the quotient a perfect square.\nand square root of the quotient will be 3 x 3 x 3 = 27\n\nGrouping the factors iq pairs, we see that 2 and 3 are left unpaired\n\u2234 Dividing 3174 by 2 x 3 i.e. 6, the quotient will be a perfect square and square root of the quotient will be = 23\n\nGrouping the factors in pairs, we find that 7 is left unpaired i\n\u2234 Dividing 1575 by 7, the quotient is a perfect square\nand square root of the quotient will be = 3 x 5 = 15\n\nQuestion 9.\nFind the greatest number of two digits which is a perfect square.\nSolution:\nThe greatest two digit number = 99 We know, 92 = 81 and 102 = 100 But 99 is in between 81 and 100\n\u2234 81 is the greatest two digit number which is a perfect square.\n\nQuestion 10.\nFind the least number of three digits which is perfect square.\nSolution:\nThe smallest three digit number =100\nWe know that 92 = 81, 102 = 100, ll2 = 121\nWe see that 100 is the least three digit number which is a perfect square.\n\nQuestion 11.\nFind the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.\nSolution:\nBy factorization:\n\nGrouping the factors in pairs, we see that 11 is left unpaired\n\u2234 The least number is 11 by which multiplying 4851, we get a perfect square.\n\nQuestion 12.\nFind the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.\nSolution:\nBy factorization,\n\nGrouping the factors in pairs, we see that 13 is left unpaired\n\u2234 Dividing 28812 by 3, the quotient will be a perfect square.\n\nQuestion 13.\nFind the smallest number by which 1152 must be divided so that it becomes a perfect square. Also find the number whose square is the resulting number.\nSolution:\nBy factorization,\n\nGrouping the factors in pairs, we see that one 2 is left unpaired.\n\u2234 Dividing 1152 by 2, we get the perfect square and square root of the resulting number 576, will be 2 x 2 x 2 x 3 = 24\n\nHope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0MCQS\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0MCQS\n\nOther Exercises\n\nChoose the correct alternative in each of the following :\n\nQuestion 1.\nSquare of $$\\left( \\frac { -2 }{ 3 } \\right)$$\n\nSolution:\n\nQuestion 2.\nCube of $$\\frac { -1 }{ 2 }$$ is\n\nSolution:\n\nQuestion 3.\nWhich of the following is not equal to\n\nSolution:\n\nQuestion 4.\nWhich of the following in not reciprocal of\n\nSolution:\n\nQuestion 5.\nWhich of the following numbers is not equal to $$\\frac { -8 }{ 27 }$$ ?\n\nSolution:\n\nQuestion 6.\n\nSolution:\n\nQuestion 7.\n\nSolution:\n\nQuestion 8.\n\nSolution:\n\nQuestion 9.\n\nSolution:\n\nQuestion 10.\n\nSolution:\n\nQuestion 11.\n\nSolution:\n\nQuestion 12.\n\nSolution:\n\nQuestion 13.\n\nSolution:\n\nQuestion 14.\n\nSolution:\n\nQuestion 15.\nFor any two non-zero rational numbers a and b,a4+b4 is equal to\n(a) (a + b)1\n(b) (a + b)0\n(c) (a + b)4\n(d) (a + b)8\nSolution:\n(c) {\u2235 a4 + b4 = (a + b)4}\n\nQuestion 16.\nFor any two rational numbers a and b, a5\u00a0x b5 is equal to\n(a) (a x b)0\n(b) (a x b)10\n\n(c) (a x b)5\n(d) (a x b)25\n\nSolution:\n(c) {\u2235 a5 x b5 = (a x b)5}\n\nQuestion 17.\nFor a non-zero rational number a, a7 + a12 is equal to\n(a) a5\n(b) a-19\n\n(c) a-5\n(d) a19\n\nSolution:\n(c) {a5 a12 = a7-12\u00a0=a-5}\n\nQuestion 18.\nFor a non-zero rational number a, (a3)-2 is equal to\n(a) a6\n(b) a-6\n(c) a-9\n(d) a1\n\nSolution:\n(b) {(a3)-2 = a3 x (-2)= a6}\n\nHope given RD Sharma Class 8 Solutions Chapter 2 Powers MCQS\u00a0are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.3\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.3\n\nOther Exercises\n\nQuestion 1.\nExpress the following numbers in standard form :\n(i) 6020000000000000\n(ii) 0.00000000000942\n(iii) 0.00000000085\n(iv) 846 X 107\n(v) 3759 x 10-4\n(vi) 0.00072984\n(vii) 0.000437 x 104\n(Viii) 4 + 100000\nSolution:\n\nQuestion 2.\nWrite the following numbers in the usual form :\n(i) 4.83 x 107\n(ii) 3.02 x 10-6\n(iii) 4.5 x 104\n\n(iv) 3 x 10-8\n(v) 1.0001 x 109\n(vi) 5.8 x 102\n(vii) 3.61492 x 106\n(viii) 3.25 x 10-7\nSolution:\n\nHope given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.3 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.2\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.2\n\nOther Exercises\n\nQuestion 1.\nWrite each of the following in exponential form :\n\nSolution:\n\nQuestion 2.\nEvaluate :\n\nSolution:\n\nQuestion 3.\nExpress each of the following as a rational number in the form $$\\frac { p }{ q } :$$\n\nSolution:\n\nQuestion 4.\nSimplify :\n\nSolution:\n\nQuestion 5.\nExpress each of the following rational numbers with a negative exponent :\n\nSolution:\n\nQuestion 6.\nExpress each of the following rational numbers with a positive exponent :\n\nSolution:\n\nQuestion 7.\nSimplify :\n\nSolution:\n\nQuestion 8.\nBy what number should 5-1 be multiplied so that the product may be equal to (-7)-1 ?\nSolution:\n\nQuestion 9.\nBy what number should $${ \\left( \\frac { 1 }{ 2 } \\right) }^{ -1 }$$ be multiplied so that the product may be equal to $${ \\left( \\frac { -4 }{ 7 } \\right) }^{ -1 }$$ ?\nSolution:\n\nQuestion 10.\nBy what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1 ?\nSolution:\n\nQuestion 11.\nBy what number should $${ \\left( \\frac { 5 }{ 3 } \\right) }^{ -2 }$$ be multiplied so that the product may be $${ \\left( \\frac { 7 }{ 3 } \\right) }^{ -1 }$$ ?\nSolution:\n\nQuestion 12.\nFind x, if\n\nSolution:\n\nQuestion 13.\n\nSolution:\n\nQuestion 14.\nFind the value of x for which 52x + 5-3 = 55.\nSolution:\n\nHope given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.2 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.\n\n## RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.1\n\nThese Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.1\n\nOther Exercises\n\nQuestion 1.\nExpress each of the following as a rational number of the form $$\\frac { p }{ q } ,$$ where p and q are integers and q\u2260 0 :\n\nSolution:\n\nQuestion 2.\nFind the values of each of the following\n\nSolution:\n\nQuestion 3.\nFind the values of each of the following :\n\nSolution:\n\nQuestion 4.\nSimplify :\n\nSolution:\n\nQuestion 5.\nSimplify :\n\nSolution:\n\nQuestion 6.\nBy what number should\u00a05-1 be multiplied so that the product may be equal to(-7)-1 ?\nSolution:\n\nQuestion 7.\nBy what number should $${ \\left( \\frac { 1 }{ 2 } \\right) }^{ -1 }$$ multiplied so that the product many be equal to $${ \\left( \\frac { -4 }{ 7 } \\right) }^{ -1 }$$\nSolution:\n\nQuestion 8.\nBy what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1 ?\nSolution:\n\nHope given RD Sharma Class 8 Solutions Chapter 2 Powers\u00a0Ex 2.1 are helpful to complete your math homework.\n\nIf you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.","date":"2022-11-27 07:59:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6004008650779724, \"perplexity\": 1364.1628251591972}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710218.49\/warc\/CC-MAIN-20221127073607-20221127103607-00838.warc.gz\"}"} | null | null |
package com.tassioauad.gamecheck.model.api.asynctask;
import android.content.Context;
import android.os.AsyncTask;
import com.tassioauad.gamecheck.R;
public abstract class GenericAsyncTask<PARAM, POGRESS, RETURN> extends AsyncTask<PARAM, POGRESS, AsyncTaskResult<RETURN>> {
private Context context;
private ApiResultListener apiResultListener;
public GenericAsyncTask(Context context) {
this.context = context;
}
public void setApiResultListener(ApiResultListener listener) {
this.apiResultListener = listener;
}
public Context getContext() {
return context;
}
public String getBaseUrl() {
return context.getString(R.string.giantbombapi_baseurl);
}
public String getApiKey() {
return context.getString(R.string.giantbombapi_key);
}
@Override
protected void onPostExecute(AsyncTaskResult<RETURN> returnAsyncTaskResult) {
if (returnAsyncTaskResult.getResult() != null) {
apiResultListener.onResult(returnAsyncTaskResult.getResult());
} else if (returnAsyncTaskResult.getError() != null) {
apiResultListener.onException(returnAsyncTaskResult.getError());
} else {
apiResultListener.onResult(returnAsyncTaskResult.getResult());
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 101 |
{"url":"https:\/\/congresso.sif.it\/talk\/284","text":"# Beyond the surface: Raman micro-SORS for in depth non-destructive analysis of fresco layers.\n\nChiriu D., Ricci P.C., Fiorino D.R., Grillo S., Carbonaro C.M.\n\nComunicazione\nVI - Fisica applicata, acceleratori e beni culturali\nGSSI Ex ISEF - Aula A - Mercoled\u00ec 25 h 10:00 - 13:00\nWe applied microscale spatially offset Raman spectroscopy (micro-SORS) to perform the analysis of the layer composition in a fresco of the San Giuseppe church in Cagliari. Standard Raman analysis of the surface evidenced the presence of expected pigments in the final plaster layer, such as hematite and gypsum, beside the contribution of calcite. The latter is observed in its anhydrous form on the surface (main Raman peak at 1024 cm${}^{-1}$) and in its hydrate form (1006 cm${}^{-1}$) from the substrate layers. With micro-SORS analysis we evaluated the thickness of the different layers of the plaster by monitoring the intensity ratio of the calcite and the anhydrous gypsum. Four different layers with separation lines at about 80 $\\mu$m, 600 $\\mu$m and 2400 $\\mu$m in depth were evidenced, corresponding to the painting layer (down to 80 $\\mu$m), the plaster (down to 600 $\\mu$m), the arriccio layer (down to 2400 $\\mu$), and the rinzaffo layer (below 2400 $\\mu$m). The cross-sectional analysis by SEM imaging of a fragment from the fresco confirmed the micro-SORS findings, promoting the technique of a new non-invasive imaging tool for cultural heritage.","date":"2021-09-16 17:52:57","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\": 0.572219729423523, \"perplexity\": 12134.175177818293}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780053717.37\/warc\/CC-MAIN-20210916174455-20210916204455-00218.warc.gz\"}"} | null | null |
I have a Sunpak 120j ttl with the Nikon ttl module. When I try to use it on my D70 the shutter locks and I get error code r10 in the panel. Any suggestions??
RXX on the LCD normally indicates how many exposures the buffer can currently hold. I'm not sure why you're getting this problem but try the camera on a different quality setting to see if you still get r10 or if the message changes.
Will sunpak 622 flash work with Nikon 5200?
I've got a D70 and never had this issue. I bought mine around the same time frame. Thanks for the info!
According to what I read the sunpak dx-8r is, with the right module, compatable with digital SLRs. Check with B&H photo to get the right module.
The Quantaray Digital NK Module has a small tab on the side of the raised boss that the 8 pins sit on that prevents it from mounting on the QTB-9500A. It's intentional- the NK Digital Module is made for newer Quantaray flashes only. You'll need the older-style AF NK Module to attach to the QTB-9500A.
Sunpack pz40X II flash - compatible with Nikon D40?
may b ur settings are not rit in the cam. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,408 |
Q: How do I display an initial value and display previous value once user has submitted the form This does not show an initial value but keeps previous data:
<input type="text" name="email" value="">
and this does not store previous value but shows an initial value of 1:
<input type="text" name="email" value="1">
Can't I have both?
What about the select element?
<select name="list">
<option value="a">a</option>
<option value="b">b</option>
<option value="c">c</option>
<option value="d">d</option>
</select>
A: You'll need to capture the old value before it's changed. One way to do that is:
*
*When a object gets focus, get its value in a temporary variable.
*If the object value is changed, then get the value in other variable.
var oldValue = null;
var newValue = null;
$(".myClass").focus(function() {
oldValue= this.value; // "this" is the <input> element
});
$( ".myClass" ).change(function() {
newValue= this.value;
});
$('.myClass').keyup(function() {
$('#newValue').html('<b>' + $(this).val() + '</b>');
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 567 |
Former Tuscarawas County funeral director accused of stealing $750K pleads guilty
Martin Feldner could face up to 38 years in prison
Published: 6:34 PM EDT June 3, 2019
Updated: 6:34 PM EDT June 3, 2019
NEW PHILADELPHIA, Ohio (AP) — A former funeral director accused of stealing more than $750,000 from an Ohio funeral home has pleaded guilty to charges including aggravated theft and forgery.
The Times-Reporter in New Philadelphia reports 55-year-old Martin Feldner pleaded guilty to nine counts last week.
He could face up to 38 years in prison. His sentencing hasn't been scheduled.
A message seeking comment was left Monday for his attorney, Kevin Cox.
Feldner was accused of embezzling from Addy Funeral Home between 2010 and 2018. The alleged theft was discovered after his employment there.
In a related civil case, Feldner has been ordered to pay the funeral home more than $2.3 million in reimbursement, damages and fees. Claims related to Feldner's wife are still active in that case. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,475 |
How Do You Incorporate an Entirely Digital Corporation? (Updated)
How Do You Incorporate an Entirely Digital Corporation? (Updated) This version of the paper was updated to reflect the passage of the Wyoming DAO statute. The presentation of the paper was made on November 5, 2021 in Houston, Texas. http://ronaldchichester.com/presentations/2021-adv-bus-law_business-legal-issues-with-virtual-currencies-part-1_chichester.pdf/view http://ronaldchichester.com/@@site-logo/Website_Logo.png
This version of the paper was updated to reflect the passage of the Wyoming DAO statute. The presentation of the paper was made on November 5, 2021 in Houston, Texas.
2021 Adv Bus Law_Business Legal Issues With Virtual Currencies-Part 1_Chichester.pdf — 211.3 KB
Distributed Autonomous Organizations | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,710 |
using System.Web.Mvc;
namespace MyBlogWeb
{
public class FilterConfig
{
public static void RegisterGlobalFilters(GlobalFilterCollection filters)
{
filters.Add(new XCLCMS.Lib.Filters.ExceptionFilter());
filters.Add(new HandleErrorAttribute());
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,647 |
Home » News » Student Project Pinwheel Cartwheels to Mobile and PC
Student Project Pinwheel Cartwheels to Mobile and PC
"Student Game Developers: Got Game?" is the question asked on the clay.io page, looking for fledgling game makers to post their creations. The challenge? Create an HTML 5 game that's mobile-friendly.
Patrick Pistor's entry for the contest is Pinwheel, a platformer in which the player controls a pinwheel. The goal is to go right, indefinitely, but it's not that easy. There's a bunch of obstacles on the way: Stationary spikes, moving sawblades, and even mines that explode after a delay. It's challenging, but rewarding. There are coins and rare gems, with the latter being especially helpful, because they revive on the spot.
The levels are always randomly generated, which goes a long way toward keeping the experience fresh. Additionally, there are multiple colors for the pinwheel, as well as season changes, offering Fall, Summer, Winter, and Spring themes.
The game is officially out now, and you can play it from Pistro's blog. He also plans to release the game to the Web through itch.io, for Windows, Mac, and Linux. Eventually, he hopes to get Pinwheel on the Firefox marketplace and Chrome Webstore.
Tags: Indie Pc Games
Luke Siuty
Luke has wide interests in games, from compelling fighting, action, and RPG titles to deeper interactive, storytelling titles that push today's genres and boundaries - especially awesome if they're related to diversity. Feel free to reach out on Twitter or via email.
The Top 5 Mods for Civilization VI in 2020 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,798 |
{"url":"http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial","text":"# Mod-Making Crash Course Tutorial\n\n## Mod-Making Crash Course Tutorial\n\nPART 1 - Instantiating your Mod\n\nNote: Please be sure to follow this tutorial first as this one directly continues from its last step:\u00a0http:\/\/bigsister.forumotion.com\/t891-setup-modding-environment-windows\n\n1. Delete the package \"com.example.examplemod\" as we no longer need this and it may get in the way.\n\n2. Right-click on the \"src\/main\/java\" folder and select New > Package.\n\n3. In the New Java Package window, in the \"Name\" field add a name for your new mod package. \u00a0You can call it what you like, but to be consistent with other mods try to use the \"com.yourname.yourmodname\" convention. \u00a0So for this tutorial, I will call this:\n\ncom.bigsis.spookjams\n\n...obviously tailor it to your own preference.\nOnce you're done, click the Finish button.\n\n4. Repeat the process in #3 to create a separate package called:\n\ncom.bigsis.spookjams.proxy\n\nThis will be important later when we create server interfaces for our mod, but for now this has to be here as a placeholder.\n\n5. Right-click on \"com.bigsis.spookjams.proxy\" and choose\u00a0New\u00a0> Interface. In the \"Name\" field of the New Java Class window, type in \"CommonProxy\" and press\u00a0Finish. No\u00a0further edits are needed in this interface class at this time.\n\n6. Right-click on \"com.bigsis.spookjams.proxy\" and choose\u00a0New\u00a0> Class. In the \"Name\" field\u00a0of the New Java Class window, type in \"ClientProxy\" and press Finish. Edit the contents of \"ClientProxy.java\" to read as follows:\n\nCode:\npackage com.bigsis.spookjams.proxy;public class ClientProxy implements CommonProxy {}\n\n7. Right-click on \"com.bigsis.spookjams.proxy\" and choose\u00a0New\u00a0>\u00a0Class. In the \"Name\" field of the\u00a0New Java Class window, type in \"ServerProxy\" and press\u00a0Finish. Edit the\u00a0contents of \"ServerProxy.java\" to read as follows:\n\nCode:\npackage com.bigsis.spookjams.proxy;public class ServerProxy implements CommonProxy {}\n\n8. Right-click on \"com.bigsis.spookjams\" and choose\u00a0New\u00a0>\u00a0Class. In the \"Name\" field of the New\u00a0Java Class window, type in \"Reference\" and press\u00a0Finish. Edit the contents of\u00a0\"Reference.java\" to read as follows:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".public class Reference \/\/This is a collection of constants used throughout the mod to help keep the code cleaner by keeping most variable information stored here in a single location. {\u00a0 \u00a0public static final String MOD_ID = \"bssjm\"; \/\/Internal name of your mod, it MUST be unique to prevent clashing with other mods.\u00a0 \u00a0public static final String NAME = \"Big Sister's SpookJam Crash Course Mod\"; \/\/The name of your mod as displayed on the Minecraft Mod Detail screen.\u00a0 \u00a0public static final String VERSION = \"0.1-wheat\"; \/\/Your version identifier; you can use numbers, decimals, alpha\/beta tags, or you can be like me and get creative (I'm naming my alpha versions after grains and pulses!).\u00a0 \u00a0public static final String MINECRAFT_VERSION = \"[1.9.4]\"; \/\/The version of Minecraft that this mod is compatible with; you can use a static value, or use special syntax to have it be compatible with other Minecraft version variants.\u00a0 \u00a0public static final String CLIENT_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ClientProxy\"; \/\/Pointer to the Minecraft client proxy; references the ClientProxy.java class inside the com.bigsis.spookjams.proxy package.\u00a0 \u00a0public static final String SERVER_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ServerProxy\"; \/\/Pointer to the Minecraft server proxy; references the ServerProxy.java class inside the com.bigsis.spookjams.proxy package.}\n\nIf the above is too hard to read because of the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/79903e21222e1fa4cf4ec34d63d7b6bf\n\n9. Right-click on \"com.bigsis.spookjams\" and choose\u00a0New\u00a0>\u00a0Class. In the \"Name\" field of the New\u00a0Java Class window, type in \"Spookjams\" and press\u00a0Finish. Edit the contents of\u00a0\"Spookjams.java\" to read as follows:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.proxy.CommonProxy; \/\/Import this to make our custom CommonProxy instance work.import net.minecraftforge.fml.common.Mod; \/\/Import this to make the @Mod annotation workimport net.minecraftforge.fml.common.Mod.EventHandler; \/\/Import this to make the @EventHandler annotation workimport net.minecraftforge.fml.common.Mod.Instance; \/\/Import this to make the @Instance annotation workimport net.minecraftforge.fml.common.SidedProxy; \/\/Import this to make the @SidedProxy annotation workimport net.minecraftforge.fml.common.event.FMLInitializationEvent; \/\/import this to make this Initialization event workimport net.minecraftforge.fml.common.event.FMLPostInitializationEvent; \/\/import this to make this Post Initialization event workimport net.minecraftforge.fml.common.event.FMLPreInitializationEvent; \/\/import this to make this Pre Initialization event work@Mod( \u00a0modid = Reference.MOD_ID, \/\/Specify the unique MOD ID of your mod, we're getting this from the Reference.java class.\u00a0name = Reference.NAME, \/\/Specify the name of your mod; also being pulled from Reference.java.\u00a0version = Reference.VERSION, \/\/Specify the version of your mod; also being pulled from Reference.java.\u00a0acceptedMinecraftVersions = Reference.MINECRAFT_VERSION \/\/Specify which Minecraft version(s) this mod works on; also being pulled from Reference.java.)public class Spookjams {\u00a0@Instance\u00a0public static Spookjams instance; \/\/Need this to allow Minecraft to instantiate this mod.\u00a0\u00a0@SidedProxy(\u00a0clientSide = Reference.CLIENT_PROXY_CLASS, \/\/Specify the Client proxy; being pulled from Reference.java.\u00a0serverSide = Reference.SERVER_PROXY_CLASS \/\/Specify the Server proxy; being pulled from Reference.java.\u00a0)\u00a0public static CommonProxy proxy; \/\/Need this to instantiate the proxy interface with Minecraft.\u00a0\u00a0@EventHandler\u00a0public void preInit(FMLPreInitializationEvent event) \/\/define what gets called first when this mod is loaded (e.g., loading item or block intialization)\u00a0{\u00a0System.out.println(\"Pre-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0}\u00a0\u00a0@EventHandler\u00a0public void init(FMLInitializationEvent event) \/\/define what gets called next after the pre-initialization (e.g., loading less critical items or entities)\u00a0{\u00a0System.out.println(\"Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0}\u00a0\u00a0@EventHandler\u00a0public void postInit(FMLPostInitializationEvent event) \/\/define what gets done after everything is loaded in the mod (e.g., loading another mod if it exists)\u00a0{\u00a0System.out.println(\"Post-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/ab7971e9de38db4e47b4dc29daeb31dc\n\n10. Going back to the Package Explorer, open the \"src\/main\/resources\" folder and open the file in there called \"mcmod.info\". \u00a0This will be a standard text file, not Java, so you will need to recreate some of your earlier work in \"Resources.java\" by hand. \u00a0The end result should be a file that looks like this:\n\nCode:\n[{\u00a0 \"modid\": \"bssjm\",\u00a0 \"name\": \"Big Sister's SpookJam Crash Course Mod\",\u00a0 \"description\": \"Ciabatta's poor excuse for an attempt to document beginner's Minecraft modding.\",\u00a0 \"version\": \"0.1-wheat\",\u00a0 \"mcversion\": \"1.9.4\",\u00a0 \"url\": \"http:\/\/bigsister.forumotion.com\/forum\",\u00a0 \"updateUrl\": \"\",\u00a0 \"authorList\": [\"Ciabatta\"],\u00a0 \"credits\": \"To all the wonderful contributors of the Big Sister Gulliver Minecraft Server Community!\",\u00a0 \"logoFile\": \"\",\u00a0 \"screenshots\": [],\u00a0 \"dependencies\": []}]\n\n11. Now you're finished! \u00a0Look for a green \"Run Client\" button in the top panel of Eclipse and click on it. \u00a0This will launch a localized instance of Minecraft 1.9.4, meanwhile the Console feedback will show traces of your new mod.\n\n12. Once in the Minecraft main menu screen, click on the Mods button and look for your fancy new mod in action! \u00a0CONGRATULATIONS!!! \u00a0You now have a brand new (if currently-empty) MINECRAFT MOD!!!!\n\nLast edited by Ciabatta on Sat Jul 23, 2016 8:01 pm; edited 5 times in total\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nEdited the main comments to illustrate a way by which you can use the Find\/Replace feature in Eclipse to remove all my comments. ^^\n\n_________________\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nI renamed some of the names to make this tutorial a bit more universal -- so instead of \"ciabatta\", it now reads \"bigsis\", and instead of that LOOOONG MOD_ID (which was getting annoying), I renamed it to \"bssjm\" (or \"Big Sister SpookJam Mod\"). ^^\n\n_________________\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nPart 2 - Creating Your First Custom Item\n\nNote: Please be sure to complete Part 1 of this tutorial first as this is a direct continuation of it:\u00a0http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15583\n\n1. Right-click over the \"src\/main\/resources\" folder and select\u00a0New\u00a0>\u00a0Package. Name this package\u00a0\"assets.bssjm\" and press\u00a0Finish.\n\n2. Right-click over the \"assets.bssjm\" package and select\u00a0New\u00a0>\u00a0Package. Name this package \"assets.bssjm.lang\" and press\u00a0Finish.\n\n3. Right-click over the \"assets.bssjm\" package and select\u00a0New\u00a0>\u00a0Package. Name this package \"assets.bssjm.models\" and press\u00a0Finish.\n\n4. Right-click over the \"assets.bssjm\" package and select\u00a0New\u00a0>\u00a0Package. Name this package \"assets.bssjm.textures\" and press\u00a0Finish.\n\n5. Right-click over the \"assets.bssjm.models\" package and select\u00a0New\u00a0>\u00a0Package. Name this package \"assets.bssjm.models.item\" and press\u00a0Finish.\n\n6. Right-click over the \"assets.bssjm.textures\" package and select\u00a0New\u00a0>\u00a0Package. Name this package \"assets.bssjm.textures.items\" and press\u00a0Finish.\n\n7. You should now have all the resources folders set up to accommodate the creation of a new custom item. \u00a0Don't worry if some of the packages disappear or bleed into one another, what matters is that the hierarchy for these packages is maintained. \u00a0If you want a less-confusing view, you can click on the View Menu\u00a0button in the Project Explorer toolbar and switch the Package Presentation to \"Hierarchical\" rather than \"Flat\", but that's a matter of preference.\n\n8. Return to the \"src\/main\/java\" folder and right-click over it to select\u00a0New\u00a0>\u00a0Package. Name this package\u00a0\"init\" and press\u00a0Finish.\n\n9. Right-click again over the \"src\/main\/java\" folder and right-click over it to select\u00a0New\u00a0Package. Name this package \"items\" and press\u00a0Finish.\n\n10. Right-click over the \"init\" package and select\u00a0New\u00a0>\u00a0Package. Name this class \"ModItems\" and press\u00a0Finish. This class will be used to initialize your new custom items and\u00a0registering them with Forge.\n\n11. Right-click over the \"items\" package and select\u00a0New\u00a0> Class. \u00a0Name this class \"ItemCheese\"\u00a0[we will be creating cheese as our first item!]; but unlike with other classes we've created so far, we will also replace the value in the Superclass\u00a0field with:\n\nnet.minecraft.item.Item\n\nOnce this is done, you can press\u00a0Finish.\n\n12. Before we edit the new classes, let's update Reference.java to accommodate our new cheese item. Go into the \"src\/main\/java\" folder, go into the \"com.bigsis.spookjams\" package,\u00a0and then open the \"Reference.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".public class Reference \/\/This is a collection of constants used throughout the mod to help keep the code cleaner by keeping most variable information stored here in a single location. {\u00a0public static final String MOD_ID = \"bssjm\"; \/\/Internal name of your mod, it MUST be unique to prevent clashing with other mods.\u00a0public static final String NAME = \"Big Sister's SpookJam Crash Course Mod\"; \/\/The name of your mod as displayed on the Minecraft Mod Detail screen.\u00a0public static final String VERSION = \"0.1-wheat\"; \/\/Your version identifier; you can use numbers, decimals, alpha\/beta tags, or you can be like me and get creative (I'm naming my alpha versions after grains and pulses!).\u00a0public static final String MINECRAFT_VERSION = \"[1.9.4]\"; \/\/The version of Minecraft that this mod is compatible with; you can use a static value, or use special syntax to have it be compatible with other Minecraft version variants.\u00a0public static final String CLIENT_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ClientProxy\"; \/\/Pointer to the Minecraft client proxy; references the ClientProxy.java class inside the com.bigsis.spookjams.proxy package.\u00a0public static final String SERVER_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ServerProxy\"; \/\/Pointer to the Minecraft server proxy; references the ServerProxy.java class inside the com.bigsis.spookjams.proxy package.\u00a0public static enum SpookjamItems \/\/An enum type is a special data type that enables for a variable to be a set of predefined constants.\u00a0{\u00a0CHEESE(\"cheese\",\"ItemCheese\"); \/\/We are declaring a CHEESE enum to be its common and system name. The declarations go first before the actual instructions on how to parse this below.\u00a0\u00a0private String unlocalizedName; \/\/Declare a blank text string to hold the common name variable of this enum.\u00a0private String registryName; \/\/Declare a blank text string to hold the system name variable of this enum.\u00a0\u00a0SpookjamItems(String unlocalizedNameTarget, String registryNameTarget) \/\/Instructions on how to parse each enum that is declared (in this example: \"CHEESE(\"cheese\",\"ItemCheese\"). Note that two parameters are allotted to this enum.\u00a0{\u00a0this.unlocalizedName = unlocalizedNameTarget; \/\/Populate unlocalizedName with the first parameter of the enum (in this example, \"cheese\").\u00a0this.registryName = registryNameTarget; \/\/Populate registryName with the second parameter of the enum (in this example, \"ItemCheese\").\u00a0}\u00a0\u00a0public String getUnlocalizedName() \/\/Declare a Get method in order to extract the common name variable of this enum, since it may get referenced again throughout the code.\u00a0{\u00a0return unlocalizedName; \/\/Have this String method return the common name variable of this enum.\u00a0}\u00a0\u00a0public String getRegistryName() \/\/Declare a Get method in order to extract the system name variable of this enum, since it may get referenced again throughout the code.\u00a0{\u00a0return registryName; \/\/Have this String method return the system name variable of this enum.\u00a0}\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/a93aa3c405c246ca3ac8971729c6d53d\n\n13. Go into the \"src\/main\/java\" folder, go into the \"com.bigsis.spookjams\" package,\u00a0go into the \"Items\" package, and then open the \"ItemCheese.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams.items; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference; \u00a0\/\/Import this to recognize our custom Reference.java class.import net.minecraft.item.Item; \/\/Need this to recognize Minecraft Item objects.public class ItemCheese extends Item {\u00a0public ItemCheese() \/\/This is the constructor for your new Cheese Item.\u00a0{\u00a0setUnlocalizedName(Reference.SpookjamItems.CHEESE.getUnlocalizedName()); \/\/Set the common name of this new item from the CHEESE enum we declared in Reference.java.\u00a0setRegistryName(Reference.SpookjamItems.CHEESE.getRegistryName()); \/\/Set the system name of this new item from the CHEESE enum we declared in Reference.java.\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/a07a051b0f6ed3e84e0fc7f46160f60b\n\n14. Go into the \"src\/main\/java\" folder, go into the \"com.bigsis.spookjams\" package,\u00a0go into the \"init\" package,\u00a0and then open the \"ModItems.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams.init; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference; \/\/Import this to recognize our custom Reference.java class.import com.bigsis.spookjams.items.ItemCheese; \/\/Import this to recognize our custom ItemCheese.java class.import net.minecraft.client.Minecraft; \/\/Imported to allow the Minecraft client to be recognizedimport net.minecraft.client.renderer.block.model.ModelResourceLocation; \/\/Imported to allow the ModelResourceLocation class to be recognized.import net.minecraft.item.Item; \/\/Imported to allow for Minecraft objects to be registered.import net.minecraftforge.fml.common.registry.GameRegistry; \/\/Imported to recognize the use of the GameRegistry class.public class ModItems{\u00a0public static Item cheese; \/\/This is the name of our new custom item.\u00a0\u00a0public static void init() \/\/This method will initialize this object.\u00a0{\u00a0cheese = new ItemCheese(); \/\/Set the new Item to be a new instance of our custom cheese object as defined in ItemCheese.java.\u00a0}\u00a0\u00a0public static void register() \/\/This will register the item into the game.\u00a0{\u00a0GameRegistry.register(cheese); \/\/Register our custom item in Minecraft using the new Item object we initiated above.\u00a0}\u00a0\u00a0public static void registerRenders() \/\/This will contain all the calls to register the render for each item.\u00a0{\u00a0registerRender(cheese); \/\/Call the private method below.\u00a0}\u00a0\u00a0private static void registerRender(Item item) \/\/This is the new item object that you will register into Minecraft.\u00a0{\u00a0Minecraft.getMinecraft().getRenderItem().getItemModelMesher().register(item, 0, new ModelResourceLocation(item.getRegistryName(), \"inventory\")); \/\/Register this new item by referencing its system name, and specifying that it will be an Inventory item.\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/419c6b3983ac587d9b2a2408d9026aa7\n\n15. Go into the \"src\/main\/java\" folder, go into the \"com.bigsis.spookjams\" package, go into the \"Proxy\" package, and then open the \"CommonProxy.java\" instance. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams.proxy;public interface CommonProxy {\u00a0public void init();}\n\n16. In the same \"Proxy\" package, and open the \"ServerProxy.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams.proxy;public class ServerProxy implements CommonProxy {\u00a0@Override\u00a0public void init() {}}\n\n17. In the same \"Proxy\" package, and open the \"ClientProxy.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams.proxy;import com.bigsis.spookjams.init.ModItems;public class ClientProxy implements CommonProxy {\u00a0@Override\u00a0public void init() {\u00a0ModItems.registerRenders(); \/\/only call this on the client side and not the server side (for now).\u00a0}}\n\n18. Go into the \"src\/main\/java\" folder, go into the \"com.bigsis.spookjams\" package,\u00a0and then open the \"Spookjams.java\" class. Edit the file to read as follows:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.init.ModItems; \/\/Import this to recognize our custom ModItems.java class.import com.bigsis.spookjams.proxy.CommonProxy; \/\/Import this to recognize our custom CommonProxy.java instance.import net.minecraftforge.fml.common.Mod; \/\/Import this to make the @Mod annotation workimport net.minecraftforge.fml.common.Mod.EventHandler; \/\/Import this to make the @EventHandler annotation workimport net.minecraftforge.fml.common.Mod.Instance; \/\/Import this to make the @Instance annotation workimport net.minecraftforge.fml.common.SidedProxy; \/\/Import this to make the @SidedProxy annotation workimport net.minecraftforge.fml.common.event.FMLInitializationEvent; \/\/import this to make this Initialization event workimport net.minecraftforge.fml.common.event.FMLPostInitializationEvent; \/\/import this to make this Post Initialization event workimport net.minecraftforge.fml.common.event.FMLPreInitializationEvent; \/\/import this to make this Pre Initialization event work@Mod( \u00a0modid = Reference.MOD_ID, \/\/Specify the unique MOD ID of your mod, we're getting this from the Reference.java class.\u00a0name = Reference.NAME, \/\/Specify the name of your mod; also being pulled from Reference.java.\u00a0version = Reference.VERSION, \/\/Specify the version of your mod; also being pulled from Reference.java.\u00a0acceptedMinecraftVersions = Reference.MINECRAFT_VERSION \/\/Specify which Minecraft version(s) this mod works on; also being pulled from Reference.java.)public class Spookjams {\u00a0@Instance\u00a0public static Spookjams instance; \/\/Need this to allow Minecraft to instantiate this mod.\u00a0\u00a0@SidedProxy(\u00a0clientSide = Reference.CLIENT_PROXY_CLASS, \/\/Specify the Client proxy; being pulled from Reference.java.\u00a0serverSide = Reference.SERVER_PROXY_CLASS \/\/Specify the Server proxy; being pulled from Reference.java.\u00a0)\u00a0public static CommonProxy proxy; \/\/Need this to instantiate the proxy interface with Minecraft.\u00a0\u00a0@EventHandler\u00a0public void preInit(FMLPreInitializationEvent event) \/\/define what gets called first when this mod is loaded (e.g., loading item or block intialization)\u00a0{\u00a0System.out.println(\"Pre-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0\u00a0ModItems.init();\u00a0ModItems.register();\u00a0}\u00a0\u00a0@EventHandler\u00a0public void init(FMLInitializationEvent event) \/\/define what gets called next after the pre-initialization (e.g., loading less critical items or entities)\u00a0{\u00a0System.out.println(\"Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0proxy.init(); \/\/initialize the client and server proxies that interface with Minecraft..\u00a0}\u00a0\u00a0@EventHandler\u00a0public void postInit(FMLPostInitializationEvent event) \/\/define what gets done after everything is loaded in the mod (e.g., loading another mod if it exists)\u00a0{\u00a0System.out.println(\"Post-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/a23ee2aa15be60b91db3fbbf89085ce1\n\n19. This concludes the last of the programming requirements for adding a new item, now it's time to get into the nitty-gritty model and textures! \u00a0First,\u00a0create an image of cheese that has a transparent background, is 16x16, and is saved in a PNG format... if you don't know how, you can also find a free sample asset of cheese, or other similar assets, here...\n\nhttp:\/\/bigsister.forumotion.com\/t896-item-texture-assets\n\n20. Inside your Windows' file explorer, copy your new cheese.png file and save it inside your mod's asset location inside the Eclipse folder. \u00a0Assuming you remember how to get into the SpookJams folder (see the tutorial on setting up the development environment in Windows), the location to save it in will be....\n\n...Spookjams\\src\\main\\resources\\assets\\bssjm\\textures\\items\n\n21. While you're in your Windows' file explorer, go into your mod's models folder location inside the Eclipse folder; the location to travel to will be....\n\n...Spookjams\\src\\main\\resources\\assets\\bssjm\\models\\item\n\nOnce there, create a new blank text file and name it \"ItemCheese.json\".\n\n22. While you're still in your Windows' file explorer, go into your mod's\u00a0lang\u00a0folder location inside the Eclipse folder; the location to travel to will be....\n\n...Spookjams\\src\\main\\resources\\assets\\bssjm\\lang\n\nOnce there, create a new blank text file and name it \"en_US.lang\".\n\n23. Return to Eclipse, go back into the \"src\/main\/resources\" folder, open the \"assets.bssjm\" package, open the \"models.item\" package, and open the \"ItemCheese.json\" file... we can edit it here in Eclipse using its native text editor. Edit ItemCheese.json so it reads as follows:\n\nCode:\n{\u00a0\u00a0 \u00a0\"parent\": \"item\/generated\",\u00a0\u00a0 \u00a0\"textures\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"layer0\": \"bssjm:items\/cheese\"\u00a0\u00a0 \u00a0},\u00a0\u00a0 \u00a0\"display\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"thirdperson\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"rotation\": [ -90, 0, 9 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"translation\": [ 0, 1, -3 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"scale\": [ 0.55, 0.55, 0.55 ]\u00a0\u00a0 \u00a0 \u00a0 \u00a0},\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"firstperson\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"rotation\": [ 0, -135, 25 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"translation\": [ 0, 4, 2],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"scale\": [ 1.7, 1.7, 1.7 ]\u00a0\u00a0 \u00a0 \u00a0 \u00a0}\u00a0\u00a0 \u00a0}}\n\n24. Go back into the \"src\/main\/resources\" folder, open the \"assets.bssjm\" package, and open the \"lang\" package, and then open the \"en_US.lang\" file. Edit it to include the following line:\n\nCode:\nitem.cheese.name=Cheese\n\n25. And you're finished! Use the Run Client button in the Eclipse toolbar to launch Minecraft, and in creative you can use this command to give yourself your new item:\n\n\/give @a bssjm:ItemCheese\n\nHave fun!\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nThe next thing on your list: Mix those two together XD\n\nThis was basically the exact same steps that I followed. And let me point this out: Once you make the first item (Which is the most complicated), the more after that will become SUPER easy.\n\nAsonja_Masenko\n\nPosts : 28\nJoin date : 2016-07-04\nAge : 17\n\nRP Character Sheet\nName: Asonja Masenko\nPersonality Trait: Quiet\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nAll I can say is that I'm astonished that I managed to name the package the correct thing before we were given a standardized name. (Well almost)\n\nPTpirahna\n\nPosts : 196\nJoin date : 2014-09-27\nAge : 15\nLocation : Watching you on the dynmap\n\nRP Character Sheet\nName: ???\nPersonality Trait: Loyal\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nPart 3 - Creating a Crafting Recipe for your First Custom Item\n\nNote: Please be sure to complete Part 2 of this tutorial first as this is a direct continuation of it:\u00a0http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15597\n\n1. Go to the \"src\/main\/java\" package, go to the \"com.bigsis.spookjams\" package, right-click the \"init\" package, and select New > Class. Name this new class \"ModCrafting\" and press Finish.\n\n2. Open the new \"ModCrafting\" class and edit the contents as follows to add a crafting recipe for cheese,... since milk is involved and milk comes in EXPENSIVE iron buckets, we also sprinkle a bit more code to get the bucket back when finished:\n\nCode:\npackage com.bigsis.spookjams.init; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import net.minecraft.init.Blocks; \/\/Imported to recognize Minecraft Block objects.import net.minecraft.init.Items; \/\/Imported to recognize Minecraft Item objects.import net.minecraft.item.ItemStack; \/\/Imported to recognize the ItemStack object.import net.minecraftforge.fml.common.registry.GameRegistry; \/\/Imported to recognize the GameRegistry object.public class ModCrafting {\u00a0 \u00a0public static void register() \u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0GameRegistry.addShapedRecipe \/\/This creates a shaped recipe, which is the type of recipe that requires specific placement on the crafting grid.\u00a0 \u00a0\u00a0 \u00a0(\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0new ItemStack(ModItems.cheese, 4), \/\/This recipe is to make cheese, specifically 4 wedges of it.\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\"BMB\", \/\/This recipe specifically calls for a horizontal row of a milk bucket in the center, flanked by two brown mushrooms; note that since we only specify one row on the grid, you can craft these items on any row. \u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0'B', Blocks.BROWN_MUSHROOM, \/\/This specifies that \"B\" stands for the Brown Mushroom \"bush\", not the block from a giant brown mushroom. Note that the use of the single quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0'M', new ItemStack(Items.MILK_BUCKET.setContainerItem(Items.BUCKET)) \/\/This specifies that \"M\" stands for a milk-filled bucket; to avoid using up a whole metal bucket just to make cheese, we also indicate that an empty bucket be returned after this crafting recipe is processed.\u00a0 Note that the use of the single quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0 \u00a0\u00a0 \u00a0);\u00a0 \u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\n\n3. Go to the \"src\/main\/java\" package, go to the \"com.bigsis.spookjams\" package, and open the \"Spookjams.java\" class. \u00a0Edit the contents of it as follows:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.init.ModCrafting; \/\/Import this to recognize our custom ModCrafting.java class.import com.bigsis.spookjams.init.ModItems; \/\/Import this to recognize our custom ModItems.java class.import com.bigsis.spookjams.proxy.CommonProxy; \/\/Import this to recognize our custom CommonProxy.java instance.import net.minecraftforge.fml.common.Mod; \/\/Import this to make the @Mod annotation workimport net.minecraftforge.fml.common.Mod.EventHandler; \/\/Import this to make the @EventHandler annotation workimport net.minecraftforge.fml.common.Mod.Instance; \/\/Import this to make the @Instance annotation workimport net.minecraftforge.fml.common.SidedProxy; \/\/Import this to make the @SidedProxy annotation workimport net.minecraftforge.fml.common.event.FMLInitializationEvent; \/\/import this to make this Initialization event workimport net.minecraftforge.fml.common.event.FMLPostInitializationEvent; \/\/import this to make this Post Initialization event workimport net.minecraftforge.fml.common.event.FMLPreInitializationEvent; \/\/import this to make this Pre Initialization event work@Mod( \u00a0 \u00a0modid = Reference.MOD_ID,\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\/\/Specify the unique MOD ID of your mod, we're getting this from the Reference.java class.\u00a0 \u00a0name = Reference.NAME,\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\/\/Specify the name of your mod; also being pulled from Reference.java.\u00a0 \u00a0version = Reference.VERSION, \u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\/\/Specify the version of your mod; also being pulled from Reference.java.\u00a0 \u00a0acceptedMinecraftVersions = Reference.MINECRAFT_VERSION\u00a0 \u00a0\/\/Specify which Minecraft version(s) this mod works on; also being pulled from Reference.java.)public class Spookjams {\u00a0 \u00a0@Instance\u00a0 \u00a0public static Spookjams instance; \/\/Need this to allow Minecraft to instantiate this mod.\u00a0 \u00a0\u00a0 \u00a0@SidedProxy(\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0clientSide = Reference.CLIENT_PROXY_CLASS,\u00a0 \u00a0\/\/Specify the Client proxy; being pulled from Reference.java.\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0serverSide = Reference.SERVER_PROXY_CLASS \u00a0 \u00a0\/\/Specify the Server proxy; being pulled from Reference.java.\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0)\u00a0 \u00a0public static CommonProxy proxy; \/\/Need this to instantiate the proxy interface with Minecraft.\u00a0 \u00a0\u00a0 \u00a0@EventHandler\u00a0 \u00a0public void preInit(FMLPreInitializationEvent event) \/\/define what gets called first when this mod is loaded (e.g., loading item or block intialization)\u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0System.out.println(\"Pre-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0ModItems.init(); \/\/initialize our custom items\u00a0 \u00a0\u00a0 \u00a0ModItems.register(); \/\/register our custom items\u00a0 \u00a0}\u00a0 \u00a0\u00a0 \u00a0@EventHandler\u00a0 \u00a0public void init(FMLInitializationEvent event) \/\/define what gets called next after the pre-initialization (e.g., loading less critical items or entities)\u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0System.out.println(\"Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0 \u00a0\u00a0 \u00a0proxy.init(); \/\/initialize the client and server proxies that interface with Minecraft.\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0\u00a0 \u00a0ModCrafting.register(); \/\/register our custom crafting recipes\u00a0 \u00a0}\u00a0 \u00a0\u00a0 \u00a0@EventHandler\u00a0 \u00a0public void postInit(FMLPostInitializationEvent event) \/\/define what gets done after everything is loaded in the mod (e.g., loading another mod if it exists)\u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0System.out.println(\"Post-Init\"); \/\/just some dummy text to serve as a placeholder for future event info -- note that this will appear in the Minecraft Launch Log to let you know when it gets called!\u00a0 \u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/b16bcd2dfca8c22f7f9ff5506ea6163e\n\n4. Congratulations, you're finished! See? that wasn't so bad! \u00a0Go ahead and press the Run Client button in the Eclipse toolbar to launch Minecraft with your crafting recipe update. When you put a single milk bucket in the middle of any of the crafting grid's rows, and then place two brown mushrooms on the left and right of it, it will let you craft 4 cheese wedges... and you get your bucket back too! \u00a0Enjoy!\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nNote: Please be sure to complete Part 3 of this tutorial first as this is a direct continuation of it:\u00a0http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15601\n\n1. This tutorial will be brief since we only have to make a quick update to the \"ItemCheese.java\" class to allow your cheese to display on the Creative Menu! \u00a0Go to the \"src\/main\/java\" folder, go to the \"com.bigsis.spookjams\" package, go to the \"items\" package, and then open the \"ItemCheese.java\" class. \u00a0Edit the file as follows, paying attention to the \"this.setCreativeTab\" line:\n\nCode:\npackage com.bigsis.spookjams.items; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference;\u00a0 \/\/Import this to recognize our custom Reference.java class.import net.minecraft.creativetab.CreativeTabs; \/\/Import this to recognize Minecraft Creative Tabs.import net.minecraft.item.Item; \/\/Need this to recognize Minecraft Item objects.public class ItemCheese extends Item {\u00a0 \u00a0public ItemCheese() \/\/This is the constructor for your new Cheese Item.\u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0setUnlocalizedName(Reference.SpookjamItems.CHEESE.getUnlocalizedName()); \/\/Set the common name of this new item from the CHEESE enum we declared in Reference.java.\u00a0 \u00a0\u00a0 \u00a0setRegistryName(Reference.SpookjamItems.CHEESE.getRegistryName()); \/\/Set the system name of this new item from the CHEESE enum we declared in Reference.java.\u00a0 \u00a0\u00a0 \u00a0this.setCreativeTab(CreativeTabs.FOOD); \/\/Specify that this item appear in the Foodstuffs category of the standard Creative Menu.\u00a0 \u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/5a43e7966cede356b65fa266b666b016\n\n2. And believe it or not, you're done! \u00a0Launch the game, go into Creative Mode, and enjoy the accessibility of your cheese item! Note that we could have always created a new Creative Tab and added the cheese there but... small steps, folks. \u00a0Small steps... ^^\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nPart 5 - Making Your First Custom Item Edible\n\nNote: Since this step obsoletes what was shown in Part 4, please be sure to complete Part 3 of this tutorial instead as this can be a direct continuation of it:\u00a0http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15601\n\n1. This tutorial will be brief since we only have to make a quick update to the \"ItemCheese.java\" class to allow your cheese to display on the Creative Menu! \u00a0Go to the \"src\/main\/java\" folder, go to the \"com.bigsis.spookjams\" package, go to the \"items\" package, and then open the \"ItemCheese.java\" class. \u00a0Edit the file as follows, noting that we're converting it into an \"ItemFood\" rather than \"Item\" object, and adding some standard hunger-filling and saturation values:\n\nCode:\npackage com.bigsis.spookjams.items; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference;\u00a0 \/\/Import this to recognize our custom Reference.java class.import net.minecraft.creativetab.CreativeTabs; \/\/Import this to recognize Minecraft Creative Tabs.import net.minecraft.item.ItemFood; \/\/Need this to recognize Minecraft Food Item objects.public class ItemCheese extends ItemFood \/\/Give this item food attributes.{\u00a0 \u00a0public ItemCheese() \/\/This is the constructor for your new Cheese Item.\u00a0 \u00a0{\u00a0 \u00a0\u00a0 \u00a0super(6, 0.7F, false); \/\/Specify how many half-hearts are provided after eating the cheese, how long its saturation duration should be (because fat is highly satiating!), and whether you can feed this to wolves (NO! dairy isn't good for canines). ;)\u00a0 \u00a0\u00a0 \u00a0setUnlocalizedName(Reference.SpookjamItems.CHEESE.getUnlocalizedName()); \/\/Set the common name of this new item from the CHEESE enum we declared in Reference.java.\u00a0 \u00a0\u00a0 \u00a0setRegistryName(Reference.SpookjamItems.CHEESE.getRegistryName()); \/\/Set the system name of this new item from the CHEESE enum we declared in Reference.java.\u00a0 \u00a0\u00a0 \u00a0\/\/The below is obsolete if we're already defining this as a food item, or if you're not planning to display this anywhere other than the Foodstuffs Creative Tab.\u00a0 \u00a0\u00a0 \u00a0\/\/this.setCreativeTab(CreativeTabs.FOOD); \/\/Specify that this item appear in the Foodstuffs category of the standard Creative Menu.\u00a0 \u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/052c52c18b22a31cf406a60d7897fa72\n\nOf interesting note, we are also disabling the Creative Tab code from Part 4, as this is already called automatically in the ItemFood object class... we may reenable it in the future if we create our own Creative Menu Tab, but for now this item WILL appear in the FoodStuffs Creative tab purely on account of being an \"ItemFood\".\n\n2. And that's all! The core Minecraft classes are already doing the heavy-lifting for you as far as the animation and other specifics are concerned. We can also add other cool nifty tricks to do things like add potion effects after eating,... we won't try this until much much later, but if you want to get a head start you can view the additional ItemFood methods by going to the \"Referenced Libraries\" folder, opening the \"forgeSrc-...\" jar, opening the \"net\" package, opening the \"item\" package, and opening the \"ItemFood.class\" file.\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nWhoops... haven't had a chance to progress this further but here are a few of the tutorials coming up next...\n\n2) Adding a composite item from new custom items\n3) Adding a new custom Creative tab\n4) Adding a new furnace recipe\n\n_________________\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nPart 6 - Adding a Second Custom Item\n\nNote: Despite being a bit redundant, it's important to see just how much work is needed to create multiple new items in Minecraft -- assuming they are simple additions, it really doesn't require all that much extra work. \u00a0In any event, be sure to complete Part 5 of this tutorial first as this is a direct continuation of it:\u00a0http:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15603\n\n1. Go to the \"src\/main\/java\" folder and then open \"Reference.java\". Update the file as follows, noting the new enum declaration for our new item (gluten-free bread!) which requires us to turn the semi-colon after the CHEESE enum into a comma:\n\nCode:\npackage com.bigsis.spookjams; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".public class Reference \/\/This is a collection of constants used throughout the mod to help keep the code cleaner by keeping most variable information stored here in a single location. {\u00a0public static final String MOD_ID = \"bssjm\"; \/\/Internal name of your mod, it MUST be unique to prevent clashing with other mods.\u00a0public static final String NAME = \"Big Sister's SpookJam Crash Course Mod\"; \/\/The name of your mod as displayed on the Minecraft Mod Detail screen.\u00a0public static final String VERSION = \"0.1-wheat\"; \/\/Your version identifier; you can use numbers, decimals, alpha\/beta tags, or you can be like me and get creative (I'm naming my alpha versions after grains and pulses!).\u00a0public static final String MINECRAFT_VERSION = \"[1.9.4]\"; \/\/The version of Minecraft that this mod is compatible with; you can use a static value, or use special syntax to have it be compatible with other Minecraft version variants.\u00a0public static final String CLIENT_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ClientProxy\"; \/\/Pointer to the Minecraft client proxy; references the ClientProxy.java class inside the com.bigsis.spookjams.proxy package.\u00a0public static final String SERVER_PROXY_CLASS = \"com.bigsis.spookjams.proxy.ServerProxy\"; \/\/Pointer to the Minecraft server proxy; references the ServerProxy.java class inside the com.bigsis.spookjams.proxy package.\u00a0public static enum SpookjamItems \/\/An enum type is a special data type that enables for a variable to be a set of predefined constants.\u00a0{\u00a0CHEESE(\"cheese\",\"ItemCheese\"), \/\/We are declaring a CHEESE enum to be its common and system name. The declarations go first before the actual instructions on how to parse this below.\u00a0GF_BREAD(\"gf_bread\",\"ItemGFBread\"); \/\/NOTE THE NEW COMMA AFTER THE CHEESE ENUM!!! \u00a0We are declaring a GF_BREAD enum to be its common and system name. The declarations go first before the actual instructions on how to parse this below.\u00a0\u00a0private String unlocalizedName; \/\/Declare a blank text string to hold the common name variable of this enum.\u00a0private String registryName; \/\/Declare a blank text string to hold the system name variable of this enum.\u00a0\u00a0SpookjamItems(String unlocalizedNameTarget, String registryNameTarget) \/\/Instructions on how to parse each enum that is declared (in this example: \"CHEESE(\"cheese\",\"ItemCheese\"). Note that two parameters are allotted to this enum.\u00a0{\u00a0this.unlocalizedName = unlocalizedNameTarget; \/\/Populate unlocalizedName with the first parameter of the enum (in this example, \"cheese\").\u00a0this.registryName = registryNameTarget; \/\/Populate registryName with the second parameter of the enum (in this example, \"ItemCheese\").\u00a0}\u00a0\u00a0public String getUnlocalizedName() \/\/Declare a Get method in order to extract the common name variable of this enum, since it may get referenced again throughout the code.\u00a0{\u00a0return unlocalizedName; \/\/Have this String method return the common name variable of this enum.\u00a0}\u00a0\u00a0public String getRegistryName() \/\/Declare a Get method in order to extract the system name variable of this enum, since it may get referenced again throughout the code.\u00a0{\u00a0return registryName; \/\/Have this String method return the system name variable of this enum.\u00a0}\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\n\n2. Go to the \"src\/main\/java\" folder, click on the \"items\" package, and right-click to select New > Class. Name the class \"ItemGFBread\" and give it a Superclass of:\n\nnet.minecraft.item.ItemFood\n\nThen open \"ItemGFBread.java\" and edit the file as follows:\n\nCode:\npackage com.bigsis.spookjams.items; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference; \/\/Import this to recognize our custom Reference.java class.import net.minecraft.item.ItemFood; \/\/Import this to recognize Minecraft ItemFood objects.public class ItemGFBread extends ItemFood { \/\/Give this item food attributes.\u00a0public ItemGFBread() \/\/This is the constructor for your new Cheese Item.\u00a0{\u00a0super(5, 0.6F, false);\/\/Specify how many half-hearts are provided after eating the gluten-free bread, how long its saturation duration should be (this should be no different than wheat-based bread!), and whether you can feed this to wolves (NO! grains are HORRIBLE for canines). \u00a0setUnlocalizedName(Reference.SpookjamItems.GF_BREAD.getUnlocalizedName()); \/\/Set the common name of this new item from an enum we declared in Reference.java.\u00a0setRegistryName(Reference.SpookjamItems.GF_BREAD.getRegistryName()); \/\/Set the system name of this new item from an enum we declared in Reference.java.\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/8b42d8326e9b4c77f99ec2f9ae37b114\n\n3. Go to the \"src\/main\/java\" folder, open the \"init\" package, and then open \"ModItems.java\". Update the file as follows, noting the addition of a new public static Item declaration and the new import line:\n\nCode:\npackage com.bigsis.spookjams.init; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import com.bigsis.spookjams.Reference; \/\/Import this to recognize our custom Reference.java class.import com.bigsis.spookjams.items.ItemCheese; \/\/Import this to recognize our custom ItemCheese.java class.import com.bigsis.spookjams.items.ItemGFBread; \/\/Import this to recognize our custom ItemGFBread.java class.import net.minecraft.client.Minecraft; \/\/Imported to allow the Minecraft client to be recognizedimport net.minecraft.client.renderer.block.model.ModelResourceLocation; \/\/Imported to allow the ModelResourceLocation class to be recognized.import net.minecraft.item.Item; \/\/Imported to allow for Minecraft objects to be registered.import net.minecraftforge.fml.common.registry.GameRegistry; \/\/Imported to recognize the use of the GameRegistry class.public class ModItems{\u00a0public static Item cheese; \/\/This is the name of our new custom item.\u00a0public static Item gf_bread; \/\/This is the name of our second custom item.\u00a0\u00a0public static void init() \/\/This method will initialize this object.\u00a0{\u00a0cheese = new ItemCheese(); \/\/Set the new Item to be a new instance of our custom cheese object as defined in ItemCheese.java.\u00a0gf_bread = new ItemGFBread(); \/\/Set the second new Item to be a new instance of our gluten-free bread object as defined in ItemGFBread.java.\u00a0}\u00a0\u00a0public static void register() \/\/This will register the item into the game.\u00a0{\u00a0GameRegistry.register(cheese); \/\/Register our custom item in Minecraft using the new Item object we initiated above.\u00a0GameRegistry.register(gf_bread); \/\/Register our second custom item.\u00a0}\u00a0\u00a0public static void registerRenders() \/\/This will contain all the calls to register the render for each item.\u00a0{\u00a0registerRender(cheese); \/\/Call the private method below.\u00a0registerRender(gf_bread); \/\/Call the private method below.\u00a0}\u00a0\u00a0private static void registerRender(Item item) \/\/This is the new item object that you will register into Minecraft.\u00a0{\u00a0Minecraft.getMinecraft().getRenderItem().getItemModelMesher().register(item, 0, new ModelResourceLocation(item.getRegistryName(), \"inventory\")); \/\/Register this new item by referencing its system name, and specifying that it will be an Inventory item.\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/6d728ea672bc5608604b88b7263a7dfa\n\n4. In the same package open \"ModCrafting.java\" and edit it as follows to give the gluten-free bread a crafting recipe:\n\nCode:\npackage com.bigsis.spookjams.init; \/\/My comments in this class are present in light-colored lines like this one that follow two forward-slashes (\"\/\/\"). To remove these, go to \"Edit > Find\/Replace...\", add \"\/\/.*\" to the Find field, and press \"Replace All\".import net.minecraft.init.Blocks; \/\/Imported to recognize Minecraft Block objects.import net.minecraft.init.Items; \/\/Imported to recognize Minecraft Item objects.import net.minecraft.item.ItemStack; \/\/Imported to recognize the ItemStack object.import net.minecraftforge.fml.common.registry.GameRegistry; \/\/Imported to recognize the GameRegistry object.public class ModCrafting {\u00a0public static void register() \u00a0{\u00a0GameRegistry.addShapedRecipe \/\/This creates a shaped recipe, which is the type of recipe that requires specific placement on the crafting grid.\u00a0(\u00a0new ItemStack(ModItems.cheese, 4), \/\/This recipe is to make cheese, specifically 4 wedges of it.\u00a0\"BMB\", \/\/This recipe specifically calls for a horizontal row of a milk bucket in the center, flanked by two brown mushrooms; note that since we only specify one row on the grid, you can craft these items on any row. \u00a0'B', Blocks.BROWN_MUSHROOM, \/\/This specifies that \"B\" stands for the Brown Mushroom \"bush\", not the block from a giant brown mushroom. Note that the use of the single quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0'M', new ItemStack(Items.MILK_BUCKET.setContainerItem(Items.BUCKET)) \/\/This specifies that \"M\" stands for a milk-filled bucket; to avoid using up a whole metal bucket just to make cheese, we also indicate that an empty bucket be returned after this crafting recipe is processed. \u00a0Note that the use of the single quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0);\u00a0GameRegistry.addShapedRecipe \u00a0(\u00a0new ItemStack(ModItems.gf_bread, 3), \/\/This recipe is to make gluten-free bread, specifically 3 loaves of it (because the ingredients are more nutrient-dense).\u00a0\"PPP\", \/\/This recipe specifically calls for a horizontal row of potatoes; note that since we only specify one row on the grid, you can craft these items on any row. \u00a0'P', Items.POTATO \/\/This specifies that \"P\" stands for \"potato\". Note that the use of the single quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0);\u00a0}}\n\nIf the above is too hard to read due to the word-wrapping, try this link:\nhttps:\/\/gist.github.com\/536e6a8546873f95f7f4b75a4857ab53\n\n5. Create an image of gluten-free bread that has a transparent background, is 16x16, and is saved in a PNG format... if you don't know how, you can also find a free sample asset of gluten-free bread from this link...\n\nhttp:\/\/bigsister.forumotion.com\/t896-item-texture-assets#15604\n\n6. Copy your new gf_bread.png file and save it inside your mod's asset location inside the Eclipse folder. \u00a0The location to save it in will be....\n\n...Spookjams\\src\\main\\resources\\assets\\bssjm\\textures\\items\n\nAlternatively, you can drag and drop gf_bread.png into the \"src\/main\/resources\" > \"textures.items\" package in Eclipse and it should allow you to import the asset directly.\n\n7. Go to the \"src\/main\/resources\" folder, open the \"assets.bssjm\" package, open the \"models.item\" package, and select the \"ItemCheese.json\" file, copy it, then paste it over itself to create a duplicate. \u00a0Name this duplicate \"ItemGFBread.json\" and edit it as follows:\n\nCode:\n{\u00a0\u00a0 \u00a0\"parent\": \"item\/generated\",\u00a0\u00a0 \u00a0\"textures\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"layer0\": \"bssjm:items\/gf_bread\"\u00a0\u00a0 \u00a0},\u00a0\u00a0 \u00a0\"display\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"thirdperson\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"rotation\": [ -90, 0, 9 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"translation\": [ 0, 1, -3 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"scale\": [ 0.55, 0.55, 0.55 ]\u00a0\u00a0 \u00a0 \u00a0 \u00a0},\u00a0\u00a0 \u00a0 \u00a0 \u00a0\"firstperson\": {\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"rotation\": [ 0, -135, 25 ],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"translation\": [ 0, 4, 2],\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"scale\": [ 1.7, 1.7, 1.7 ]\u00a0\u00a0 \u00a0 \u00a0 \u00a0}\u00a0\u00a0 \u00a0}}\n\n8. Go back into the \"src\/main\/resources\" folder, open the \"assets.bssjm\" package, and open the \"lang\" package, and then open the \"en_US.lang\" file. Edit it to include the following line:\n\n9. And you're done! Launch the game, and enjoy your cheese and gluten-free bread! Yum!\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nPart 7 - Creating a Furnace Recipe\n\nNote: Be sure to complete Part 6 of this tutorial first as this is a direct continuation of it:\nhttp:\/\/bigsister.forumotion.com\/t895-mod-making-crash-course-tutorial#15634\n\n1. Prior to beginning, I used Part 6 of this tutorial to create some new items for the game, specifically a boiled egg for this tutorial...\n\nI also created a milk chocolate bar more for fun and to reuse a decommissioned art asset:\n\nIt was also a good opportunity to practice the crafting lessons in Part 3 of this tutorial. \u00a0The actual crafting recipe for this milk chocolate bar is as follows [check the comments for more explanation]:\n\nCode:\n\u00a0GameRegistry.addShapedRecipe\u00a0(\u00a0new ItemStack(ModItems.chocolatebar, 1), \/\/This recipe is to make a milk \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0chocolate bar -- inferior to \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0dark chocolate, heh. \u00a0\u00a0\"SCS\", \/\/This represents the first row of recipe items, \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0consisting of two sugars flanking a cocoa bean. \u00a0\"SCS\", \/\/This represents the second row of recipe items, \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0consisting of two sugars flanking a cocoa bean.\u00a0\"SMS\", \/\/This represents the third row of recipe items, \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0consisting of two sugars flanking a milk bucket. \u00a0'S', Items.SUGAR, \/\/This specifies that \"S\" stands for sugar [boo!].\u00a0'C', new ItemStack(Items.DYE, 1, 3), \/\/This specifies that \"C\" stands \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0for cacao beans (one Dye with \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0metadata #3).\u00a0'M', new ItemStack(Items.MILK_BUCKET.setContainerItem(Items.BUCKET)) \u00a0\u00a0 \u00a0 \u00a0\/\/This specifies that \"M\" stands for a milk-filled bucket; \u00a0\u00a0 \u00a0 \u00a0 \u00a0to avoid using up a whole metal bucket just to make cheese, \u00a0\u00a0 \u00a0 \u00a0 \u00a0we also indicate that an empty bucket be returned after this \u00a0\u00a0 \u00a0 \u00a0 \u00a0crafting recipe is processed. \u00a0Note that the use of the single \u00a0\u00a0 \u00a0 \u00a0 \u00a0quote it intentional... it MUST BE IN SINGLE QUOTES!\u00a0);\n\nAnd yields the following.....\n\nThere is no need to go more in-depth into how these new items were created... just review the old lessons and you have a general understanding of how they're added.\n\n2. Now that we have our new assets, and especially our new ItemBoiledEgg, go ahead into\u00a0the \"src\/main\/java\" folder, open the \"init\" package, and then open \"ModCrafting.java\". The actual syntax for adding new Furnace Recipes is pretty simple, you only need to add the following:\n\nCode:\nGameRegistry.addSmelting \/\/this creates a smelting recipe, \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 which is a simple one-to-one exchange(\u00a0Items.EGG, \/\/Indicates the item you add into the furnace to smelt; \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0in this example, an egg.\u00a0new ItemStack(ModItems.boiledegg, 1), \/\/This specifies what item \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 you get after it's \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 smelted -- in this case, \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 one boiled egg.\u00a00.2F \/\/This is the amount of experience you get smelting \u00a0\u00a0 \u00a0 \u00a0 \u00a0this item; the lowest is 0.1F [cobblestone], and \u00a0\u00a0 \u00a0 \u00a0 \u00a0greatest is 1.0F [diamond ore]);\n\n3. And done! \u00a0Yep, that's it! \u00a0Load up Minecraft and enjoy your new boiled eggs!!!\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:\n\n## Re: Mod-Making Crash Course Tutorial\n\nIf you'd like the files for Part 7 of this tutorial, you can download the source code for version 0.02 \"soy\" here:\n\n_________________\n\nCiabatta\n\nPosts : 3885\nJoin date : 2014-01-03\nAge : 27\n\nRP Character Sheet\nName: Ciabatta Sylvia\nPersonality Trait: Busy\nCharacter Description:","date":"2017-10-19 19:45:57","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\": 0.18815335631370544, \"perplexity\": 10112.277897862243}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187823462.26\/warc\/CC-MAIN-20171019194011-20171019214011-00805.warc.gz\"}"} | null | null |
El Territori d'Idaho va ser un territori organitzat incorporat als Estats Units que va existir del 3 de març de 1863 al 3 de juliol de 1890, quan va ser admès dins la Unió com l'Estat d'Idaho. Cobria el territori dels actuals estats d'Idaho i Montana, i quasi tot l'actual estat de Wyomin, a excepció d'un cantó a l'extrem sud-est de l'estat. La primera capital va ser Lewiston (Idaho), fins que el 1866 va passar-ho a ser Boise.
L'any 1887 va estar a punt de ser dividida entre els estats de Washington al nord i de Nevada al sud, però el President Grover Cleveland va refusar de signar-ne l'autorització. Dos anys més tard, la Universitat d'Idaho es va establir a Moscow, al nord, en comptes d'Eagle Rock (que era com es coneixia l'actual ciutat d'Idaho Falls), al sud, cosa que va ajudar a alleugerir els resentiments dels habitants del nord per haver perdut la capitalitat del territori. L'any següent va esdevenir el 43è estat dels Estats Units.
Idaho | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,128 |
# **_Acclaim for_**
# **_The Spell of the Sensuous_**
"Forges a thoroughly articulate passage between science and mysticism.... Speculative, learned, and always 'lucid and precise' as the eye of the vulture that confronted him once on a cliff ledge, Abram has one of those rare minds which, like the mind of a musician or a great mathematician, fuses dreaminess with smarts."
— _Village Voice Literary Supplement_
"This is a landmark book. Scholars will doubtless recognize its brilliance, but they may overlook the most important part of Abram's achievement: he has written the best instruction manual yet for becoming fully human. I walked outside when I was done and the world was a different place."
—Bill McKibben,
author of _The End of Nature_
"Abram manages, almost magically, to stir in us a long-lost memory: deep in our bones, in our blood, in the air we breathe, we know that the world lives and speaks to us.... He shows that it is possible to reawaken the animistic dimension of perception and feeling without renouncing rationality and intellectual analysis.... A joy to read and a brilliant gift to our rapidly darkening world."
— _Shambhala Sun_
"This is a major work of research and intuitive brilliance, an archive of clear ideas. At the end of our century of precarious ecology, _The Spell of the Sensuous_ strikes the deepest notes of celebration and alertness—an indispensable book!"
—Howard Norman,
author of _The Bird Artist_
"A tour-de-force of sustained intelligence, broad scholarship, and a graceful prose style that has produced one of the most interesting books about nature published during the past decade."
— _Terra Nova_
"The wind, the rain, the mountains and rivers, the woodlands and meadows and all their inhabitants; we need these perhaps even more for our psyche than for our physical survival. No one that I know of has presented all this with the literary skill as well as the understanding that we find in this work of David Abram. It should be one of the most widely read and discussed books of these times."
—Thomas Berry,
author of _The Dream of the Earth_
"Abram's _Spell_ must be praised. It's so well done, well written, well thought. I know of no work more valuable for shifting our thinking and feeling about the place of humans in the world."
—James Hillman,
author of _The Soul's Code_
"Important and highly original, a fresh look at the world we live in but don't see."
—Elizabeth Marshall Thomas,
author of _The Hidden Life of Dogs_
"A masterpiece—combining poetic passion with intellectual rigor and daring. Electric with energy, it offers us a new approach to scholarly inquiry: as a fully embodied human animal. It opens pathways and vistas that will be fruitfully explored for years, indeed for generations, to come."
—Joanna Macy,
translator of Rainer Maria Rilke's
_The Book of Hours_
" _The Spell of the Sensuous_ does more than place itself on the cutting edge where ecology meets philosophy, psychology, and history. It magically subverts the dichotomies of culture and nature, body and mind, opening a vista of organic being and human possibility that is often imagined but seldom described. Reader beware, the message is spell-binding. One cannot read this book without risk of entering into an altered state of perceptual possibility."
—Max Oelschlager,
author of _The Idea of Wilderness_
"This book by David Abram lights up the landscape of language, flesh, mind, history, mapping us back into the world."
—Gary Snyder,
author of _Mountains and Rivers Without End_
"Nobody writes about the ecological depths of the human and more-than-human world with more love and lyrical sensitivity than David Abram."
—Theodore Roszak,
author of _Where the Wasteland Ends_
"Disclosing the sentience of all nature, and revealing the unsuspected effect of the more-than-human on our language and our lives, in unprecedented fashion, Abram generates true philosophy for the twenty-first century."
—Lynn Margulis, originator of the Gaia Hypothesis, author of _What Is Life?_
# David Abram
# **_The Spell of the Sensuous_**
David Abram, Ph.D., is an ecologist and philosopher whose writings have had a deepening influence upon the environmental movement in North America and abroad. A summa cum laude graduate of Wesleyan University, he holds a doctorate in philosophy from the State University of New York at Stony Brook and has been the recipient of fellowships from the Watson and Rockefeller Foundations and a Lannan Literary Award for Nonfiction. He is an accomplished sleight-of-hand magician and has lived and traded magic with indigenous magicians in Indonesia, Nepal, and the Americas. This is his first book.
**FIRST VINTAGE BOOKS EDITION, FEBRUARY 1997**
_Copyright © 1996 by David Abram_
All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Vintage Books, a division of Random House, Inc., New York, and simultaneously in Canada by Random House of Canada Limited, Toronto. Originally published in hardcover by Pantheon Books, a division of Random House, Inc., New York, in 1996.
Permissions acknowledgments are on this page.
The Library of Congress has cataloged the Pantheon edition as follows:
Abram, David.
The spell of the sensuous: perception and language in a more-than-human world / David Abram.
p. cm.
Includes bibliographical references
eISBN: 978-0-307-83055-5
1. Philosophy of nature. 2. Body, Human (Philosophy). 3. Sense (Philosophy). 4. Perception (Philosophy). 5. Human ecology. I. Title.
BD581.A25 1996 95-31466
128—dc20
Random House Web address: <http://www.randomhouse.com/>
v3.1
to the endangered and vanishing ones
# **_Contents_**
#
_Cover_
_About the Author_
_Title Page_
_Copyright_
_Dedication_
**_Preface and Acknowledgments_**
_Epigraph_
[1. **_The Ecology of Magic_
A PERSONAL INTRODUCTION TO THE INQUIRY**](Abra_9780307830555_epub_c01_r1.htm)
[2. **_Philosophy on the Way to Ecology_
A TECHNICAL INTRODUCTION TO THE INQUIRY**](Abra_9780307830555_epub_c02_r1.htm)
PART I: **_Edmund Husserl and Phenomenology_**
PART II: **_Maurice Merleau-Ponty and the Participatory Nature of Perception_**
3. **_The Flesh of Language_**
4. **_Animism and the Alphabet_**
5. **_In the Landscape of Language_**
6. **_Time, Space, and the Eclipse of the Earth_**
PART I: **_Abstraction_**
PART II: **_The Living Present_**
**_7. The Forgetting and Remembering of the Air_**
**_Coda: Turning Inside Out_**
**_Notes_**
**_Bibliography_**
# **_Preface and Acknowledgments_**
#
Humans are tuned for relationship. The eyes, the skin, the tongue, ears, and nostrils—all are gates where our body receives the nourishment of otherness. This landscape of shadowed voices, these feathered bodies and antlers and tumbling streams—these breathing shapes are our family, the beings with whom we are engaged, with whom we struggle and suffer and celebrate. For the largest part of our species' existence, humans have negotiated relationships with every aspect of the sensuous surroundings, exchanging possibilities with every flapping form, with each textured surface and shivering entity that we happened to focus upon. All could speak, articulating in gesture and whistle and sigh a shifting web of meanings that we felt on our skin or inhaled through our nostrils or focused with our listening ears, and to which we replied—whether with sounds, or through movements, or minute shifts of mood. The color of sky, the rush of waves—every aspect of the earthly sensuous could draw us into a relationship fed with curiosity and spiced with danger. Every sound was a voice, every scrape or blunder was a meeting—with Thunder, with Oak, with Dragonfly. And from all of these relationships our collective sensibilities were nourished.
Today we participate almost exclusively with other humans and with our own human-made technologies. It is a precarious situation, given our age-old reciprocity with the many-voiced landscape. We still _need_ that which is other than ourselves and our own creations. The simple premise of this book is that we are human only in contact, and conviviality, with what is not human.
Does such a premise imply that we must renounce all our complex technologies? It does not. But it does imply that we must renew our acquaintance with the sensuous world in which our techniques and technologies are all rooted. Without the oxygenating breath of the forests, without the clutch of gravity and the tumbled magic of river rapids, we have no distance from our technologies, no way of assessing their limitations, no way to keep ourselves from turning into them. We need to know the textures, the rhythms and tastes of the bodily world, and to distinguish readily between such tastes and those of our own invention. Direct sensuous reality, in all its more-than-human mystery, remains the sole solid touchstone for an experiential world now inundated with electronically-generated vistas and engineered pleasures; only in regular contact with the tangible ground and sky can we learn how to orient and to navigate in the multiple dimensions that now claim us.
THIS BOOK HAS BEEN WRITTEN WITH TWO GOALS IN MIND. I HAVE hoped, first, to provide a set of powerful conceptual tools for my colleagues in the broad world of environmental activism—for conservationists, wilderness advocates, community organizers, bioregionalists, nature writers, conservation biologists, ecopsychologists, and all others who are already struggling to make sense of, and to alleviate, our current estrangement from the animate earth. Yet I have also wished to provoke some new thinking within the institutional realm of scholars, scientists, and educators—many of whom have been strangely silent in response to the rapid deterioration of wild nature, the steady vanishing of other species, and the consequent flattening of our human relationships.
In light of these twin aims, I have tried to maintain a high standard of theoretical and scholarly precision, without, however, masking the passion, the puzzlement, and the pleasure that flow from my own engagement with the living land.
The reader will discover, for instance, that there are _two_ introductory chapters to the book. There is, first, a "Personal Introduction," which details some of the unusual adventures that first led me to raise the various questions addressed in this work. This chapter focuses upon my encounters and reflections while living as an itinerant sleight-of-hand magician among traditional, indigenous magicians in rural Asia. Second, there is a "Technical Introduction," outlining the theoretical approach brought to bear upon the questions addressed herein. More specifically, this chapter discusses the development, in the twentieth century, of the tradition of "phenomenology"—the study of direct experience. Originally intended to provide a solid foundation for the empirical sciences, the careful study of perceptual experience unexpectedly began to make evident the hidden centrality of the earth in all human experience; indeed, phenomenological research began to suggest that the human mind was thoroughly dependent upon (and thoroughly influenced by) our forgotten relation with the encompassing earth.
While sensorial experience, philosophical reflection, and empirical information are thoroughly entwined throughout this book, those readers who have little patience with philosophical matters should feel free to leap across the technical introduction (Chapter 2)—perhaps touching briefly down to explore those subsections whose titles provoke their curiosity. Others may wish to dance across parts of Chapter 3, which necessarily contains a few somewhat technical sections regarding the bodily nature of language. Toward the end of Chapter 3 a very brief summary will set the stage for what follows.
MANY COMRADES LENT THEIR SUPPORT TO THIS PROJECT. AMONG those whose curiosity and kindness helped engender this book are the bioregional animateur Chris Wells, ecological cellist Nelson Denman, seeress Heather Rowntree, dreamtracker R. P. Harbour, Julia Meeks, Francis Huxley, Sam Hitt, Vicki Dean, Rich Ryan, Stella Reed, and the rest of the All-Species clan of northern New Mexico.
The various reflections in this work were honed in passionate conversations with friends in diverse places, among them David Rothenberg, Arne Naess, Rachel Wiener, Bill Boaz, Gary Nabhan, Ivan Illich, Christopher Manes, Drew Leder, Max Oelschlager, Lynn Margulis, Dorion Sagan, James Hillman, Chellis Glendinning, Laura Sewall, Rick Boothby, Baird Callicott, Starhawk, Rex and Lisa Weyler, Valerie Gremillion, Tom Jay, and the greathearted Thomas Berry. Mountain-wizard Dolores LaChapelle and letter-scribe Amy Hannon gave essential encouragement in the earliest stages. Among those who read through parts of the earliest manuscript, Peter Manchester, Anthony Weston, Paul Shepard, and John Elder all offered fine insights.
Philosopher Edward Casey provided fellowship and guidance, as did the wild salmon-sage Freeman House. Historian Donald Worster provided encouragement and inspiration. The Buddhist scholar-poet Stan Lombardo offered unexpected hospitality, as did prairie-stewards Ken Lassman and Caryn Goldberg. Christian Gronau and Aileen Douglas shared their keen insights into the worlds of other animals. Rachel Bagby provided soul sustenance.
My editors were both a pleasure to work with. Jack Shoemaker deserves my warm thanks for his immediate enthusiasm with the book, and for taking time out from the bustle of setting up a new publishing house in order to read and refine the manuscript. Dan Frank provided patient guidance through the publishing maze, and many keen-sighted suggestions. He has my gratitude, as does his assistant Claudine O'Hearn, Thanks, as well, to my agent Ned Leavitt.
Generous grants from the Foundation for Deep Ecology and from the Levinson Foundation, as well as a year-long fellowship from the Rockefeller Foundation, greatly aided the researching and writing of this book.
Few people are gifted with great artists for parents, as I have been. Blanche Abram and Irv Abram, pianist and painter, provided much tactical help during the crafting of this work. I thank them for their encouragement, and for the intuition of beauty that they carefully granted to their children.
Finally, I extend a gratitude beyond words to my closest friend and ally, Grietje Laga, whose graceful intelligence deepens all my thoughts, and whose gentle magic ceaselessly returns me to my senses. Her company has made this whole adventure ever so much more wonderful.
**As** **THE CRICKETS' SOFT AUTUMN HUM**
**IS TO US**
**SO ARE WE TO THE TREES**
**AS ARE THEY**
**TO THE ROCKS AND THE HILLS**
**_—Gary Snyder_**
# **1**
# **_The Ecology of Magic_**
# A PERSONAL INTRODUCTION TO THE INQUIRY
LATE ONE EVENING I STEPPED OUT OF MY LITTLE HUT IN THE rice paddies of eastern Bali and found myself falling through space. Over my head the black sky was rippling with stars, densely clustered in some regions, almost blocking out the darkness between them, and more loosely scattered in other areas, pulsing and beckoning to each other. Behind them all streamed the great river of light with its several tributaries. Yet the Milky Way churned beneath me as well, for my hut was set in the middle of a large patchwork of rice paddies, separated from each other by narrow two-foot-high dikes, and these paddies were all filled with water. The surface of these pools, by day, reflected perfectly the blue sky, a reflection broken only by the thin, bright green tips of new rice. But by night the stars themselves glimmered from the surface of the paddies, and the river of light whirled through the darkness underfoot as well as above; there seemed no ground in front of my feet, only the abyss of star-studded space falling away forever.
I was no longer simply beneath the night sky, but also _above_ it—the immediate impression was of weightlessness. I might have been able to reorient myself, to regain some sense of ground and gravity, were it not for a fact that confounded my senses entirely: between the constellations below and the constellations above drifted countless fireflies, their lights flickering like the stars, some drifting up to join the clusters of stars overhead, others, like graceful meteors, slipping down from above to join the constellations underfoot, and all these paths of light upward and downward were mirrored, as well, in the still surface of the paddies. I felt myself at times falling through space, at other moments floating and drifting. I simply could not dispel the profound vertigo and giddiness; the paths of the fireflies, and their reflections in the water's surface, held me in a sustained trance. Even after I crawled back to my hut and shut the door on this whirling world, I felt that now the little room in which I lay was itself floating free of the earth.
Fireflies! It was in Indonesia, you see, that I was first introduced to the world of insects, and there that I first learned of the great influence that insects—such diminutive entities—could have upon the human senses. I had traveled to Indonesia on a research grant to study magic—more precisely, to study the relation between magic and medicine, first among the traditional sorcerers, or _dukuns_ , of the Indonesian archipelago, and later among the _dzankris_ , the traditional shamans of Nepal. One aspect of the grant was somewhat unique: I was to journey into rural Asia not outwardly as an anthropologist or academic researcher, but as a magician in my own right, in hopes of gaining a more direct access to the local sorcerers. I had been a professional sleight-of-hand magician for five years back in the United States, helping to put myself through college by performing in clubs and restaurants throughout New England. I had, as well, taken a year off from my studies in the psychology of perception to travel as a street magician through Europe and, toward the end of that journey, had spent some months in London, England, exploring the use of sleight-of-hand magic in psychotherapy, as a means of engendering communication with distressed individuals largely unapproachable by clinical healers. The success of this work suggested to me that sleight-of-hand might lend itself well to the curative arts, and I became, for the first time, interested in the relation, largely forgotten in the West, between folk medicine and magic.
It was this interest that led to the aforementioned grant, and to my sojourn as a magician in rural Asia. There, my sleight-of-hand skills proved invaluable as a means of stirring the curiosity of the local shamans. For magicians—whether modern entertainers or indigenous, tribal sorcerers—have in common the fact that they work with the malleable texture of perception. When the local sorcerers gleaned that I had at least some rudimentary skill in altering the common field of perception, I was invited into their homes, asked to share secrets with them, and eventually encouraged, even urged, to participate in various rituals and ceremonies.
But the focus of my research gradually shifted from questions regarding the application of magical techniques in medicine and ritual curing toward a deeper pondering of the relation between traditional magic and the animate natural world. This broader concern seemed to hold the keys to the earlier questions. For none of the several island sorcerers that I came to know in Indonesia, nor any of the _dzankris_ with whom I lived in Nepal, considered their work as ritual healers to be their major role or function within their communities. Most of them, to be sure, _were_ the primary healers or "doctors" for the villages in their vicinity, and they were often spoken of as such by the inhabitants of those villages. But the villagers also sometimes spoke of them, in low voices and in very private conversations, as witches (or "lejaks" in Bali), as dark magicians who at night might well be practicing their healing spells backward (or while turning to the left instead of to the right) in order to afflict people with the very diseases that they would later work to cure by day. Such suspicions seemed fairly common in Indonesia, and often were harbored with regard to the most effective and powerful healers, those who were most renowned for their skill in driving out illness. For it was assumed that a magician, in order to expel malevolent influences, must have a strong understanding of those influences and demons—even, in some areas, a close rapport with such powers. I myself never consciously saw any of those magicians or shamans with whom I became acquainted engage in magic for harmful purposes, nor any convincing evidence that they had ever done so. (Few of the magicians that I came to know even accepted money in return for their services, although they did accept gifts in the way of food, blankets, and the like.) Yet I was struck by the fact that none of them ever did or said anything to counter such disturbing rumors and speculations, which circulated quietly through the regions where they lived. Slowly, I came to recognize that it was through the agency of such rumors, and the ambiguous fears that such rumors engendered in the village people, that the sorcerers were able to maintain a basic level of privacy. If the villagers did not entertain certain fears about the local sorcerer, then they would likely come to obtain his or her magical help for every little malady and disturbance; and since a more potent practitioner must provide services for several large villages, the sorcerer would be swamped from morning to night with requests for ritual aid. By allowing the inevitable suspicions and fears to circulate unhindered in the region (and sometimes even encouraging and contributing to such rumors), the sorcerer ensured that _only_ those who were in real and profound need of his skills would dare to approach him for help.
This privacy, in turn, left the magician free to attend to what he acknowledged to be his primary craft and function. A clue to this function may be found in the circumstance that such magicians rarely dwell at the heart of their village; rather, their dwellings are commonly at the spatial periphery of the community or, more often, out beyond the edges of the village—amid the rice fields, or in a forest, or a wild cluster of boulders. I could easily attribute this to the just-mentioned need for privacy, yet for the magician in a traditional culture it seems to serve another purpose as well, providing a spatial expression of his or her symbolic position with regard to the community. For the magician's intelligence is not encompassed _within_ the society; its place is at the edge of the community, mediating _between_ the human community and the larger community of beings upon which the village depends for its nourishment and sustenance. This larger community includes, along with the humans, the multiple nonhuman entities that constitute the local landscape, from the diverse plants and the myriad animals—birds, mammals, fish, reptiles, insects—that inhabit or migrate through the region, to the particular winds and weather patterns that inform the local geography, as well as the various landforms—forests, rivers, caves, mountains—that lend their specific character to the surrounding earth.
The traditional or tribal shaman, I came to discern, acts as an intermediary between the human community and the larger ecological field, ensuring that there is an appropriate flow of nourishment, not just from the landscape to the human inhabitants, but from the human community back to the local earth. By his constant rituals, trances, ecstasies, and "journeys," he ensures that the relation between human society and the larger society of beings is balanced and reciprocal, and that the village never takes more from the living land than it returns to it—not just materially but with prayers, propitiations, and praise. The scale of a harvest or the size of a hunt are always negotiated between the tribal community and the natural world that it inhabits. To some extent every adult in the community is engaged in this process of listening and attuning to the other presences that surround and influence daily life. But the shaman or sorcerer is the exemplary voyager in the intermediate realm between the human and the more-than-human worlds, the primary strategist and negotiator in any dealings with the Others.
And it is only as a result of her continual engagement with the animate powers that dwell beyond the human community that the traditional magician is able to alleviate many individual illnesses that arise _within_ that community. The sorcerer derives her ability to cure ailments from her more continuous practice of "healing" or balancing the community's relation to the surrounding land. Disease, in such cultures, is often conceptualized as a kind of systemic imbalance within the sick person, or more vividly as the intrusion of a demonic or malevolent presence into his body. There are, at times, malevolent influences within the village or tribe itself that disrupt the health and emotional well-being of susceptible individuals within the community. Yet such destructive influences within the human community are commonly traceable to a disequilibrium between that community and the larger field of forces in which it is embedded. Only those persons who, by their everyday practice, are involved in monitoring and maintaining the relations _between_ the human village and the animate landscape are able to appropriately diagnose, treat, and ultimately relieve personal ailments and illnesses arising _within_ the village. Any healer who was not simultaneously attending to the intertwined relation between the human community and the larger, more-than-human field, would likely dispel an illness from one person only to have the same problem arise (perhaps in a new guise) somewhere else in the community. Hence, the traditional magician or medicine person functions primarily as an intermediary between human and nonhuman worlds, and only secondarily as a healer. Without a continually adjusted awareness of the relative balance or imbalance between the human group and its nonhuman environ, along with the skills necessary to modulate that primary relation, any "healer" is worthless—indeed, not a healer at all. The medicine person's primary allegiance, then, is not to the human community, but to the earthly web of relations in which that community is embedded—it is from this that his or her power to alleviate human illness derives—and this sets the local magician apart from other persons.
The primacy for the magician of nonhuman nature—the centrality of his relation to other species and to the earth—is not always evident to Western researchers. Countless anthropologists have managed to overlook the ecological dimension of the shaman's craft, while writing at great length of the shaman's rapport with "supernatural" entities. We can attribute much of this oversight to the modern, civilized assumption that the natural world is largely determinate and mechanical, and that that which is regarded as mysterious, powerful, and beyond human ken must therefore be of some other, nonphysical realm _above_ nature, "supernatural."
The oversight becomes still more comprehensible when we realize that many of the earliest European interpreters of indigenous lifeways were Christian missionaries. For the Church had long assumed that only human beings have intelligent souls, and that the other animals, to say nothing of trees and rivers, were "created" for no other reason than to serve humankind. We can easily understand why European missionaries, steeped in the dogma of institutionalized Christianity, assumed a belief in supernatural, otherworldly powers among those tribal persons whom they saw awestruck and entranced by nonhuman (but nevertheless natural) forces. What is remarkable is the extent to which contemporary anthropology still preserves the ethnocentric bias of these early interpreters. We no longer describe the shamans' enigmatic spirit-helpers as the "superstitious claptrap of heathen primitives"—we have cleansed ourselves of at least _that_ much ethnocentrism; yet we still refer to such enigmatic forces, respectfully now, as "supernaturals"—for we are unable to shed the sense, so endemic to scientific civilization, of nature as a rather prosaic and predictable realm, unsuited to such mysteries. Nevertheless, that which is regarded with the greatest awe and wonder by indigenous, oral cultures is, I suggest, none other than what we view as nature itself. The deeply mysterious powers and entities with whom the shaman enters into a rapport are ultimately the same forces—the same plants, animals, forests, and winds—that to literate, "civilized" Europeans are just so much scenery, the pleasant backdrop of our more pressing human concerns.
The most sophisticated definition of "magic" that now circulates through the American counterculture is "the ability or power to alter one's consciousness at will." No mention is made of any _reason_ for altering one's consciousness. Yet in tribal cultures that which we call "magic" takes its meaning from the fact that humans, in an indigenous and oral context, experience their own consciousness as simply one form of awareness among many others. The traditional magician cultivates an ability to shift out of his or her common state of consciousness precisely in order to make contact with the other organic forms of sensitivity and awareness with which human existence is entwined. Only by temporarily shedding the accepted perceptual logic of his culture can the sorcerer hope to enter into relation with other species on their own terms; only by altering the common organization of his senses will he be able to enter into a rapport with the multiple nonhuman sensibilities that animate the local landscape. It is this, we might say, that defines a shaman: the ability to readily slip out of the perceptual boundaries that demarcate his or her particular culture—boundaries reinforced by social customs, taboos, and most importantly, the common speech or language—in order to make contact with, and learn from, the other powers in the land. His magic is precisely this heightened receptivity to the meaningful solicitations—songs, cries, gestures—of the larger, more-than-human field.
Magic, then, in its perhaps most primordial sense, is the experience of existing in a world made up of multiple intelligences, the intuition that every form one perceives—from the swallow swooping overhead to the fly on a blade of grass, and indeed the blade of grass itself—is an _experiencing_ form, an entity with its own predilections and sensations, albeit sensations that are very different from our own.
To be sure, the shaman's ecological function, his or her role as intermediary between human society and the land, is not always obvious at first blush, even to a sensitive observer. We see the sorcerer being called upon to cure an ailing tribesman of his sleeplessness, or perhaps simply to locate some missing goods; we witness him entering into trance and sending his awareness into other dimensions in search of insight and aid. Yet we should not be so ready to interpret these dimensions as "supernatural," nor to view them as realms entirely "internal" to the personal psyche of the practitioner. For it is likely that the "inner world" of our Western psychological experience, like the supernatural heaven of Christian belief, originates in the loss of our ancestral reciprocity with the animate earth. When the animate powers that surround us are suddenly construed as having less significance than ourselves, when the generative earth is abruptly defined as a determinate object devoid of its own sensations and feelings, then the sense of a wild and multiplicitous otherness (in relation to which human existence has always oriented itself) must migrate, either into a supersensory heaven beyond the natural world, or else into the human skull itself—the only allowable refuge, in this world, for what is ineffable and unfathomable.
But in genuinely oral, indigenous cultures, the sensuous world itself remains the dwelling place of the gods, of the numinous powers that can either sustain or extinguish human life. It is not by sending his awareness out beyond the natural world that the shaman makes contact with the purveyors of life and health, nor by journeying into his personal psyche; rather, it is by propelling his awareness laterally, outward into the depths of a landscape at once both sensuous and psychological, the living dream that we share with the soaring hawk, the spider, and the stone silently sprouting lichens on its coarse surface.
The magician's intimate relationship with nonhuman nature becomes most evident when we attend to the easily overlooked background of his or her practice—not just to the more visible tasks of curing and ritual aid to which she is called by individual clients, or to the larger ceremonies at which she presides and dances, but to the content of the prayers by which she prepares for such ceremonies, and to the countless ritual gestures that she enacts when alone, the daily propitiations and praise that flow from her toward the land and _its_ many voices.
ALL THIS ATTENTION TO NONHUMAN NATURE WAS, AS I HAVE MENTIONED, very far from my intended focus when I embarked on my research into the uses of magic and medicine in Indonesia, and it was only gradually that I became aware of this more subtle dimension of the native magician's craft. The first shift in my preconceptions came rather quietly, when I was staying for some days in the home of a young "balian," or magic practitioner, in the interior of Bali. I had been provided with a simple bed in a separate, one-room building in the balian's family compound (most compound homes, in Bali, are comprised of several separate small buildings, for sleeping and for cooking, set on a single enclosed plot of land), and early each morning the balian's wife came to bring me a small but delicious bowl of fruit, which I ate by myself, sitting on the ground outside, leaning against the wall of my hut and watching the sun slowly climb through the rustling palm leaves. I noticed, when she delivered the fruit, that my hostess was also balancing a tray containing many little green plates: actually, they were little boat-shaped platters, each woven simply and neatly from a freshly cut section of palm frond. The platters were two or three inches long, and within each was a little mound of white rice. After handing me my breakfast, the woman and the tray disappeared from view behind the other buildings, and when she came by some minutes later to pick up my empty bowl, the tray in her hands was empty as well.
The second time that I saw the array of tiny rice platters, I asked my hostess what they were for. Patiently, she explained to me that they were offerings for the household spirits. When I inquired about the Balinese term that she used for "spirit," she repeated the same explanation, now in Indonesian, that these were gifts for the spirits of the family compound, and I saw that I had understood her correctly. She handed me a bowl of sliced papaya and mango, and disappeared around the corner. I pondered for a minute, then set down the bowl, stepped to the side of my hut, and peered through the trees. At first unable to see her, I soon caught sight of her crouched low beside the corner of one of the other buildings, carefully setting what I presumed was one of the offerings on the ground at that spot. Then she stood up with the tray, walked to the other visible corner of the same building, and there slowly and carefully set another offering on the ground. I returned to my bowl of fruit and finished my breakfast. That afternoon, when the rest of the household was busy, I walked back behind the building where I had seen her set down the two offerings. There were the little green platters, resting neatly at the two rear corners of the building. But the mounds of rice that had been within them were gone.
The next morning I finished the sliced fruit, waited for my hostess to come by for the empty bowl, then quietly headed back behind the buildings. Two fresh palm-leaf offerings sat at the same spots where the others had been the day before. These were filled with rice. Yet as I gazed at one of these offerings, I abruptly realized, with a start, that one of the rice kernels was actually moving.
Only when I knelt down to look more closely did I notice a line of tiny black ants winding through the dirt to the offering. Peering still closer, I saw that two ants had already climbed onto the offering and were struggling with the uppermost kernel of rice; as I watched, one of them dragged the kernel down and off the leaf, then set off with it back along the line of ants advancing on the offering. The second ant took another kernel and climbed down with it, dragging and pushing, and fell over the edge of the leaf, then a third climbed onto the offering. The line of ants seemed to emerge from a thick clump of grass around a nearby palm tree. I walked over to the other offering and discovered another line of ants dragging away the white kernels. This line emerged from the top of a little mound of dirt, about fifteen feet away from the buildings. There was an offering on the ground by a corner of my building as well, and a nearly identical line of ants. I walked into my room chuckling to myself: the balian and his wife had gone to so much trouble to placate the household spirits with gifts, only to have their offerings stolen by little six-legged thieves. What a waste! But then a strange thought dawned on me: what if the ants were the very "household spirits" to whom the offerings were being made?
I soon began to discern the logic of this. The family compound, ike most on this tropical island, had been constructed in the vicinity of several ant colonies. Since a great deal of cooking took place in the compound (which housed, along with the balian and his wife and children, various members of their extended family), and also much preparation of elaborate offerings of foodstuffs for various rituals and festivals in the surrounding villages, the grounds and the buildings at the compound were vulnerable to infestations by the sizable ant population. Such invasions could range from rare nuisances to a periodic or even constant siege. It became apparent that the daily palm-frond offerings served to preclude such an attack by the natural forces that surrounded (and underlay) the family's land. The daily gifts of rice kept the ant colonies occupied—and, presumably, satisfied. Placed in regular, repeated locations at the corners of various structures around the compound, the offerings seemed to establish certain boundaries between the human and ant communities; by honoring this boundary with gifts, the humans apparently hoped to persuade the insects to respect the boundary and not enter the buildings.
Yet I remained puzzled by my hostess's assertion that these were gifts "for the spirits." To be sure, there has always been some confusion between our Western notion of "spirit" (which so often is defined in contrast to matter or "flesh"), and the mysterious presences to which tribal and indigenous cultures pay so much respect. I have already alluded to the gross misunderstandings arising from the circumstance that many of the earliest Western students of these other customs were Christian missionaries all too ready to see occult ghosts and immaterial phantoms where the tribespeople were simply offering their respect to the local winds. While the notion of "spirit" has come to have, for us in the West, a primarily anthropomorphic or human association, my encounter with the ants was the first of many experiences suggesting to me that the "spirits" of an indigenous culture are primarily those modes of intelligence or awareness that do _not_ possess a human form.
As humans, we are well acquainted with the needs and capacities of the human body—we _live_ our own bodies and so know, from within, the possibilities of our form. We cannot know, with the same familiarity and intimacy, the lived experience of a grass snake or a snapping turtle; we cannot readily experience the precise sensations of a hummingbird sipping nectar from a flower or a rubber tree soaking up sunlight. And yet we do know how it feels to sip from a fresh pool of water or to bask and stretch in the sun. Our experience may indeed be a variant of these other modes of sensitivity; nevertheless, we cannot, as humans, precisely experience the living sensations of another form. We do not know, with full clarity, their desires or motivations; we cannot know, or can never be sure that we know, what they know. That the deer does experience sensations, that it carries knowledge of how to orient in the land, of where to find food and how to protect its young, that it knows well how to survive in the forest without the tools upon which we depend, is readily evident to our human senses. That the mango tree has the ability to create fruit, or the yarrow plant the power to reduce a child's fever, is also evident. To humankind, these Others are purveyors of secrets, carriers of intelligence that we ourselves often need: it is these Others who can inform us of unseasonable changes in the weather, or warn us of imminent eruptions and earthquakes, who show us, when foraging, where we may find the ripest berries or the best route to follow back home. By watching them build their nests and shelters, we glean clues regarding how to strengthen our own dwellings, and their deaths teach us of our own. We receive from them countless gifts of food, fuel, shelter, and clothing. Yet still they remain Other to us, inhabiting their own cultures and displaying their own rituals, never wholly fathomable.
Moreover, it is not only those entities acknowledged by Western civilization as "alive," not only the other animals and the plants that speak, as spirits, to the senses of an oral culture, but also the meandering river from which those animals drink, and the torrential monsoon rains, and the stone that fits neatly into the palm of the hand. The mountain, too, has its thoughts. The forest birds whirring and chattering as the sun slips below the horizon are vocal organs of the rain forest itself.
Bali, of course, is hardly an aboriginal culture; the complexity of its temple architecture, the intricacy of its irrigation systems, the resplendence of its colorful festivals and crafts all bespeak the influence of various civilizations, most notably the Hindu complex of India. In Bali, nevertheless, these influences are thoroughly intertwined with the indigenous animism of the Indonesian archipelago; the Hindu gods and goddesses have been appropriated, as it were, by the more volcanic, eruptive spirits of the local terrain.
Yet the underlying animistic cultures of Indonesia, like those of many islands in the Pacific, are steeped as well in beliefs often referred to by ethnologists as "ancestor worship," and some may argue that the ritual reverence paid to one's long-dead human ancestors (and the assumption of their influence in present life), easily invalidates my assertion that the various "powers" or "spirits" that move through the discourse of indigenous, oral peoples are ultimately tied to nonhuman (but nonetheless sentient) forces in the enveloping landscape.
This objection rests upon certain assumptions implicit in Christian civilization, such as the assumption that the "spirits" of dead persons necessarily retain their human form, and that they reside in a domain outside of the physical world to which our senses give us access. However, most indigenous tribal peoples have no such ready recourse to an immaterial realm outside earthly nature. Our strictly human heavens and hells have only recently been abstracted from the sensuous world that surrounds us, from this more-than-human realm that abounds in its own winged intelligences and cloven-hoofed powers. For almost all oral cultures, the enveloping and sensuous earth remains the dwelling place of both the living _and_ the dead. The "body"—whether human or otherwise—is not yet a mechanical object in such cultures, but is a magical entity, the mind's own sensuous aspect, and at death the body's decomposition into soil, worms, and dust can only signify the gradual reintegration of one's ancestors and elders into the living landscape, from which all, too, are born.
Each indigenous culture elaborates this recognition of metamorphosis in its own fashion, taking its clues from the particular terrain in which it is situated. Often the invisible atmosphere that animates the visible world—the subtle presence that circulates both within us and between all things—retains within itself the spirit or breath of the dead person until the time when that breath will enter and animate another visible body—a bird, or a deer, or a field of wild grain. Some cultures may burn, or "cremate," the body in order to more completely return the person, as smoke, to the swirling air, while that which departs as flame is offered to the sun and stars, and that which lingers as ash is fed to the dense earth. Still other cultures may dismember the body, leaving certain parts in precise locations where they will likely be found by condors, or where they will be consumed by mountain lions or by wolves, thus hastening the re-incarnation of that person into a particular animal realm within the landscape. Such examples illustrate simply that death, in tribal cultures, initiates a metamorphosis wherein the person's presence does not "vanish" from the sensible world (where would it go?) but rather remains as an animating force within the vastness of the landscape, whether subtly, in the wind, or more visibly, in animal form, or even as the eruptive, ever to be appeased, wrath of the volcano. "Ancestor worship," in its myriad forms, then, is ultimately another mode of attentiveness to nonhuman nature; it signifies not so much an awe or reverence of human powers, but rather a reverence for those forms that awareness takes when it is _not_ in human form, when the familiar human embodiment dies and decays to become part of the encompassing cosmos.
This cycling of the human back into the larger world ensures that the other forms of experience that we encounter—whether ants, or willow trees, or clouds—are never absolutely alien to ourselves. Despite the obvious differences in shape, and ability, and style of being, they remain at least distantly familiar, even familial. It is, paradoxically, this perceived kinship or consanguinity that renders the difference, or otherness, so eerily potent.
SEVERAL MONTHS AFTER MY ARRIVAL IN BALI, I LEFT THE VILLAGE in which I was staying to visit one of the pre-Hindu sites on the island. I arrived on my bicycle early in the afternoon, after the bus carrying tourists from the coast had departed. A flight of steps took me down into a lush, emerald valley, lined by cliffs on either side, awash with the speech of the river and the sighing of the wind through high, unharvested grasses. On a small bridge crossing the river I met an old woman carrying a wide basket on her head and holding the hand of a little, shy child; the woman grinned at me with the red, toothless smile of a beetle nut chewer. On the far side of the river I stood in front of a great moss-covered complex of passageways, rooms, and courtyards carved by hand out of the black volcanic rock.
I noticed, at a bend in the canyon downstream, a further series of caves carved into the cliffs. These appeared more isolated and remote, unattended by any footpath I could discern. I set out through the grasses to explore them. This proved much more difficult than I anticipated, but after getting lost in the tall grasses, and fording the river three times, I at last found myself beneath the caves. A short scramble up the rock wall brought me to the mouth of one of them, and I entered on my hands and knees. It was a wide but low opening, perhaps only four feet high, and the interior receded only about five or six feet into the cliff. The floor and walls were covered with mosses, painting the cave with green patterns and softening the harshness of the rock; the place, despite its small size—or perhaps because of it—had an air of great friendliness. I climbed to two other caves, each about the same size, but then felt drawn back to the first one, to sit cross-legged on the cushioning moss and gaze out across the emerald canyon. It was quiet inside, a kind of intimate sanctuary hewn into the stone. I began to explore the rich resonance of the enclosure, first just humming, then intoning a simple chant taught to me by a balian some days before. I was delighted by the overtones that the cave added to my voice, and sat there singing for a long while. I did not notice the change in the wind outside, or the cloud shadows darkening the valley, until the rains broke—suddenly and with great force. The first storm of the monsoon!
I had experienced only slight rains on the island before then, and was startled by the torrential downpour now sending stones tumbling along the cliffs, building puddles and then ponds in the green landscape below, swelling the river. There was no question of returning home—I would be unable to make my way back through the flood to the valley's entrance. And so, thankful for the shelter, I re-crossed my legs to wait out the storm. Before long the rivulets falling along the cliff above gathered themselves into streams, and two small waterfalls cascaded across the cave's mouth. Soon I was looking into a solid curtain of water, thin in some places, where the canyon's image flickered unsteadily, and thickly rushing in others. My senses were all but overcome by the wild beauty of the cascade and by the roar of sound, my body trembling inwardly at the weird sense of being sealed into my hiding place.
And then, in the midst of all this tumult, I noticed a small, delicate activity. Just in front of me, and only an inch or two to my side of the torrent, a spider was climbing a thin thread stretched across the mouth of the cave. As I watched, it anchored another thread to the top of the opening, then slipped back along the first thread and joined the two at a point about midway between the roof and the floor. I lost sight of the spider then, and for a while it seemed that it had vanished, thread and all, until my focus rediscovered it. Two more threads now radiated from the center to the floor, and then another; soon the spider began to swing between these as on a circular trellis, trailing an ever-lengthening thread which it affixed to each radiating rung as it moved from one to the next, spiraling outward. The spider seemed wholly undaunted by the tumult of waters spilling past it, although every now and then it broke off its spiral dance and climbed to the roof or the floor to tug on the radii there, assuring the tautness of the threads, then crawled back to where it left off. Whenever I lost the correct focus, I waited to catch sight of the spinning arachnid, and then let its dancing form gradually draw the lineaments of the web back into visibility, tying my focus into each new knot of silk as it moved, weaving my gaze into the ever-deepening pattern.
And then, abruptly, my vision snagged on a strange incongruity: another thread slanted across the web, neither radiating nor spiraling from the central juncture, violating the symmetry. As I followed it with my eyes, pondering its purpose in the overall pattern, I began to realize that it was on a different plane from the rest of the web, for the web slipped out of focus whenever this new line became clearer. I soon saw that it led to its own center, about twelve inches to the right of the first, another nexus of forces from which several threads stretched to the floor and the ceiling. And then I saw that there was a _different_ spider spinning this web, testing its tautness by dancing around it like the first, now setting the silken cross weaves around the nodal point and winding outward. The two spiders spun independently of each other, but to my eyes they wove a single intersecting pattern. This widening of my gaze soon disclosed yet another spider spiraling in the cave's mouth, and suddenly I realized that there were _many_ overlapping webs coming into being, radiating out at different rhythms from myriad centers poised—some higher, some lower, some minutely closer to my eyes and some farther—between the stone above and the stone below.
I sat stunned and mesmerized before this ever-complexifying expanse of living patterns upon patterns, my gaze drawn like a breath into one converging group of lines, then breathed out into open space, then drawn down into another convergence. The curtain of water had become utterly silent—I tried at one point to hear it, but could not. My senses were entranced.
I had the distinct impression that I was watching the universe being born, galaxy upon galaxy....
NIGHT FILLED THE CAVE WITH DARKNESS. THE RAIN HAD NOT stopped. Yet, strangely, I felt neither cold nor hungry—only remarkably peaceful and at home. Stretching out upon the moist, mossy floor near the back of the cave, I slept.
When I awoke, the sun was staring into the canyon, the grasses below rippling with bright blues and greens. I could see no trace of the webs, nor their weavers. Thinking that they were invisible to my eyes without the curtain of water behind them, I felt carefully with my hands around and through the mouth of the cave. But the webs were gone. I climbed down to the river and washed, then hiked across and out of the canyon to where my cycle was drying in the sun, and headed back to my own valley.
I have never, since that time, been able to encounter a spider without feeling a great strangeness and awe. To be sure, insects and spiders are not the only powers, or even central presences, in the Indonesian universe. But they were _my_ introduction to the spirits, to the magic afoot in the land. It was from them that I first learned of the intelligence that lurks in nonhuman nature, the ability that an alien form of sentience has to echo one's own, to instill a reverberation in oneself that temporarily shatters habitual ways of seeing and feeling, leaving one open to a world all alive, awake, and aware. It was from such small beings that my senses first learned of the countless worlds within worlds that spin in the depths of this world that we commonly inhabit, and from them that I learned that my body could, with practice, enter sensorially into these dimensions. The precise and minuscule craft of the spiders had so honed and focused my awareness that the very webwork of the universe, of which my own flesh was a part, seemed to be being spun by their arcane art. I have already spoken of the ants, and of the fireflies, whose sensory likeness to the lights in the night sky had taught me the fickleness of gravity. The long and cyclical trance that we call malaria was also brought to me by insects, in this case mosquitoes, and I lived for three weeks in a feverish state of shivers, sweat, and visions.
I had rarely before paid much attention to the natural world. But my exposure to traditional magicians and seers was shifting my senses; I became increasingly susceptible to the solicitations of nonhuman things. In the course of struggling to decipher the magicians' odd gestures or to fathom their constant spoken references to powers unseen and unheard, I began to _see_ and to _hear_ in a manner I never had before. When a magician spoke of a power or "presence" lingering in the corner of his house, I learned to notice the ray of sunlight that was then pouring through a chink in the roof, illuminating a column of drifting dust, and to realize that that column of light was indeed a power, influencing the air currents by its warmth, and indeed influencing the whole mood of the room; although I had not consciously seen it before, it had already been structuring my experience. My ears began to attend, in a new way, to the songs of birds—no longer just a melodic background to human speech, but meaningful speech in its own right, responding to and commenting on events in the surrounding earth. I became a student of subtle differences: the way a breeze may flutter a single leaf on a whole tree, leaving the other leaves silent and unmoved (had not that leaf, then, been brushed by a magic?); or the way the intensity of the sun's heat expresses itself in the precise rhythm of the crickets. Walking along the dirt paths, I learned to slow my pace in order to _feel_ the difference between one nearby hill and the next, or to taste the presence of a particular field at a certain time of day when, as I had been told by a local _dukun_ , the place had a special power and proffered unique gifts. It was a power communicated to my senses by the way the shadows of the trees fell at that hour, and by smells that only then lingered in the tops of the grasses without being wafted away by the wind, and other elements I could only isolate after many days of stopping and listening.
And gradually, then, other animals began to intercept me in my wanderings, as if some quality in my posture or the rhythm of my breathing had disarmed their wariness; I would find myself face-to-face with monkeys, and with large lizards that did not slither away when I spoke, but leaned forward in apparent curiosity. In rural Java, I often noticed monkeys accompanying me in the branches overhead, and ravens walked toward me on the road, croaking. While at Pangandaran, a nature preserve on a peninsula jutting out from the south coast of Java ("a place of many spirits," I was told by nearby fishermen), I stepped out from a clutch of trees and found myself looking into the face of one of the rare and beautiful bison that exist only on that island. Our eyes locked. When it snorted, I snorted back; when it shifted its shoulders, I shifted my stance; when I tossed my head, it tossed _its_ head in reply. I found myself caught in a nonverbal conversation with this Other, a gestural duet with which my conscious awareness had very little to do. It was as if my body in its actions was suddenly being motivated by a wisdom older than my thinking mind, as though it was held and moved by a logos, deeper than words, spoken by the Other's body, the trees, and the stony ground on which we stood.
ANTHROPOLOGY'S INABILITY TO DISCERN THE SHAMAN'S ALLEGIANCE to nonhuman nature has led to a curious circumstance in the "developed world" today, where many persons in search of spiritual understanding are enrolling in workshops concerned with "shamanic" methods of personal discovery and revelation. Psychotherapists and some physicians have begun to specialize in "shamanic healing techniques." "Shamanism" has thus come to connote an alternative form of therapy; the emphasis, among these new practitioners of popular shamanism, is on personal insight and curing. These are noble aims, to be sure, yet they are secondary to, and derivative from, the primary role of the indigenous shaman, a role that cannot be fulfilled without long and sustained exposure to wild nature, to its patterns and vicissitudes. Mimicking the indigenous shaman's curative methods without his intimate knowledge of the wider natural community cannnot, if I am correct, do anything more than trade certain symptoms for others, or shift the locus of dis-ease from place to place within the human community. For the source of stress lies in the relation _between_ the human community and the natural landscape.
Western industrial society, of course, with its massive scale and hugely centralized economy, can hardly be seen in relation to any particular landscape or ecosystem; the more-than-human ecology with which it is directly engaged is the biosphere itself. Sadly, our culture's relation to the earthly biosphere can in no way be considered a reciprocal or balanced one: with thousands of acres of nonregenerating forest disappearing every hour, and hundreds of our fellow species becoming extinct each month as a result of our civilization's excesses, we can hardly be surprised by the amount of epidemic illness in our culture, from increasingly severe immune dysfunctions and cancers, to widespread psychological distress, depression, and ever more frequent suicides, to the accelerating number of household killings and mass murders committed for no apparent reason by otherwise coherent individuals.
From an animistic perspective, the clearest source of all this distress, both physical and psychological, lies in the aforementioned violence needlessly perpetrated by our civilization on the ecology of the planet; only by alleviating the latter will we be able to heal the former. While this may sound at first like a simple statement of faith, it makes eminent and obvious sense as soon as we acknowledge our thorough dependence upon the countless other organisms with whom we have evolved. Caught up in a mass of abstractions, our attention hypnotized by a host of human-made technologies that only reflect us back to ourselves, it is all too easy for us to forget our carnal inherence in a more-than-human matrix of sensations and sensibilities. Our bodies have formed themselves in delicate reciprocity with the manifold textures, sounds, and shapes of an animate earth—our eyes have evolved in subtle interaction with _other_ eyes, as our ears are attuned by their very structure to the howling of wolves and the honking of geese. To shut ourselves off from these other voices, to continue by our lifestyles to condemn these other sensibilities to the oblivion of extinction, is to rob our own senses of their integrity, and to rob our minds of their coherence. We are human only in contact, and conviviality, with what is not human.
ALTHOUGH THE INDONESIAN ISLANDS ARE HOME TO AN ASTONISHING diversity of birds, it was only when I went to study among the Sherpa people of the high Himalayas that I was truly initiated into the avian world. The Himalayas are young mountains, their peaks not yet rounded by the endless action of wind and ice, and so the primary dimension of the visible landscape is overwhelmingly vertical. Even in the high ridges one seldom attains a view of a distant horizon; instead one's vision is deflected upward by the steep face of the next mountain. The whole land has surged skyward in a manner still evident in the lines and furrows of the mountain walls, and this ancient dynamism readily communicates itself to the sensing body.
In such a world those who dwell and soar in the sky are the primary powers. They alone move easily in such a zone, swooping downward to become a speck near the valley floor, or spiraling into the heights on invisible currents. The wingeds, alone, carry the immediate knowledge of what is unfolding on the far side of the next ridge, and hence it is only by watching them that one can be kept apprised of climatic changes in the offing, as well as of subtle shifts in the flow and density of air currents in one's own valley. Several of the shamans that I met in Nepal had birds as their close familiars. Ravens are constant commentators on village affairs. The smaller, flocking birds perform aerobatics in unison over the village rooftops, twisting and swerving in a perfect sympathy of motion, the whole flock appearing like a magic banner that floats and flaps on air currents over the village, then descends in a heap, only to be carried aloft by the wind a moment later, rippling and swelling.
For some time I visited a Sherpa _dzankri_ whose rock home was built into one of the steep mountainsides of the Khumbu region in Nepal. On one of our walks along the narrow cliff trails that wind around the mountain, the _dzankri_ pointed out to me a certain boulder, jutting out from the cliff, on which he had "danced" before attempting some especially difficult cures. I recognized the boulder several days later when hiking back down toward the _dzankri_ 's home from the upper yak pastures, and I climbed onto the rock, not to dance but to ponder the pale white and red lichens that gave life to its surface, and to rest. Across the dry valley, two lammergeier condors floated between gleaming, snow-covered peaks. It was a ringing blue Himalayan day, clear as a bell. After a few moments I took a silver coin out of my pocket and aimlessly began a simple sleight-of-hand exercise, rolling the coin over the knuckles of my right hand. I had taken to practicing this somewhat monotonous exercise in response to the endless flicking of prayer-beads by the older Sherpas, a practice usually accompanied by a repetitively chanted prayer: _"Om Mani Padme Hum"_ (O the Jewel in the Lotus). But there was no prayer accompanying my revolving coin, aside from my quiet breathing and the dazzling sunlight. I noticed that one of the two condors in the distance had swerved away from its partner and was now floating over the valley, wings outstretched. As I watched it grow larger, I realized, with some delight, that it was heading in my general direction; I stopped rolling the coin and stared. Yet just then the lammergeier halted in its flight, motionless for a moment against the peaks, then swerved around and headed back toward its partner in the distance. Disappointed, I took up the coin and began rolling it along my knuckles once again, its silver surface catching the sunlight as it turned, reflecting the rays back into the sky. Instantly, the condor swung out from its path and began soaring back in a wide arc. Once again, I watched its shape grow larger. As the great size of the bird became apparent, I felt my skin begin to crawl and come alive, like a swarm of bees all in motion, and a humming grew loud in my ears. The coin continued rolling along my fingers. The creature loomed larger, and larger still, until, suddenly, it was there—an immense silhouette hovering just above my head, huge wing feathers rustling ever so slightly as they mastered the breeze. My fingers were frozen, unable to move; the coin dropped out of my hand. And then I felt myself stripped naked by an alien gaze infinitely more lucid and precise than my own. I do not know for how long I was transfixed, only that I felt the air streaming past naked knees and heard the wind whispering in my feathers long after the Visitor had departed.
I RETURNED TO A NORTH AMERICA WHOSE ONLY INDIGENOUS species of condor was on the brink of extinction, mostly as a result of lead poisoning from bullets in the carrion it consumes. But I did not think about this. I was excited by the new sensibilities that had stirred in me—my newfound awareness of a more-than-human world, of the great potency of the land, and particularly of the keen intelligence of other animals, large and small, whose lives and cultures interpenetrate our own. I startled neighbors by chattering with squirrels, who swiftly climbed down the trunks of their trees and across lawns to banter with me, or by gazing for hours on end at a heron fishing in a nearby estuary, or at gulls opening clams by dropping them from a height onto the rocks along the beach.
Yet, very gradually, I began to lose my sense of the animals' own awareness. The gulls' technique for breaking open the clams began to appear as a largely automatic behavior, and I could not easily feel the attention that they must bring to each new shell. Perhaps each shell was entirely the same as the last, and _no_ spontaneous attention was really necessary....
I found myself now observing the heron from outside its world, noting with interest its careful high-stepping walk and the sudden dart of its beak into the water, but no longer feeling its tensed yet poised alertness with my own muscles. And, strangely, the suburban squirrels no longer responded to my chittering calls. Although I wished to, I could no longer focus my awareness on engaging in their world as I had so easily done a few weeks earlier, for my attention was quickly deflected by internal, verbal deliberations of one sort or another—by a conversation I now seemed to carry on entirely within myself. The squirrels had no part in this conversation.
It became increasingly apparent, from books and articles and discussions with various people, that other animals were not as awake and aware as I had assumed, that they lacked any real language and hence the possibility of thought, and that even their seemingly spontaneous responses to the world around them were largely "programmed" behaviors, "coded" in the genetic material now being mapped by biologists. Indeed, the more I spoke _about_ other animals, the less possible it became to speak _to_ them. I gradually came to discern that there was no common ground between the unlimited human intellect and the limited sentience of other animals, no medium through which we and they might communicate with and reciprocate one another.
As the expressive and sentient landscape slowly faded behind my more exclusively human concerns, threatening to become little more than an illusion or fantasy, I began to feel—particularly in my chest and abdomen—as though I were being cut off from vital sources of nourishment. I was indeed reacclimating to my own culture, becoming more attuned to its styles of discourse and interaction, yet my bodily senses seemed to be losing their acuteness, becoming less awake to subtle changes and patterns. The thrumming of crickets, and even the songs of the local blackbirds, readily faded from my awareness after a few moments, and it was only by an effort of will that I could bring them back into the perceptual field. The flight of sparrows and of dragonflies no longer sustained my focus very long, if indeed they gained my attention at all. My skin quit registering the various changes in the breeze, and smells seemed to have faded from the world almost entirely, my nose waking up only once or twice a day, perhaps while cooking, or when taking out the garbage.
In Nepal, the air had been filled with smells—whether in the towns, where burning incense combined with the aromas of roasting meats and honeyed pastries and fruits for trade in the open market, and the stench of organic refuse rotting in the ravines, and sometimes of corpses being cremated by the river; or in the high mountains, where the wind carried the whiffs of countless wildflowers, and of the newly turned earth outside the villages where the fragrant dung of yaks was drying in round patties on the outer walls of the houses, to be used, when dry, as fuel for the household fires, and where smoke from those many home fires always mingled in the outside air. And sounds as well: the chants of aspiring monks and adepts blended with the ringing of prayer bells on near and distant slopes, accompanied by the raucous croaks of ravens, and the sigh of the wind pouring over the passes, and the flapping of prayer flags, and the distant hush of the river cascading through the far-below gorge.
There the air was a thick and richly textured presence, filled with invisible but nonetheless tactile, olfactory, and audible influences. In the United States, however, the air seemed thin and void of substance or influence. It was not, here, a sensuous medium—the felt matrix of our breath and the breath of the other animals and plants and soils—but was merely an absence, and indeed was constantly referred to in everyday discourse as mere empty space. Hence, in America I found myself lingering near wood fires and even garbage dumps—much to the dismay of my friends—for only such an intensity of smells served to remind my body of its immersion in an enveloping medium, and with this experience of being immersed in a world of influences came a host of body memories from my year among the shamans and village people of rural Asia.
I BEGAN TO FIND OTHER WAYS, AS WELL, OF TAPPING THE VERY DIFFERENT sensations and perceptions that I had grown accustomed to in the "undeveloped world," by living for extended periods on native Indian reservations in the southwestern desert and along the northwestern coast, or by hiking off for weeks at a time into the North American wilderness. Intermittently, I began to wonder if my culture's assumptions regarding the lack of awareness in other animals and in the land itself was less a product of careful and judicious reasoning than of a strange inability to clearly perceive other animals—a real inability to clearly see, or focus upon, anything outside the realm of human technology, or to hear as meaningful anything other than human speech. The sad results of our interactions with the rest of nature were being reported in every newspaper—from the depletion of topsoil due to industrial farming techniques to the fouling of groundwater by industrial wastes, from the rapid destruction of ancient forests to, worst of all, the ever-accelerating extinction of our fellow species—and these remarkable and disturbing occurrences, all readily traceable to the ongoing activity of "civilized" humankind, did indeed suggest the possibility that there was a perceptual problem in my culture, that modern, "civilized" humanity simply did not perceive surrounding nature in a clear manner, if we have even been perceiving it at all.
The experiences that shifted the focus of my research in rural Indonesia and Nepal had shown me that nonhuman nature can be perceived and experienced with far more intensity and nuance than is generally acknowledged in the West. What was it that made possible the heightened sensitivity to extrahuman reality, the profound attentiveness to other species and to the Earth that is evidenced in so many of these cultures, and that had so altered my awareness that my senses now felt stifled and starved by the patterns of my own culture? Or, reversing the question, what had made possible the absence of this attentiveness in the modern West? For Western culture, too, has its indigenous origins. If the relative attunement to environing nature exhibited by native cultures is linked to a more primordial, participatory mode of perception, how had Western civilization come to be so exempt from this sensory reciprocity? How, that is, have we become so deaf and so blind to the vital existence of other species, and to the animate landscapes they inhabit, that we now so casually bring about their destruction?
To be sure, our obliviousness to nonhuman nature is today held in place by ways of speaking that simply deny intelligence to other species and to nature in general, as well as by the very structures of our civilized existence—by the incessant drone of motors that shut out the voices of birds and of the winds; by electric lights that eclipse not only the stars but the night itself; by air "conditioners" that hide the seasons; by offices, automobiles, and shopping malls that finally obviate any need to step outside the purely human world at all. We consciously encounter nonhuman nature only as it has been circumscribed by our civilization and its technologies: through our domesticated pets, on the television, or at the zoo (or, at best, in carefully managed "nature preserves"). The plants and animals we consume are neither gathered nor hunted—they are bred and harvested in huge, mechanized farms. "Nature," it would seem, has become simply a stock of "resources" for human civilization, and so we can hardly be surprised that our civilized eyes and ears are somewhat oblivious to the existence of perspectives that are not human at all, or that a person either entering into or returning to the West from a nonindustrial culture would feel startled and confused by the felt absence of nonhuman powers.
Still, the current commodification of "nature" by civilization tells us little or nothing of the perceptual shift that made possible this reduction of the animal (and the earth) to an object, little of the process whereby our senses first relinquished the power of the Other, the vision that for so long had motivated our most sacred rituals, our dances, and our prayers.
But can we even hope to catch a glimpse of this process, which has given rise to so many of the habits and linguistic prejudices that now structure our very thinking? Certainly not if we gaze toward that origin from within the midst of the very civilization it engendered. But perhaps we may make our stand along the _edge_ of that civilization, like a magician, or like a person who, having lived among another tribe, can no longer wholly return to his own. He lingers half within and half outside of his community, open as well, then, to the shifting voices and flapping forms that crawl and hover beyond the mirrored walls of the city. And even there, moving along those walls, he may hope to find the precise clues to the mystery of how those walls were erected, and how a simple boundary became a barrier, only if the moment is timely—only, that is, if the margin he frequents is a temporal as well as a spatial edge, and the temporal structure that it bounds is about to dissolve, or metamorphose, into something else.
# **2**
# **_Philosophy on the Way to Ecology_**
# A TECHNICAL INTRODUCTION TO THE INQUIRY
## PART I:
EDMUND HUSSERL AND PHENOMENOLOGY
IT IS NATURAL THAT WE TURN TO THE TRADITION OF PHENOMENOLOGY in order to understand the strange difference between the experienced world, or worlds, of indigenous, vernacular cultures and the world of modern European and North American civilization. For phenomenology is the Western philosophical tradition that has most forcefully called into question the modern assumption of a single, wholly determinable, objective reality.
This assumption has its source in René Descartes's well-known separation of the thinking mind, or subject, from the material world of things, or objects. Actually, Galileo had already asserted that only those properties of matter that are directly amenable to mathematical measurement (such as size, shape, and weight) are real; the other, more "subjective" qualities such as sound, taste, and color are merely illusory impressions, since the "book of nature" is written in the language of mathematics alone. In his words:
This grand book the universe... is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.
Yet it was only after the publication of Descartes's _Meditations_ , in 1641, that material reality came to be commonly spoken of as a strictly mechanical realm, as a determinate structure whose laws of operation could be discerned only via mathematical analysis. By apparently purging material reality of subjective experience, Galileo cleared the ground and Descartes laid the foundation for the construction of the objective or "disinterested" sciences, which by their feverish and forceful investigations have yielded so much of the knowledge and so many of the technologies that have today become commonplace in the West. The chemical table of the elements, automobiles, smallpox vaccines, "close-up" images of the outer planets—so much that we have come to assume and depend upon has emerged from the bold experimentalization of the world by the objective sciences.
Yet these sciences consistently overlook our ordinary, everyday experience of the world around us. Our direct experience is necessarily subjective, necessarily relative to our own position or place in the midst of things, to our particular desires, tastes, and concerns. The everyday world in which we hunger and make love is hardly the mathematically determined "object" toward which the sciences direct themselves. Despite all the mechanical artifacts that now surround us, the world in which we find ourselves before we set out to calculate and measure it is not an inert or mechanical object but a living field, an open and dynamic landscape subject to its own moods and metamorphoses.
My life and the world's life are deeply intertwined; when I wake up one morning to find that a week-long illness has subsided and that my strength has returned, the world, when I step outside, fairly sparkles with energy and activity: swallows are swooping by in vivid flight; waves of heat rise from the newly paved road smelling strongly of tar; the old red barn across the field juts into the sky at an intense angle. Likewise, when a haze descends upon the valley in which I dwell, it descends upon my awareness as well, muddling my thoughts, making my muscles yearn for sleep. The world and I reciprocate one another. The landscape as I directly experience it is hardly a determinate object; it is an ambiguous realm that responds to my emotions and calls forth feelings from me in turn. Even the most detached scientist must begin and end her study in this indeterminate field of experience, where shifts of climate or mood may alter his experiment or her interpretation of "the data": the scientist, too, must take time off from his measurements and analyses to eat, to defecate, to converse with friends, to interact straightforwardly with a familiar world that is never explicitly thematized and defined. Indeed, it is precisely from his experience in this preconceptual and hence ambiguous world that an individual is first drawn to become a scientist, to adopt the ways of speaking and seeing that are acknowledged as appropriate by the scientific community, to affect the proper disinterested or objective attitude with regard to a certain range of natural events. The scientist does not randomly choose a specific discipline or specialty, but is drawn to a particular field by a complex of subjective experiences and encounters, many of which unfold far from the laboratory and its rarefied atmosphere. Further, the scientist never completely succeeds in making himself into a pure spectator of the world, for he cannot cease to live in the world as a human among other humans, or as a creature among other creatures, and his scientific concepts and theories necessarily borrow aspects of their character and texture from his untheorized, spontaneously lived experience.
Indeed, the ostensibly "value-free" results of our culture's investigations into biology, physics, and chemistry ultimately come to display themselves in the open and uncertain field of everyday life, whether embedded in social policies with which we must come to terms or embodied in new technologies with which we all must grapple. Thus, the living world—this ambiguous realm that we experience in anger and joy, in grief and in love—is both the soil in which all our sciences are rooted and the rich humus into which their results ultimately return, whether as nutrients or as poisons. Our spontaneous experience of the world, charged with subjective, emotional, and intuitive content, remains the vital and dark ground of all our objectivity.
And yet this ground goes largely unnoticed or unacknowledged in scientific culture. In a society that accords priority to that which is predictable and places a premium on certainty, our spontaneous, preconceptual experience, when acknowledged at all, is referred to as "merely subjective." The fluid realm of direct experience has come to be seen as a secondary, derivative dimension, a mere consequence of events unfolding in the "realer" world of quantifiable and measurable scientific "facts." It is a curious inversion of the actual, demonstrable state of affairs. Subatomic quanta are now taken to be more primordial and "real" than the world we experience with our unaided senses. The living, feeling, and thinking organism is assumed to derive, somehow, from the mechanical body whose reflexes and "systems" have been measured and mapped, the living person now an epiphenomenon of the anatomized corpse. That it takes living, sensing subjects, complete with their enigmatic emotions and unpredictable passions, to conceive of those subatomic fields, or to dissect and anatomize the body, is readily overlooked, or brushed aside as inconsequential.
Nevertheless, the ambiguity of experience is already a part of any phenomenon that draws our attention. For whatever we perceive is necessarily entwined with our own subjectivity, already blended with the dynamism of life and sentience. The living pulse of subjective experience cannot finally be stripped from the things that we study (in order to expose the pure unadulterated "objects") without the things themselves losing all existence for us. Such conundrums are commonly consigned to psychology, to that science that studies subjective awareness and perception. And so perhaps by turning to psychology we can expect to find a recognition and avowal of the pre-objective dimension that permeates and sustains every reality that we know, and hence an understanding of the manner in which subjective experience both supports and sets limits to the positive sciences.
In psychology, however, we discover nothing of the sort. Instead, we find a discipline that is itself modeled on the positivism of the "hard" sciences, a science wherein the psyche has itself been reified into an "object," a thing to be studied like any other thing in the determinate, objective world. Much of cognitive science strives to model the computational processes that ostensibly underlie mental experience. While for Galileo and Descartes perceptual qualities like color and taste were illusory, unreal properties because of their ambiguous and indeterminate character, mathematical indices have at last been found for _these_ qualities as well, or rather such qualities are now studied only to the extent that they can be rendered, by whatever process of translation, into _quantities_. Here as elsewhere, the everyday world—the world of our direct, spontaneous experience—is still assumed to derive from an impersonal, objective dimension of pure "facts" that we glimpse only through our instruments and equations.
IT WAS HIS FRUSTRATION WITH SUCH ASSUMPTIONS, AND WITH THE early discipline of psychology—which, far from directing attention toward the fluid region of direct experience, was already at the start of the twentieth century solidifying the "mind" into another "object" in the mathematized and mechanical universe—that led Edmund Husserl to inaugurate the philosophical discipline of phenomenology. Phenomenology, as he articulated it in the early 1900s, would turn toward "the things themselves," toward the world as it is experienced in its felt immediacy. Unlike the mathematics-based sciences, phenomenology would seek not to explain the world, but to describe as closely as possible the way the world makes itself evident to awareness, the way things first arise in our direct, sensorial experience. By thus returning to the taken-for-granted realm of subjective experience, not to explain it but simply to pay attention to its rhythms and textures, not to capture or control it but simply to become familiar with its diverse modes of appearance—and ultimately to give voice to its enigmatic and ever-shifting patterns—phenomenology would articulate the ground of the other sciences. It was Husserl's hope that phenomenology, as a rigorous "science of experience," would establish the other sciences at last upon a firm footing—not, perhaps, as solid as the fixed and finished "object" upon which those sciences _pretend_ to stand, but the only basis possible for a knowledge that necessarily emerges from our lived experience of the things around us. In the words of the French phenomenologist Maurice Merleau-Ponty:
All my knowledge of the world, even my scientific knowledge, is gained from my own particular point of view, or from some experience of the world without which the symbols of science would be meaningless. The whole universe of science is built upon the world as directly experienced, and if we want to subject science itself to rigorous scrutiny and arrive at a precise assessment of its meaning and scope, we must begin by reawakening the basic experience of the world, of which science is the second-order expression.... To return to things themselves is to return to that world which precedes knowledge, of which knowledge always _speaks_ , and in relation to which every scientific schematization is an abstract and derivative sign-language, as is geography in relation to the countryside in which we have learnt beforehand what a forest, a prairie or a river is.
### Intersubjectivity
In the early stages of his project, Husserl spoke of the world of experience (the "phenomenal" world) as a thoroughly subjective realm. In order to explore this realm philosophically, he insisted that it be viewed as a wholly mental dimension, an immaterial field of appearances. That which experiences this dimension—the experiencing self, or subject—was similarly described by Husserl as a pure consciousness, a "transcendental" mind or ego.
Perhaps by designating subjective reality as a nonmaterial, transcendental realm, Husserl hoped to isolate this qualitative dimension from the apparently mechanical world of material "facts" that was then being constructed by the objective sciences (and thus to protect this realm from being colonized by those technological methods of inquiry). Yet his insistence upon the mental character of phenomenal reality led critics to attack Husserl's method as being inherently solipsistic—an approach that seals the philosopher inside his own solitary experience, rendering him ultimately unable to recognize anyone or anything outside of his own mind.
Husserl struggled long and hard to answer this important criticism. How does our subjective experience enable us to recogize the reality of other selves, other experiencing beings? The solution seemed to implicate the body—one's own as well as that of the other—as a singularly important structure within the phenomenal field. The body is that mysterious and multifaceted phenomenon that seems always to accompany one's awareness, and indeed to be the very location of one's awareness within the field of appearances. Yet the phenomenal field also contains many _other_ bodies, other forms that move and gesture in a fashion similar to one's own. While one's own body is experienced, as it were, only from within, these other bodies are experienced from outside; one can vary one's distance from these bodies and can move around them, while this is impossible in relation to one's own body.
Despite this difference, Husserl discerned that there was an inescapable affinity, or affiliation, between these other bodies and one's own. The gestures and expressions of these other bodies, viewed from without, echo and resonate one's own bodily movements and gestures, experienced from within. By an associative "empathy," the embodied subject comes to recognize these other bodies as other centers of experience, other subjects.
In this manner, carefully describing the ways in which the subjective field of experience, mediated by the body, opens onto other subjectivities—other selves besides one's own self—Husserl sought to counter the charge of solipsism that had been directed against his phenomenology. The field of appearances, while still a thoroughly subjective realm, was now seen to be inhabited by _multiple_ subjectivities; the phenomenal field was no longer the isolate haunt of a solitary ego, but a collective landscape, constituted by other experiencing subjects as well as by oneself.
There remain, however, many phenomena in the experiential field that are not collective or commonly shared. When daydreaming, for example, my attention is carried by phenomena whose contours and movements I am able to alter at will, a whole phantasmagoria of images that nevertheless lack the solidity of bodies. Such forms offer very little resistance to my gaze. They are not, that is, held in place by gazes other than my own—these are entirely _my_ images, _my_ phantasies and fears, _my_ dreamings. And so I am brought, like Husserl, to recognize at least two regions of the experiential or phenomenal field: one of phenomena that unfold entirely for me—images that arise, as it were, on this side of my body—and another region of phenomena that are, evidently, responded to and experienced by other embodied subjects as well as by myself. These latter phenomena are still subjective—they appear to me within a field of experience colored by my mood and my current concerns—and yet I cannot alter or dissipate them at will, for they seem to be buttressed by many involvements besides my own. That tree bending in the wind, this cliff wall, the cloud drifting overhead: these are not merely subjective; they are _intersubjective_ phenomena—phenomena experienced by a multiplicity of sensing subjects.
HUSSERL'S NOTION OF _INTERSUBJECTIVITY_ SUGGESTED A REMARKABLE new interpretation of the so-called "objective world." For the conventional contrast between "subjective" and "objective" realities could now be reframed as a contrast within the subjective field of experience itself—as the felt contrast between subjective and intersubjective phenomena.
The sciences are commonly thought to aim at clear knowledge of an objective world utterly independent of awareness or subjectivity. Considered experientially, however, the scientific method enables the achievement of greater intersubjectivity, greater knowledge of that which is or can be experienced by many different selves or subjects. The striving for objectivity is thus understood, phenomenologically, as a striving to achieve greater consensus, greater agreement or consonance among a plurality of subjects, rather than as an attempt to avoid subjectivity altogether. The pure "objective reality" commonly assumed by modern science, far from being the concrete basis underlying all experience, was, according to Husserl, a theoretical construction, an unwarranted idealization of intersubjective experience.
The "real world" in which we find ourselves, then—the very world our sciences strive to fathom—is not a sheer "object," not a fixed and finished "datum" from which all subjects and subjective qualities could be pared away, but is rather an intertwined matrix of sensations and perceptions, a collective field of experience lived through from many different angles. The mutual inscription of others in my experience, and (as I must assume) of myself in their experiences, effects the interweaving of our individual phenomenal fields into a single, ever-shifting fabric, a single phenomenal world or "reality."
And yet, as we know from our everyday experience, the phenomenal world is remarkably stable and solid; we are able to count on it in so many ways, and we take for granted much of its structure and character. This experienced solidity is precisely sustained by the continual encounter with others, with other embodied subjects, other centers of experience. The encounter with other perceivers continually assures me that there is more to any thing, or to the world, than I myself can perceive at any moment. Besides that which I directly see of a particular oak tree or building, I know or intuit that there are also those facets of the oak or building that are visible to the other perceivers that I see. I sense that that tree is much more than what I directly see of it, since it is also what the others whom I see perceive of it; I sense that as a perceivable presence it already existed before I came to look at it, and indeed that it will not dissipate when I turn away from it, since it remains an experience for others—not just for other persons, but (as we shall see later in this chapter) for other sentient organisms, for the birds that nest in its branches and for the insects that move along its bark, and even, finally, for the sensitive cells and tissues of the oak itself, quietly drinking sunlight through its leaves. It is this informing of my perceptions by the evident perceptions and sensations of other bodily entities that establishes, for me, the relative solidity and stability of the world.
### The Life-World
Although Husserl at first wrote of the nonmaterial, mental character of experienced reality, his growing recognition of inter subjective experience, and of the body's importance for such experience, ultimately led him to recognize a more primary, corporeal dimension, midway between the transcendental "consciousness" of his earlier analyses and the utterly objective "matter" assumed by the natural sciences. This was the intersubjective world of life, the _Lebenswelt_ , or "life-world."
The life-world is the world of our immediately lived experience, _as_ we live it, prior to all our thoughts about it. It is that which is present to us in our everyday tasks and enjoyments—reality as it engages us before being analyzed by our theories and our science. The life-world is the world that we count on without necessarily paying it much attention, the world of the clouds overhead and the ground underfoot, of getting out of bed and preparing food and turning on the tap for water. Easily overlooked, this primordial world is always already there when we begin to reflect or philosophize. It is not a private, but a collective, dimension—the common field of our lives and the other lives with which ours are entwined—and yet it is profoundly ambiguous and indeterminate, since our experience of this field is always relative to our situation within it. The life-world is thus the world as we organically experience it in its enigmatic multiplicity and open-endedness, prior to conceptually freezing it into a static space of "facts"—prior, indeed, to conceptualizing it in any complete fashion. All of our concepts and representations, scientific and otherwise, necessarily draw nourishment from this indeterminate realm, as the physicist analyzing data is still nourished by the air that she is breathing, by the feel of the chair that supports her and the light flooding in through the window, without her being particularly conscious of these participations.
The life-world is thus peripherally present in any thought or activity we undertake. Yet whenever we attempt to _explain_ this world conceptually, we seem to forget our active participation within it. Striving to represent the world, we inevitably forfeit its direct presence. It was Husserl's genius to realize that the assumption of objectivity had led to an almost total eclipse of the life-world in the modern era, to a nearly complete forgetting of this living dimension in which all of our endeavors are rooted. In their striving to attain a finished blueprint of the world, the sciences had become frightfully estranged from our direct human experience. Their many specialized and technical discourses had lost any obvious relevance to the sensuous world of our ordinary engagements. The consequent impoverishment of language, the loss of a common discourse tuned to the qualitative nuances of living experience, was leading, Husserl felt, to a clear crisis in European civilization. Oblivious to the quality-laden life-world upon which they themselves depend for their own meaning and existence, the Western sciences, and the technologies that accompany them, were beginning to blindly overrun the experiential world—even, in their errancy, threatening to obliterate the world-of-life entirely.
IT SHOULD BE EVIDENT THAT THE LIFE-WORLD MAY BE QUITE different for different cultures. The world that a people experiences and comes to count on is deeply influenced by the ways they live and engage that world. The members of any given culture necessarily inhabit an experienced world very different from that of another culture with a very different language and way of life. Even the scientifically disclosed "objective universe" of contemporary Western civilization cannot genuinely be separated from the particular institutions, technologies, and ways of life endemic to this society since the seventeenth century.
If the worlds experienced by humans are so diverse, how much more diverse, still, must be the life-worlds of other animals—of wolves, or owls, or a community of bees! And yet, despite this multiplicity, it would seem that there are basic structures of the life-world that are shared, elements that are common to different cultures and even, we may suspect, to different species. Husserl's writings seem to suggest that the life-world has various layers, that underneath the layer of the diverse cultural life-worlds there reposes a deeper, more unitary life-world, always already there beneath all our cultural acquisitions, a vast and continually overlooked dimension of experience that nevertheless supports and sustains all our diverse and discontinuous worldviews.
Husserl sheds light on this most primordial, most deeply intersubjective dimension of the life-world in a series of notes written in 1934. The notes describe a set of phenomenological investigations into the contemporary understanding of _space_. Underneath the modern, scientific conception of space as a mathematically infinite and homogenous void, Husserl discloses the experienced spatiality of the _earth_ itself. The encompassing earth, he suggests, provides the most immediate, bodily awareness of space, from which all later _conceptions_ of space are derived. While according to contemporary physics the earth is but one celestial body among many others "in" space, phenomenologically considered _all_ bodies (including our own) are first located relative to the ground of the earth, whereas the earth itself is not "in" space, since it is earth that, from the first, _provides_ space. To our most immediate sensorial experience, "bodies are given as having the sense of being earthly bodies, and space is given as having the sense of being earth-space." Further, while contemporary science maintains that "in reality" the earth is in motion (around its own axis, and around the sun), Husserl maintains that the very concepts of "motion" and "rest" derive all their meaning from our primary, bodily experience of being in motion or at rest relative to the "absolute" rest of the "earth-basis."
Husserl's notes on these matters were found in an envelope on which he had written a few summary words: " _Overthrow of the Copernican Theory_... The original ark, earth, does not move." Such a remarkable assertion illustrates well the radical nature of Husserl's thought. He suggests in these notes that there is a profound instability in the scientific worldview, resulting from the continual clash between our scientific convictions and our spontaneous experience. After the investigations of Copernicus, Kepler, and Galileo, the sun came to be conceived as the center of the phenomenal world. Yet this conception simply did not agree with our spontaneous sensory _perception_ , which remained the experience of a radiant orb traversing the sky of a stable earth. A profound schism was thus brought about between our intellectual convictions and the most basic conviction of our senses, between our mental _concepts_ and our bodily _percepts_. (Descartes's philosophical disjunction of the mind from the body was surely prompted by this already existing state of affairs—it was necessary, for the maintenance of the new, Copernican worldview, that the rational intellect hold itself apart from the experiencing body.) Nevertheless, our very words have continued to betray the intellect and to prevent the clean ascendancy of the Copernican system: we still say "the sun rises" and "the sun sets" whether we are farmers or physicists. It is in this sense, writing from the perspective of the experiencing body, that Husserl is able to claim that _earth_ , "the original ark," _does not move_.
Finally, Husserl seems to suggest that the earth lies at the heart of our notions of time as well as of space. He writes of the earth as our "primitive home" and our "primitive history." Every unique cultural history is but an episode in this larger story; every culturally constructed notion of time presupposes our deep history as carnal beings present to a single earth.
The earth is thus, for Husserl, the secret depth of the life-world. It is the most unfathomable region of experience, an enigma that exceeds the structurations of any particular culture or language. In his words, the earth is the encompassing "ark of the world," the common "root basis" of all relative life-worlds. Husserl's late insights into the importance of the earth for all human cognition were, as we shall see, to have profound implications for the subsequent unfolding of phenomenological philosophy.
EDMUND HUSSERL'S WORK WAS IN NO SENSE A REJECTION OF SCIENCE. It was a plea that science, for its own integrity and meaningfulness, must acknowledge that it is rooted in the same world that we all engage in our everyday lives and with our unaided senses—that, for all its technological refinements, quantitative science remains an expression of, and hence must be guided by, the qualitative world of our common experience. The true task of phenomenology, as Husserl saw it at the end of his career, lay in the careful demonstration of the manner in which every theoretical and scientific practice grows out of and remains supported by the forgotten ground of our directly felt and lived experience, and has value and meaning only in reference to this primordial and open realm.
Originally an attempt to certify theoretical awareness by placing it on a firm footing, Husserl's project culminated in the still ongoing attempt to rejuvenate the full-blooded world of our sensorial experience, and, consequently, in the dawning recognition of Earth as the forgotten basis of all our awareness.
I now turn to the work of the phenomenologist Maurice Merleau-Ponty, in order to show how Husserl's legacy was taken up and transformed in a manner that endowed this philosophy with a particular power and relevance for the ecological questions that now confront us.
## PART II: MAURICE MERLEAU-PONTY
AND THE PARTICIPATORY NATURE OF PERCEPTION
Maurice Merleau-Ponty set out to radicalize Husserl's phenomenology, both by clarifying the inconsistencies lodged in this philosophy by Husserl's own ambivalences, and further, by disclosing a more eloquent way of speaking, a style of language which, by virtue of its fluidity, its carnal resonance, and its careful avoidance of abstract terms, might itself draw us into the sensuous depths of the life-world.
### The Mindful Life of the Body
We have seen, for instance, that the physical body came to play an increasingly important role in Husserl's philosophy. Only by acknowledging the embodied nature of the experiencing self was Husserl able to avoid the pitfalls of solipsism. It is as visible, animate bodies that other selves or subjects make themselves evident in my subjective experience, and it is only as a body that I am visible and sensible to others. The body is precisely my insertion in the common, or intersubjective, field of experience.
Nevertheless, the body remained a mere appearance, albeit a unique and pivotal one, in Husserl's thought. The body was, to be sure, the very locus of the experiencing subject, or self, in the phenomenal world—in the manifold of appearances—but the self was still affirmed, by Husserl, as a transcendental ego, ultimately separable from the phenomena (including the body) that it posits and ponders. Despite his growing recognition of the living body's centrality in all experience, and despite his disclosure of the thoroughly incarnate, intersubjective realm of our preconceptual life, Husserl was unable to drop the transcendental, idealist aspirations of his early philosophy.
It is precisely this lingering assumption of a self-subsistent, disembodied, transcendental ego that Merleau-Ponty rejects. If this body is my very presence in the world, if it is the body that alone enables me to enter into relations with other presences, if without these eyes, this voice, or these hands I would be unable to see, to taste, and to touch things, or to be touched by them—if without this body, in other words, there would be no possibility of experience—then the body itself is the true subject of experience. Merleau-Ponty begins, then, by identifying the subject—the experiencing "self"—with the bodily organism.
It is indeed a radical move. Most of us are accustomed to consider the self, our innermost essence, as something incorporeal. Yet consider: Without this body, without this tongue or these ears, you could neither speak nor hear another's voice. Nor could you have anything to speak about, or even to reflect on, or to think, since without any contact, any encounter, without any glimmer of sensory experience, there could be nothing to question or to know. The living body is thus the very possibility of contact, not just with others but with oneself—the very possibility of reflection, of thought, of knowledge. The common notion of the experiencing self, or mind, as an immaterial phantom ultimately independent of the body can only be a mirage: Merleau-Ponty invites us to recognize, at the heart of even our most abstract cogitations, the sensuous and sentient life of the body itself.
This breathing body, as it experiences and inhabits the world, is very different from that objectified body diagrammed in physiology textbooks, with its separable "systems" (the circulatory system, the digestive system, the respiratory system, etc.) laid bare on each page. The body I here speak of is very different from the body we have been taught to see and even to feel, very different, finally, from that complex machine whose broken parts or stuck systems are diagnosed by our medical doctors and "repaired" by our medical technologies. Underneath the anatomized and mechanical body that we have learned to conceive, prior indeed to all our conceptions, dwells the body as it actually experiences things, this poised and animate power that initiates all our projects and suffers all our passions.
The living, attentive body—which Merleau-Ponty called the "body subject"—is this very being that, pondering a moment ago, suddenly took up this pen and scribbled these thoughts. It is the very power I have to look and to see things, or to turn away and look elsewhere, the ability to cry and to laugh, or to howl at night with the wolves, to find and gather food whether in a forest or a market, the power to walk upon the ground and to imbibe the swirling air. Yet "I" do not deploy these powers like a commander piloting a ship, for I am, in my depths, indistinguishable from them, as my sadness is indistinguishable from a certain heaviness of my bodily limbs, or as my delight is only artificially separable from the widening of my eyes, from the bounce in my step and the heightened sensitivity of my skin. Indeed, facial expressions, gestures, and spontaneous utterances like sighs and cries seem to immediately incarnate feelings, moods, and desires without "my" being able to say which came first—the corporeal gesture or its purportedly "immaterial" counterpart.
To acknowledge that "I am this body" is not to reduce the mystery of my yearnings and fluid thoughts to a set of mechanisms, or my "self" to a determinate robot. Rather it is to affirm the uncanniness of this physical form. It is not to lock up awareness within the density of a closed and bounded object, for as we shall see, the boundaries of a living body are open and indeterminate; more like membranes than barriers, they define a surface of metamorphosis and exchange. The breathing, sensing body draws its sustenance and its very substance from the soils, plants, and elements that surround it; it continually contributes itself, in turn, to the air, to the composting earth, to the nourishment of insects and oak trees and squirrels, ceaselessly spreading out of itself as well as breathing the world into itself, so that it is very difficult to discern, at any moment, precisely where this living body begins and where it ends. Considered phenomenologically—that is, as we actually experience and _live_ it—the body is a creative, shape-shifting entity. Certainly, it has its finite character and style, its unique textures and temperaments that distinguish it from other bodies; yet these mortal limits in no way close me off from the things around me or render my relations to them wholly predictable and determinate. On the contrary, my finite bodily presence alone is what enables me to freely engage the things around me, to choose to affiliate with certain persons or places, to insinuate myself in other lives. Far from restricting my access to things and to the world, the body is my very means of entering into relation with all things.
To be sure, by disclosing the body itself as the very subject of awareness, Merleau-Ponty demolishes any hope that philosophy might eventually provide a complete picture of reality (for any such total account of "what is" requires a mind or consciousness that stands somehow _outside_ of existence, whether to compile the account or, finally, to receive and comprehend it). Yet by this same move he opens, at last, the possibility of a truly authentic phenomenology, a philosophy which would strive, not to explain the world as if from outside, but to give voice to the world from our experienced situation _within_ it, recalling us to our participation in the here-and-now, rejuvenating our sense of wonder at the fathomless things, events and powers that surround us on every hand.
ULTIMATELY, TO ACKNOWLEDGE THE LIFE OF THE BODY, AND TO affirm our solidarity with this physical form, is to acknowledge our existence as one of the earth's animals, and so to remember and rejuvenate the organic basis of our thoughts and our intelligence. According to the central current of the Western philosophical tradition, from its source in ancient Athens up until the present moment, human beings alone are possessed of an incorporeal intellect, a "rational soul" or mind which, by virtue of its affinity with an eternal or divine dimension outside the bodily world, sets us radically apart from, or above, all other forms of life. In Aristotle's writings, for instance, while plants are endowed with a _vegetal soul_ (which enables nourishment, growth, and reproduction), and while animals possess, in addition to the vegetal soul, an _animal soul_ (which provides sensation and locomotion), these souls remain inseparable from the earthly world of generation and decay. Humans, however, possess along with these other souls a _rational soul_ , or intellect, which alone provides access to the less corruptible spheres and has affinities with the divine "Unmoved Mover" himself. In Descartes's hands, two thousand years later, this hierarchical continuum of living forms, commonly called "the Great Chain of Being," was polarized into a thorough dichotomy between mechanical, unthinking matter (including all minerals, plants, and animals, as well as the human body) and pure, thinking mind (the exclusive province of humans and God). Since humans alone are a mixture of extended matter and thinking mind, we alone are able to feel and to experience our body's mechanical sensations. Meanwhile, all other organisms, consisting solely of extended matter, are in truth nothing more than automatons, incapable of actual experience, unable to feel pleasure or suffer pain. Hence, we humans need have no scruples about manipulating, exploiting, or experimenting upon other animals in any manner we see fit.
Curiously, such arguments for human specialness have regularly been utilized by human groups to justify the exploitation not just of other organisms, but of other _humans_ as well (other nations, other races, or simply the "other" sex); armed with such arguments, one had only to demonstrate that these others were not _fully_ human, or were "closer to the animals," in order to establish one's right of dominion. According to Aristotle, for example, women are deficient in the rational soul, and hence "the relation of male to female is naturally that of the superior to the inferior—of the ruling to the ruled." Such justifications for social exploitation draw their force from the prior hierarchicalization of the natural landscape, from that hierarchical ordering that locates "humans," by virtue of our incorporeal intellect, above and apart from all other, "merely corporeal," entities.
Such hierarchies are wrecked by any phenomenology that takes seriously our immediate sensory experience. For our senses disclose to us a wild-flowering proliferation of entities and elements, in which humans are thoroughly immersed. While this diversity of sensuous forms certainly displays some sort of reckless order, we find ourselves in the midst of, rather than on top of, this order. We may cast our gaze downward to watch the field mice and the insects that creep along the bending grasses, or to glimpse the snakes that slither into hollows deep underfoot, yet, at the same moment, hawks soaring on great winds gaze down upon _our_ endeavors. Melodious feathered beings flit like phantoms among the high branches of the trees, while other animate powers, known only by their traces, move within the hidden depths of the forest. In the waters that surge in waves against the distant edge of the land, still stranger powers, multihued and silent, move in crowds among alien forests of coral and stone.... Does the human intellect, or "reason," really spring us free from our inherence in the depths of this wild proliferation of forms? _Or on the contrary, is the human intellect rooted in, and secretly borne by, our forgotten contact with the multiple nonhuman shapes that surround us?_
### The Body's Silent conversation with Things
For Merleau-Ponty, all of the creativity and free-ranging mobility that we have come to associate with the human intellect is, in truth, an elaboration, or recapitulation, of a profound creativity already underway at the most immediate level of sensory perception. The sensing body is not a programmed machine but an active and open form, continually improvising its relation to things and to the world. The body's actions and engagements are never wholly determinate, since they must ceaselessly adjust themselves to a world and a terrain that is itself continually shifting. If the body were truly a set of closed or predetermined mechanisms, it could never come into genuine contact with anything outside of itself, could never perceive anything really new, could never be genuinely startled or surprised. All of its experiences, and all its responses, would already have been anticipated from the beginning, already programmed, as it were, into the machine. But could we even, then, call them experiences? For is not experience, or more precisely, _perception_ , the constant thwarting of such closure?
Consider a spider weaving its web, for instance, and the assumption still held by many scientists that the behavior of such a diminutive creature is thoroughly "programmed in its genes." Certainly, the spider has received a rich genetic inheritance from its parents and its predecessors. Whatever "instructions," however, are enfolded within the living genome, they can hardly predict the specifics of the microterrain within which the spider may find itself at any particular moment. They could hardly have determined in advance the exact distances between the cave wall and the branch that the spider is now employing as an anchorage point for her current web, or the exact strength of the monsoon rains that make web-spinning a bit more difficult on this evening. And so the genome could not explicitly have commanded the order of every flexion and extension of her various limbs as she weaves this web into its place. However complex are the inherited "programs," patterns, or predispositions, they must still be adapted to the immediate situation in which the spider finds itself. However determinate one's genetic inheritance, it must still, as it were, be woven into the present, an activity that necessarily involves both a receptivity to the specific shapes and textures of that present and a spontaneous creativity in adjusting oneself (and one's inheritance) to those contours. It is this open activity, this dynamic blend of receptivity and creativity by which every animate organism necessarily orients itself to the world (and orients the world around itself), that we speak of by the term "perception."
BUT LET US NOW PONDER THE EVENT OF PERCEPTION AS WE OURSELVES experience and live it. The human body with its various predilections is, to be sure, our _own_ inheritance, our own rootedness in an evolutionary history and a particular ancestry. Yet it is also our insertion in a world that exceeds our grasp in every direction, our means of contact with things and lives that are still unfolding, open and indeterminate, all around us. Indeed, from the perspective of my bodily senses, there is no thing that appears as a completely determinate or finished object. Each thing, each entity that my body sees, presents some face or facet of itself to my gaze while withholding other aspects from view.
The clay bowl resting on the table in front of me meets my eyes with its curved and grainy surface. Yet I can only see one side of that surface—the other side of the bowl is invisible, hidden by the side that faces me. In order to view that other side, I must pick up the bowl and turn it around in my hands, or else walk around the wooden table. Yet, having done so, I can no longer see the first side of the bowl. Surely I know that it still exists; I can even _feel_ the presence of that aspect which the bowl now presents to the lamp on the far side of the table. Yet I myself am simply unable to see the whole of this bowl all at once.
Moreover, while examining its outer surface I have caught only a glimpse of the smooth and finely glazed _inside_ of the bowl. When I stand up to look down into that interior, which gleams with curved reflections from the skylight overhead, I can no longer see the unglazed outer surface. This earthen vessel thus reveals aspects of its presence to me only by withholding other aspects of itself for further exploration. There can be no question of ever totally exhausting the presence of the bowl with my perception; its very existence as a bowl ensures that there are dimensions wholly inaccessible to me—most obviously the patterns hidden _between_ its glazed and unglazed surfaces, the interior density of its clay body. If I break it into pieces, in hopes of discovering these interior patterns or the delicate structure of its molecular dimensions, I will have destroyed its integrity as a bowl; far from coming to know it completely, I will simply have wrecked any possibility of coming to know it further, having traded the relation between myself and the bowl for a relation to a collection of fragments.
Even a single facet of this bowl resists being plumbed by my gaze once and for all. For, like myself, the bowl is a temporal being, an entity shifting and changing in time, although the rhythm of its changes may be far slower than my own. Each time that I return to gaze at the outward surface of the bowl, my eyes and my mood have shifted, however slightly; informed by my previous encounters with the bowl, my senses now more attuned to its substance, I continually discover new and unexpected aspects. But this is in part because the bowl has changed as well, as a result perhaps of shifts in the light pouring through the window, of dust and of wear—as a result, even, of my own earlier explorations. When I look now at its unglazed outer surface, where before I had seen a homogeneous expanse of bright grey, I now see various faint smudges, some of them ancient and some of them recent—the record of the many hands that have held it through the seasons. Each spot invites me to peer at it more closely, to distinguish that smudge from the others, to try to discern which are the traces of my own hands, and which are of hands larger, or more delicate, and which may be the trace even of those hands that first threw this fine and useful bowl on some potter's wheel years ago.
As this bowl awaits the further involvement of my eyes and my hands, so also every other object in this room invites the participation of my senses—the wooden dresser with its stuffed drawers, the plants on the windowsill quietly turning toward the sun, the individual glasses and dishes stashed above the old sink with its hidden and clattering pipes, and the ancient pinewood table that I now write upon, its coffee stains and countless knife scratches cutting across the curving grain of the wood, and those pens and pencils that beckon to my fingers, and the books that call to me from the shelves, one always asking to be read more deeply, another chanting to me of my childhood, another merely waiting, coldly it seems, to be returned to the library. Like the bowl, each presence presents some facet that catches my eye while the rest of it lies hidden behind the horizon of my current position, each one inviting me to focus my senses upon it, to let the other objects fall into the background as I enter into its particular depth. When my body thus responds to the mute solicitation of another being, that being responds in turn, disclosing to my senses some new aspect or dimension that in turn invites further exploration. By this process my sensing body gradually attunes itself to the style of this other presence—to the _way_ of this stone, or tree, or table—as the other seems to adjust itself to my own style and sensitivity. In this manner the simplest thing may become a world for me, as, conversely, the thing or being comes to take its place more deeply in _my_ world.
Perception, in Merleau-Ponty's work, is precisely this reciprocity, the ongoing interchange between my body and the entities that surround it. It is a sort of silent conversation that I carry on with things, a continuous dialogue that unfolds far below my verbal awareness—and often, even, _independent_ of my verbal awareness, as when my hand readily navigates the space between these scribed pages and the coffee cup across the table without my having to think about it, or when my legs, hiking, continually attune and adjust themselves to the varying steepness of the mountain slopes behind this house without my verbal consciousness needing to direct those adjustments. Whenever I quiet the persistent chatter of words within my head, I find this silent or wordless dance always already going on—this improvised duet between my animal body and the fluid, breathing landscape that it inhabits.
### The Animateness of the Perceptual world
Where does perception originate? I cannot say truthfully that my perception of a particular wildflower, with its color and its fragrance, is determined or "caused" entirely by the flower—since other persons may experience a somewhat different fragrance, as even I, in a different moment or mood, may see the color differently, and indeed since any bumblebee that alights on that blossom will surely have a very different perception of it than I do. But neither can I say truthfully that my perception is "caused" solely by myself—by my physiological or neural organization—or that it exists entirely "in my head." For without the actual existence of this other entity, of this flower rooted not in my brain but in the soil of the earth, there would be no fragrant and colorful perception at all, neither for myself nor for any others, whether human or insect.
Neither the perceiver nor the perceived, then, is wholly passive in the event of perception:
[M]y gaze pairs off with colour, and my hand with hardness and softness, and in this transaction between the subject of sensation and the sensible it cannot be held that one acts while the other suffers the action, or that one confers significance on the other. Apart from the probing of my eye or my hand, and before my body synchronizes with it, the sensible is nothing but a vague beckoning.
There is thus a solicitation of my body by the sensible, and a questioning of the sensible by my body, a reciprocal encroachment:
... [a sensible quality, like the color blue,] which is on the point of being felt sets a kind of muddled problem for my body to solve. I must find the attitude which will provide it with the means of becoming determinate, of showing up as blue; I must find the reply to a question which is obscurely expressed. And yet I do so only when I am invited by it; my attitude is never sufficient to make me really see blue or really touch a hard surface. The sensible gives back to me what I lent to it, but this is only what I took from it in the first place. As I contemplate the blue of the sky... I abandon myself to it and plunge into this mystery, it 'thinks itself within me,' I am the sky itself as it is drawn together and unified, and as it begins to exist for itself; my consciousness is saturated with this limitless blue....
In the act of perception, in other words, I enter into a sympathetic relation with the perceived, which is possible only because neither my body nor the sensible exists outside the flux of time, and so each has its own dynamism, its own pulsation and style. Perception, in this sense, is an attunement or synchronization between my own rhythms and the rhythms of the things themselves, their own tones and textures:
... in so far as my hand knows hardness and softness, and my gaze knows the moon's light, it is as a certain way of linking up with the phenomenon and communicating with it. Hardness and softness, roughness and smoothness, moonlight and sunlight, present themselves in our recollection not pre-eminently as sensory contents but as certain kinds of symbioses, certain ways the outside has of invading us and certain ways we have of meeting this invasion....
In this ceaseless dance between the carnal subject and its world, at one moment the body leads, at another the things. In one luminous passage, which suggests the profound intimacy of the body's preconceptual relation to the sensible things or powers that surround it, Merleau-Ponty writes of perception in terms of an almost magical invocation enacted by the body, and the body's subsequent "possession" by the perceived:
The relations of sentient to sensible are comparable with those of the sleeper to his slumber: sleep suddenly comes when a certain voluntary attitude suddenly receives from outside the confirmation for which it was waiting. I am breathing deeply and slowly in order to summon sleep, and suddenly it is as if my mouth were connected to some great lung outside myself which alternately calls forth and forces back my breath. A certain rhythm of respiration, which a moment ago I voluntarily maintained, now becomes my very being, and sleep, until now aimed at..., suddenly becomes my situation. In the same way I give ear, or look, in the expectation of a sensation, and suddenly the sensible takes possession of my ear or my gaze, and I surrender a part of my body, even my whole body, to this particular manner of vibrating and filling space known as blue or red....
What are we to make of these strange ways of speaking? In these and other passages throughout Merleau-Ponty's major work, _Phenomenology of Perception_ , the sensible thing, commonly considered by our philosophical tradition to be passive and inert, is consistently described in the active voice: the sensible "beckons to me," "sets a problem for my body to solve," "responds" to my summons and "takes possession of my senses," and even "thinks itself within me." The sensible world, in other words, is described as active, animate, and, in some curious manner, alive: it is not I, when asleep, who breathes, but "some great lung outside myself which alternately calls forth and forces back my breath"; a color is "a manner of vibrating and filling space"; a thing is an "entity," an "Other" which at one moment "holds itself aloof from us" and at another moment actively "expresses itself" directly to our senses, so that we may ultimately describe perception as a mutual interaction, an intercourse, "a coition, so to speak, of my body with things."
Are such animistic turns of phrase to be attributed simply to some sort of poetic license that Merleau-Ponty has introduced into his philosophy? Are they evidence, that is, merely of an idiosyncratic style of writing, as some critics have asserted? I think not. Merleau-Ponty writes of the perceived things as entities, of sensible qualities as powers, and of the sensible itself as a field of animate presences, in order to acknowledge and underscore their active, dynamic contribution to perceptual experience. To describe the animate life of particular things is simply the most precise and parsimonious way to articulate the things _as we spontaneously experience them_ , prior to all our conceptualizations and definitions.
Our most immediate experience of things, according to Merleau-Ponty, is necessarily an experience of reciprocal encounter—of tension, communication, and commingling. From within the depths of this encounter, we know the thing or phenomenon only as our interlocutor—as a dynamic presence that confronts us and draws us into relation. We conceptually immobilize or objectify the phenomenon only by mentally absenting ourselves from this relation, by forgetting or repressing our sensuous involvement. To define another being as an inert or passive object is to deny its ability to actively engage us and to provoke our senses; _we thus block our perceptual reciprocity with that being_. By linguistically defining the surrounding world as a determinate set of objects, we cut our conscious, speaking selves off from the spontaneous life of our sensing bodies.
If, on the other hand, we wish to describe a particular phenomenon without repressing our direct experience, then we cannot avoid speaking of the phenomenon as an active, animate entity with which we find ourselves engaged. It is for this reason that Merleau-Ponty so consistently uses the active voice to describe things, qualities, and even the enveloping world itself. To the sensing body, _no_ thing presents itself as utterly passive or inert. _Only by affirming the animateness of perceived things do we allow our words to emerge directly from the depths of our ongoing reciprocity with the world_.
### Perception as Participation
If we wish to choose a single term to characterize the event of perception, as it is disclosed by phenomenological attention, we may borrow the term "participation," used by the early French anthropologist Lucien Lévy-Bruhl. The brilliant forerunner of today's "cognitive" and "symbolic" schools of anthropology, Lévy-Bruhl used the word "participation" to characterize the animistic logic of indigenous, oral peoples—for whom ostensibly "inanimate" objects like stones or mountains are often thought to be alive, for whom certain names, spoken aloud, may be felt to influence at a distance the things or beings that they name, for whom particular plants, particular animals, particular places and persons and powers may all be felt to _participate_ in one another's existence, influencing each other and being influenced in turn.
For Lévy-Bruhl participation was thus a perceived relation between diverse phenomena; Merleau-Ponty's work, however, suggests that participation is a defining attribute of perception itself. By asserting that perception, phenomenologically considered, is inherently participatory, we mean that perception always involves, at its most intimate level, the experience of an active interplay, or coupling, between the perceiving body and that which it perceives. Prior to all our verbal reflections, at the level of our spontaneous, sensorial engagement with the world around us, we are _all_ animists.
SOME INSIGHT INTO THE PARTICIPATORY NATURE OF PERCEPTION may be gleaned by considering the craft of the sleight-of-hand magician. For the conjuror depends upon this active participation between the body and the world for the creation of his magic. Working, for instance, with a silver dollar, he uses his sleights to enhance the animation of the object, generating ambiguous gaps and lacunae in the visible trajectory of the coin. The spectators' eyes, already drawn by the coin's fluid dance across the magician's fingers, spontaneously fill in those gaps with impossible events, and it is this spontaneous involvement of the spectators' own senses that enables the coin to vanish and reappear, or to pass through the magician's hand.
After flourishing a silver dollar in my right hand, for example, spinning it a few times to catch the audience's attention, I may suddenly hide that coin behind the hand, clipping it between two fingers so that it is no longer visible to their gaze. If, an instant later, I reach into the air on the other side of my body with my left hand, and bring into view _another_ silver dollar that had been clipped behind _that_ hand, the audience will commonly perceive something quite wondrous. They will _not_ perceive that one coin has been momentarily hidden while a wholly different coin, in another place, has been brought out of hiding, although this would surely be the most obvious and rational interpretation. Rather, they will perceive that a single coin, having vanished from my right hand, has traveled invisibly through the air and reappeared in my left hand! For the perceiving body does not calculate logical probabilities; it gregariously participates in the activity of the world, lending its imagination to things in order to see them more fully. The invisible journey of the coin is contributed, quite spontaneously, by the promiscuous creativity of the senses. The magician induces us to assist in the metamorphosis of his objects, and then startles us with what we ourselves have created!
From the magician's, or the phenomenologist's, perspective, that which we call _imagination_ is from the first an attribute of the senses themselves; imagination is not a separate mental faculty (as we so often assume) but is rather the way the senses themselves have of throwing themselves beyond what is immediately given, in order to make tentative contact with the other sides of things that we do not sense directly, with the hidden or invisible aspects of the sensible. And yet such sensory anticipations and projections are not arbitrary; they regularly respond to suggestions offered by the sensible itself. The magician, for instance, may make the magic palpable for the audience by following the invisible coin's journey with the focus of his own eyes, and by imaginatively "feeling" the coin depart from the one hand and arrive in the palm of the other; the audience's senses, responding to subtle shifts in the magician's body as well as to the coin, will then find the effect irresistible. In other words, it is when the magician lets _himself_ be captured by the magic that his audience will be most willing to join him.
Of course, there are those few who simply will not see any magic, either at a performance or in the world at large; armored with countless explanations and analyses, they "see" only how the trick must have been accomplished. Commonly, they will claim to have "caught sight of the wires," or to have seen me clandestinely "throw the coin into the other hand" although I myself have done no such thing. Encouraged by a cultural discourse that disdains the unpredictable and puts a premium on detached objectivity, such persons attempt to halt the participation of their senses in the phenomenon. Yet they can do so only by imaginatively projecting other phenomena (wires, or threads, or mirrors), or by looking away.
In truth, since the act of perception is always open-ended and unfinished, we are never wholly locked into any particular instance of participation. As the spectator can turn away from the magician's magic, we are always somewhat free to break our participation with any particular phenomenon. It is thus that, caught up in contemplation of a blade of grass, I may nevertheless shift my attention to the grove of trees nearby, or my focus may suddenly be usurped by a fly that lands upon my nose. Similarly, we may readily break our fascination with a television commercial in order to notice how it plays upon our emotions and our desires. But we suspend this participation only on behalf of other participations already going on—with the other persons in the room, with the hard and uncomfortable chair on which we sit, with our own thoughts and analyses. We always retain the ability to alter or suspend any particular instance of participation. Yet we can never suspend the flux of participation itself.
### Synaesthesia—The Fusion of the Senses
Until now we have spoken of perception in primarily visual terms. Yet perception involves touching as well, and hearing and smelling and tasting. By the term "perception" we mean the concerted activity of _all_ the body's senses as they function and flourish together. Indeed, if I attend closely to my nonverbal experience of the shifting landscape that surrounds me, I must acknowledge that the so-called separate senses are thoroughly blended with one another, and it is only after the fact that I am able to step back and isolate the specific contributions of my eyes, my ears, and my skin. As soon as I attempt to distinguish the share of any one sense from that of the others, I inevitably sever the full participation of my sensing body with the sensuous terrain.
When, for instance, I perceive the wind surging through the branches of an aspen tree, I am unable, at first, to distinguish the sight of those trembling leaves from their delicate whispering. My muscles, too, feel the torsion as those branches bend, ever so slightly, in the surge, and this imbues the encounter with a certain tactile tension. The encounter is influenced, as well, by the fresh smell of the autumn wind, and even by the taste of an apple that still lingers on my tongue.
Yet already, in this brief attempt to acknowledge the contribution of the various senses, I have had to remove myself from that "primary layer of sense experience that precedes its division among the separate senses." Although contemporary neuroscientists study "synaesthesia"—the overlap and blending of the senses—as though it were a rare or pathological experience to which only certain persons are prone (those who report "seeing sounds," "hearing colors," and the like), our primordial, preconceptual experience, as Merleau-Ponty makes evident, is _inherently_ synaesthetic. The intertwining of sensory modalities seems unusual to us only to the extent that we have become estranged from our direct experience (and hence from our primordial contact with the entities and elements that surround us):
... Synaesthetic perception is the rule, and we are unaware of it only because scientific knowledge shifts the center of gravity of experience, so that we have unlearned how to see, hear, and generally speaking, feel, in order to deduce, from our bodily organization and the world as the _physicist_ conceives it, what we are to see, hear, and feel.
Nevertheless, we still speak of "cool" or "warm" colors, of "loud" clothing, of "hard" or "brittle" sounds. The speaking body readily transposes qualities from one sensory domain into another, according to a logic we easily understand but cannot easily explain.
Many Westerners become conscious of this overlapping of the senses only when their allegiance to the presumably impartial, analytic logic of their culture temporarily breaks down. Merleau-Ponty discusses the effect upon European researchers of mescaline, the psychoactive component of the peyote cactus, a plant traditionally used in ceremonial practice by indigenous tribes in Mexico and North America:
The influence of mescalin, by weakening the attitude of impartiality and surrendering the subject to his vitality, should [if we are correct] favor forms of synaesthetic experience. And indeed, under mescalin, the sound of a flute gives a bluish-green colour, [and] the tick of a metronome, in darkness, is translated as grey patches, the spatial intervals between them corresponding to the intervals of time between the ticks, the size of the patch to the loudness of the tick, and its height to the pitch of the sound. A subject under mescalin finds a piece of iron, strikes the windowsill with it and exclaims: "This is magic"; the trees are growing greener.... Seen in the perspective of the objective [Cartesian] world, with its opaque qualities, the phenomenon of synaesthetic experience is paradoxical....
Seen, however, from the perspective of the life-world—from the perspective, that is, of our pretheoretical awareness—such experiences are recognized as amplifications or intensifications of quite ordinary phenomena that are always going on.
This is not to deny that the senses are distinct modalities. It is to assert that they are divergent modalities of a single and unitary living body, that they are complementary powers evolved in complex interdependence with one another. Each sense is a unique modality of this body's existence, yet in the activity of perception these divergent modalities necessarily intercommunicate and overlap. It is thus that a raven soaring in the distance is not, for me, a mere visual image; as I follow it with my eyes, I inevitably feel the stretch and flex of its wings with my own muscles, and its sudden swoop toward the nearby trees is a visceral as well as a visual experience for me. The raven's loud, guttural cry, as it swerves overhead, is not circumscribed within a strictly audible field—it echoes _through_ the visible, immediately animating the visible landscape with the reckless style or mood proper to that jet black shape. My various senses, diverging as they do from a single, coherent body, coherently _converge_ , as well, in the perceived thing, just as the separate perspectives of my two eyes converge upon the raven and convene there into a single focus. My senses connect up with each other in the things I perceive, or rather each perceived thing gathers my senses together in a coherent way, and it is this that enables me to experience the thing itself as a center of forces, as another nexus of experience, as an Other.
Hence, just as we have described perception as a dynamic participation between my body and things, so we now discern, within the act of perception, a participation between the various sensory systems of the body itself. Indeed, these events are not separable, for the intertwining of my body with the things it perceives is effected only through the interweaving of my senses, and vice versa. The relative divergence of my bodily senses (eyes in the front of the head, ears toward the back, etc.) and their curious bifurcation (not one but _two_ eyes, one on each side, and similarly two ears, two nostrils, etc.), indicates that this body is a form destined to the world; it ensures that my body is a sort of open circuit that completes itself only in things, in others, in the encompassing earth.
### The Recuperation of the Sensuous is the Rediscovery of the Earth
In the autumn of 1985, a strong hurricane ripped across suburban Long Island, where I was then living as a student. For several days afterward much of the populace was without electricity; power lines were down, telephone lines broken, and the roads were strewn with toppled trees. People had to walk to their jobs, and to whatever shops were still open. We began encountering each other on the streets, "in person" instead of by telephone. In the absence of automobiles and their loud engines, the rhythms of crickets and birdsong became clearly audible. Flocks were migrating south for the winter, and many of us found ourselves simply listening, with new and childlike curiosity, to the ripples of song in the still-standing trees and the fields. And at night the sky was studded with stars! Many children, their eyes no longer blocked by the glare of houselights and streetlamps, saw the Milky Way for the first time, and were astonished. For those few days and nights our town became a community aware of its place in an encompassing cosmos. Even our noses seemed to come awake, the fresh smells from the ocean somehow more vibrant and salty. The breakdown of our technologies had forced a return to our senses, and hence to the natural landscape in which those senses are so profoundly embedded. We suddenly found ourselves inhabiting a sensuous world that had been waiting, for years, at the very fringe of our awareness, an intimate terrain infused by birdsong, salt spray, and the light of stars.
AS WE REACQUAINT OURSELVES WITH OUR BREATHING BODIES, then the perceived world itself begins to shift and transform. When we begin to consciously frequent the wordless dimension of our sensory participations, certain phenomena that have habitually commanded our focus begin to lose their distinctive fascination and to slip toward the background, while hitherto unnoticed or overlooked presences begin to stand forth from the periphery and to engage our awareness. The countless human artifacts with which we are commonly involved—the asphalt roads, chain-link fences, telephone wires, buildings, lightbulbs, ballpoint pens, automobiles, street signs, plastic containers, newspapers, radios, television screens—all begin to exhibit a common style, and so to lose some of their distinctiveness; meanwhile, organic entities—crows, squirrels, the trees and wild weeds that surround our house, humming insects, streambeds, clouds and rainfalls—all these begin to display a new vitality, each coaxing the breathing body into a unique dance. Even boulders and rocks seem to speak their own uncanny languages of gesture and shadow, inviting the body and its bones into silent communication. In contact with the native forms of the earth, one's senses are slowly energized and awakened, combining and recombining in ever-shifting patterns.
For these other shapes and species have coevolved, like ourselves, with the rest of the shifting earth; their rhythms and forms are composed of layers upon layers of earlier rhythms, and in engaging them our senses are led into an inexhaustible depth that echoes that of our own flesh. The patterns on the stream's surface as it ripples over the rocks, or on the bark of an elm tree, or in a cluster of weeds, are all composed of repetitive figures that _never exactly repeat themselves_ , of iterated shapes to which our senses may attune themselves even while the gradual drift and metamorphosis of those shapes draws our awareness in unexpected and unpredictable directions.
In contrast, the mass-produced artifacts of civilization, from milk cartons to washing machines to computers, draw our senses into a dance that endlessly reiterates itself _without variation_. To the sensing body these artifacts are, like all phenomena, animate and even alive, but their life is profoundly constrained by the specific "functions" for which they were built. Once our bodies master these functions, the machine-made objects commonly teach our senses nothing further; they are unable to surprise us, and so we must continually acquire _new_ built objects, new technologies, the latest model of this or that if we wish to stimulate ourselves.
Of course, our human-made artifacts inevitably retain an element of more-than-human otherness. This unknowability, this otherness, resides most often in the materials from which the object is made. The tree trunk of the telephone pole, the clay of the bricks from which the building is fashioned, the smooth metal alloy of the car door we lean against—all these still carry, like our bodies, the textures and rhythms of a pattern that we ourselves did not devise, and their quiet dynamism responds directly to our senses. Too often, however, this dynamism is stifled within mass-produced structures closed off from the rest of the earth, imprisoned within technologies that plunder the living land. The superstraight lines and right angles of our office architecture, for instance, make our animal senses wither even as they support the abstract intellect; the wild, earth-born nature of the materials—the woods, clays, metals, and stones that went into the building—are readily forgotten behind the abstract and calculable form.
It is thus that so much of our built environment, and so many of the artifacts that populate it, seem sadly superfluous and dull when we identify with our bodies and taste the world with our animal senses. (Of course, this is not to say that these artifacts are innocuous: many of them are exceedingly loud, even blaring, for what they lack in variation and nuance they must make up in clamorous insistence, monopolizing the perceptual field.) Whenever we assume the position and poise of the human animal—Merleau-Ponty's body-subject—then the entire material world itself seems to come awake and to speak, yet organic, earth-born entities speak far more eloquently than the rest. Like suburbanites after a hurricane, we find ourselves alive in a living field of powers far more expressive and diverse than the strictly human sphere to which we are accustomed.
SO THE RECUPERATION OF THE INCARNATE, SENSORIAL DIMENSION of experience brings with it a recuperation of the living landscape in which we are corporeally embedded. As we return to our senses, we gradually discover our sensory perceptions to be simply our part of a vast, interpenetrating webwork of perceptions and sensations borne by countless other bodies—supported, that is, not just by ourselves, but by icy streams tumbling down granitic slopes, by owl wings and lichens, and by the unseen, imperturbable wind.
This intertwined web of experience is, of course, the "life-world" to which Husserl alluded in his final writings, yet now the life-world has been disclosed as a profoundly _carnal_ field, as this very dimension of smells and tastes and chirping rhythms warmed by the sun and shivering with seeds. It is, indeed, nothing other than the biosphere—the matrix of earthly life in which we ourselves are embedded. Yet this is not the biosphere as it is conceived by an abstract and objectifying science, not that complex assemblage of planetary mechanisms presumably being mapped and measured by our remote-sensing satellites; it is, rather, the biosphere as it is experienced and _lived from within_ by the intelligent body—by the attentive human animal who is entirely a part of the world that he, or she, experiences.
### Matter as Flesh
In his final work, _The Visible and the Invisible_ (a work interrupted by his sudden death in 1961), Merleau-Ponty was striving for a new way of speaking that would express this consanguinity of the human animal and the world it inhabits. Here he writes less about "the body" (which in his earlier work had signified primarily the _human_ body) and begins to write instead of the collective "Flesh," which signifies both _our_ flesh and "the flesh of the world." By "the Flesh" Merleau-Ponty means to indicate an elemental power that has had no name in the entire history of Western philosophy. The Flesh is the mysterious tissue or matrix that underlies and gives rise to both the perceiver and the perceived as interdependent aspects of its own spontaneous activity. It is the reciprocal presence of the sentient in the sensible and of the sensible in the sentient, a mystery of which we have always, at least tacitly, been aware, since we have never been able to affirm one of these phenomena, the perceivable world or the perceiving self, without implicitly affirming the existence of the other. We are unable even to _imagine_ a sensible landscape that would not at the same time be sensed (since in imagining any landscape we inevitably envisage it from a particular perspective, and thus implicate our own senses, and indeed our own sentience, in that landscape), and are similarly unable to fully imagine a sensing self, or sentience, that would not be situated in some field of sensed phenomena.
Nevertheless, conventional scientific discourse privileges the sensible field in abstraction from sensory experience, and commonly maintains that subjective experience is "caused" by an objectifiable set of processes in the mechanically determined field of the sensible. Meanwhile, New Age spiritualism regularly privileges pure sentience, or subjectivity, in abstraction from sensible matter, and often maintains that material reality is itself an illusory effect caused by an immaterial mind or spirit. Although commonly seen as opposed world-views, both of these positions assume a qualitative difference between the sentient and the sensed; by prioritizing one or the other, both of these views perpetuate the distinction between human "subjects" and natural "objects," and hence neither threatens the common conception of sensible nature as a purely passive dimension suitable for human manipulation and use. While both of these views are unstable, each bolsters the other; by bouncing from one to the other—from scientific determinism to spiritual idealism and back again—contemporary discourse easily avoids the possibility that both the perceiving being and the perceived being are _of the same stuff_ , that the perceiver and the perceived are interdependent and in some sense even reversible aspects of a common animate element, or Flesh, that is _at once both sensible and sensitive_.
We readily experience this paradox in relation to other persons; this stranger who stands before me and is an object for my gaze suddenly opens his mouth and speaks to me, forcing me to acknowledge that he is a sentient subject like myself, and that I, too, am an object for his gaze. Each of us, in relation to the other, is both subject and object, sensible and sentient. Why, then, might this not also be the case in relation to another, nonhuman entity—a mountain lion, for instance, that I unexpectedly encounter in the northern forest? Indeed, such a meeting brings home to me even more forcefully that I am not just a sentient subject but also a sensible object, even an _edible_ object, in the eyes (and nose) of the other. Even an ant crawling along my arm, visible to my eyes and tactile to my skin, displays at the same time its own sentience, responding immediately to my movements, even to the chemical changes of my mood. In relation to the ant I feel myself as a dense and material object, as capricious in my actions as the undulating earth itself. Finally, then, why might not this "reversibility" of subject and object extend to every entity that I experience? Once I acknowledge that my own sentience, or subjectivity, does not preclude my visible, tactile, objective existence for others, I find myself forced to acknowledge that _any_ visible, tangible form that meets my gaze may also be an experiencing subject, sensitive and responsive to the beings around it, and to me.
### Touching and Being Touched: The Reciprocity of the sensuous
In order to demonstrate, empirically, his notion of the Flesh, Merleau-Ponty provides what may be the most direct illustration of that which we have termed "participation." He calls attention to the obvious but easily overlooked fact that my hand is able to touch things only because my hand is itself a touchable thing, and thus is entirely a part of the tactile world that it explores. Similarly, the eyes, with which I see things, are themselves visible. With their gleaming surfaces, their colors and hues, they are included _within_ the visible field that they see—they are themselves part of the visible, like the bark of a cedar, or a piece of sandstone, or the blue sky.
To touch the coarse skin of a tree is thus, at the same time, to experience one's own tactility, to feel oneself touched _by_ the tree. And to see the world is also, at the same time, to experience oneself as visible, to feel oneself _seen_. Clearly, a wholly immaterial mind could neither see things nor touch things—indeed, could not experience anything at all. _We_ can experience things—can touch, hear, and taste things—only because, as bodies, we are ourselves included in the sensible field, and have our own textures, sounds, and tastes. We can perceive things at all only because we ourselves are entirely a part of the sensible world that we perceive! We might as well say that we are organs of this world, flesh of its flesh, and that the world is perceiving itself _through_ us.
Walking in a forest, we peer into its green and shadowed depths, listening to the silence of the leaves, tasting the cool and fragrant air. Yet such is the transitivity of perception, the reversibility of the flesh, that we may suddenly feel that the trees are looking at us—we feel ourselves exposed, watched, observed from all sides. If we dwell in this forest for many months, or years, then our experience may shift yet again—we may come to feel that we are a part of this forest, consanguineous with it, and that our experience of the forest is nothing other than the forest experiencing itself.
Such are the exchanges and metamorphoses that arise from the simple fact that our sentient bodies are entirely continuous with the vast body of the land, that "the presence of the world is precisely the presence of its flesh to my flesh."
MERLEAU-PONTY'S NOTION OF THE FLESH OF THE WORLD, ALONG with his related discoveries regarding the reciprocity of perception, bring his work into startling consonance with the worldviews of many indigenous, oral cultures. According to cultural anthropologist Richard Nelson, in his exhaustive study of the ecology of the Koyukon Indians of north central Alaska:
[t]raditional Koyukon people live in a world that watches, in a forest of eyes. A person moving through nature—however wild, remote, even desolate the place may be—is never truly alone. The surroundings are aware, sensate, personified. They feel. They can be offended. And they must, at every moment, be treated with the proper respect.
Such a mode of experience, which seems so strange and confused to our civilized ways of thinking, becomes understandable as soon as we acknowledge, underneath our conventional assumptions, the reciprocal nature of direct perception—the fact that to touch is also to feel oneself being touched, that to see is also to feel oneself seen. Nelson's description suggests, as well, that such perceptual reciprocity, when consciously acknowledged, may profoundly influence one's behavior. If the surroundings are experienced as sensate, attentive, and watchful, then I must take care that my actions are mindful and respectful, even when I am far from other humans, lest I offend the watchful land itself.
It may be that the new "environmental ethic" toward which so many environmental philosophers aspire—an ethic that would lead us to respect and heed not only the lives of our fellow humans but also the life and well-being of the rest of nature—will come into existence not primarily through the logical elucidation of new philosophical principles and legislative strictures, but through a renewed attentiveness to this perceptual dimension that underlies all our logics, through a rejuvenation of our carnal, sensorial empathy with the living land that sustains us.
Such a recuperation is, perhaps, already underway. Many individuals today experience a profound anguish that only deepens with each report of more ancient forests cleared, of new oil spills, of the ever-accelerating loss of species. It is an anguish that seems to come from the earth itself, from this vast Flesh in which our own sentient flesh is embedded. In the words of a Koyukon elder: "The country knows. If you do wrong things to it, the whole country knows. It feels what's happening to it."
THE INFLUENCE OF A KIND OF PERCEPTUAL RECIPROCITY UPON oneself and one's actions is evident as well in these words spoken by Old Torlino, a Navajo elder, before telling part of the creation story:
_I am ashamed before the earth;_
_I am ashamed before the heavens;_
_I am ashamed before the dawn;_
_I am ashamed before the evening twilight;_
_I am ashamed before the blue sky;_
_I am ashamed before the sun_.
_I am ashamed before that standing within me which speaks with me_.
_Some of these things are always looking at me_.
_I am never out of sight_.
_Therefore I must tell the truth_.
_I hold my word tight to my breast. 27_
The final lines of this prayer/incantation call our attention to speaking itself as a form of behavior that can be mindful or callous, truthful or dishonest, in the face of a sentient cosmos. Spoken words here are real presences, entities that may be cherished—"held tight to my breast"—or flung carelessly into the world. These phrases from the Navajo, like the Koyukon words before them, provide evidence not only of a different way of seeing, but also of a way of speaking very different from that to which so many of us are accustomed. The practice of language among indigenous peoples would seem to carry a very different significance than it does in the modern West. Enacted primarily in song, prayer, and story, among oral peoples language functions not simply to dialogue with other humans but also to converse with the more-than-human cosmos, to renew reciprocity with the surrounding powers of earth and sky, to invoke kinship even with those entities which, to the civilized mind, are utterly insentient and inert. Hence, a Lakota medicine person may address a stone as "Tunkashila"—"Grandfather." Likewise, among the Omaha, a rock may be addressed with the respect and reverence that one pays to an ancient elder:
unmoved
from time without
end
you rest
there in the midst of the paths
in the midst of the winds
you rest
covered with the droppings of birds
grass growing from your feet
your head decked with the down of birds
you rest
in the midst of the winds
you wait
Aged one.
Here words do not speak _about_ the world; rather they speak _to_ the world, and to the expressive presences that, with us, inhabit the world. In multiple and diverse ways, taking (as we shall see) a unique form in each indigenous culture, spoken language seems to give voice to, and thus to enhance and accentuate, the sensorial affinity between humans and the environing earth.
This would appear, at least at first, to be in direct contradiction to the character of linguistic discourse in the "developed" or "civilized" world, where language functions largely to _deny_ reciprocity with nature—by defining the rest of nature as inert, mechanical, and determinate—and where, in consequence, our sensorial participation with the land around us must remain mute, inchoate, and in most cases wholly unconscious. In indigenous, oral cultures, in other words, language seems to encourage and augment the participatory life of the senses, while in Western civilization language seems to deny or deaden that life, promoting a massive distrust of sensorial experience while valorizing an abstract realm of ideas hidden behind or beyond the sensory appearances.
How can we account for this divergence? In what manner can we make sense of this difference in the character of language, and in the relation between language and perception? Before attempting a precise answer to this question, we must come to a clearer understanding of just what is meant, in this context, by "language."
# **3**
# **_The Flesh of Language_**
The rain surrounded the cabin... with a whole world of meaning, of secrecy, of rumor. Think of it: all that speech pouring down, selling nothing, judging nobody, drenching the thick mulch of dead leaves, soaking the trees, filling the gullies and crannies of the wood with water, washing out the places where men have stripped the hillside.... Nobody started it, nobody is going to stop it. It will talk as long as it wants, the rain. As long as it talks I am going to listen.
—THOMAS MERTON
EVERY ATTEMPT TO DEFINITIVELY SAY _WHAT LANGUAGE IS_ is subject to a curious limitation. For the only medium with which we can define language is language itself. We are therefore unable to circumscribe the whole of language within our definition. It may be best, then, to leave language undefined, and to thus acknowledge its open-endedness, its mysteriousness. Nevertheless, by paying attention to this mystery we may develop a conscious familiarity with it, a sense of its texture, its habits, its sources of sustenance.
Merleau-Ponty, as we have seen, spent much of his life demonstrating that the event of perception unfolds as a reciprocal exchange between the living body and the animate world that surrounds it. He showed, as well, that this exchange, for all its openness and indeterminacy, is nevertheless highly articulate. (Although it confounds the causal logic that we attempt to impose upon it, perceptual experience has its own coherent structure; it seems to embody an open-ended logos that we enact from within rather than the abstract logic we deploy from without.) The disclosure that preverbal perception is already an exchange, and the recognition that this exchange has its own coherence and articulation, together suggested that perception, this ongoing reciprocity, is the very soil and support of that more conscious exchange we call language.
Already in the _Phenomenology of Perception_ , Merleau-Ponty had begun to work out a notion of human language as a profoundly carnal phenomenon, rooted in our sensorial experience of each other and of the world. In a famous chapter entitled "The Body as Expression, and Speech," he wrote at length of the gestural genesis of language, the way that communicative meaning is first incarnate in the gestures by which the body spontaneously expresses feelings and responds to changes in its affective environment. The gesture is spontaneous and immediate. It is not an arbitrary sign that we mentally attach to a particular emotion or feeling; rather, the gesture _is_ the bodying-forth of that emotion into the world, it _is_ that feeling of delight or of anguish in its tangible, visible aspect. When we encounter such a spontaneous gesture, we do not first see it as a blank behavior, which we then mentally associate with a particular content or significance; rather, the bodily gesture speaks directly to our own body, and is thereby understood without any interior reflection:
Faced with an angry or threatening gesture, I have no need, in order to understand it, to [mentally] recall the feelings which I myself experienced when I used these gestures on my own account.... I do not see anger or a threatening attitude as a psychic fact hidden behind the gesture, I read anger in it. The gesture _does not make me think of anger_ , it is anger itself.
Active, living speech is just such a gesture, a vocal gesticulation wherein the meaning is inseparable from the sound, the shape, and the rhythm of the words. Communicative meaning is always, in its depths, affective; it remains rooted in the sensual dimension of experience, born of the body's native capacity to resonate with other bodies and with the landscape as a whole. Linguistic meaning is not some ideal and bodiless essence that we arbitrarily assign to a physical sound or word and then toss out into the "external" world. Rather, meaning sprouts in the very depths of the sensory world, in the heat of meeting, encounter, participation.
We do not, as children, first enter into language by consciously studying the formalities of syntax and grammar or by memorizing the dictionary definitions of words, but rather by actively making sounds—by crying in pain and laughing in joy, by squealing and babbling and playfully mimicking the surrounding soundscape, gradually entering through such mimicry into the specific melodies of the local language, our resonant bodies slowly coming to echo the inflections and accents common to our locale and community.
_We thus learn our native language not mentally but bodily_. We appropriate new words and phrases first through their expressive tonality and texture, through the way they feel in the mouth or roll off the tongue, and it is this direct, felt significance—the _taste_ of a word or phrase, the way it influences or modulates the body—that provides the fertile, polyvalent source for all the more refined and rarefied meanings which that term may come to have for us.
... the meaning of words must be finally induced by the words themselves, or more exactly, their conceptual meaning must be formed by a kind of subtraction from a _gestural meaning_ , which is immanent in speech.
Language, then, cannot be genuinely studied or understood in isolation from the sensuous reverberation and resonance of active speech. James M. Edie attempts to summarize this aspect of Merleau-Ponty's thought in this manner:
... Merleau-Ponty's first point is that words, even when they finally achieve the ability to carry referential and, eventually, conceptual levels of meaning, never completely lose that primitive, strictly phonemic, level of 'affective' meaning which is not translatable into their conceptual definitions. There is, he argues, an affective tonality, a mode of conveying meaning beneath the level of thought, beneath the level of the words themselves... which is contained in the words _just insofar as they are patterned sounds_ , as just the sounds which this particular historical language uniquely uses, and which are much more like a melody—a 'singing of the world'—than fully translatable, conceptual thought. Merleau-Ponty is almost alone among philosophers of language in his sensitivity to this level of meaning....
Edie here emphasizes Merleau-Ponty's originality with regard to language, and asserts that Merleau-Ponty gave special attention to "what no philosopher from Plato on down ever had any interest in" (namely, the gestural significance of spoken sounds). Yet this assertion is true only if one holds a very restricted view of the philosophical tradition. The expressive, gestural basis of language had already been emphasized in the first half of the eighteenth century by the Italian philosopher Giambattista Vico (1668–1744), who in his _New Science_ wrote of language as arising from expressive gestures, and suggested that the earliest and most basic words had taken shape from expletives uttered in startled response to powerful natural events, or from the frightened, stuttering mimesis of such events—like the crack and rumble of thunder across the sky. Shortly thereafter, in France, Jean-Jacques Rousseau (1712–1778) wrote of gestures and spontaneous expressions of feeling as the earliest forms of language, while in Germany, Johann Gottfried Herder (1744–1803) argued that language originates in our sensuous receptivity to the sounds and shapes of the natural environment.
In his embodied philosophy of language, then, Merleau-Ponty is the heir of a long-standing, if somewhat heretical, lineage. Linguistic meaning, for him, is rooted in the felt experience induced by specific sounds and sound-shapes as they echo and contrast with one another, each language a kind of song, a particular way of "singing the world."
### Toward an Ecology of Language
The more prevalent view of language, at least since the scientific revolution, and still assumed in some manner by most linguists today, considers any language to be a set of arbitrary but conventionally agreed upon words, or "signs," linked by a purely formal system of syntactic and grammatical rules. Language, in this view, is rather like a _code;_ it is a way of _representing_ actual things and events in the perceived world, but it has no internal, nonarbitrary connections to that world, and hence is readily separable from it.
If we agree with Merleau-Ponty's assertion that active speech is the generative core of all language, how can we possibly account for the overwhelming prevalence of a view that considers language to be an ideal or formal system readily detachable from the material act of speaking? Merleau-Ponty suggests that such a view of language could arise only at a time when the fresh creation of meaning has become a rare occurrence, a time when people commonly speak in conventional, ready-made ways "which demand from us no real effort of expression and... demand from our listeners no real effort of comprehension"—at a time, in short, when meaning has become impoverished.
Yet there is another, more overt reason for the dominance of the idea that language is an arbitrary, or strictly conventional, set of signs. As we noted earlier, European philosophy has consistently occupied itself with the question of human specialness. Ever since Aristotle, philosophers have been concerned to demonstrate, in the most convincing manner possible, that human beings are significantly different from all other forms of life. It was not enough to demonstrate that human beings were unique, for each species is evidently unique in its way; rather, it was necessary to show that the human form was _uniquely_ unique, that our noble gifts set us definitively apart from, and above, the rest of the animate world. Such demonstrations were, we may suspect, needed to justify the increasing manipulation and exploitation of nonhuman nature by, and for, (civilized) humankind. The necessity for such philosophical justification became especially urgent in the wake of the scientific revolution, when our capacity to manipulate other organisms increased a hundredfold. Descartes's radical separation of the immaterial human mind from the wholly mechanical world of nature did much to fill this need, providing a splendid rationalization for the vivisection experiments that soon began to proliferate, as well as for the steady plundering and despoilment of nonhuman nature in the New World and the other European colonies.
But in the latter half of the nineteenth century, the publication of Darwin's _Origin of Species_ and _The Descent of Man_ introduced a profound tension into the anthropocentric trajectory of European philosophy and science. If humans are animals evolved like other animals, if in truth we are descended by "natural selection" from primates, if indeed fish are our distant ancestors and mice are our cousins, then our own traits and capacities must be, to some degree, continuous with those found in the rest of the earthly environment.
Most scientists, however, while accepting Darwin's theories, were reluctant to relinquish the assumption of human specialness—the assumption that alone justifies so many of the cultural and research practices to which we have now become accustomed. In earlier centuries we could ascribe our superiority to the dispensation of God, who had "created" us as his representatives on earth, or who had bequeathed to humans alone the divine capacity for awareness and intelligence. After Darwin, however, we no longer had such easy recourse to extraworldly dispensation; it became necessary to find new, more naturalistic evidence for the superiority of humankind.
In our own time it is _language_ , conceived as an exclusively human property, that is most often used to demonstrate the excellence of humankind relative to all other species. Other animals have been shown to build complex dwellings, even to use tools. But language, it is widely asserted, remains the special provenance of the human species. To be sure, most other animals manage to communicate with each other, often employing a repertoire of gestures, from "marking" territory with chemical secretions, to the facial expressions of many mammal species, to the host of rattles, cries, howls, and growls that sound across the fields and forests—to say nothing of the complex melodic songs employed, most obviously by birds, as well as by various marine-dwelling mammals like orcas and humpback whales. One of the founding events of the science of ethology, earlier in this century, was the discovery of the intricate "waggledance" whereby individual bees communicate the precise direction and distance of a newfound food source to the rest of the hive. Yet each of these communicative arrays—these "dances," "songs," and gestures, both vocal and visual—may be said to remain within the sphere of felt, bodily expression. The meanings here, it is assumed, are tied to the expressive nature of the gestures themselves, and to the direct sensations induced by these movements—to the immediacy of instinct and bodily urge.
In everyday human discourse, on the other hand, we readily locate a dimension of significance beyond the merely expressive power of the words, a layer of abstract meanings fixed solely, it would seem, by convention. Thus, the term "Wow!" may at first be a simple expression of wonder, but it may also come to designate, if we so choose, a particular type of hairdo, or a shade of blue, or a specific tactic to be used when debating with fishermen. It is this second layer of agreed-upon meanings that is identified with "language in the proper sense" by most philosophers and scientists since the Enlightenment. Only by isolating this secondary layer of conventional meanings from the felt significance carried by the tone, rhythm, and resonance of spoken expressions can we conceive of language as a code—as a determinate and mappable structure composed of arbitrary signs linked by purely formal rules. And only thus, by conceiving language as a purely abstract phenomenon, can we claim it as an exclusively human attribute. Only by overlooking the sensuous, evocative dimension of human discourse, and attending solely to the denotative and conventional aspect of verbal communication, can we hold ourselves apart from, and outside of, the rest of animate nature.
If Merleau-Ponty is right, however, then the denotative, conventional dimension of language can never be truly severed from the sensorial dimension of direct, affective meaning. If we are not, in truth, immaterial minds merely housed in earthly bodies, but are from the first material, corporeal beings, then it is the sensuous, gestural significance of spoken sounds—their direct bodily resonance—that makes verbal communication possible at all. It is this expressive potency—the soundful influence of spoken words upon the sensing body—that supports all the more abstract and conventional meanings that we assign to those words. Although we may be oblivious to the gestural, somatic dimension of language, having repressed it in favor of strict dictionary definitions and the abstract precision of specialized terminologies, this dimension remains subtly operative in all our speaking and writing—if, that is, our words have any significance whatsoever. For meaning, as we have said, remains rooted in the sensory life of the body—it cannot be completely cut off from the soil of direct, perceptual experience without withering and dying.
Yet to affirm that linguistic meaning is primarily expressive, gestural, and poetic, and that conventional and denotative meanings are inherently secondary and derivative, is to renounce the claim that "language" is an exclusively human property. If language is always, in its depths, physically and sensorially resonant, then it can never be definitively separated from the evident expressiveness of birdsong, or the evocative howl of a wolf late at night. The chorus of frogs gurgling in unison at the edge of a pond, the snarl of a wildcat as it springs upon its prey, or the distant honking of Canadian geese veeing south for the winter, all reverberate with affective, gestural significance, the same significance that vibrates through our own conversations and soliloquies, moving us at times to tears, or to anger, or to intellectual insights we could never have anticipated. Language as a bodily phenomenon accrues to _all_ expressive bodies, not just to the human. Our own speaking, then, does not set us outside of the animate landscape but—whether or not we are aware of it—inscribes us more fully in its chattering, whispering, soundful depths.
If, for instance, one comes upon two human friends unexpectedly meeting for the first time in many months, and one chances to hear their initial words of surprise, greeting, and pleasure, one may readily notice, if one pays close enough attention, a tonal, melodic layer of communication beneath the explicit denotative meaning of the words—a rippling rise and fall of the voices in a sort of musical duet, rather like two birds singing to each other. Each voice, each side of the duet, mimes a bit of the other's melody while adding its own inflection and style, and then is echoed by the other in turn—the two singing bodies thus tuning and attuning to one another, rediscovering a common register, _remembering_ each other. It requires only a slight shift in focus to realize that this melodic singing is carrying the bulk of communication in this encounter, and that the explicit meanings of the actual words ride on the surface of this depth like waves on the surface of the sea.
It is by a complementary shift of attention that one may suddenly come to hear the familiar song of a blackbird or a thrush in a surprisingly new manner—not just as a pleasant melody repeated mechanically, as on a tape player in the background, but as active, meaningful speech. Suddenly, subtle variations in the tone and rhythm of that whistling phrase seem laden with expressive intention, and the two birds singing to each other across the field appear for the first time as attentive, conscious beings, earnestly engaged in the same world that we ourselves engage, yet from an astonishingly different angle and perspective.
Moreover, if we allow that spoken meaning remains rooted in gesture and bodily expressiveness, we will be unable to restrict our renewed experience of language solely to animals. As we have already recognized, in the untamed world of direct sensory experience _no_ phenomenon presents itself as utterly passive or inert. To the sensing body _all_ phenomena are animate, actively soliciting the participation of our senses, or else withdrawing from our focus and repelling our involvement. Things disclose themselves to our immediate perception as vectors, as styles of unfolding—not as finished chunks of matter given once and for all, but as dynamic ways of engaging the senses and modulating the body. Each thing, each phenomenon, has the power to reach us and to influence us. Every phenomenon, in other words, is potentially expressive. At the end of his chapter "The Body as Expression, and Speech," Merleau-Ponty writes:
It is the body which points out, and which speaks.... This disclosure [of the body's immanent expressiveness]... extends, as we shall see, to the whole sensible world, and our gaze, prompted by the experience of our own body, will discover in all other "objects" the miracle of expression.
Thus, at the most primordial level of sensuous, bodily experience, we find ourselves in an expressive, gesturing landscape, in a world that _speaks_.
__We regularly talk of howling winds, and of chattering brooks. Yet these are more than mere metaphors. Our own languages are continually nourished by these other voices—by the roar of waterfalls and the thrumming of crickets. It is not by chance that, when hiking in the mountains, the English terms we spontaneously use to describe the surging waters of the nearby river are words like "rush," "splash," "gush," "wash." For the sound that unites all these words is that which the water itself chants as it flows between the banks. If language is not a purely mental phenomenon but a sensuous, bodily activity born of carnal reciprocity and participation, then our discourse has surely been influenced by many gestures, sounds, and rhythms besides those of our single species. Indeed, if human language arises from the perceptual interplay between the body and the world, then this language "belongs" to the animate landscape as much as it "belongs" to ourselves.
IN 1945, MERLEAU-PONTY BEGAN READING THE WORK OF THE SWISS linguist Ferdinand de Saussure (1857-1913), whose posthumously published _Course in General Linguistics_ signaled the emergence of scientific linguistics in the twentieth century. Merleau-Ponty was intrigued by Saussure's theoretical distinction between _la langue—_ language considered as a system of terminological, syntactic, and semantic rules, and _la parole_ —the concrete act of speech itself.
Language considered as a formal system of rules and conventions is that aspect of language which, alone, is susceptible to objective, scientific study. By isolating this aspect of language, Saussure effectively cleared the way for the rigorous, scientific analysis of language systems. Yet the proper way to understand the relation _between_ the formal structure of language and the expressive act of speaking (between _la langue_ and _la parole_ ) remained enigmatic, and it was this enigma that most fascinated Merleau-Ponty.
For Saussure, _la langue_ —language considered as a purely structural system—was not a mechanical structure that could readily be taken apart into its separable components, but more an organic, living system, each of whose parts is internally related to all the others. Saussure described the structure of any language as a thoroughly interdependent matrix, a webwork wherein each term has meaning only by virtue of its relation to other terms within the system. In English, for instance, the sounded word "red" draws its precise meaning from its situation in a network of like-sounding terms, including, for instance, "read," "rod," "reed," and "raid," and in a whole complex of color terms, such as "orange," "yellow," "purple," "brown"; as well as from its participation in a still wider nexus of related terms like "blood," "rose," "sunset," "fire," "blush," "angry," "hot," each of which holds significance only in relation to a constellation of still other words, expanding thus outward to every term within the language. By describing any particular language as a _system of differences_ , Saussure indicated that meaning is found not in the words themselves but in the intervals, the contrasts, the participations _between_ the terms. As Merleau-Ponty states:
What we have learned from Saussure is that, taken singly, signs do not signify anything, and that each one of them does not so much express a meaning as mark a divergence of meaning between itself and other signs.
This does not mean that it is necessary to know, explicitly, the whole of a language in order to speak it. Rather, the weblike nature of language ensures that the whole of the system is implicitly present in every sentence, in every phrase. In order to learn a community's language, suggests Merleau-Ponty, it is necessary simply to begin speaking, to enter the language with one's body, to begin to move within it. The language in its entirety is invoked by the child in his first attempts at speech. "[Then] the whole of the spoken language surrounding the child snaps him up like a whirlwind, tempts him by its internal articulations...."
The enigma that is language, constituted as much by silence as by sounds, is not an inert or static structure, but an evolving bodily field. It is like a vast, living fabric continually being woven by those who speak. Merleau-Ponty here distinguishes sharply between genuine, expressive speech and speech that merely repeats established formulas. The latter is hardly "speech" at all; it does not really carry meaning in the weave of its words but relies solely upon the memory of meanings that once lived there. It does not alter the already existing structures of the language, but rather treats the language as a finished institution. Nevertheless, those preexisting structures must at some moment have been created, and this can only have been effected by active, expressive speech. Indeed, all truly meaningful speech is inherently creative, using established words in ways they have never quite been used before, and thus altering, ever so slightly, the whole webwork of the language. Wild, living speech takes up, from within, the interconnected matrix of the language and _gestures_ with it, subjecting the whole structure to a "coherent deformation."
At the heart of any language, then, is the poetic productivity of expressive speech. A living language is continually being made and remade, woven out of the silence by those who speak.... And this silence is that of our wordless participations, of our perceptual immersion in the depths of an animate, expressive world.
Thus, Saussure's distinction between the structure of language and the activity of speech is ultimately undermined by Merleau-Ponty, the two dimensions blended back together into a single, ever-evolving matrix. While individual speech acts are surely guided by the structured lattice of the language, that lattice is nothing other than the sedimented result of all previous acts of speech, and will itself be altered by the very expressive activity it now guides. Language is not a fixed or ideal form, but an evolving medium we collectively inhabit, a vast topological matrix in which the speaking bodies are generative sites, vortices where the matrix itself is continually being spun out of the silence of sensorial experience.
What Merleau-Ponty retains from Saussure is Saussure's notion of any language as an interdependent, weblike system of relations. But since our expressive, speaking bodies are for Merleau-Ponty necessary parts of this system—since the web of language is for him a carnal medium woven in the depths of our perceptual participation with the things and beings around us—Merleau-Ponty comes in his final writings to affirm that it is first the sensuous, perceptual world that is relational and weblike in character, and hence that the organic, interconnected structure of any language is an extension or echo of the deeply interconnected matrix of sensorial reality itself. Ultimately, it is not human language that is primary, but rather the sensuous, perceptual life-world, whose wild, participatory logic ramifies and elaborates itself in language.
Since the mid-nineteenth century, the study of our earthly environment has increasingly yielded a view of nature as a realm of complexly interwoven relationships, a field of subtle interdependencies from which, in John Muir's words, no single phenomenon can be picked out without "finding it hitched to everything else." The character of an individual fruit tree simply cannot be understood without reference to the others of its species, to the insects that fertilize it and to the animals that consume its fruit and so disperse its seeds. Yet a single one of those animals can hardly be comprehended without learning of the _other_ plants or animals that it eats throughout the year, and of the predators that prey upon _it_ —without, in other words, acknowledging the host of other organisms upon which that animal depends, and which depend upon it. We have at last come to realize that neither the soils, the oceans, nor the atmosphere can be comprehended without taking into account the participation of innumerable organisms, from the lichens that crumble rocks, and the bacterial entities that decompose organic detritus, to all the respiring plants and animals exchanging vital gases with the air. The notion of earthly nature as a densely interconnected organic network—a "biospheric web" wherein each entity draws its specific character from its relations, direct and indirect, to all the others—has today become commonplace, and it converges neatly with Merleau-Ponty's late description of sensuous reality, "the Flesh," as an intertwined, and actively intertwining, lattice of mutually dependent phenomena, both sensorial and sentient, of which our own sensing bodies are a part.
It is this dynamic, interconnected reality that provokes and sustains all our speaking, lending something of its structure to all our various languages. The enigmatic nature of language echoes and "prolongs unto the invisible" the wild, interpenetrating, interdependent nature of the sensible landscape itself.
Ultimately, then, it is not the human body alone but rather the whole of the sensuous world that provides the deep structure of language. As we ourselves dwell and move within language, so, ultimately, do the other animals and animate things of the world; if we do not notice them there, it is only because language has forgotten its expressive depths. "Language is a life, is our life and the life of the things...." It is no more true that _we_ speak than that the things, and the animate world itself, _speak within us:_
__That the things have us and that it is not we who have the things.... That it is being that speaks within us and not we who speak of being. 15
From such reflections we may begin to suspect that the complexity of human language is related to the complexity of the earthly ecology—not to any complexity of our species considered apart from that matrix. Language, writes Merleau-Ponty, "is the very voice of the trees, the waves, and the forests."
As technological civilization diminishes the biotic diversity of the earth, language itself is diminished. As there are fewer and fewer songbirds in the air, due to the destruction of their forests and wetlands, human speech loses more and more of its evocative power. For when we no longer hear the voices of warbler and wren, our own speaking can no longer be nourished by their cadences. As the splashing speech of the rivers is silenced by more and more dams, as we drive more and more of the land's wild voices into the oblivion of extinction, our own languages become increasingly impoverished and weightless, progressively emptied of their earthly resonance.
### Word Magic
Merleau-Ponty's work on language is admittedly fragmentary and unfinished, cut short by his sudden death. Yet it provides the most extensive investigation we have, as yet, into the living _experience_ of language—the way the expressive medium discloses itself to us when we do not pretend to stand outside it, but rather accept our inherence _within_ it, as speaking animals. When we attend to our experience not as intangible minds but as sounding, speaking bodies, we begin to sense that we are heard, even listened to, by the numerous other bodies that surround us. Our sensing bodies respond to the eloquence of certain buildings and boulders, to the articulate motions of dragonflies. We find ourselves alive in a listening, speaking world.
Here (as we saw earlier with regard to perception) Merleau-Ponty's work resonates, and brings us close to, the spoken beliefs of many indigenous, oral peoples.
In such indigenous cultures the solidarity between language and the animate landscape is palpable and evident. According to Ogotemmêli, an elder of the Dogon tribe of Mali, spoken language was originally a swirling garment of vapour and breath worn by the encompassing earth itself. Later this undulating garment was stolen by the jackal, an animal whose movements, ever since, have disclosed the prophetic speech of the world to seers and diviners. Many tribes, like the Swampy Cree of Manitoba, hold that they were given spoken language by the animals. For the Inuit (Eskimo), as for numerous other peoples, humans and animals all originally spoke the same language. According to Nalungiaq, an Inuit woman interviewed by ethnologist Knud Rasmussen early in the twentieth century:
_In the very earliest time_
_when both people and animals lived on earth_ ,
_a person could become an animal if he wanted to_
_and an animal could become a human being_.
_Sometimes they were people_
_and sometimes animals_
_and there was no difference_.
_All spoke the same language_.
_That was the time when words were like magic_.
_The human mind had mysterious powers_.
_A word spoken by chance_
_might have strange consequences_.
_It would suddenly come alive_
_and what people wanted to happen could happen—_
_all you had to do was say it_.
_Nobody could explain this:_
_That's the way it was._ 20
Despite this originary language common to both people and animals, the various animals and other natural forms today speak their own unique dialects. But nevertheless _all speak_ , all have the power of language. Moreover, traces of the primordial common language remain, and just as a human may suddenly understand the subtle gestures of a deer, or the guttural speech of a raven, so the other entities hear, and may understand, our own talking.
Owls often make it difficult to speak Cree with them. They can cause stuttering, and when stuttering is going on they are attracted to it. It is said that stuttering is laughable to owls. Yet this can work to the Cree's advantage as well, for if you think an owl is causing trouble in your village, then go stutter in the woods. There's a good chance an owl will arrive. Then you can confront this owl, question it, argue with it, perhaps solve the problem.
Most indigenous hunting peoples carefully avoid speaking about the hunt beforehand, or referring directly to the species that they are hunting, lest they offend the listening animals themselves. After the kill, however, they will speak directly to the dying animal, praising it, promising respect, and thanking it for offering itself to them.
Yet it is those who are recognized as shamans, or medicine persons, who most fully remember the primordial sacred language, and who are thus able to slip, at will, out of the purely human discourse in order to converse directly with the other powers. As Mircea Eliade writes:
The existence of a specific secret language has been verified among the Lapps, the Ostyak, the Chukchee, the Yakut, and the Tungus. During his trance the Tungus shaman is believed to understand the language of all nature....
Very often this secret language is actually the "animal language" or originates in animal cries. In South America the neophyte must learn, during his initiation period, to imitate the voices of animals. The same is true of North America. The Pomo and the Menomini shamans, among others, imitate bird songs. During séances among the Yakut, the Yukagir, the Chukchee, the Goldi, the Eskimo, and others, wild animal cries and bird calls are heard....
Many words used during the séance have their origin in the cries of birds or other animals.... "Magic" and "song"—especially song like that of birds—are frequently expressed by the same term. The Germanic word for magic formula is _galdr_ , derived from the verb _galan_ , "to sing," a term applied especially to bird calls.
We will later explore at length specific instances of this affinity between language and the animate landscape as it is embodied not only in myths and magical practices but in the everyday discourse of several contemporary indigenous tribes. Here it is enough to mention that Merleau-Ponty's view of language as a thoroughly incarnate medium, of speech as rhythm and expressive gesture, and hence of spoken words and phrases as active sensuous presences afoot in the material landscape (rather than as ideal forms that represent, but are not a part of, the sensuous world)—goes a long way toward helping us understand the primacy of language and word magic in native rituals of transformation, metamorphosis, and healing. _Only if words are felt, bodily presences, like echoes or waterfalls, can we understand the power of spoken language to influence, alter, and transform the perceptual world_. As this is expressed in a Modoc song:
_I_
_the song_
_I walk here_ 24
To neglect this dimension—to overlook the power that words or spoken phrases have to influence the body, and hence to modulate our sensory experience of the world around us—is to render even the most mundane, communicative capacity of language incomprehensible.
WE MAY VERY BRIEFLY SUMMARIZE THE GENERAL RESULTS OF Merleau-Ponty's phenomenological investigations, or at least our own interpretation of those results, as follows: (1) The event of perception, experientially considered, is an inherently interactive, _participatory_ event, a reciprocal interplay between the perceiver and the perceived. (2) Perceived things are encountered by the perceiving body as animate, living powers that actively draw us into relation. Our spontaneous, pre-conceptual experience yields no evidence for a dualistic division between animate and "inanimate" phenomena, only for relative distinctions between diverse forms of animateness. (3) The perceptual reciprocity between our sensing bodies and the animate, expressive landscape both engenders and supports our more conscious, linguistic reciprocity with others. The complex interchange that we call "language" is rooted in the non-verbal exchange always already going on between our own flesh and the flesh of the world. (4) Human languages, then, are informed not only by the structures of the human body and the human community, but by the evocative shapes and patterns of the more-than-human terrain. Experientially considered, language is no more the special property of the human organism than it is an expression of the animate earth that enfolds us.
Such, at any rate, are the sort of descriptions at which we arrive when we carefully attend to perception and to language as we directly experience them.
Here, however, this philosophy encounters an impasse that threatens to dissipate its conclusions and to invalidate all its efforts. Specifically, if sensory perception is inherently participatory, and if, as Merleau-Ponty has maintained, perception (broadly considered) is the inescapable source of all experience, how can we possibly account for the apparent absence of participation in the modern world? "What right have I," asks Merleau-Ponty, "to call 'immediate' this original that can be forgotten to such an extent?" If our primordial experience is inherently animistic, if our "immediate" awareness discloses a field of phenomena that are all potentially animate and expressive, how can we ever account for the _loss_ of such animateness from the world around us? How can we account for our culture's experience of other animals as senseless automata, or of trees as purely passive fodder for lumber mills? If perception, in its depths, is wholly participatory, how could we ever have broken out of those depths into the inert and determinate world we now commonly perceive?
We may suspect, at first, that the apparent loss of participation has something to do with language. For language, although it is rooted in perception, nevertheless has a profound capacity to turn back upon, and influence, our sensorial experience. While the reciprocity of perception engenders the more explicit reciprocity of speech and language, perception always remains vulnerable to the decisive influence of language, as a mother remains especially sensitive to the actions of her child. It was this influence that led the American linguist Edward Sapir to formulate his hypothesis of linguistic determination, suggesting that one's perception is largely determined by the language that one speaks:
We see and hear and otherwise experience very largely as we do because the language habits of our community predispose certain choices of interpretation.
Certainly, the perceptual style of any community is both reflected in, and profoundly shaped by, the common language of the community. Yet the influence of language alone can hardly explain the shift from a participatory to a nonparticipatory world. Indeed, if we accept the phenomenological position sketched at length in this chapter, then the turn toward language for a solution can only confront us with a problem analogous to that which meets us with regard to perception. If human discourse is experienced by indigenous, oral peoples to be participant with the speech of birds, of wolves, and even of the wind, how could it ever have become severed from that vaster life? How could we ever have become so _deaf_ to these other voices that nonhuman nature now seems to stand mute and dumb, devoid of any meaning besides that which we choose to give it?
If perception, in its depths, is truly participatory, why do we not experience the rest of the world as animate and alive? If our own language is truly dependent upon the existence of other, nonhuman voices, why do we now experience language as an exclusively human property or possession? These two questions are in fact the same query asked from two different angles. Moreover, this query is the very same that arose at the end of the first chapter, the same that I there posed with regard to the felt shift in my own experience of nonhuman nature upon returning to the West from my sojourn in rural Asia. The question, however, is now set in a more methodic context; it is backed up by a whole tradition of philosophical inquiry. It should now be evident, as well, that the question has more than a purely personal relevance. Nonhuman nature seems to have withdrawn from both our speaking and our senses. What event could have precipitated this double withdrawal, constricting our ways of speaking even as it muffled our ears and set a veil before our eyes?
# **4**
# **_Animism and the Alphabet_**
Lifting a brush, a burin, a pen, or a stylus
is like releasing a bite or lifting a claw.
–GARY SNYDER
THE QUESTION REGARDING THE ORIGINS OF THE ECOLOGICAL crisis, or of modern civilization's evident disregard for the needs of the natural world, has already provoked various responses from philosophers. There are those who suggest that a generally exploitative relation to the rest of nature is part and parcel of being human, and hence that the human species has from the start been at war with other organisms and the earth. Others, however, have come to recognize that long-established indigenous cultures often display a remarkable solidarity with the lands that they inhabit, as well as a basic respect, or even reverence, for the other species that inhabit those lands. Such cultures, much smaller in scale (and far less centralized) than modern Western civilization, seem to have maintained a relatively homeostatic or equilibrial relation with their local ecologies for vast periods of time, deriving their necessary sustenance from the land without seriously disrupting the ability of the earth to replenish itself. The fecundity and flourishing diversity of the North American continent led the earliest European explorers to speak of this terrain as a primeval and unsettled wilderness—yet this continent had been continuously inhabited by human cultures for at least ten thousand years. That indigenous peoples can have gathered, hunted, fished, and settled these lands for such a tremendous span of time without severely degrading the continent's wild integrity readily confounds the notion that humans are innately bound to ravage their earthly surroundings. In a few centuries of European settlement, however, much of the native abundance of this continent has been lost—its broad animal populations decimated, its many-voiced forests overcut and its prairies overgrazed, its rich soils depleted, its tumbling clear waters now undrinkable.
European civilization's neglect of the natural world and its needs has clearly been encouraged by a style of awareness that disparages sensorial reality, denigrating the visible and tangible order of things on behalf of some absolute source assumed to exist entirely beyond, or outside of, the bodily world. Some historians and philosophers have concluded that the Jewish and Christian traditions, with their otherworldly God, are primarily responsible for civilization's negligent attitude toward the environing earth. They cite, as evidence, the Hebraic God's injunction to humankind in Genesis: "Be fertile and increase, fill the earth and master it; and rule the fish of the sea, the birds of the sky, and all the living things that creep on earth."
Other thinkers, however, have turned toward the Greek origins of our philosophical tradition, in the Athens of Socrates and Plato, in their quest for the roots of our nature-disdain. A long line of recent philosophers, stretching from Friedrich Nietzsche down to the present, have attempted to demonstrate that Plato's philosophical derogation of the sensible and changing forms of the world—his claim that these are mere simulacra of eternal and pure ideas existing in a nonsensorial realm beyond the apparent world—contributed profoundly to civilization's distrust of bodily and sensorial experience, and to our consequent estrangement from the earthly world around us.
So the ancient Hebrews, on the one hand, and the ancient Greeks on the other, are variously taken to task for providing the mental context that would foster civilization's mistreatment of nonhuman nature. Each of these two ancient cultures seems to have sown the seeds of our contemporary estrangement—one seeming to establish the spiritual or religious ascendancy of humankind over nature, the other effecting a more philosophical or rational dissociation of the human intellect from the organic world. Long before the historical amalgamation of Hebraic religion and Hellenistic philosophy in the Christian New Testament, these two bodies of belief already shared—or seem to have shared—a similar intellectual distance from the nonhuman environment.
In every other respect these two traditions, each one originating out of its own specific antecedents, and in its own terrain and time, were vastly different. In every other respect, that is, but one: they were both, from the start, profoundly informed by writing. Indeed, they both made use of the strange and potent technology which we have come to call "the alphabet."
WRITING, LIKE HUMAN LANGUAGE, IS ENGENDERED NOT ONLY within the human community but between the human community and the animate landscape, born of the interplay and contact between the human and the more-than-human world. The earthly terrain in which we find ourselves, and upon which we depend for all our nourishment, is shot through with suggestive scrawls and traces, from the sinuous calligraphy of rivers winding across the land, inscribing arroyos and canyons into the parched earth of the desert, to the black slash burned by lightning into the trunk of an old elm. The swooping flight of birds is a kind of cursive script written on the wind; it is this script that was studied by the ancient "augurs," who could read therein the course of the future. Leaf-miner insects make strange hieroglyphic tabloids of the leaves they consume. Wolves urinate on specific stumps and stones to mark off their territory. And today you read these printed words as tribal hunters once read the tracks of deer, moose, and bear printed in the soil of the forest floor. Archaeological evidence suggests that for more than a million years the subsistence of humankind has depended upon the acuity of such hunters, upon their ability to read the traces—a bit of scat here, a broken twig there—of these animal Others. These letters I print across the page, the scratches and scrawls you now focus upon, trailing off across the white surface, are hardly different from the footprints of prey left in the snow. We read these traces with organs honed over millennia by our tribal ancestors, moving instinctively from one track to the next, picking up the trail afresh whenever it leaves off, hunting the _meaning_ , which would be the _meeting_ with the Other.
The multiform meanings of the Chinese word for writing, _wen_ , illustrate well this interpenetration of human and nonhuman scripts:
The word _wen_ signifies a conglomeration of marks, the simple symbol in writing. It applies to the veins in stones and wood, to constellations, represented by the strokes connecting the stars, to the tracks of birds and quadrapeds on the ground (Chinese tradition would have it that the observation of these tracks suggested the invention of writing), to tattoos and even, for example, to the designs that decorate the turtle's shell ("The turtle is wise," an ancient text says—gifted with magico-religious powers—"for it carries designs on its back"). The term _wen_ has designated, by extension, literature....
Our first writing, clearly, was our own tracks, our footprints, our handprints in mud or ash pressed upon the rock. Later, perhaps, we found that by copying the distinctive prints and scratches made by other animals we could gain a new power; here was a method of identifying with the other animal, taking on its expressive magic in order to learn of its whereabouts, to draw it near, to make it appear. Tracing the impression left by a deer's body in the snow, or transferring that outline onto the wall of the cave: these are ways of placing oneself in distant contact with the Other, whether to invoke its influence or to exert one's own. Perhaps by multiplying its images on the cavern wall we sought to ensure that the deer itself would multiply, be bountiful in the coming season....
All of the early writing systems of our species remain tied to the mysteries of a more-than-human world. The petroglyphs of pre-Columbian North America abound with images of prey animals, of rain clouds and lightning, of eagle and snake, of the paw prints of bear. On rocks, canyon walls, and caves these figures mingle with human shapes, or shapes part human and part Other (part insect, or owl, or elk.)
Some researchers assert that the picture writing of native North America is not yet "true" writing, even where the pictures are strung together sequentially—as they are, obviously, in many of the rock inscriptions (as well as in the calendrical "winter counts" of the Plains tribes). For there seems, as yet, no strict relation between image and utterance.
In a much more conventionalized pictographic system, like the Egyptian hieroglyphics (which first appeared during the First Dynasty, around 3000 B.C.E. and remained in use until the second century C.E.), stylized images of humans and human implements are still interspersed with those of plants, of various kinds of birds, as well as of serpents, felines, and other animals. Such pictographic systems, which were to be found as well in China as early as the fifteenth century B.C.E., and in Mesoamerica by the middle of the sixth century B.C.E., typically include characters that scholars have come to call "ideograms." An ideogram is often a pictorial character that refers not to the visible entity that it explicitly pictures but to some quality or other phenomenon readily associated with that entity. Thus—to invent a simple example—a stylized image of a jaguar with its feet off the ground might come to signify "speed." For the Chinese, even today, a stylized image of the sun and moon together signifies "brightness"; similarly, the word for "east" is invoked by a stylized image of the sun rising behind a tree.
The efficacy of these pictorially derived systems necessarily entails a shift of sensory participation away from the voices and gestures of the surrounding landscape toward our own human-made images. However, the glyphs which constitute the bulk of these ancient scripts continually remind the reading body of its inherence in a more-than-human field of meanings. As signatures not only of the human form but of other animals, trees, sun, moon, and landforms, they continually refer our senses beyond the strictly human sphere.
Yet even a host of pictograms and related ideograms will not suffice for certain terms that exist in the local discourse. Such terms may refer to phenomena that lack any precise visual association. Consider, for example, the English word "belief." How might we signify this term in a pictographic, or ideographic, manner? An image of a phantasmagorical monster, perhaps, or one of a person in prayer. Yet no such ideogram would communicate the term as readily and precisely as the simple image of a bumblebee, followed by the figure of a leaf. We could, that is, resort to a visual pun, to images of things that have nothing overtly to do with belief but which, when named in sequence, carry the same _sound_ as the spoken term "belief" ("bee-leaf"). And indeed, such pictographic puns, or _rebuses_ , came to be employed early on by scribes in ancient China and in Mesoamerica as well as in the Middle East, to record certain terms that were especially amorphous or resistant to visual representation. Thus, for instance, the Sumerian word _ti_ , which means "life," was written in cuneiform with the pictorial sign for "arrow," which in Sumerian is also called _ti_.
An important step has been taken here. With the rebus, a pictorial sign is used to directly invoke a particular sound of the human voice, rather than the outward reference of that sound. The rebus, with its focus upon the sound of a name rather than the thing named, inaugurated the distant possibility of a _phonetic_ script (from the Greek _phonein_ : "to sound"), one that would directly transcribe the sound of the speaking voice rather than its outward intent or meaning.
However, many factors impeded the generalization of the rebus principle, and thus prevented the development of a fully phonetic writing system. For example, a largely pictographic script can easily be utilized, for communicative purposes, by persons who speak very different dialects (and hence cannot understand one another's speech). The same image or ideogram, readily understood, would simply invoke a different sound in each dialect. Thus a pictographic script allows for commerce between neighboring and even distant linguistic communities—an advance that would be lost if rebuslike signs alone were employed to transcribe the spoken sounds of one community. (This factor helps explain why China, a vast society comprised of a multitude of distinct dialects, has never developed a fully phonetic script.)
Another factor inhibiting the development of a fully phonetic script was the often elite status of the scribes. Ideographic scripts must make use of a vast number of stylized glyphs or characters, since every term in the language must, at least in principle, have its own written character. (In 1716 a dictionary of Chinese—admittedly an extreme example—listed 40,545 written characters! Today a mere 8,000 characters are in use.) Complete knowledge of the pictographic system, therefore, could only be the province of a few highly trained individuals. Literacy, within such cultures, was in fact the literacy of a caste, or cult, whose sacred knowledge was often held in great esteem by the rest of society. It is unlikely that the scribes would willingly develop innovations that could simplify the new technology and so render literacy more accessible to the rest of the society, for this would surely lessen their own importance and status.
... it is clear that ancient writing was in the hands of a small literate elite, the scribes, who manifested great conservatism in the practice of their craft, and, so far from being interested in its simplification, often chose to demonstrate their virtuosity by a proliferation of signs and values....
Nevertheless, in the ancient Middle East the rebus principle was eventually generalized—probably by scribes working at a distance from the affluent and established centers of civilization—to cover all the common sounds of a given language. Thus, "syllabaries" appeared, wherein every basic sound-syllable of the language had its own conventional notation or written character (often rebuslike in origin). Such writing systems employed far fewer signs than the pictographic scripts from which they were derived, although the number of signs was still very much larger than the alphabetic script we now take for granted.
The innovation which gave rise to the alphabet was itself developed by Semitic scribes around 1500 B.C.E. It consisted in recognizing that almost every syllable of their language was composed of one or more silent consonantal elements plus an element of sounded breath—that which we would today call a vowel. The silent consonants provided, as it were, the bodily framework or shape through which the sounded breath must flow. The original Semitic _aleph-beth_ , then, established a character, or letter, for each of the consonants of the language. The vowels, the sounded breath that must be added to the written consonants in order to make them come alive and to speak, had to be chosen by the reader, who would vary the sounded breath according to the written context.
By this innovation, the _aleph-beth_ was able to greatly reduce the necessary number of characters for a written script to just twenty-two—a simple set of signs that could be readily practiced and learned in a brief period by anyone who had the chance, even by a young child. The utter simplicity of this technical innovation was such that the early Semitic _aleph-beth_ , in which were written down the various stories and histories that were later gathered into the Hebrew Bible, was adopted not only by the Hebrews but by the Phonecians (who presumably carried the new technology across the Mediterranean to Greece), the Aramaeans, the Greeks, the Romans, and indeed eventually gave rise (directly or indirectly) to virtually every alphabet known, including that which I am currently using to scribe these words.
With the advent of the _aleph-beth_ , a new distance opens between human culture and the rest of nature. To be sure, pictographic and ideographic writing already involved a displacement of our sensory participation from the depths of the animate environment to the flat surface of our walls, of clay tablets, of the sheet of papyrus. However, as we noted above, the written images themselves often related us back to the other animals and the environing earth. The pictographic glyph or character still referred, implicitly, to the animate phenomenon of which it was the static image; it was that worldly phenomenon, in turn, that provoked from us the sound of its name. _The sensible phenomenon and its spoken name were, in a sense, still participant with one another_ —the name a sort of emanation of the sensible entity. With the phonetic _aleph-beth_ , however, the written character no longer refers us to any sensible phenomenon out in the world, or even to the name of such a phenomenon (as with the rebus), but solely to a gesture to be made by the human mouth. There is a concerted shift of attention away from any outward or worldly reference of the pictorial image, away from the sensible phenomenon that had previously called forth the spoken utterance, to the shape of the utterance itself, now invoked directly by the written character. _A direct association is established between the pictorial sign and the vocal gesture, for the first time completely bypassing the thingpictured_. The evocative phenomena—the entities imaged—are no longer a necessary part of the equation. Human utterances are now elicited, directly, by human-made signs; _the larger, more-than-human life-world is no longer a part of the semiotic, no longer a necessary part of the system_.
Or is it? When we ponder the early Semitic _aleph-beth_ , we readily recognize its pictographic inheritance. _Aleph_ , the first letter, is written thus: _Aleph_ is also the ancient Hebrew word for "ox." The shape of the letter, we can see, was that of an ox's head with horns; turned over, it became our own letter _A_. The name of the Semitic letter _mem_ is also the Hebrew word for "water"; the letter, which later became our own letter _M_ , was drawn as a series of waves: . The letter _ayin_ , which also means "eye" in Hebrew, was drawn as a simple circle, the picture of an eye; it is this letter, made over into a vowel by the Greek scribes, that eventually became our letter O. The Hebrew letter _qoph_ , which is also the Hebrew term for "monkey," was drawn as a circle intersected by a long, dangling, tail . Our letter _Q_ retains a sense of this simple picture.
These are a few examples. By thus comparing the names of the letters with their various shapes, we discern that the letters of the early _aleph-beth_ are still implicitly tied to the more-than-human field of phenomena. But these ties to other animals, to natural elements like water and waves, and even to the body itself, are far more tenuous than in the earlier, predominantly nonphonetic scripts. These traces of sensible nature linger in the new script only as vestigial holdovers from the old—they are no longer necessary participants in the transfer of linguistic knowledge. The other animals, the plants, and the natural elements—sun, moon, stars, waves—are beginning to lose their own voices. In the Hebrew Genesis, the animals do not speak their own names to Adam; rather, they are _given_ their names by this first man. Language, for the Hebrews, was becoming a purely _human_ gift, a human power.
IT WAS ONLY, HOWEVER, WITH THE TRANSFER OF PHONETIC WRITING to Greece, and the consequent transformation of the Semitic _aleph-beth_ into the Greek "alphabet," that the progressive abstraction of linguistic meaning from the enveloping life-world reached a type of completion. The Greek scribes took on, with slight modifications, both the shapes of the Semitic letters and their Semitic names. Thus _aleph_ —the name of the first letter, and the Hebrew word for "ox"—became _alpha;_ _beth_ —the name of the second letter, as well as the word for "house"—became _beta; gimel_ —the third letter, and the word for "camel," became _gamma_ , etc. But while the Semitic names had older, nongrammatological meanings for those who spoke a Semitic tongue, the Greek versions of those names had no nongrammatological meaning whatsoever for the Greeks. That is, while the Semitic name for the letter was also the name of the sensorial entity commonly imaged by or associated with the letter, the Greek name had no sensorial reference at all. While the Semitic name had served as a reminder of the worldy origin of the letter, the Greek name served only to designate the human-made letter itself. The pictorial (or iconic) significance of many of the Semitic letters, which was memorialized in their spoken names, was now readily lost. The indebtedness of human language to the more-than-human perceptual field, an indebtedness preserved in the names and shapes of the Semitic letters, could now be entirely forgotten.
### The Rapper's Rhythm
"... I'm a lover of learning, and trees and open country won't teach me anything, whereas men in the town do." These words are pronounced by Socrates, the wise and legendary father of Western philosophy, early in the course of the _Phaedrus_ —surely one of the most eloquent and lyrical of the Platonic dialogues. Written by Socrates' most illustrious student, Plato, these words inscribe a new and curious assumption at the very beginning of the European philosophical tradition.
It is difficult to reconcile Socrates' assertion—that trees and the untamed country have nothing to teach—with the Greece that we have come to know through Homer's epic ballads. In the Homeric songs the natural landscape itself bears the omens and signs that instruct human beings in their endeavors; the gods speak directly ****through the patterns of clouds, waves, and the flight of birds. Zeus rouses storms, sends thunderclaps, dispatches eagles to swoop low over the heads of men, disrupting their gatherings. Athena herself may take the shape of a seahawk, or may stir a wind to fill a ship's sails. Proteus, "the ancient of the salt sea, who serves under Poseidon," can readily transform into any beast, or into a flaming fire, or into water itself. Indeed, the gods seem indistinguishable at times from the natural elements that display their power: Poseidon, "the blue-maned god who makes the islands tremble," is the very life and fury of the sea itself; Helios, "lord of high noon," is not distinct from the sun (the fiery sun here a willful intelligence able even to father children: Circe, the sorceress, is his daughter). Even "fair Dawn, with her spreading fingertips of rose," is a living power. Human events and emotions are not yet distinct from the shifting moods of the animate earth—an army's sense of relief is made palpable in a description of thick clouds dispersing from the land; Nestor's anguish is likened to the darkening of the sea before a gale; the inward release of Penelope's feelings on listening to news of her husband is described as the thawing of the high mountain snows by the warm spring winds, melting the frozen water into streams that cascade down the slopes—as though the natural landscape was the proper home of those emotions, or as though a common psyche moved between humans and clouds and trees. When Odysseus, half-drowned by Poseidon's wrath and nearly dashed to pieces on the rocky coast of Phaiákia, spies the mouth of a calm river between the cliffs, he prays directly to the spirit of that river to have mercy and offer him shelter—and straightaway the tide shifts, and the river draws him into safety. Here, then, is a land that is everywhere alive and awake, animated by a multitude of capricious but willful forces, at times vengeful and at other times tender, yet always in some sense responsive to human situations. The diverse forms of the earth still speak and offer guidance to humankind, albeit in gestures that we cannot always directly understand. 17
This participatory and animate earth contrasts vividly with the dismissive view of nature espoused by Socrates in the _Phaedrus_. To make sense of this contrast, it is necessary to realize that the Homeric epics, probably written down in the seventh century B.C.E., are essentially orally evolved creations, oral poems that had been sung and resung, shifting and complexifying, long before they were written down and thus frozen in the precise form in which we now know them. The Platonic dialogues, on the other hand, written in the first half of the fourth century B.C.E., are thoroughly lettered constructions, composed in a literate context by a manifestly literate author. And indeed they inscribe for the first time many of the mental patterns or thought styles that today we of literate culture take for granted.
The Greek alphabet was first invented—or, rather, adapted from the Semitic _aleph-beth_ —several centuries before Plato, probably during the eighth century B.C.E. The new technology did not spread rapidly through Greece; rather, it encountered remarkable resistance in the form of a highly developed and ritualized oral culture. That is, the traditions of prealphabetic Greece were actively preserved in numerous oral stories regularly recited and passed along from generation to generation by the Greek bards, or "rhapsodes." The chanted tales carried within their nested narratives much of the accumulated knowledge of the culture. Since they were not written down, they were never wholly fixed, but would shift incrementally with each telling to fit the circumstances or needs of a particular audience, gradually incorporating new practical knowledge while letting that which was obsolete fall away. The sung stories, along with the numerous ceremonies to which they were linked, were in a sense the living encyclopedias of the culture—carrying and preserving the collected knowledge and established customs of the community—and they themselves were preserved through constant repetition and ritual reenactment. There was thus little overt need for the new technology of reading and writing. According to literary historian Eric Havelock, for the first two or three centuries after its appearance in Greece, "[t]he alphabet was an interloper, lacking social standing and achieved use. The elite of society were all reciters and performers."
The alphabet, after all, had not here developed gradually, as it had across the Mediterranean, out of a series of earlier scripts, and there was thus no already existing context of related inscriptions and scribal practices for it to latch onto. Moreover, the oral techniques for preserving and transmitting knowledge, and the sensorial habits associated with those techniques, were, as we shall see, largely incompatible with the sensorial patterns demanded by alphabetic literacy.
In a culture as thoroughly and complexly oral as Greek culture in this period, the alphabet could take root only by allying itself, at first, with the oral tradition. Thus, the first large written texts to appear in Greece—namely, the _Iliad_ and the _Odyssey_ —are, paradoxially, "oral texts." That is, they are not written compositions, as had long been supposed, but rather alphabetic transcriptions of orally chanted poems. Homer, as an oral bard, or rhapsode (from the Greek _rhapsoidein_ , which meant "to stitch song together"), improvised the precise form of the poems by "stitching together" an oral tapestry from a vast fund of memorized epithets and formulaic phrases, embellishing and elaborating a cycle of stories that had already been variously improvised or "stitched together" by earlier bards since the Trojan War itself.
_We owe our recognition of the oral nature of the Homeric epics_ to the pioneering research undertaken by the Harvard classicist Milman Parry and his assistant Albert Lord, in the 1930s. Parry had noticed the existence of certain stock phrases—such as "the wine-dark sea," "there spoke clever Odysseus," or "when Dawn spread out her fingertips of rose"—that are continually repeated throughout the poems. Careful study revealed that the poems were composed almost entirely of such expressions (in the twenty-seven thousand hexameters there are twenty-nine thousand repetitions of phrases with two or more words). Moreover, Homer's choice of one particular epithet or formula rather than another seemed at times to be governed less by the exact meaning of the phrase than by the metrical exigencies of the line; the bard apparently called upon one specific formula after another in order to fit the driving meter of the chant, in a trance of rhythmic improvisation. This is not at all to minimize Homer's genius, but simply to indicate that his poetic brilliance was performative as much as creative—less the genius of an author writing a great novel than that of an inspired and eloquent rap artist.
The reliance of the Homeric texts upon repeated verbal formulas and stock epithets—this massive dependence upon that which we today refer to, disparagingly, as "clichés"—offered Parry and subsequent researchers a first insight into the very different world of a European culture without writing. In a literate society, like our own, any verbal discovery or realization can be preserved simply by being written down. Whenever we wish to know how to accomplish a certain task, we need only find the book wherein that knowledge is inscribed. When we wish to ponder a particular historical encounter, we simply locate the text wherein that encounter is recorded. Oral cultures, however, lacking the fixed and permanent record that we have come to count on, can preserve verbal knowledge only by constantly repeating it. Practical knowledge must be embedded in spoken formulas that can be easily recalled—in prayers and proverbs, in continually recited legends and mythic stories. The rhythmic nature of many such spoken formulas is a function of their mnemonic value; such pulsed phrases are much easier for the pulsing, breathing body to assimilate and later recall than the strictly prosaic statements that appear only after the advent of literacy. (For example, the phrase "an apple a day keeps the doctor away" is vastly easier to remember than the phrase "one should always eat fruit in order to stay healthy"). The discourse of nonwriting cultures is, of necessity, largely comprised of such formulaic and rhythmic phrases, which readily spring to the tongue in appropriate situations.
Parry's insights regarding the orally composed nature of the Homeric epics remained somewhat speculative until he was able to meet and observe representatives of an actual bardic tradition still in existence in Eastern Europe. In the 1930s, Parry and his student Albert Lord traveled to Serbia, where they befriended a number of nonliterate Slavic singers whose craft was still rooted in the ancient oral traditions of the Balkans. These singers (or _guslars_ ) chanted their long stories—for which there existed no written texts—in coffeehouses and at weddings, accompanying themselves on a simple stringed instrument called a _gusla_. Parry and Lord recorded many of these epic songs on early phonographic disks, and so were later able to compare the metrical structure of these chanted stories with the structure and phrasing of the Homeric poems. The parallels were clear and remarkable.
When one hears the Southern Slavs sing their tales he has the overwhelming feeling that, in some way, he is hearing Homer. This is no mere sentimental feeling that comes from his seeing a way of life and a cast of thought that are strange to him.... When the hearer looks closely to see why he should seem to be hearing Homer he finds precise reasons: he is ever hearing the same ideas that Homer expresses, and is hearing them expressed in phrases which are rhythmically the same, and which are grouped in the same order.
Parry carefully documented these strong parallels, and after his early death his research into oral modes of composition was carried on by Albert Lord. Among other things, Lord's research indicated that learning to read and write thoroughly disabled the oral poet, ruining his capacity for oral improvisation.
WHEN THE HOMERIC EPICS WERE RECORDED IN WRITING, THEN THE art of the rhapsodes began to lose its preservative and instructive function. The knowledge embedded in the epic stories and myths was now captured for the first time in a visible and fixed form, which could be returned to, examined, and even questioned. Indeed, it was only then, under the slowly spreading influence of alphabetic technology, that "language" was beginning to separate itself from the animate flux of the world, and so becoming a ponderable presence in its own right.
It is only as language is written down that it becomes possible to think about it. The acoustic medium, being incapable of visualization, did not achieve recognition as a phenomenon wholly separable from the person who used it. But in the alphabetized document the medium became objectified. There it was, reproduced perfectly in the alphabet... no longer just a function of "me" the speaker but a document with an independent existence.
The scribe, or author, could now begin to dialogue with his own visible inscriptions, viewing and responding to his own words even as he wrote them down. _A new power of reflexivity was thus coming into existence, borne by the relation between the scribe and his scripted text_.
We can witness the gradual spread of this new power in the written fragments of the pre-Socratic philosophers of the sixth and fifth centuries B.C.E. These thinkers are still under the sway of the oral-poetic mode of discourse—their teachings are commonly couched in an aphoristic or poetic form, and their attention is still turned toward the sensuous terrain that surrounds them. Nevertheless, they seem to stand at a new distance from the natural order, their thoughts inhabiting a different mode of temporality from the flux of nature, which they now question and strive to understand. The written fragments of Heraclitus or of Empedocles give evidence of a radically new, literate reflection combined with a more traditional, oral preoccupation with a sensuous nature still felt to be mysteriously animate and alive, filled with immanent powers. In the words of the pre-Socratic philosopher Thales, "all things are full of gods."
It was not until the early fourth century B.C.E. that such numinous powers, or gods, were largely expelled from the natural surroundings. For it was only at this time that alphabetic literacy became a collective reality in Greece. Indeed, it was only during Plato's lifetime (428–348 B.C.E.) that the alphabet was incorporated within Athenian life to the extent that we might truthfully speak of Athenian Greece as a "literate" culture:
Plato, in the early fourth century B.C., stands on the threshold between the oral and written cultures of Greece. The earliest epigraphic and iconographic indications of young boys being taught to write date from Plato's childhood. In his day, people had already been reciting Homer from the text for centuries. But the art of writing was still primarily a handicraft.... In the fifth century B.C., craftsmen began to acquire the art of carving or engraving letters of the alphabet. But writing was still not a part of recognized instruction: the most a person was expected to be able to write and spell was his own name....
Plato was teaching, then, precisely at the moment when the new technology of reading and writing was shedding its specialized "craft" status and finally spreading, by means of the Greek curriculum, into the culture at large. The significance of this conjunction has not been well recognized by Western philosophers, all of whom stand—to a greater or lesser extent—within Plato's lineage. Plato, or rather the association between the literate Plato and his mostly nonliterate teacher Socrates (469?–399 B.C.E.), may be recognized as the hinge on which the sensuous, mimetic, profoundly embodied style of consciousness proper to orality gave way to the more detached, abstract mode of thinking engendered by alphabetic literacy. Indeed, it was Plato who carefully developed and brought to term the collective thought-structures appropriate to the new technology.
### An Eternity of Unchanging Ideas
Although Socrates himself may have been able to write little more than his own name, he made brilliant use of the new reflexive capacity introduced by the alphabet. Eric Havelock has suggested that the famed "Socratic dialectic"—which, in its simplest form, consisted in asking a speaker to explain what he has said—was primarily a method for disrupting the mimetic thought patterns of oral culture. The speaker's original statement, if it concerned important matters of morality and social custom, would necessarily have been a memorized formula, a poetic or proverbial phrase, which presented a vivid example of the matter being discussed. By asking the speaker to explain himself or to repeat his statement in different terms, Socrates forced his interlocutors to separate themselves, for the first time, from their own words—to separate themselves, that is, from the phrases and formulas that had become habitual through the constant repetition of traditional teaching stories. Prior to this moment, spoken discourse was inseparable from the endlessly repeated stories, legends, and myths that provided many of the spoken phrases one needed in one's daily actions and interactions. To speak was to live within a storied universe, and thus to feel one's closeness to those protagonists and ancestral heroes whose words often seemed to speak through one's own mouth. Such, as we have said, is the way culture preserves itself in the absence of written records. But Socrates interrupted all this. By continually asking his interlocutors to repeat and explain what they had said in other words, by getting them thus to listen to and ponder their own speaking, Socrates stunned his listeners out of the mnemonic trance demanded by orality, and hence out of the sensuous, storied realm to which they were accustomed. Small wonder that some Athenians complained that Socrates' conversation had the numbing effect of a stingray's electric shock.
Prior to the spread of writing, ethical qualities like "virtue," "justice," and "temperance" were thoroughly entwined with the specific situations in which those qualities were exhibited. The terms for such qualities were oral utterances called forth by particular social situations; they had no apparent existence independent of those situations. As utterances, they slipped back into the silence immediately after they were spoken; they had no permanent presence to the senses. "Justice" and "temperance" were thus experienced as living occurrences, as _events_. Arising in specific situations, they were inseparable from the particular persons or actions that momentarily embodied them.
Yet as soon as such utterances were recorded in writing, they acquired an autonomy and a permanence hitherto unknown. Once written down, "virtue" was seen to have an unchanging, visible form independent of the speaker—and independent as well of the corporeal situations and individuals that exhibited it.
Socrates clearly aligned his method with this shift in the perceptual field. Whenever, in Plato's dialogues, Socrates asks his interlocutor to give an account of what "virtue," or "justice," or "courage" actually is, questioning them regarding the real meaning of the qualitative terms they unthinkingly employ in their speaking, they confidently reply by recounting particular instances of the quality under consideration, enumerating specific examples of "justice," yet never defining "justice" itself. When Socrates invites Meno to say what "virtue" is, Meno readily enumerates so many different instances or embodiments of virtue that Socrates retorts sardonically: "I seem to be in luck. I only asked you for one thing, virtue, but you have given me a whole swarm of virtues." In keeping with older, oral modes of discourse, Socrates' fellow Athenians cannot abstract these spoken qualities from the lived situations that seem to exemplify these terms and call them forth. Socrates, however, has little interest in these multiple embodiments of "virtue," except in so far as they all partake of some common, unchanging element, which he would like to abstract and ponder on its own. In every case Socrates attempts to induce a reflection upon the quality as it exists in itself, independent of particular circumstances. The specific embodiments of "justice" that we may encounter in the material world are necessarily variable and fleeting; genuine knowledge, claims Socrates, must be of what is eternal and unchanging.
Socrates, then, is clearly convinced that there is a fixed, unchanging essence of "justice" that unites all the just instances, as there is an eternal essence of "virtue," of "beauty," of "goodness," "courage," and all the rest. Yet Socrates' conviction would not be possible without the alphabet. For only when a qualitative term is written down does it become ponderable as a fixed form independent of both the speakers and of situations.
Not all writing systems foster this thorough abstraction of a spoken quality from its embeddedness in corporeal situations. The ideographic script of China, as we have seen, still retains pictorial ties to the phenomenal world of sensory experience. Thus, the Chinese ideograph for "red" is itself a juxtaposition of lived examples; it is composed of abbreviated pictorial images of a rose, a cherry, iron rust, and a flamingo. And indeed, according to some observers, if one asks a cultured person in China to explain a general quality like "red," or "loyalty," or "happiness," she will likely reply by describing various instances or examples of that quality, much like Socrates' interlocutors. It was not writing per se, but phonetic writing, and the Greek alphabet in particular, that enabled the abstraction of previously ephemeral qualities like "goodness" and "justice" from their inherence in situations, promoting them to a new realm independent of the flux of ordinary experience. For the Greek alphabet had effectively severed all ties between the written letters and the sensible world from which they were derived; it was the first writing system able to render almost any human utterance in a fixed and lasting form.
While Socrates focused his teaching on the moral qualities, his disciple Plato recognized that not just ephemeral qualities but _all_ general terms, from "table" to "cloud," could now be pondered as eternal, unchanging forms. In retrospect, we can see that the alphabet had indeed granted a new autonomy and permanence to all such terms. Besides the various meandering rivers, for instance, that one could view, or wade through, in the sensible world, there was also the singular notion "river," which now had its own visibility; "river" itself could now be pondered apart from all those material rivers that were liable to change their course or to dry up from one season to the next. For Plato, as for his teacher, genuine knowledge must be of what is unchanging and eternal—there can be no "true" knowledge of a particular river, but only of the pure Idea (or _eidos_ ) "river." That Plato often used the Greek term _eidos_ (which meant "visible shape or form") to refer to such unchanging essences is itself, I believe, an indication of the affinity between these eternal essences and the unchanging, visible shapes of the alphabet.
_For the letters of the alphabet, like the Platonic Ideas, do not exist in the world of ordinary vision_. The letters, and the written words that they present, are not subject to the flux of growth and decay, to the perturbations and cyclical changes common to other visible things; they seem to hover, as it were, in another, strangely timeless dimension. Further, the letters defer and dissimulate their common visibility, each one dissolving into sound even as we look at it, trading our eyes for our ears, so that we seem not to be _seeing_ so much as _hearing_ something. Alphabetic writing deflects our attention from its visible aspect, effectively vanishing behind the current of human speech that it provokes.
As we have already seen, the process of learning to read and to write with the alphabet engenders a new, profoundly reflexive, sense of self. The capacity to view and even to dialogue with one's own words after writing them down, or even in the process of writing them down, enables a new sense of autonomy and independence from others, and even from the sensuous surroundings that had earlier been one's constant interlocutor. The fact that one's scripted words can be returned to and pondered at any time that one chooses, regardless of when, or in what situation, they were first recorded, grants a timeless quality to this new reflective self, a sense of the relative independence of one's verbal, speaking self from the breathing body with its shifting needs. The literate self cannot help but feel its own transcendence and timelessness relative to the fleeting world of corporeal experience.
This new, seemingly autonomous, reflective awareness is called, by Socrates, the _psyche_ , a term he thus twists from its earlier, Homeric significance as the invisible breath that animates the living body and that remains, as kind of wraith or ghost, after the body's death. (The term _psychê_ was derived from an older Greek term, _psychein_ , which meant "to breathe" or "to blow".) For Plato, as for Socrates, the _psychê_ is now that aspect of oneself that is refined and strengthened by turning away from the ordinary sensory world in order to contemplate the intelligible Ideas, the pure and eternal forms that, alone, truly exist. The Socratic-Platonic _psychê_ , in other words, is none other than the literate intellect, that part of the self that is born and strengthened in relation to the written letters.
PLATO HIMSELF EFFECTS A POWERFUL CRITIQUE OF THE INFLUENCE of writing in the _Phaedrus_ , that dialogue from which I quoted earlier in this chapter. In the course of that dialogue, Socrates relates to the young Phaedrus a curious legend regarding the Egyptian king Thamus. According to this story, Thamus was approached directly by the god Thoth—the divine inventor of geometry, mathematics, astronomy, and writing—who offers writing as a gift to the king so that Thamus may offer it, in turn, to the Egyptian people. But Thamus, after considering both the beneficent and the baneful aspects of the god's inventions, concludes that his people will be much better off _without_ writing, and so he refuses the gift. Against Thoth's claim that writing will make people wiser and improve their memory, the king asserts that the very opposite is the case:
If men learn this, it will implant forgetfulness in their souls; they will cease to exercise memory because they rely on that which is written, calling things to remembrance no longer from within themselves, but by means of external marks.
Moreover—according to the king—spoken teachings, once written down, easily find their way into the hands of those who will misunderstand those teachings while nevertheless thinking that they understand them. Thus, the written letters bring not wisdom but only "the conceit of wisdom," making men seem to know much when in fact they know little.
Plato's Socrates clearly agrees with the king's judgment, and it is evident that Plato wishes the reader to take these criticisms of writing quite seriously. Later in the same dialogue we read that "a written discourse on any subject is bound to contain much that is fanciful," and that in any case "nothing that has ever been written whether in verse or prose merits much serious attention." Certainly, it is strange to read such strong remarks against writing from a thinker whose numerous written texts are among the most widely distributed and worshipfully read in the Western world. Here is Plato, from whom virtually all Western philosophers draw their literary ancestry, disparaging writing as nothing more than a pastime! What are we to make of these statements?
Such doubts about the alphabet, and such assertions regarding its potentially debilitating effects, must have been legion in Athens just before or during the time that Plato was writing. It is remarkable that Plato held to such criticisms despite the fact that he was an inveterate participant in the alphabetic universe. Given his multiple and diverse writings, which constitute what is probably the first large corpus of prose by a single author in the history of the alphabet, it seems clear that Plato did not intend his own criticisms to dissuade his students and readers from writing, or from reading him further. Rather, it is as though he meant to build into the very body of his writings a caution that they not be given too much weight. Not because he was uncertain about the genuine and serious worth of his philosophy, but simply because he had strong reservations about the written word and its ability to convey the full meaning of a philosophy that was as much a practice—involving direct, personal interaction and instruction—as it was a set of static formulations and reflections. Writing, according to Socrates, can at best serve as a _reminder_ to a reader who already knows those things that have been written. It is possible that Plato wrote his various dialogues to serve just such a restricted function; to act as reminders, for the students of his academy, of the methods and insights that they first learned in direct, face-to-face dialogue with their teacher.
Nevertheless Plato, despite his cautions, did not recognize the extent to which the very content of his teaching—with its dependence upon the twin notions of a purely rational _psychê_ and a realm of eternal, unchanging Ideas—was already deeply under the influence of alphabetic writing. In the early fourth century B.C.E., when literacy was gradually spreading throughout Athenian society, it was certainly possible to witness the impact that writing was having upon the dissemination of particular teachings. An astute observer might discern as well the debilitating effects of writing upon the collective practice of memory, as what had previously been accomplished through the memorized repetition of ritual poems, songs, and stories was transferred to an external and fixed artifact. But it was hardly possible to discern the pervasive influence of letters upon patterns of perception and contemplation in general. Similarly, today we are simply unable to discern with any clarity the manner in which our own perceptions and thoughts are being shifted by our sensory involvement with electronic technologies, since any thinking that seeks to discern such a shift is itself subject to the very effect that it strives to thematize. Nevertheless, we may be sure that the shapes of our consciousness _are_ shifting in tandem with the technologies that engage our senses—much as we can now begin to discern, in retrospect, how the distinctive shape of Western philosophy was born of the meeting between the human senses and the alphabet in ancient Greece.
### Of Tongues in Trees
Socrates' critique of writing, in the _Phaedrus_ , is occasioned by a written text carried by the young Phaedrus at the very beginning of the dialogue, when Socrates encounters him on his way out of the city. Phaedrus has just heard a friend of his, Lysias, declaiming a newly written speech on the topic of love; impressed by Lysias's speech, Phaedrus has obtained a copy of the speech and is going for a walk outside the city walls to ponder the text at his leisure. Socrates, always eager for philosophical discourse, agrees to accompany Phaedrus into the open country where they may together consider Lysias's text and discuss its merits. It is summer; the two men walk along the Ilissus River, wade across it, then settle on the grass in the shade of a tall, spreading plane tree. Socrates compliments Phaedrus for leading them to this pleasant glen, and Phaedrus replies, with some incredulity, that Socrates seems wholly a stranger to the country, like one who had hardly ever set foot outside the city walls. It is then that Socrates explains himself: "You must forgive me, dear friend. I'm a lover of learning, and trees and open country won't teach me anything, whereas men in the town do."
We have already seen how peculiar this statement seems in relation to the world of the Homeric poems. How much more bizarre Socrates' words would seem to the members of an oral society still less exposed to the influence of literate traders than was Homeric Greece—to a culture, in other words, whose gods were not yet as anthropomorphic as even frothy-haired Poseidon and eruptive Hephaestus. The claim that "trees and open country won't teach anything" would have scant coherence within an indigenous hunting community, for the simple reason that such communities necessarily take their most profound teachings or instructions directly from the more-than-human earth. Whether among the Plains Indians of North America, the bushmen of the Kalahari Desert, or the Pintupi of the Australian outback, the elders and "persons of high degree" within such hunting communities continually defer to the animate powers of the surrounding landscape—to those nonhuman powers from which they themselves draw their deepest inspiration.
When a young person within such a culture is chosen, by whatever circumstance, to become a seer or shaman for the community, he or she may be trained by an elder seer within the tribe. Yet the most learned and powerful shaman will be one who has first learned his or her skills directly from the land itself—from a specific animal or plant, from a river or a storm—during a prolonged sojourn out beyond the boundaries of the human society. Indeed, among many of the tribes once indigenous to North America, a boy could gain the insight necessary to enter the society of grown men only by undertaking a solitary quest for vision—only by rendering himself vulnerable to the wild forces of the land and, if need be, crying to those forces for a vision. The initiatory "Walkabout" undertaken by Aboriginal Australians is again just such an act whereby oral peoples turn toward the more-than-human earth for the teachings that must vitalize and sustain the human community.
In indigenous, oral cultures, nature itself is articulate; it _speaks_. The human voice in an oral culture is always to some extent participant with the voices of wolves, wind, and waves—participant, that is, with the encompassing discourse of an animate earth. There is no element of the landscape that is definitively void of expressive resonance and power: any movement may be a gesture, any sound may be a voice, a meaningful utterance.
Socrates' claim that trees have nothing to teach is a vivid indicator of the extent to which the human senses in Athens had already withdrawn from direct participation with the natural landscape. To directly perceive any phenomenon is to enter into relation with it, to feel oneself in a living interaction with another being. To define the phenomenon as an inert object, to deny the ability of a tree to inform and even instruct one's awareness, is to have turned one's senses away from that phenomenon. It is to ponder the tree from outside of its world, or, rather, from outside of the world in which both oneself and the tree are active participants.
Yet even here Plato seems to waver and vacillate. Indeed, just as the _Phaedrus_ is the prime locus of Plato's apparent ambivalence with regard to his own practice of writing, so it is also the locus of a profound ambivalence with regard to nature, or to the expressive power of the natural world. Although the dialogue opens with Socrates' disparagement of trees and the open countryside, it is significant that the dialogue itself takes place in the midst of that very countryside. Unlike the other Platonic dialogues, the _Phaedrus_ alone occurs outside the walls of the city, out beyond the laws and formalities that enclose and isolate the human community from the more-than-human earth. Socrates and Phaedrus have themselves embarked, as it were, on a kind of vision quest, stepping outside the city norms in order to test their citified knowledge against the older knowings embedded in the land. Plato is here, in a sense, putting philosophy itself to the test, by opening and exposing it to the nonhuman powers that for so long had compelled the awe and attention of humankind. In direct contrast to _The Republic_ , in which Plato vilifies the ancient gods and effectively banishes the oral poets and storytellers from the Utopian city that he envisions, in the _Phaedrus_ , Plato brings philosophy itself outside the city, there to confront and come to terms with the older, oral ways of knowing which, although they may be banished from the city, nevertheless still dwell in the surrounding countryside. It is only outside the city walls that Plato will allow himself to question and critique the practice of writing to which he (and all later philosophy) is indissolubly tied. And it is only outside those walls that he will allow himself to fully acknowledge and offer respect to the oral, animistic universe that is on the wane.
Thus, shortly after his assertion that trees can teach him nothing, Socrates allows himself to be goaded into making an impromptu speech by an oath that Phaedrus swears upon the spirit of the very tree beneath which they sit Trees, it would seem, still retain a modicum of efficacious power. Later in the dialogue Socrates himself will remind Phaedrus that, according to tradition, "the first prophetic utterances came from an oak tree."
Not just trees but animals, too, have—in the _Phaedrus_ —magical powers. Socrates initiates the discussion of writing by speculating that the cicadas chirping and "conversing with one another" in the tree overhead are probably observing the two of them as well; he maintains that the cicadas will intercede with the Muses on their behalf if he and Phaedrus continue to converse on philosophical matters. And he proceeds to recount a story that describes how the cicadas, who were originally persons, were transformed into their present form:
The story is that once upon a time these creatures were men—men of an age before there were any Muses—and that when the latter came into the world, and music made its appearance, some of the people of those days were so thrilled with pleasure that they went on singing and quite forgot to eat and drink until they actually died without noticing it. From them in due course sprang the race of cicadas, to which the Muses have granted the boon of needing no sustenance right from their birth, but of singing from the very first, without food or drink, until the day of their death, after which they go and report to the Muses how they severally are paid honor among mankind and by whom....
Any student of indigenous, oral cultures will hear a ring of familiarity in this tale. The story of the cicadas is identical in its character to the stories of the "Distant Time" told today by the Koyukon Indians of Alaska, identical to stories from that mysterious realm "long ago, in the future" which are told by the Inuit (or eastern Eskimo), or to the "Dreamtime" stories told by Aboriginal Australians. We may recall, in this context, these Inuit words quoted toward the end of the last chapter: "In the very earliest time, when both people and animals lived on earth, a person could become an animal if he wanted to, and an animal could become a human being...." Here is a typical Distant Time story told by the Koyukon:
When the burbot [ling cod] was human, he decided to leave the land and become a water animal. So he started down the bank, taking a piece of bear fat with him. But the other animal people wanted him to stay and tried to hold him back, stretching him all out of shape in the process. This is why the burbot has such a long, stretched-out body, and why its liver is rich and oily like the bear fat its ancestor carried to the water long ago.
Like all oral stories of the Distant Time or Dreamtime, Socrates' myth of the cicadas is a functional myth; it serves to explain certain observed characteristics of the cicadas, like their endless humming and buzzing, and their apparent lack of any need for nourishment ("when music appeared, some of the people of those days were so thrilled with pleasure that they went on singing, and quite forgot to eat and drink"). Anthropologists have tended to view such stories from the Dreamtime or Distant Time as confused attempts at causal explanation by the primitive mind. Here, however, in the light of our discussion regarding orality and literacy, such stories can be seen to serve a far more practical function.
Without a versatile writing system, there is simply no way to preserve, in any fixed, external medium, the accumulated knowledge regarding particular plants (including where to find them, which parts of them are edible, which poisonous, how they are best prepared, what ailments they may cure or exacerbate), and regarding specific animals (how to recognize them, what they eat, how best to track or hunt them), or even regarding the land itself (how best to orient oneself in the surrounding terrain, what landforms to avoid, where to find water or fuel). Such practical knowledge must be preserved, then, in spoken formulations that can be easily remembered, modified when new facts are learned, and retold from generation to generation. Yet not all verbal formulations are amenable to simple recall—most verbal forms that we are conversant with today are dependent upon a context of writing. To us, for instance, a simple mental list of the known characteristics of a particular plant or animal would seem the easiest and most obvious formulation. Yet such lists have no value in an oral culture; without a visible counterpart that can be brought to mind and scanned by the mind's eye, spoken lists cannot be readily recalled and repeated. Without writing, knowledge of the diverse properties of particular animals, plants, and places can be preserved only by being woven into _stories_ , into vital tales wherein the specific characteristics of the plant are made evident through a narrated series of events and interactions. Stories, like rhymed poems or songs, readily incorporate themselves into our felt experience; the shifts of action echo and resonate our own encounters—in hearing or telling the story we vicariously _live_ it, and the travails of its characters embed themselves into our own flesh. The sensuous, breathing body is, as we have seen, a dynamic, ever-unfolding form, more a process than a fixed or unchanging object. As such, it cannot readily appropriate inert "facts" or "data" (static nuggets of "information" abstracted from the lived situations in which they arise). Yet the living body can easily assimilate other dynamic or eventful processes, like the unfolding of a story, appropriating each episode or event as a variation of its own unfolding.
And the more lively the story—the more vital or stirring the encounters within it—the more readily it will be incorporated. Oral memorization calls for lively, dynamic, often violent, characters and encounters. If the story carries knowledge about a particular plant or natural element, then that entity will often be cast, like all of the other characters, in a fully animate form, capable of personlike adventures and experiences, susceptible to the kinds of setbacks or difficulties that we know from our own lives. In this manner the character or personality of a medicinal plant will be easily remembered, its poisonous attributes will be readily avoided, and the precise steps in its preparation will be evident from the sequence of events in the very legend that one chants while preparing it. One has only to recite the appropriate story, from the Distant Time, about a particular plant, animal, or element in order to recall the accumulated cultural knowledge regarding that entity and its relation to the human community.
In this light, that which we literates misconstrue as a naïve attempt at causal explanation may be recognized as a sophisticated mnemonic method whereby precise knowledge is preserved and passed along from generation to generation. The only causality proper to such stories is a kind of cyclical causality alien to modern thought, according to which persons may influence events in the enveloping natural order and yet are themselves continually under the influence of those very events. By invoking a dimension or a time when all entities were in human form, or when humans were in the shape of other animals and plants, these stories affirm human kinship with the multiple forms of the surrounding terrain. They thus indicate the respectful, mutual relations that must be maintained with natural phenomena, the reciprocity that must be practiced in relation to other animals, plants, and the land itself, in order to ensure one's own health and to preserve the well-being of the human community.
This facet of respectful consideration, and its attendant circular causality, is also present in Socrates' tale of the cicadas. By relating the tale to Phaedrus, Socrates indicates, although not without a sense of irony, the respect that is properly due to such insects, who might confer a boon upon the two of them in return. Later, indeed, Socrates will attribute his own loquacious eloquence in this dialogue to the inspiration of the cicadas, "those mouthpieces of the Muses."
It seems clear that in the _Phaedrus_ , Plato accords much more consideration to the oral-poetic universe, with its surplus of irrational, sensuous, and animistic powers, than he does in other dialogues. The _Phaedrus_ seems to attempt a reconciliation of the transcendent, bodiless world of eternal Ideas proposed in this and other dialogues with the passionate, feeling-toned world of natural magic that still lingered in the common language of his day. But this conciliatory affirmation of the animistic, sensuous universe is effected only within the context of a more subtle devaluation. This is most obviously evident in the allegory at the heart of the dialogue, wherein Socrates gives his own account of love, or "eros." According to Socrates, the divine madness of love is to be honored and praised, for it is love that can most powerfully awaken the soul from its slumber in the bodily world. The lover's soul is stirred by the sensuous beauty of the beloved into remembering, however faintly, the more pure, genuine beauty of the eternal, bodiless Ideas which it once knew. Thus reminded of its own transcendent nature, the previously dormant soul begins to sprout wings, and soon aspires to rise beyond this world of ceaseless "becoming" toward that changeless eternal realm beyond the stars:
It is there that true being dwells, without color or shape, that cannot be touched; reason alone, the soul's pilot, can behold it, and all true knowledge is knowledge thereof.
In this dialogue, then, the bodily desire for sensuous contact and communion with other bodies and with the bodily earth is honored, but only as an incitement or spur toward the more genuine union of the reasoning soul with the eternal forms of "justice," "temperance," "virtue," and the like, which—according to Plato—lie beyond the sensory world entirely.
We have seen that this affinity between the reasoning soul or _psychê_ and the changeless Ideas is inseparable from the relation between the new, literate intellect and the visible letters of the alphabet (which, although not outside of the sensory world, do present an entirely new and stable order of phenomena, relative to which all other phenomenal forms may come to seem remarkedly fleeting, ambiguous, and derivative). Just as Plato's apparent criticisms of alphabetic writing in the _Phaedrus_ take place within the context of a much broader espousal of the detached (or disembodied) reflection that writing engenders, so in the same dialogue his apparent affirmation of oral-animistic modes of experience is accomplished only in the context of a broader disparagement. The erotic, participatory world of the sensing body is conjured forth only to be subordinated to the incorporeal world toward which, according to Plato, it points. The literate intellect here certifies its dominion by claiming the sensuous life of the body-in-nature as its subordinate ally. What was previously a threat to the literate mind's clean ascendance is now disarmed by being given a place within the grand project of transcendence. Hence, even and especially in this most pastoral of dialogues, in which the rational intellect seems almost balanced by the desiring body, and in which trees that "can teach nothing" seem balanced by watchful cicadas, we may still discern the seeds of nature's eventual eclipse behind a world of letters, numbers, and texts.
### Synaesthesia and the Encounter with the Other
It is remarkable that none of the major twentieth-century scholars who have directed their attention to the changes wrought by literacy have seriously considered the impact of writing—and, in particular, phonetic writing—upon the human experience of the wider natural world. Their focus has generally centered upon the influence of phonetic writing on the structure and deployment of human language, on patterns of cognition and thought, or upon the internal organization of human societies. Most of the major research, in other words, has focused upon the alphabet's impact on processes either internal to human society or presumably "internal" to the human mind. Yet the limitation of such research—its restriction within the bounds of human social interaction and personal interiority—itself reflects an anthropocentric bias wholly endemic to alphabetic culture. In the absence of phonetic literacy, neither society, nor language, nor even the experience of "thought" or consciousness, can be pondered in isolation from the multiple nonhuman shapes and powers that lend their influence to all our activities (we need think only of our ceaseless involvement with the ground underfoot, with the air that swirls around us, with the plants and animals that we consume, with the daily warmth of the sun and the cyclic pull of the moon). Indeed, in the absence of formal writing systems, human communities come to know themselves primarily as they are reflected back by the animals and the animate landscapes with which they are directly engaged. This epistemological dependence is readily evidenced, on every continent, by the diverse modes of identification commonly categorized under the single term "totemism."
It is exceedingly difficult for us literates to experience anything approaching the vividness and intensity with which surrounding nature spontaneously presents itself to the members of an indigenous, oral community. Yet as we saw in the previous chapters, Merleau-Ponty's careful phenomenology of perceptual experience had begun to disclose, underneath all of our literate abstractions, a deeply participatory relation to things and to the earth, a felt reciprocity curiously analogous to the animistic awareness of indigenous, oral persons. If we wish to better comprehend the remarkable shift in the human experience of nature that was occasioned by the advent and spread of phonetic literacy, we would do well to return to the intimate analysis of sensory perception inaugurated by Merleau-Ponty. For without a clear awareness of what reading and writing amounts to when considered at the level of our most immediate, bodily experience, any "theory" regarding the impact of literacy can only be provisional and speculative.
Although Merleau-Ponty himself never attempted a phenomenology of reading or writing, his recognition of the importance of synaesthesia—the overlap and intertwining of the senses—resulted in a number of experiential analyses directly pertinent to the phenomenon of reading. For reading, as soon as we attend to its sensorial texture, discloses itself as a profoundly synaesthetic encounter. Our eyes converge upon a visible mark, or a series of marks, yet what they find there is a sequence not of images but of sounds, something heard; the visible letters, as we have said, trade our eyes for our ears. Or, rather, the eye and the ear are brought together at the surface of the text—a new linkage has been forged between seeing and hearing which ensures that a phenomenon apprehended by one sense is instantly transposed into the other. Further, we should note that this sensory transposition is mediated by the human mouth and tongue; it is not just any kind of sound that is experienced in the act of reading, but specifically human, vocal sounds—those which issue from the human mouth. It is important to realize that the now common experience of "silent" reading is a late development in the story of the alphabet, emerging only during the Middle Ages, when spaces were first inserted between the words in a written manuscript (along with various forms of punctuation), enabling readers to distinguish the words of a written sentence without necessarily sounding them out audibly. Before this innovation, to read was necessarily to read aloud, or at the very least to mumble quietly; after the twelfth century it became increasingly possible to internalize the sounds, to listen inwardly to phantom words (or the inward echo of words once uttered).
Alphabetic reading, then, proceeds by way of a new synaesthetic collaboration between the eye and the ear, between seeing and hearing. To discern the consequences of this new synaesthesia, we need to examine the centrality of synaesthesia in our perception of others and of the earth.
The experiencing body (as we saw in chapter 2) is not a self-enclosed object, but an open, incomplete entity. This openness is evident in the arrangement of the senses: I have these multiple ways of encountering and exploring the world—listening with my ears, touching with my skin, seeing with my eyes, tasting with my tongue, smelling with my nose—and all of these various powers or pathways continually open outward from the perceiving body, like different paths diverging from a forest. Yet my experience of the world is not fragmented; I do not commonly experience the visible appearance of the world as in any way separable from its audible aspect, or from the myriad textures that offer themselves to my touch. When the local tomcat comes to visit, I do not have distinctive experiences of a visible cat, an audible cat, and an olfactory cat; rather, the tomcat is precisely the place where these separate sensory modalities join and dissolve into one another, blending as well with a certain furry tactility. Thus, my divergent senses meet up with each other in the surrounding world, converging and commingling in the things I perceive. We may think of the sensing body as a kind of open circuit that completes itself only in things, and in the world. The differentiation of my senses, as well as their spontaneous convergence in the world at large, ensures that I am a being destined for relationship: it is primarily through my engagement with what is _not_ me that I effect the integration of my senses, and thereby experience my own unity and coherence.
Indeed, the synaesthetic flowing together of different senses into a dynamic and unified experience is already operative within the single system of vision itself. For ordinary vision is a blending of two unique vistas, two perspectives, _two eyes_. Even here, within a single sensory system, we discern an originary openness or divergence—between, in this case, the two sides of my body, each with its own access to the visible—and it is only via the convergence and meeting of these two perspectives at some point out in front of my body that the visible world becomes present to me in all its depth. The double images common to unfocused vision have only a flimsy reality: if I let my eyes focus upon a shelf across the room, and meanwhile hold my index finger up in front of my face, I find that two images of my finger float before me like insubstantial phantoms and that the shelf, despite its greater distance, is much more substantial and present to my awareness than is my finger. Only when I break my focus upon the shelf and let my eyes reunite at the finger does this appendage with its delicate hairs and gnarly knuckles become fully present.
Ordinary seeing, then, involves the convergence of two views into a single dynamic vision; divergent parts of myself are drawn together by the object, and I thus meet up with myself _over there_ , at that tree or that spider upon which I focus. Vision itself, in other words, is already a kind of synaesthesia, a collaboration of different sensory channels or organs.
When we attend carefully to our perceptual experience, we discover that the convergence of the eyes often prompts the added collaboration of the other senses. When, for instance, I gaze through the window toward a blackbird in a nearby bush—my two eyes drawn together by the bird's jerking body as it plucks red berries from the branches—other senses are quite naturally drawn into that same focus. Certain tactile sensations, for instance, may accompany the blackbird's movements, and if I have been watching carefully I may notice, as it squoonches each new berry in its beak, a slightly acidic taste burst within my mouth. Or rather, strangely, I seem to feel this burst of taste over there, in _its_ mouth, yet I feel its mouth only with my own.
Similarly, when I watch a stranger learning to ride a bicycle for the first time, my own body, although it is standing solidly on the ground, inadvertently experiences the uncertain equilibrum of the rider, and when that bicycle teeters and falls I feel the harsh impact of the asphalt against my own leg and shoulder. My tactile and proprioceptive senses are, it would seem, caught up over there where my eyes have been focused; the momentary shock and subsequent throbbing in my limbs make me wince. My hearing, as well, had been focused by the crash; the other ambient sounds to which I'd been listening just before (birds, children playing) have no existence for me now, only this stranger's pained breathing as he slowly shoves the bicycle aside and accepts the hand I am offering, pulling himself to his feet. He shakes his head, laughs a bit, then grins—all in a manner that readily communicates to my body that he's okay—and then turns to inspect the bicycle.
The diversity of my sensory systems, and their spontaneous convergence in the things that I encounter, ensures this interpenetration or interweaving between my body and other bodies—this magical participation that permits me, at times, to feel what others feel. The gestures of another being, the rhythm of its voice, and the stiffness or bounce in its spine all gradually draw my senses into a unique relation with one another, into a coherent, if shifting, organization. And the more I linger with this other entity, the more coherent the relation becomes, and hence the more completely I find myself face-to-face with another intelligence, another center of experience.
In the encounter with the cyclist, as in my experience of the blackbird, the visual focus induced and made possible the participation of the other senses. In different situations, other senses may initiate the synaesthesia: our ears, when we are at an orchestral concert; or our nostrils, when a faint whiff of burning leaves suddenly brings images of childhood autumns; our skin, when we are touching or being touched by a lover. Nonetheless, the dynamic conjunction of the eyes has a particularly ubiquitous magic, opening a quivering depth in whatever we focus upon, ceaselessly inviting the other senses into a concentrated exchange with stones, squirrels, parked cars, persons, snow-capped peaks, clouds, and termite-ridden logs. This power—the synaesthetic magnetism of the visual focus—will prove crucial for our understanding of literacy and its perceptual effects.
The most important chapter of Merleau-Ponty's last, unfinished work is entitled "The Intertwining—The Chiasm." The word "chiasm," derived from an ancient Greek term meaning "crisscross," is in common use today only in the field of neurobiology: the "optic chiasm" is that anatomical region, between the right and left hemispheres of the brain, where neuronal fibers from the right eye and the left eye cross and interweave. As there is a chiasm between the two eyes, whose different perspectives continually conjoin into a single vision, so—according to Merleau-Ponty—there is a chiasm between the various sense modalities, such that they continually couple and collaborate with one another. Finally, this interplay of the different senses is what enables the chiasm between the body and the earth, the reciprocal participation—between one's own flesh and the encompassing flesh of the world—that we commonly call perception.
Phonetic reading, of course, makes use of a _particular_ sensory conjunction—that between seeing and hearing. And indeed, among the various synaesthesias that are common to the human body, the confluence (or chiasm) between seeing and hearing is particularly acute. For vision and hearing are the two "distance" senses of the human organism. In contrast to touch and proprioception (inner-body sensations), and unlike the chemical senses of taste and smell, seeing and hearing regularly place us in contact with things and events unfolding at a substantial distance from our own visible, audible body.
My visual gaze explores the reflective surfaces of things, their outward color and contour. By following the play of light and shadow, the dance of colors, and the gradients of repetitive patterns, the eyes—themselves gleaming surfaces—keep me in contact with the multiple outward facets, or faces, of the things arrayed about me. The ears, meanwhile, are more inward organs; they emerge from the depths of my skull like blossoms or funnels, and their participation tells me less about the outer surface than the interior substance of things. For the audible resonance of beings varies with their material makeup, as the vocal calls of different animals vary with the size and shape of their interior cavities and hollows. I feel their expressive cries resound in my skull or my chest, echoing their sonorous qualities with my own materiality, and thus learn of their inward difference from myself. Looking and listening bring me into contact, respectively, with the outward surfaces and with the interior voluminosity of things, and hence where these senses come together, I experience, over there, the complex interplay of inside and outside that is characteristic of my own self-experience. It is thus at those junctures in the surrounding landscape where my eyes and my ears are drawn together that I most readily feel myself confronted by another power like myself, another life.
If a native hunter is tracking, alone, in the forest, and a whooping cry reaches his ears from the leafy canopy, he will likely halt in his steps, silencing his breathing in order to hear that sound, when it comes again, more precisely. His eyes scan the cacophony of branches overhead with an unfocused gaze, attentive to minute movements on the periphery of the perceptual field. A slight rustle of branches draws his eyes into a more precise focus, his attention now restricted to a small patch of the canopy, yet still open, questioning, listening. When the cry comes again, the eyes, led by the ears, swiftly converge upon the source of that sound, and suddenly a monkey's form becomes evident, half-hidden from the leaves, its tail twirled around a limb, its body poised, watching. As the tribesman's searching eyes are drawn into a common focus with his listening ears, this conjunction, this chiasm, rebounds upon his own tactile and proprioceptive sensations—he feels himself suddenly confronted, caught up in a dynamic exchange with another entity, another carnal intelligence.
Indeed, the synaesthesia between the human eyes and ears is especially concentrated in our relation to other animals, since for a million years these "distance" senses were most tightly coupled at such moments of extreme excitement, when closing in on prey, or when escaping from predators. When backing slowly away from a mother grizzly protecting her cubs, or when watching intently the movements of an aroused rattlenake in order to avoid its numbing strike—these are moments when visual and auditory foci are virtually indistinguishable. For these senses are functioning here as a single, hyperattentive organ; we feel ourselves listening with our eyes and watching with our ears, ready to respond with our whole body to any change in the Other's behavior.
Yet our ears and our eyes are drawn together not only by animals, but by numerous other phenomena within the landscape. And, strangely, _wherever_ these two senses converge, we may suddenly feel ourselves in relation with another expressive power, another center of experience. Trees, for instance, can seem to speak to us when they are jostled by the wind. Different forms of foliage lend each tree a distinctive voice, and a person who has lived among them will easily distinguish the various dialects of pine trees from the speech of spruce needles or Douglas fir. Anyone who has walked through cornfields knows the uncanny experience of being scrutinized and spoken to by whispering stalks. Certain rock faces and boulders request from us a kind of auditory attentiveness, and so draw our ears into relation with our eyes as we gaze at them, or with our hands as we touch them—for it is only through a mode of listening that we can begin to sense the interior voluminosity of the boulder, its particular density and depth. There is an expectancy to the ears, a kind of patient receptivity that they lend to the other senses whenever we place ourselves in a mode of listening—whether to a stone, or a river, or an abandoned house. That so many indigenous people allude to the articulate speech of trees or of mountains suggests the ease with which, in an oral culture, one's auditory attention may be joined with the visual focus in order to enter into a living relation with the expressive character of things.
Far from presenting a distortion of their factual relation to the world, the animistic discourse of indigenous, oral peoples is an inevitable counterpart of their immediate, synaesthetic engagement with the land that they inhabit. The animistic proclivity to perceive the angular shape of a boulder (while shadows shift across its surface) as a kind of meaningful gesture, or to enter into felt conversations with clouds and owls—all of this could be brushed aside as imaginary distortion or hallucinatory fantasy if such active participation were not the very structure of perception, if the creative interplay of the senses in the things they encounter was not our sole way of linking ourselves to those things and letting the things weave themselves into our experience. Direct, prereflective perception is inherently synaesthetic, participatory, and animistic, disclosing the things and elements that surround us not as inert objects but as expressive subjects, entities, powers, potencies.
And yet most of us seem, today, very far from such experience. Trees rarely, if ever, speak to us; animals no longer approach us as emissaries from alien zones of intelligence; the sun and the moon no longer draw prayers from us but seem to arc blindly across the sky. How is it that these phenomena _no longer address us_ , no longer compel our involvement or reciprocate our attention? If participation is the very structure of perception, how could it ever have been brought to a halt? To freeze the ongoing animation, to block the wild exchange between the senses and the things that engage them, would be tantamount to freezing the body itself, stopping it short in its tracks. And yet our bodies still move, still live, still breathe. If we no longer experience the enveloping earth as expressive and alive, this can only mean that the animating interplay of the senses has been transferred to another medium, another locus of participation.
_I T IS THE WRITTEN TEXT THAT PROVIDES THIS NEW LOCUS._ FOR TO read is to enter into a profound participation, or chiasm, with the inked marks upon the page. In learning to read we must break the spontaneous participation of our eyes and our ears in the surrounding terrain (where they had ceaselessly converged in the synaesthetic encounter with animals, plants, and streams) in order to recouple those senses upon the flat surface of the page. As a Zuñi elder focuses her eyes upon a cactus and hears the cactus begin to speak, so we focus our eyes upon these printed marks and immediately hear voices. We hear spoken words, witness strange scenes or visions, even experience other lives. As nonhuman animals, plants, and even "inanimate" rivers once spoke to our tribal ancestors, so the "inert" letters on the page now speak to us! _This is a form of animism that we take for granted, but it is animism nonetheless—as mysterious as a talking stone_.
And indeed, it is only when a culture shifts its participation to these printed letters that the stones fall silent. Only as our senses transfer their animating magic to the written word do the trees become mute, the other animals dumb.
But let us be more precise, recalling the distinction between different forms of writing discussed at the start of this chapter. As we saw there, pictographic, ideographic, and even rebuslike writing still makes use of, or depends upon, our sensorial participation with the natural world. As the tracks of moose and bear refer beyond themselves to those entities of whom they are the trace, so the images in early writing systems draw their significance not just from ourselves but from sun, moon, vulture, jaguar, serpent, lightning—from all those sensorial, never strictly human powers, of which the written images were a kind of track or tracing. To be sure, these signs were now inscribed by human hands, not by the hooves of deer or the clawed paws of bear; yet as long as they presented images of paw prints and of clouds , of sun and of serpent , these characters still held us in relation to a more-than-human field of discourse. Only when the written characters lost all explicit reference to visible, natural phenomena did we move into a new order of participation. Only when those images came to be associated, alphabetically, with purely human-made sounds, and even the names of the letters lost all worldly, extrahuman significance, could speech or language come to be experienced as an exclusively human power. For only then did civilization enter into the wholly self-reflexive mode of animism, or magic, that still holds us in its spell:
We know what the animals do, what are the needs of the beaver, the bear, the salmon, and other creatures, because long ago men married them and acquired this knowledge from their animal wives. Today the priests say we lie, but we know better. The white man has been only a short time in this country and knows very little about the animals; we have lived here thousands of years and were taught long ago by the animals themselves. The white man _writes everything down in a book_ so that it will not be forgotten; but our ancestors _married_ animals, learned all their ways, and passed on this knowledge from one generation to another.
THAT ALPHABETIC READING AND WRITING WAS ITSELF EXPERIENCED as a form of magic is evident from the reactions of cultures suddenly coming into contact with phonetic writing. Anthropological accounts from entirely different continents report that members of indigenous, oral tribes, after seeing the European reading from a book or from his own notes, came to speak of the written pages as "talking leaves," for the black marks on the flat, leaflike pages seemed to talk directly to the one who knew their secret.
The Hebrew scribes never lost this sense of the letters as living, animate powers. Much of the Kabbalah, the esoteric body of Jewish mysticism, is centered around the conviction that each of the twenty-two letters of the Hebrew _aleph-beth_ is a magic gateway or guide into an entire sphere of existence. Indeed, according to some kabbalistic accounts, it was by combining the letters that the Holy One, Blessed Be He, created the ongoing universe. The Jewish kabbalists found that the letters, when meditated upon, would continually reveal new secrets; through the process of _tzeruf_ , the magical permutation of the letters, the Jewish scribe could bring himself into sucessively greater states of ecstatic union with the divine. Here, in other words, was an intensely concentrated form of animism—a participation conducted no longer with the sculpted idols and images worshiped by other tribes but solely with the visible letters of the _aleph-beth_.
Perhaps the most succinct evidence for the potent magic of written letters is to be found in the ambiguous meaning of our common English word "spell." As the roman alphabet spread through oral Europe, the Old English word "spell," which had meant simply to recite a story or tale, took on the new double meaning: on the one hand, it now meant to arrange, in the proper order, the written letters that constitute the name of a thing or a person; on the other, it signified a magic formula or charm. Yet these two meanings were not nearly as distinct as they have come to seem to us today. For to assemble the letters that make up the name of a thing, in the correct order, was precisely to effect a magic, to establish a new kind of influence over that entity, to summon it forth! To spell, to correctly arrange the letters to form a name or a phrase, seemed thus at the same time to _cast a spell_ , to exert a new and lasting power over the things spelled. Yet we can now realize that to learn to spell was also, and more profoundly, to step under the influence of the written letters ourselves, to cast a spell upon our own senses. It was to exchange the wild and multiplicitous magic of an intelligent natural world for the more concentrated and refined magic of the written word.
THE BULGARIAN SCHOLAR TZVETAN TODOROV HAS WRITTEN AN illuminating study of the Spanish conquest of the Americas, based on extensive study of documents from the first months and years of contact between European culture and the native cultures of the American continent. The lightning-swift conquest of Mexico by Cortéz has remained a puzzle for historians, since Cortéz, leading only a few hundred men, managed to seize the entire kingdom of Montezuma, who commanded _several hundred thousand_. Todorov concludes that Cortéz's astonishing and rapid success was largely a result of the discrepancy between the different forms of participation engaged in by the two societies. The Aztecs, whose writing was highly pictorial, necessarily felt themselves in direct communication with an animate, more-than-human environment. "Everything happens as if, for the Aztecs, [written] signs automatically and necessarily proceed from the world they designate..."; the Aztecs are unable to use their spoken words, or their written characters, to hide their true intentions, since these signs belong to the world around them as much as to themselves. To be duplicitous with signs would be, for the Aztecs, to go against the order of nature, against the encompassing speech or logos of an animate world, in which their own tribal discourse was embedded.
The Spaniards, however, suffer no such limitation. Possessed of an _alphabetic_ writing system, they experience themselves not in communication with the sensuous forms of the world, but solely with one another. The Aztecs must answer, in their actions as in their speech, to the whole sensuous, natural world that surrounds them; the Spanish need answer only to themselves.
In contact with this potent new magic, with these men who participate solely with their own self-generated signs, whose speech thus seems to float free of the surrounding landscape, and who could therefore be duplicitous and _lie_ even in the presence of the sun, the moon, and the forest, the Indians felt their own rapport with those sensuous powers, or gods, beginning to falter:
The testimony of the Indian accounts, which is a description rather than an explanation, asserts that everything happened because the Mayas and the Aztecs lost control of communication. The language of the gods has become unintelligible, or else these gods fell silent. "Understanding is lost, wisdom is lost" [from the Mayan account of the Spanish invasion].... As for the Aztecs, they describe the beginning of their own end as a silence that falls: the gods no longer speak to them.
In the face of aggression from this new, entirely self-reflexive form of magic, the native peoples of the Americas—like those of Africa and, later, of Australia—felt their own magics wither and become useless, unable to protect them.
# **5**
# **_In the Landscape of language_**
Tired of all who come with words, words but no language
I went to the snow-covered island.
The wild does not have words.
The unwritten pages spread themselves out in all directions!
I come across the marks of roe-deer's hooves in the snow.
Language, but no words.
TOMAS TRANSTROMER
THE FIRST PART OF THIS BOOK RAISED THIS QUESTION: HOW did Western civilization become so estranged from nonhuman nature, so oblivious to the presence of other animals and the earth, that our current lifestyles and activities contribute daily to the destruction of whole ecosystems—whole forests, river valleys, oceans—and to the extinction of countless species? Or, more specifically, how did civilized humankind lose all sense of reciprocity and relationship with the animate natural world, that rapport that so influences (and limits) the activities of most indigenous, tribal peoples? How did civilization break out of, and leave behind, the animistic or participatory mode of experience known to all native, place-based cultures?
In the last chapter, however, we showed that animism was never, in truth, left behind. The participatory proclivity of the senses was simply transferred from the depths of the surrounding life-world to the visible letters of the alphabet. Only by concentrating the synaesthetic magic of the senses upon the written letters could these letters begin to come alive and to speak. "Written words," says Socrates, "seem to talk to you as though they were intelligent...." Indeed, today it is virtually impossible for us to look at a printed word _without_ seeing, or rather hearing, what "it says." For our senses are now coupled, synaesthetically, to these printed shapes as profoundly as they were once wedded to cedar trees, ravens, and the moon. As the hills and the bending grasses once spoke to our tribal ancestors, so these written letters and words now speak to us.
We have seen as well that iconic writing systems—those that employ pictographic, ideographic, and/or rebuslike characters—necessarily rely, to some extent, upon our original sensory participation with the enveloping natural field. Only with the emergence of the phonetic alphabet, and its appropriation by the ancient Greeks, did the written images lose all evident ties to the larger field of expressive beings. Each image now came to have a strictly _human_ referent: each letter was now associated purely with a gesture or sound of the human mouth. Such images could no longer function as windows opening on to a more-than-human field of powers, but solely as mirrors reflecting the human form back upon itself. The senses that engaged or participated with this new writing found themselves locked within a discourse that had become exclusively human. Only thus, with the advent and spread of phonetic writing, did the rest of nature begin to lose its voice.
The highly anthropocentric (human-centered) mode of experience endemic to alphabetic culture spread throughout Europe in the course of two millennia, receiving a great boost from the calligraphic innovations introduced in the monastic scriptoria (the rooms where monks copied manuscripts) by the English monk Alcuin (732–804) during the reign of Charlemagne, and a major thrust from the invention of movable type by Johann Gutenberg (c. 1394–1468), in the fifteenth century. The printing press, and the dissemination of uniformly printed texts that it made possible, ushered in the Enlightenment and the profoundly detached view of "nature" that was to prevail in the modern period. In recent centuries the industrial and technological practices made possible by this new distance from the natural world have carried alphabetic awareness throughout the globe, infiltrating even those cultures that had retained iconic, ideographic writing systems.
Nevertheless, there remain, on the edges and even in the midst of this ever-expanding monoculture, small-scale local cultures or communities where the traditional oral, indigenous modes of experience still prevail—cultures that have never fully transferred their sensory participation to the written word. They have not yet closed themselves within an exclusively human field of meanings, and so still dwell within a landscape that is alive, aware, and expressive. To such peoples, that which we term "language" remains as much a property of the animate landscape as of the humans who dwell and speak within that terrain. Indeed, the linguistic discourse of such cultures is commonly bound, in specific and palpable ways, to the expressive earth.
In this chapter, then, we will glance at just a few of the very diverse ways in which the common discourse of an oral culture may open, directly, onto the evocative sounds, shapes, and gestures of the surrounding ecology.
### The Language of the Birds
Whenever we of literate culture seek to engage and understand the discourse of oral cultures, we must strive to free ourselves from our habitual impulse to _visualize_ any language as a static structure that could be diagrammed, or a set of rules that could be ordered and listed. Without a formal writing system, the language of an oral culture cannot be objectified as a separable entity by those who speak it, and this lack of objectification influences not only the way in which oral cultures experience the field of discursive meanings, but also the very character and structure of that field. In the absence of any written analogue to speech, the sensible, natural environment remains the primary visual counterpart of spoken utterance, the visible accompaniment of all spoken meaning. The land, in other words, is the sensible site or matrix wherein meaning occurs and proliferates. In the absence of writing, we find ourselves situated in the field of discourse as we are embedded in the natural landscape; indeed, the two matrices are not separable. We can no more stabilize the language and render its meanings determinate than we can freeze all motion and metamorphosis within the land.
IF WE LISTEN, FIRST, TO THE SOUNDS OF AN ORAL LANGUAGE—TO the rhythms, tones, and inflections that play through the speech of an oral culture—we will likely find that these elements are attuned, in multiple and subtle ways, to the contour and scale of the local landscape, to the depth of its valleys or the open stretch of its distances, to the visual rhythms of the local topography. But the human speaking is necessarily tuned, as well, to the various nonhuman calls and cries that animate the local terrain. Such attunement is simply imperative for any culture still dependent upon foraging for its subsistence. Minute alterations in the weather, changes in the migratory patterns of prey animals, a subtle shift in the focus of a predator—sensitivity to such subtleties is a necessary element of all oral, subsistence cultures, and this sensitivity is inevitably reflected not just in the content but in the very shapes and patterns of human discourse.
Hunting, for an indigenous, oral community, entails abilities and sensitivities very different from those associated with hunting in technological civilization. Without guns or gunpowder, a native hunter must often come much closer to his wild prey if he is to take its life. Closer, that is, not just physically but emotionally, empathically entering into proximity with the other animal's ways of sensing and experiencing. The native hunter, in effect, must _apprentice_ himself to those animals that he would kill. Through long and careful observation, enhanced at times by ritual identification and mimesis, the hunter gradually develops an instinctive knowledge of the habits of his prey, of its fears and its pleasures, its preferred foods and favored haunts. Nothing is more integral to this practice than learning the communicative signs, gestures, and cries of the local animals. Knowledge of the sounds by which a monkey indicates to the others in its band that it has located a good source of food, or the cries by which a particular bird signals distress, or by which another attracts a mate, enables the hunter to anticipate both the large-scale and small-scale movements of various animals. A familiarity with animal calls and cries provides the hunter, as well, with an expanded set of senses, an awareness of events happening beyond his field of vision, hidden by the forest leaves or obscured by the dark of night. Moreover, the skilled human hunter often can generate and mimic such sounds himself, and it is this that enables him to enter most directly into the society of other animals.
One of the most revealing twentieth-century accounts of a relatively intact indigenous community is that recorded by F. Bruce Lamb from the spoken recollections of the Peruvian doctor Manuel Córdova-Rios. Córdova-Rios was captured in 1907, when he was fifteen years old, by a small tribe of Amahuaca Indians living deep in the Amazonian rain forest (between the headwaters of the Juruá, Purús, Madre de Dios, and Inuya rivers)—probably the remnant of a larger tribe decimated by the incursion of the rubber-tapping industry into the forest. He was carefully trained by the headman of this small tribe to become his successor, and was for six years meticulously tutored in the ways of the hunt, in the medicinal and magical powers of the rain forest plants, and in the traditional preparation and use of extracts from the ayahuasca vine to attain, when necessary, a clairvoyant state of fusion with the enveloping jungle ecosystem.
Curiously, the tribe's language, which remained largely meaningless to Córdova-Rios for six months or more, became understandable to his ears only as his senses became attuned to the subtleties of the rain forest ecology in which the culture was embedded. He did, eventually, become headman of the tribe, yet he fled the rain forest the following year after a series of attempts on his life by a neighboring band.
Córdova-Rios's descriptions of the various hunts in which he participated make vividly evident the extent to which these people's senses were directly coupled to the enveloping forest:
They reacted to the faintest signals of sound and smell, intuitively relating them to all other conditions of the environment and then interpreting them to achieve the greatest possible capture of game.... Many of the best hunters seemed to know by some special extra sense just where to find the game they sought, or they had developed some special method of drawing game to them. Knowing how to imitate and to use the signals the animals made to communicate between their kind in various situations helped in locating game and drawing it within sighting range of an astute hunter.
In the course of Córdova-Rios's account, we read careful descriptions of hunters sequestered in the foliage of high fruit trees luring partridges toward them with mimicked bird calls signaling the discovery of an abundant food source. We read of one hunter who, upon hearing a band of monkeys moving through the dense forest canopy overhead, utters a cry that would be made by a baby monkey if it had fallen to the ground. This call stops the roving monkeys and brings them down beneath the thick foliage into the hunter's arrow range; the hunter shoots two of them to feed his family. Later Córdova-Rios's native comrades teach him, through imitation, the principal vocal signals of a species of wild pig that they are hunting.
Through ancestral stories and tales of recent hunts, the hunters continually exchange knowledge among themselves regarding the nuanced meanings of particular calls made by various creatures, a knowledge gleaned from ever-renewed encounters with those animals in the wild. In many instances knowledge of the specific alarm cries of birds and other animals alert the human hunters to the presence of dangerous predators, like the jaguar, that they themselves must avoid.
A typical example of such interspecies linguistic savvy is an encounter reported by a man named Raci to the other members of a hunting expedition, including Córdova-Rios, as the various hunters lie in their hammocks at night, recounting for each other, in detail, their individual efforts of the day:
It was time to start back and I had no game. Just as I turned to come back toward camp a small ground-sleeping tinamou [a type of jungle partridge] sent out his sad call, close to where I was, and he was answered by another. You know why their evening call is so sad? They don't like to sleep alone and at sunset each one wanders around aimlessly calling and calling until an answer comes back from somewhere, and then the two move closer and closer together, guided by the calls. And so they find a sleeping partner. I answered the call and found I was between the two birds. So I backed up between the buttresses of a big tree where the ground could be seen for a good distance in front of me, and I started calling the birds to me. You know that it is dangerous to call the tinamou without the protection of a big tree. The jaguar sometimes comes in answer to the call! The tinamou is also his favorite bird.
One bird was nearby and soon had my arrow in his body. He fluttered his wings and kicked a few turns, but was soon with me at the base of the tree. I broke his leg and put a long streak of his blood under each of my eyes to bring good luck.
Every collective hunting expedition is preceded by careful ritual preparations, during which the hunters eat only certain foods, erasing their human odors by soaking themselves in various herbal baths and immersing themselves in the smoke of burning leaves. The expeditions themselves are accompanied by reverent chants to particular forest spirits. The various practices of the tribe, according to Córdova-Rios, embody clear knowledge of the limits beyond which a species of animal must not be hunted; overhunting of a single type of animal or bird is known to bring poor luck upon the hunter or even upon the whole village. Córdova-Rios, for instance, is taught that if he kills the leader of a band of wild pigs (which leaves the pigs disorganized and all too easy to prey upon until a new leader takes over), he must never again kill a leader of the same band.
Meanwhile Xumu, the tribal headman, oversaw the hunting engagements of the group as a whole. Each of the men was assigned by him to an individual hunting territory, and they all reported daily to Xumu regarding the shifting locations of the various bands of monkeys and wild pigs, of the jaguar and other forest inhabitants. Kept apprised in this manner of systemic events unfolding throughout the forest (to a distance of several days journey in all directions from the village) the headman was able by his instructions to appropriately modulate the hunting activities of the small tribe, continually modifying these activities in response to the living gestures of the forest itself.
Córdova-Rios's narrative provides vivid evidence of the extent to which, in the Amazon rain forest, human and nonhuman life-worlds interpenetrate and inform one another. Analogous forms of interaction may be found in every hunting and foraging culture. For subsistence hunting, once again, entails that the human tribesman enter into a profound sensorial rapport with other animals. And this participation, as Córdova-Rios makes evident, necessarily extends into the vocal dimension, wherein animal cries and communicative calls are pondered, mimicked, and replied to by human hunters, becoming as it were part of the tribal vocabulary. Tribespeople traveling through the forest at some distance from one another, for example, often use mimicked animal cries and bird calls to communicate _among themselves_ , as a means of calling out to each other without drawing the attention of certain animals, or of rival human bands that might be lingering in the area. It would be startling if these constantly employed calls, cries, hoots, riffs, and whistles had no influence on the everyday speech of the tribe as a whole. On the contrary, in the absence of any formal writing system that might stabilize the local language and render it impervious to the shifting sounds of the animate landscape, the spoken discourse of oral, foraging peoples remains uniquely responsive to the multiple sounds and rhythms of the nonhuman surroundings, and especially attuned to the vocal gestures and cries of the local animals.
We have learned from Saussure that a human language is structured not so much as a collection of terms, each of which possesses a determinate meaning, but as a complexly ramified web, wherein the knots, or terms, hold their specific place or meaning only by virtue of their direct and indirect relations to all other terms within the language. If such is indeed the case, then even just a few terms or phrases borrowed directly from the vocal speech sounds of other animals would serve to subtly influence all the ratios of the language, rooting the language, as it were, in a particular ecology, a particular terrain. Once again, no indigenous, oral language can be genuinely understood in separation from the more-than-human earth that sustains it, of which the language itself is a kind of internal articulation.
Saussure himself, however, denied the possibility of such intimacy between language and the land; his resolute insistence upon the arbitrariness of the relation between spoken sounds and that which they signify led him to downplay the influence of mimicry, onomatopoeia, and sound symbolism within the life of any language. Nevertheless, more recent research on the echoic and gestural significance of spoken sounds has demonstrated that a subtle sort of onomatopoeia is constantly at work in language: certain meanings inevitably gravitate toward certain sounds, and vice versa. (Every poet is aware of this primordial depth in language, whereby particular sensations are invoked by the sounds themselves, and whereby the shape, rhythm, and texture of particular phrases conjure the expressive character of particular phenomena.)
THE INTERTWINING OF HUMAN SPEECH WITH THE CALLS AND CRIES of the local earth is evident even when we turn away from the tropics toward an oral culture of the far north, like that of the Koyukon Indians of northwestern Alaska. The Koyukon inhabit a vast expanse of wild country extending well north of the Arctic Circle, with camps and villages set along the Yukon and Koyukuk rivers. Their language belongs to the Athapaskan family of languages spoken by native peoples scattered throughout much of northwestern North America and in pockets as far south as Arizona. The ancestors of the Koyukon people may have inhabited Alaska as early as ten thousand years ago, although archaeologists have been unable to date the Athapaskan emergence into North America with any precision. The Koyukon, first encountered by Europeans in the mid-nineteenth century, have in the twentieth century slowly abandoned their traditional pattern of scattered seminomadism, moving into a few settlements built near trading posts or Catholic missions. Yet they still travel widely, using their villages more as home bases from which to journey on foraging expeditions for fish, land animals (for clothing as well as food), berries, and other wild provisions.
According to anthropologist and ethnobiologist Richard Nelson, who has lived and worked closely with the Koyukon people, language to them is as much the province of other animals as it is the domain of humankind. The Koyukon assume that nonhuman animals
communicate among themselves, and [that] they understand human behavior and language. They are constantly aware of what people say and do.... But animals do not use human language among themselves. They communicate with sounds which are considered their own form of language.
In Koyukon belief, the other animals and the plants once shared a common language with human beings. This was in the Distant Time ( _Kk'adonts'idnee_ ), a time during which all living beings "shared one society and went through dreamlike transmutations from animals or plants to humans, and sometimes back again." We will postpone until the next chapter the question of whether the stories told of the Distant Time by the Koyukon people depict an originary time "long ago" in the past—as they are often interpreted according to the linear-historical view of time first imported into the Koyukon territory by Catholic missionaries—or whether the Distant Time is more coherently understood as a unique dimension or _modality_ of time, one that is more integral to the living present than it is to the historical past. In any case, and despite the apparent differentiation of animal and human languages since, or outside of, the Distant Time, the various discourses of humans and animals still overlap and interpenetrate in the everyday experience of Koyukon persons.
Caribou, for instance, are said to "sing through" human beings when in their vicinity, granting the tribespeople songs that certain persons remember upon waking from sleep. When those persons sing these songs later, their success in finding and hunting caribou is ensured. Tribal elders, meanwhile, listen closely to the rippling cries and wails of the loon as a source of inspiration in composing their own songs and chants. When a revered Koyukon elder lay near death, Nelson watched an old woman visiting from another village as she approached the near shore of a lake and began to sing Koyukon "spring songs" to a pair of loons that had been lingering there.
Shortly the loons swam toward her until they rested in the water some fifty yards away, and there they answered her, filling the air with eerie and wonderful voices. When I spoke with her later, she said that loons will often answer spring songs this way. For several days people talked of how beautiful the songs had been that morning.
The lilting cries of the common loon are linguistically meaningful to the Koyukon. According to one man, "Sometimes people will hunt the loon, but me, I don't like to kill it. I like to listen to it all I can and pick up the words it knows." The speech of the rare yellow-billed loon is still more powerful than that of the common loon to the Koyukon: "... it says the same words, but its voice is just a little different."
The assumption that nature is all aware, and that the sounds made by animals are at least as meaningful as those made by humans, leads the Koyukon to listen attentively to subtle nuances and variations in the calls of local birds. The Koyukon names for birds are often highly onomatopoeic, so that in speaking their names one is also echoing their cries. The Arctic tern ( _k'idagaas'_ ), the northern phalarope ( _tiyee_ ), the rusty blackbird ( _ts'uhutlts'eegga_ ), the blackpoll warbler ( _k'oot'anh_ ), the slate-colored junco ( _k'it'otlt'ahga_ )—all have such names. Written transcription, however, cannot convey the remarkable aptness of these names, which when spoken in Koyukon have a lilting, often whistlelike quality. The interpenetration of human and nonhuman utterances is particularly vivid in the case of numerous bird songs that seem to enunciate whole phrases or statements in Koyukon.
Many bird calls are interpreted as Koyukon words.... What is striking about these words is how perfectly they mirror the call's pattern, so that someone [outside the tribe] who knows birdsongs can readily identify the species when the words are spoken in Koyukon. Not only the rhythm comes through, but also some of the tone, the "feel" that goes with it.
As we ponder such correspondences, we come to realize that the sounds and rhythms of the Koyukon language have been deeply nourished by these nonhuman voices.
Hence the whirring, flutelike phrases of the hermit thrush, which sound in the forest thickets at twilight, speak the Koyukon words _sook'eeyis deeyo_ —"it is a fine evening." The thrushes also sometimes speak the phrase _nahutl-eeyh_ —literally, "a sign of the spirit is perceived." The thrush first uttered these words in the Distant Time, when it sensed a ghost nearby, and even today the call may be heard as a warning.
In fact, many of the phrases spoken by birds are understood by reference to events that happened in the Distant Time, events that contemporary Koyukon persons know of through the innumerable Distant Time stories that are told and retold from one generation to another.
Once, during the Distant Time, a starving man struggled in deep spring snow, trying to reach a camp called "Ts'eetee Tlot." He was carrying a headband decorated with elongated, ivory-colored dentalium shells that reached the north country through trade from distant places on the coast. It was a hard spring. The man became weaker and weaker, until finally he collapsed in the snow and died. At that moment he was transformed into a white-crowned sparrow, and then he flew on toward his destination. When he reached the camp he sang: _Dzo do'o sik'its'eetee tlot_. "Here is Tse'eetee Tlot, but it is too late." Anyone who listens to a white-crowned sparrow today can still hear these melancholy words. And anyone who looks closely will see the white stripes on its head, remnants of the dentalium shell band he carried to his death long ago.
Another bird commonly seen in the boreal forest is the Bohemian waxwing as it hurries in small flocks from one tree to another, uttering high, wispy trills. The Koyukon call the waxwing _diltsooga_ —"he squeaks."
According to a Distant Time story, the waxwing had a very jealous wife who once dragged him around by the hair, giving him the crest that now adorns his crown and making him cry out until his voice became nothing but a squeak.
Meanwhile, the lesser yellowlegs, a shorebird, sometimes flies straight up, then utters a piercing call as it descends: " _Siyeets, siyeets,siyeets,"_ which means "My breath, my breath, my breath" in Koyukon. Sometimes a person will shout back to it— _"Siyeets!"_ —hoping to receive from the bird some indication or omen of how long his or her life (her span of breath) will be.
Many birds offer such vocal prophesies to the Koyukon. Once, Nelson's principal Koyukon instructor, along with her grandfather, heard a grey jay speak in an uncommonly human voice:
Rain was falling, and the bird sat on a branch overhead, looking soggy and disheveled. Suddenly it spoke in clear words, "My brother... my brother, what is going to happen?" The old man, a shaman, was startled by the voice and worried by its message. Afterward the rain poured down for nine days, flooding bears from their dens and creating general havoc. And then people knew what the bird had meant.
However, the preeminent prophet or seer among birds is the great horned owl, which is called by the Koyukon _nigoodzagha_ (small ears) or _nodneeya_ (tells you things). The horned owl dwells in the north country year-round, rarely seen but often heard, and is sometimes hunted for food. According to the Koyukon, when the _nodneeya_ speaks to human persons, it utters only what is certain:
When it is about to speak prophetically, the bird first makes a muffled squawking sound—then it hoots in tones and patterns that can be interpreted. The most terrifying words it can say are "Soon you will cry" ( _"Adakk'ut daa'tohtsah"_ ), meaning that someone close to you will die. It may even seal the forecast tightly with a name, and not long afterward its omen will be fulfilled.
Once, some years ago, people heard a horned owl clearly intone the Koyukon words "Black bears will cry." For the next two seasons, the wild berry crops failed and many bears found it hard to survive.
The owl's augury is not always foreboding. Sometimes it seems to call repeatedly in Koyukon, "You will eat the belly of something," foretelling good luck in one's hunting. It can also predict imminent storms. According to one Koyukon elder: "When the owl makes a kind of grunting sound, like this, _Mmmmm... Mmmmm_ , it means stormy weather is coming. Owl's call, that's the only weather report we used to have!"
Meanwhile, the robins, when they sing their lilting phrases, are experienced by the Koyukon as making a short speech: _"Dodo Silinh k'oolkkoy ts'eega, tilzoot tilzoot silnee silnee"_ —"Down there, my brother-in-law tells me to eat pike entrails." Yet the tribespeople, ever attentive to shifts in the surrounding environment, have noticed that the robin's song is itself shifting. One of them remarked to Nelson: "Even the birds are changing. The robins don't say their song plainly anymore—they only say it halfway, like a kid would when its learning."
Another conspicuous bird in the Koyukon bioregion is the fox sparrow, whose loud and oft-heard call, " _Sitsoo sidziy huldaghudla gheeyits_ ," is a sorrowful lament, understood only by reference to a vivid Distant Time story:
In the Distant Time there was a beautiful woman who lived with her husband and grandmother. Once, when her husband was away, the old woman pretended to search through her granddaughter's hair for lice but instead she thrust a bone awl into her ear and broke it off, killing her. Then she took her scalp and put it on her own head, disguising herself as the wife. She also put a bone needle into her navel and twisted it to tighten the loose, flabby skin on her belly. Finally she put on the younger woman's clothes; and disguised this way she fooled the husband into thinking she was his wife.
But when she carried game from his canoe she could not move nimbly, so she had to excuse herself by saying that work made her feel stiff. After they went to bed, however, the husband recognized who she was. He remained quiet until the next morning, and then he killed the old woman and dragged her body into the woods, where he also found his wife lying dead.
Then the young woman's body became a little bird that flew into the air, singing: _sitsoo sidziy huldaghudla gheeyits_ , "Grandmother poked a bone awl in my ear." Nowadays the fox sparrow still sings this way....
The telling of Distant Time stories is central to the Koyukon way of life. Some of the story cycles are so long that their telling consumes many evenings, even several weeks of evenings. By describing the emergence of the world into its evident form, and by thus articulating the formal relations that exist between the various entities in the enveloping cosmos (i cluding humans and other land animals, birds, fish, the various trees and plants, conspicuous landforms, bodies of water and weather patterns—all of whom, in that time out of time, shared a common society and spoke a common tongue), the Distant Time stories make explicit the proper etiquette that must be maintained by the Koyukon people when dealing with the diverse presences that surround them, the kinships that must be celebrated and the taboos that must be respected if the human community and the land are to support and sustain one another.
Distant Time stories are told only during the late fall and the first half of the long northern winter. Indeed, scholars of native lore have found this to be an almost continentwide rule: throughout North America, at least prior to 1900, native communities listened to their most sacred stories only at night and only during the winter. For the spoken stories themselves carry a magic, a power to influence not only persons but the living land itself; in the dark winter night a story well told may hasten the coming of spring. (Thus, a Koyukon teller may conclude a story with a phrase such as "I thought that winter had just begun, but now I have chewed off part of it.") The dark of winter, when some of the most powerful animals are hibernating, when other animals have gone south and the land itself is sleeping, is also the _safest_ time to recount the stories; during the summer, when most of the animals are out and about, the animals and other natural powers may get upset at hearing themselves and their Distant Time exploits referred to so directly.
_For since the other animals themselves speak, they can also hear and understand our own talking_. We must be careful what we say about animals, especially when they are nearby. The Koyukon people take great care to avoid speaking of certain animals directly, using elaborate circumlocutions so as not to offend them. It is for this reason that at night the red squirrel is never spoken of by its ordinary names, but is referred to by the indirect appellation _dikink k'alyee_ —"the one that is on the side of a tree." Women, because they have an excess of spiritual power, must avoid calling the otter by its real name, lest they frighten it, and so refer to the animal only indirectly as _biziya_ —"shiny black." The lynx, another profoundly potent animal to the Koyukon, is called by the women _nodooya_ , a vague circumlocution that means "something going around." To speak carelessly or to disregard such taboos, which hold for many of the forest animals, would invite bad luck for oneself and one's family.
Such roundabout ways of speaking are particularly important during the hunt, when the slightest disrespect for the hunted animal may ensure failure, not just in the present but in future hunts as well. "Hunting black bears in their dens required many gestures of respect, beginning with the etiquette of speech." Preparing for such an encounter, the hunter cannot speak of his intentions directly, and afterward, even if successful, he must not tell what he has done. Later, in the evening perhaps, he might obliquely tell someone, "I found something in a hole." To speak any more directly would offend the powerful being that he has killed.
As the anthropologist Richard Nelson spends more time with the Koyukon, the efficacy of such spoken etiquette begins to influence even his own solitary experience. At home on the Alaskan coast, preparing for a trip back to Koyukon country, he decides to catch a halibut to bring his native friends. Never even considering that he might not be successful, he mentions to a friend that he will take the whole fish to their village so that they can see what it looks like. But
[a]s the words came out, I knew Koyukon people would never talk as if catching a fish was a foregone conclusion. That day I spent hours in places where I'd done well all summer, and caught nothing except one quillback and a lingcod so small I didn't have the heart to keep it. When I arrived at the [Koyukon] village and told Sarah Stevens, she shook her head like a mother gently scolding her child. "The most you should say is that you'll _try_ to catch a fish, or better yet, don't say anything at all. Otherwise it sounds like you're bragging, and the animals always stay away from people who talk like that."
Of course, it is not only when speaking of other animals that one must be mindful, but also when alluding to the forest trees, to the rivers, even to the winds and the weather. Nelson, stung by the winter cold, reminds himself of the Koyukon elders' advice "about accepting the weather as it comes and avoiding remarks that might offend it. This is especially true of cold, which has great power and is easily provoked to numbing fits of temper."
All things can hear and understand our speaking, for all things are capable of speech. Even the crackling sounds made by the new ice on the lakes are a kind of earthly utterance, laden with meaning:
In falltime you'll hear the lakes make loud, cracking noises after they freeze. It means they're asking for snow to cover them up, to protect them from the cold....
Such deference in the face of natural elements—the clear sense that the animate terrain is not just speaking to us but also _listening_ to us—bears out Merleau-Ponty's thesis of perceptual reciprocity; to listen to the forest is also, primordially, to feel oneself listened to _by_ the forest, just as to gaze at the surrounding forest is to feel oneself exposed and visible, to feel oneself watched _by_ the forest.
Much as humans communicate not only with audible utterances but with visible movements and gestures, so the land also speaks to the Koyukon through visible gestures and signs. The way a raven flies in the wind, swerving or gliding upside down, may indicate success or failure in the hunt; the movements of other animals may indicate the presence of danger, or the approach of a storm, or that the spring thaw will come early this year. The assumption, common to alphabetic culture, that "reading omens" is a superstitious and utterly irrational activity, prevents us from recognizing the practical importance, for foraging peoples, of such careful attention to the behavior of the natural surroundings. This watching and interpreting of the world's gestures, as if every movement bears a meaning, accords with a worldview that simply has no notion of pure meaninglessness. No event for the Koyukon is ever wholly accident or chance, but neither is any event entirely predetermined. Rather like the trickster, Raven, who first gave it its current form, the sensuous world is a spontaneous, playful, and dangerous mystery in which we participate, an animate and articulate field of powers ever responsive to human actions and spoken words.
### The Storied Earth
We have begun exploring some evidence for the thesis that language, in indigenous oral cultures, is experienced not as the exclusive property of humankind, but as a property of the sensuous life-world. We've been pondering, that is, some of the ways in which the human discourse within indigenous, oral communities responds directly to the felt expressiveness of other species, of the elements, of the intelligent, animate earth. I have drawn some obvious examples from an equatorial culture embedded in the Amazonian jungle and from a society of the subarctic taiga, or boreal forest. Let's now shift our attention away from forests, whether equatorial or subarctic, toward the arid, desert ecology of the American southwest—in particular, toward the terrain inhabited by the Western Apache of Arizona.
The Apache languages are, like Koyukon, part of the vast Athapaskan family of languages, but the Apachean peoples split off from the northern Athapaskans around one thousand years ago, and eventually established themselves in the American Southwest. In turning from Koyukon culture to Apache culture, we move from an indigenous community that, by virtue of its semiarctic location, has until recently been well insulated from the full impact of European civilization, to a native society that, at least since being confined to the Fort Apache Indian Reservation in 1872, has been surrounded and circumscribed by an ever-expanding population of European settlers. Yet the Apache, despite multiple generations of confrontation, confinement, and forced assimilation, have retained many of their distinct lifeways and linguistic practices. Keith Basso is a linguistic anthropologist who has worked with the Western Apache from 1959 until the present, living intermittently at Cibecue (from the Apache phrase _deeschii′bikoh_ —"valley with elongated red bluffs"), a village of about eleven hundred people that has been inhabited by the Apache for centuries.
As he became conversant in the Apache language, and attuned to the rhythms of life in the village, Basso began to notice the remarkable frequency with which place-names typically arise in Western Apache discourse. The Apache seem to take great pleasure simply in uttering the native names of various locations within the Cibecue valley. For instance, while stringing a fence with two Apache cowboys, Basso noticed one of them talking quietly to himself. When he listened more closely, Basso discovered that the man was reciting a long series of place-names—"punctuated only by spurts of tobacco juice"—that went on for almost ten minutes. Later, when Basso asked him what he'd been doing, the man replied that he often "talked names" to himself. "I like to," he told the anthropologist. "I ride that way in my mind." Another Apache told Basso that his people like to pronounce place-names "because those names are good to say."
The evident pleasure derived from saying these names is clearly linked to the precision with which Apache place-names depict the actual places that they name. Basso himself mapped 104 square kilometers in and around Cibecue, and within this area recorded the Apache names of 296 locations. He found that all but a few of these place-names take the form of complete sentences, each name invoking its place through a succinct yet precise visual description. Here are a few such names: "big cottonwood trees stand spreading here and there"; "coarse textured rocks lie above in a compact cluster"; "water flows down on top of a regular succession of flat rocks." Upon pronouncing, or hearing, such a name, Apache persons straightaway feel themselves in the presence of that place; hence, when reciting a series of place-names, the Apache experience themselves "traveling in their minds." It would seem that the spoken place-names, by their precision, effect a direct sensorial bond between Apache persons and particular places, and we may suspect that the benefit drawn from speaking these names aloud derives not so much from the names themselves but from the nourishing power of the actual locations to which the names draw those who speak them. Place-names, that is, seem to take their particular power and magic from the actual places that they designate.
The experiential importance of geographic place for the Western Apache, and the consequent influence of particular locations in the surrounding landscape upon their everyday language, is especially evident with regard to the ethics and etiquette of contemporary Apache society. For, in a manner entirely alien to alphabetic civilization, the land itself is the ever-vigilant guardian of right behavior within traditional Apache culture. According to Mrs. Annie Peaches, a seventy-seven-year-old Apache woman:
The land is always stalking people. The land makes people live right. The land looks after us. The land looks after people.
The moral efficacy of the landscape—this power of the land to ensure mindful and respectful behavior in the community—is mediated by a whole class of stories that are regularly recounted within the village. These narratives tell of persons who underwent misfortune as a consequence of violating Apache standards for right behavior; they tell of individuals who, as a result of acting impulsively or in open defiance of Apache custom, suffered humiliation, illness, or death. Unlike the long cosmological myths told only by medicine persons, and unlike the sagas of the contemporary world told primarily for entertainment, these tales—called _′agodzaahi_ (literally, "that which has happened")—are typically very brief; they can usually be told in less than five minutes. More significantly, _′agodzaahi_ tales always begin and end with a statement that indicates, with a place-name, exactly _where_ the events in the story actually occurred. Here is an example of such a story:
It happened at "whiteness spreads out descending to water."
Long ago, a boy went out to hunt deer. He rode on horseback. Pretty soon he saw one [a deer], standing on the side of a canyon. Then he went closer and shot it. He killed it. Then the deer rolled all the way down to the bottom of the canyon.
Then the boy went down there. It was a buck, fat and muscular. There he butchered it. The meat was heavy, so he had to carry it up in pieces. He had a hard time reaching the top of the canyon with each piece.
Now it was getting dark. One hindquarter was still lying at the bottom of the canyon. "I have enough meat already," he thought. So he left the hindquarter where it was lying. He left it there.
Then he packed his horse and started to ride home. Then the boy got dizzy and nearly fell off his horse. Then his nose twitched uncontrollably, like Deer's nose does. Then pain shot up behind his eyes. Then he became scared.
Now he went back to the canyon. It was dark when he got there. He walked down to where the hindquarter was lying—but it was gone! Then he returned to his horse. He rode fast to where he was living with his relatives.
The boy was sick for a long time. The people prayed for him on four separate occasions. He got better slowly.
Some time after that, when the boy had grown to manhood, he always had bad luck in hunting. No deer would present themselves to him. He said to his children: "Look at me now. I failed to be careful when I was a boy and now I have a hard time getting meat for you to eat."
It happened at "whiteness spreads out descending to water."
This tale of "that which has happened" illustrates the misfortunes that might befall a hunter who neglects the respect that must be continually maintained with his animal prey, or, more broadly, the strife that attends those who fail to observe the proper etiquette in their interactions with the natural world. Yet many _′agodzaahi_ tales deal solely with the right relations that must be sustained between individual persons and the larger tribal community:
It happened at "men stand above here and there."
Long ago, a man killed a cow off the reservation. The cow belonged to a Whiteman. The man was arrested by a policeman living at Cibecue at "men stand above here and there." The policeman was an Apache. The policeman took the man to the head Army officer at Fort Apache. There, at Fort Apache, the head Army officer questioned him. "What do you want?" he said. The policeman said, "I need cartridges and food." The policeman said nothing about the man who had killed the Whiteman's cow. That night some people spoke to the policeman. "It is best to report on him," they said to him. The next day the policeman returned to the head Army officer. "Now what do you want?" he said. The policeman said, "Yesterday I was going to say 'HELLO' and 'GOODBYE' [to you] but I forgot to do it." Again he said nothing about the man he arrested. Someone was working with words on his mind. The policeman returned with the man to Cibecue. He released him at "men stand above here and there."
It happened at "men stand above here and there."
This particular story demonstrates the confusion that befalls an Apache person who acts too much like a white man. In the early years of the reservation, disease and malnutrition took the lives of many tribespeople. And so it is perfectly understandable to the Apache people that one of them would have killed a white man's cow for food. It was _not_ acceptable, however, that _another_ Apache would arrest him with the intent of taking him to jail. In other words, it is wrong to join with outsiders against members of one's own community, or to flaunt one's disrespect for the tribe by taking on the attitudes and mannerisms of white men or women. Hence, the policeman in the story found himself unable to turn in the man that he had arrested, although he twice attempted to do so. Unable to speak his purpose, he was humiliated and made to look foolish before the head officer. Finally, he released the man at the same place where he had arrested him.
Now let us see how the actual place where these events unfolded contributes to the operative potency of the _′agodzaahi_ tales. The telling of any such tale today is always prompted by a misdeed committed by someone in the community; the _′agodzaahi_ story, precisely told, acts as a remedial response to that misdeed. Thus, when an Apache person offends the community by a certain action, one of his or her elders will wait for an appropriate moment—perhaps at a community gathering—and will then "shoot" the person by recounting an appropriate _′agodzaahi_ story. Although the offender is not identified or named aloud, he or she will know, if the "arrow" (the tale) has been well chosen and well aimed, that he is the target; he will feel the story penetrate deep beneath his skin and sap his strength, making him feel ill and weak. Then the story will begin to work on him from within, making him want to change his ways, to "replace himself," to live right. And so his behavior will change. Yet the story will stay with him. For he will continually encounter the place in the land where it all happened. Perhaps, if that location is near his home, he will see it everyday. _The place_ , it is said, _will keep "stalking" him. 46_
Basso himself relates an example of such a story "going to work" on a person. In June 1977 he was present at a birthday party in Cibecue that was also attended by a young woman who two weeks earlier had gone to a girls' puberty ceremonial with her hair rolled up in a set of oversized pink plastic curlers. Although such ornamentation was no doubt considered fashionable at the off-reservation boarding school where the young woman lived, it was a clear affront to Apache custom to appear thus adorned at a traditional ceremony. Two weeks later, Basso recalls, in the midst of casual conversation at the birthday party, the young woman's maternal grandmother suddenly narrated a version of the above _′agodzaahi_ tale regarding the Apache policeman who had behaved overmuch like a white man. Shortly after hearing the story, the young woman stood up and silently walked away from the party. When Basso, uncertain of what had happened, asked her grandmother if the woman was ill, the grandmother replied simply, "No. I shot her with an arrow."
Two summers later Basso again met the young woman and, while helping her home with some groceries, asked if she remembered that party and why she had left so suddenly. The woman then told him that she had thrown the curlers away after hearing the story about the policeman. When Basso pointed out, as they passed it, the place where the story's events occurred ("men stand above here and there"), the woman "said nothing for several moments. Then she smiled and spoke softly in her own language: 'I know that place. It stalks me every day.' "
In this uniquely oral form of community censure, a topographic place becomes the guarantor of corrected behavior, the visible presence that reminds one of past foibles and that ensures one's subsequent attentiveness. The telling of _′agodzaahi_ tales establishes an almost familial bond between the persons at whom the stories are aimed and particular sites or features of the natural landscape. According to an Apache elder,
[i]t doesn't matter if you get old—that place will keep on stalking you like the one who shot you with the story. Maybe that person will die. Even so, that place will keep on stalking you. It's like that person is still alive.
Hence, Apache persons often associate places with particular ancestors. Indeed, the earthly places seem to _speak_ to certain persons in the voices of those grandparents who first "shot" them with stories, or even to speak in the voices of those long-dead ancestors whose follies and exploits are related in the _′agodzaahi_ tales. The ancestral wisdom of the community resides, as it were, in the stories, but the stories—and even the ancestors themselves—reside in the land.
We used to survive only off the land. Now its no longer that way. Now we live only with money, so we need jobs. But the land still looks after us. We know the names of the places where everything happened. So we stay away from badness.
Yet to move away from the land is ultimately to lose contact with the actual sites invoked by the place-names, and so to lose touch with the spoken stories that reside in those places.
One time I went to L.A., training for mechanic. It was no good, sure no good. I start drinking, hang around bars all the time. I start getting into trouble with my wife, fight sometimes with her. It was _bad_. I forget about this country here around Cibecue. I forget all the names and stories. I don't hear them in my mind anymore. I forget how to live right, forget how to be strong.
Basso, the anthropologist, presents a largely functional explanation for the native association of moral teachings with geographical sites. "Mountains and arroyos," he writes, "step in symbolically for grandmothers and uncles." Persons must be continually attentive to maintaining right behavior, especially with regard to those situations in which they were once careless and impulsive, and yet the grandmothers and uncles who originally corrected such behavior necessarily grow old and perish. Since earthly sites readily outlast one's human elders, and indeed maintain their basic character across many generations, such places are perfectly suited to "step in" as ever-present symbolic reminders of the moral lessons learned in the past.
Yet Basso's suggestion that the sites in the land serve a "symbolic" function (that they have come to "symbolize" moral teachings) implies an unwarranted degree of arbitrariness to the association between moral lessons and the natural landscape, by implying that the association is more conceptual or pragmatic than it is organic and unavoidable. The suggestion masks the extent to which the places themselves may be felt to be the active instigators of those painful lessons, the ultimate authors of those events and hence those stories. Note, here, Basso's own stress on the primacy of place in Western Apache storytelling:
Nothing is considered more basic to the effective telling of a Western Apache "story" or "narrative"... than identifying the geographical locations at which events in the story unfold. For unless Apache listeners are able to picture a physical setting for narrated events (unless, as one of my consultants said, "your mind can travel to the place and really see it"), the events themselves will be difficult to imagine. This is because events in the narrative will seem to "happen nowhere" ( _dohwaa′agodzaa da_ ), and such an idea, Apaches assert, is both preposterous and disquieting. Placeless events are an impossibility, everything that happens must happen somewhere. The location of an event is an integral aspect of the event itself, and therefore identifying the event's location is essential to properly depicting—and effectively picturing—the event's occurrence.
Basso makes evident here the central importance of place in the Western Apache experience of phenomena. Yet he provides no indication of why the Apache should put so much more stress on geographical location than we do. Surely for non-native persons, as well, "all things that happen must happen somewhere." Yet most of us do not insist on identifying the precise location of every event we hear about. Why, then, do the Apache, and native cultures in general, give so much importance to places?
The answer should by now be obvious. To members of a nonwriting culture, places are never just passive settings. Remember that in oral cultures the human eyes and ears have not yet shifted their synaesthetic participation from the animate surroundings to the written word. Particular mountains, canyons, streams, boulder-strewn fields, or groves of trees have not yet lost the expressive potency and dynamism with which they spontaneously present themselves to the senses. A particular place in the land is never, for an oral culture, just a passive or inert setting for the human events that occur there. _It is an active participant in those occurrences_. Indeed, by virtue of its underlying and enveloping presence, the place may even be felt to be the source, the primary power that expresses itself through the various events that unfold there.
It is precisely for this reason that stories are not told without identifying the earthly sites where the events in those stories occur. For the Western Apache, as for other traditionally oral peoples, human events and encounters simply cannot be isolated from the places that engender them. Thus, anthropologist Harry Hojier, speaking of another Athapaskan group—the Diné, or Navajo—notes:
Even the most minute occurrences are described by Navajos in close conjunction with their physical settings, suggesting that unless narrated events are _spatially anchored_ their significance is somehow reduced and cannot be properly assessed.
Yet here again the professional anthropologist subtly misses the primary reason for this conjunction. By suggesting that narrated events must be "spatially anchored" he allows us to assume a purely external relation between events and their geographical settings; he implies that the events could be conceived as floating free of any locale before dropping anchor and binding themselves to the land. If, however, the place is itself an active element in the genesis of the event, then the metaphor of _a root_ is far more precise than that of an anchor; to an oral culture, experienced events remain rooted in the particular soils, the particular ecologies, the particular places that give rise to them.
FROM THE DISTANT TIME STORIES OF THE KOYUKON PEOPLE, AND from the _′agodzaahi_ tales of the Western Apache, we begin to discern that storytelling is a primary form of human speaking, a mode of discourse that continually weds the human community to the land. Among the Koyukon, the Distant Time stories serve, among other things, to preserve a link between human speech and the spoken utterances of other species, while for the Western Apache the ′ _agodzaahi_ narratives express a deep association between moral behavior and the land and, when heard, are able to effect a lasting kinship between persons and particular places.
The telling of stories, like singing and praying, would seem to be an almost ceremonial act, an ancient and necessary mode of speech that tends the earthly rootedness of human language. For narrated events, as Basso reminds us, always happen _somewhere_. And for an oral culture, that locus is never merely incidental to those occurrences. The events belong, as it were, to the place, and to tell the story of those events is to let the place itself speak through the telling.
Yet there remains another reason for the profound association between storytelling and the more-than-human terrain. It resides in the encompassing, enveloping wholeness of a story in relation to the characters that act and move within it. A story envelops its protagonists much as we ourselves are enveloped by the terrain. In other words, we are situated in the land in much the same way that characters are situated in a story. Indeed, for the members of a deeply oral culture this relation may be experienced as something more than a mere analogy: along with the other animals, the stones, the trees, and the clouds, we ourselves are characters within a huge story that is visibly unfolding all around us, participants within the vast imagination, or Dreaming, of the world.
### Dreamtime
With this thought we bring ourselves very close to the Dreamtime beliefs common to the Aboriginal peoples of Australia. Their diverse cultures—Pintupi, Pitjantjatjara, Aranda, Kaititj, Warumungu, Walbiri, and a host of others—may well be the oldest human cultures of any still in existence, cultures that have evolved in some of the harshest of human environments for tens of thousands of years (the earliest Aboriginal remains discovered in Australia are between forty thousand and sixty thousand years old), only to be decimated in our own time through contact with alphabetic civilization. The astonishing endurance of the Aboriginal peoples must be attributed, at least partially, to their minimal involvement with technologies. Their relation to the sustaining landscape was direct and intimate, unencumbered by unnecessary mediations. They relied upon only the simplest of tools—primarily the boomerang, the hunting spear, and the digging stick—and thus avoided dependence upon specialized resources while maintaining the greatest possible mobility in the face of climatic changes. Meanwhile, the isolation of their continent, as well as its outwardly inhospitable character, clearly protected these peoples from onslaught by more ambitious or expansionist nations—until, that is, the British arrived on their coast in 1788.
What, then, _is_ the Dreamtime—the _Jukurrpa_ , or _Alcheringa—_ that plays such a prominent part in the mythology of Aboriginal Australia?
It is a kind of time out of time, a time hidden beyond or even _within_ the evident, manifest presence of the land, a magical temporality wherein the powers of the surrounding world first took up their current orientation with regard to one another, and hence acquired the evident shapes and forms by which we now know them. It is that time before the world itself was entirely awake (a time that still exists just below the surface of wakeful awareness)—that dawn when the totem Ancestors first emerged from their slumber beneath the ground and began to sing their way across the land in search of food, shelter, and companionship.
The earth itself was still in a malleable, half-awake state, and as Kangaroo Dreaming Man (the ancestral progenitor not only of kangaroos but of all humans who are born of Kangaroo Dreaming), Frilled Lizard Man, Tortoise Woman, Little Wallaby Man, Emu Woman, and innumerable other Ancestors wandered, singing, across its surface, they shaped that surface by their actions, forming plains where they lay down, creeks or waterholes where they urinated, forests where they kicked up dust, and so on.
Gabidji, Little Wallaby, came from the West to Ooldea Soak. He came across the large western sand-ridge, close to a black desert-oak tree. He was carrying a _malu-meri_ or _buda_ skin waterbag, which was full. He crossed the ridge and came to Yuldi. There he put his _buda_ at the base of a large sand dune to the south, and urinated in a depression which became the present-day Ooldea Soak ("That's the water we drink now!" said the people in 1941.) He stayed there for a while, and then went on to another sandhill to the north; from there he looked out toward the east. That sandhill was named Bimbali. He returned to pick up his _buda_ , and then he spilt a little water, and that became the lake. However, he was not sure whether he should go farther and finally decided to return to Ooldea. He left his _buda_ there and it was metamorphosed as the large southern sandhill. "That's why there is always water there." He camped for a while, then decided to go east again....
Eventually, having found an appropriate location, or simply exhausted from the work of world-shaping, each of the Ancestors went "back in" (becoming _djang_ , in Gunwinggu terminology), transforming himself (or herself) into some physical aspect of the land, and/or metamorphosing into the plant or animal species from which he takes his name.
[Leech Man] looked this way, that way, as he was coming. He saw a good place. He said, "I do this, because it's a good place. I'll settle down, I'll stay always." That man who was eating fish, Naberg-gaidmi, asked him, "What are you?," and he said, "I'm turning into Leech, I'm going to stay in one place. I'm going to become a rock, a little rock, and stay here, with a flat head, a short head. I'm Leech _djang_ , Leech Dreaming!" he said. "I'm Leech!" and he said, "Here I sit. This is my creek flowing, this is mine, where I'm staying. I'm _djang_ , Dreaming!"
Each Ancestor thus leaves in his wake a meandering trail of geographic sites, perceivable features in the land that are the result of particular events and encounters in that Ancestor's journey, culminating in that place where the Ancestor went "back in," metamorphosing entirely into some aspect of the world we now experience.
These meandering trails, or Dreaming tracks, are auditory as well as visible and tactile phenomena, for the Ancestors were singing the names of things and places into the land as they wandered through it. Indeed, each ancestral track is a sort of musical score that winds across the continent, the score of a vast, epic song whose verses tell of the Ancestor's many adventures, of how the various sites along her path came into being (and hence, indirectly, of what food plants, water sources, or sheltering rocks may be found at those sites). The distance between two significant sites along the Ancestor's track can be measured, or spoken of, as a stretch of song, for the song unfolds in an unbroken chain of couplets across the land, one couplet "for each pair of the Ancestor's footfalls." The song is thus a kind of auditory route map through the country; in order to make her way through the land, an Aboriginal person has only to chant the local stanzas of the appropriate Dreaming, the appropriate Ancestor's song.
The Australian continent is crisscrossed by thousands of such meandering "songlines" or "ways through," most of them passing through multiple tribal areas. A given song may thus sing its way through twenty or more different languages before reaching the place where the Ancestor went "back in." Yet while the language changes, the basic melody of the song remains the same, so that a person of the Barking Lizard Clan will readily recognize distant stretches of the Barking Lizard songline when he hears them, even though those stanzas are being sung in a language entirely alien to his ears.... Knowledge of distant parts of one's song cycle—albeit in one's own language—apparently enables a person to vividly experience certain stretches of the land even before he or she has actually visited those places. Rehearsing a long part of a song cycle together while sitting around a campfire at night, Aboriginal persons apparently feel themselves journeying across the land in their collective imagination—much as the Apache man "talking names" to himself is "riding in his mind."
Every Ancestor, while chanting his or her way across the land during the Dreamtime, also deposited a trail of "spirit children" along the line of his footsteps. These "life cells" are children not yet born: they lie in a kind of potential state within the ground, waiting. While sexual intercourse between a woman and a man is thought, by traditional Aboriginal persons, to _prepare_ the woman for conception, the actual conception is assumed to occur much later, when the already pregnant woman is out on her daily round gathering roots and edible grubs, and she happens to step upon (or even near) a song couplet. The "spirit child" lying beneath the ground at that spot slips up into her at that moment, "works its way into her womb, and impregnates the foetus with song." Wherever the woman finds herself when she feels the _quickening_ —the first kick within her womb—she knows that a spirit child has just leapt into her body from the earth. And so she notes the precise _place_ in the land where the quickening occurred, and reports this to the tribal elders. The elders then examine the land at that spot, discerning which Ancestor's songline was involved, and precisely which stanzas of that Ancestor's song will belong to the child.
In this manner every Aboriginal person, at birth, inherits a particular stretch of song as his private property, a stretch of song that is, as it were, his title to a stretch of land, to his conception site. This land is that part of the Dreaming from whence his life comes—it is that place on the earth where he most belongs, and his essence, his deepest self, is indistinguishable from that terrain:
_Nyunymanu:_
_dingo [wild dog] dreaming place_
_Paddy Anatari's country_.
_Old man squints between wrinkles_
_drawn into a smile in the broad, red land_.
_Played a child; walked every foot in its sand_.
_"You see that rock over there?"_
_(The top had been rubbed smooth and_
_flat soft, as if it were cut by a diamond, but_
_its been done by another rock_
_cupped in hundreds of hands:_
_increase site for birthing of dingo pup)_
_and_
_Paddy Anatari strokes the rock again_ ,
_and again. He says:_
_"You see this rock?_
_This rock's_ me!"
The sung verses that are the tribesman's birthright, of which he is now the primary caretaker, provide him also with a kind of passport to the other lands or territories that are crossed by the same Dreaming. He is recognized as an offspring of that Ancestor whose songline he owns a part of, a descendant of the Dreamtime Being whose sacred life and power still dwells within the shapes of those lands. If, for instance, the Ancestor who walked there was Wallaby Man, then the person is said to have a Wallaby Dreaming, to be a member of the Wallaby Clan (a wallaby is a marsupial animal resembling a small kangaroo). He has allegiances to all other Wallaby Dreaming persons, both within and outside of his own tribe. He has responsibilities to the wallabies themselves; he cannot hunt them for food, since they are his brothers and sisters. And he has a profound responsibility to the land along the Wallaby Dreaming track, or songline, a responsibility to keep the land as it should be— _the way it was when it was first sung into existence_.
According to tradition, he might do this by periodically going "Walkabout," by making a ritual journey along the Dreaming track, walking in the footsteps of the clan Ancestor. As he walks, he chants the Ancestor's verses, without altering a single word, singing the land into view—and in this manner "recreates the Creation."
Finally, just as each Dreamtime Ancestor metamorphosed him-or-herself, at the end of her journey, into some aspect or feature within the contemporary landscape, so also each Aboriginal person intends, at the end of his or her life, to sing himself back into the land. A traditional Pitjantjatjara or Pintupi man will return to his conception site—to his particular stretch of the Ancestral songline—to die, so that his vitality will be able to rejoin the dreaming earth at that place.
The Dreamtime is not, like the Western, biblical notion of Genesis, a finished event; it is not, like the common scientific interpretation of the "Big Bang," an event that happened once and for all in the distant past. Rather, it is an ongoing process—the perpetual emerging of the world from an incipient, indeterminate state into full, waking reality, from invisibility to visibility, from the secret depths of silence into articulate song and speech. That Native Australians chose the English term "Dreaming" to translate this cosmological notion indicated their sense that the ordinary act of dreaming participates directly in the time of the clan Ancestors, and hence that that time is not entirely elsewhere, not entirely sealed off from the perceivable present. Rather, the Dreaming lies in the same relation to the open presence of the earth around us as our own dream life lies in relation to our conscious or waking experience. It is a kind of depth, ambiguous and metamorphic.
_[See there,] That tree is a digging stick_
_left by the giant woman who was looking_
_for honey ants;_
_That rock, a dingo's nose;_
_There, on that mountain, is the footprint_
_left by Tjangara on his way to Ulamburra;_
_Here, the rockhole of Warnampi—very dangerous—_
_and the cave where the nyi-nyi women escaped_
_the anger of marapulpa—the spider_.
_Wati Kutjarra—the two brothers—travelled this way_.
_There, you can see, one was tired_
_from too much lovemaking—the mark of his penis_
_dragging on the ground;_
_Here, the bodies of the honey ant men_
_where they crawled from the sand—_
_no, they are not dead—they keep coming_
_from the ground, moving toward the water at Warumpi—_
_it has been like this for many years:_
_the Dreaming does not end; it is not like the whiteman's way_.
_What happened once happens again and again_.
_This is the Law_ ,
_This is the power of the Song_.
_Through the singing we keep everything alive;_
_through the songs... the spirits keep us alive_.
What happened once happens again and again. The Dreaming, the imaginative life of the land itself, must be continually renewed, and as an Aboriginal man walks along his Ancestor's Dreaming track, singing the country into visibility, he virtually _becomes_ the journeying Ancestor, and thus the storied earth is born afresh.
This identification, this bleeding of the Dreamtime into the here and now, happens not just during the solitary Walkabout, but also and especially during the collective rituals held at specific Dreaming sites, rituals wherein the Ancestors' encounters and adventures at those locations are not just sung but also _enacted_ by the elders. Even an "open," greatly abbreviated version of such an enactment can display an astonishing degree of participation with the animal Ancestor (such "open" versions, or sketches, may be performed for strangers). Author Bruce Chatwin witnesses one such sketch by a late-night campfire in the outback. In response to a question from Chatwin's fellow researcher, about the significance of a nearby hill, one of the Aboriginal men
got to his feet and began to mime (with words of pidgin thrown in) the travels of the Lizard Ancestor.
It was a song of how the lizard and his lovely young wife had walked from northern Australia to the Southern Sea, and of how a southerner had seduced the wife and sent him home with a substitute.
I don't know what species of lizard he was supposed to be: whether he was a "jew-lizard" or a "road-runner" or one of those rumpled, angry looking lizards with ruffs around their necks. All I do know is that the man in blue made the most lifelike lizard you could ever hope to imagine.
He was male and female, seducer and seduced. He was glutton, he was cuckold, he was weary traveller. He would claw his lizard-feet sideways, then freeze and cock his head. He would lift his lower lid to cover the iris, and flick out his lizard-tongue. He puffed his neck into goiters of rage; and at last, when it was time for him to die, he writhed and wriggled, his movements growing fainter and fainter....
Then his jaw locked, and that was the end.
The man in blue waved towards the hill and, with the triumphant cadence of someone who has told the best of all possible stories, shouted: "That... that is where he is!"
The nearby hill, in other words, is that place where the Lizard Ancestor had metamorphosed back into the earth—his spirit power, or life, now inseparable from the life of the hill itself.
The enactment of such stories, songs, and ceremonies is done less for the human persons than for the land itself—upon which, of course, the humans depend. In the words of anthropologist Helen Payne:
The maintenance of a site requires both physical caring—for example the rubbing of rocks or clearing of debris—and the performance of [ritual] items aimed at caring for the spirit housed at it. Without these maintenance processes the site remains, but is said to lose the spirit held within it. It is then said to die and all those who share physical features and spiritual connections with it are then also thought to die. Thus, to endure the well-being of life, sites must be cared for and rites performed to keep alive the dreaming powers entrapped within them.
Or as Bruce Chatwin writes, "an unsung land is a dead land."
On certain occasions, traditionally, the elders of a particular clan would decide that it was time to sing their song cycle in all of its intricacies from start to finish. Messages would be sent up and down the Dreaming track, summoning all of the song-owners to gather at one of the important water holes along the Dreaming. Once assembled, each clan member in turn would sing his stretch of the Ancestor's footprints. The precise sequence of the chanted verses was essential; to sing one's stanzas out of order was thought to rupture the coherence of the earth itself.
It is important to realize that in Aboriginal Australia (as throughout indigenous North America) there is a high degree of differentiation between women's knowledge and men's knowledge, women's rituals and men's rituals. The power and importance of women's rites within native Australian cultures has only recently been recognized by nonaboriginal researchers, perhaps because most of the early ethnologists were male, and hence had little or no access to women's sacred knowledge. It is now apparent, as well, that Aboriginal women's song knowledge is more closely guarded than that of the men. In recent years a certain amount of innovation has occurred both in the songs sung by women and those sung by men, especially in response to changes in the landscape, and in Aboriginal society, brought about by industrial civilization. Lost segments of a song cycle, for instance, may be redreamed by qualified persons. Nevertheless, the song knowledge of women (at least in central Australia) has tended to be more conservative, more resistant to change than that of the men. Another difference is this: while men's secret ceremonies seem to focus almost exclusively on renewing the vitality of the particular sites and species being celebrated, women's closed ceremonies often involve, as well, utilizing the songs to _tap_ the magic power of those sites—drawing upon the power in the land for various practical purposes. Such purposes include the curing of illness (whether the sick person is female or male), as well as the practice of "love magic"—whereby the women elders influence, for the good of the community as a whole, the flows of desire between particular persons.
### Place and Memory
In Australia, then, among the least technological of human cultures, we find the most intimate possible relation between land and human language. Language here is inseparable from song and story, and the songs and stories, in turn, are inseparable from the shapes and features of the land. The chanting of any part of a song cycle links the human singer to one of the animals or plants or powers within the landscape, to Crocodile Man or Pandanus Tree Woman or Thunderstorm Man—to whatever more-than-human being first chanted those verses as he or she wandered across the dreaming earth. But it also binds the human singer to the land itself, to the specific hills, rocks, and streambeds that are the visible correlate of those sung stanzas.
The lived affinity between language and the land is well illustrated by an anecdote that American poet Gary Snyder tells, from a visit that he made to Australia in the fall of 1981. Snyder was traveling through part of the central desert in the back of a pickup truck, accompanied by a Pintupi elder named Jimmy Tjungurrayi. As the truck rolled down the road, the old aborigine began to speak very rapidly to Snyder, telling him a Dreamtime story about some Wallaby people and their encounter with some Lizard girls at a mountain they could see from the road. As soon as that story ended, he launched into
another story about another hill over here and another story over there. I couldn't keep up. I realized after about half an hour of this that these were tales meant to be told while _walking_ , and that I was experiencing a speeded-up version of what might be leisurely told over several days of foot travel.
A similar tale is told by Chatwin. He was traveling in a Land Cruiser with several friends, including an Aboriginal man nicknamed Limpy whom they were driving to a particular place on his songline. Limpy, whose clan Ancestor was the Native Cat, or _tjilpa_ (a small marsupial with a long, banded tail), had never been to this place along the Native Cat songline, yet he now wished to go there in order to see some distant relatives who were dying there. During the course of seven hours driving through the back country, bumping across shallow rivers and under gum trees, the Aboriginal man sat motionless in the front seat, squeezed between the driver, Arkady, and another passenger, except for a short burst of action when the truck crossed part of his songline. Later,
[w]e came to the confluence of two streams: that is, we met the stream we had crossed higher up on the main road. This lesser stream was the route of the Tjilpa Men, and we were joining it at right angles.
As Arkady turned the wheel to the left, Limpy bounced into action. Again he shoved his head through both windows. His eyes rolled wildly over the rocks, the cliffs, the palms, the water. His lips moved at the speed of a ventriloquist's and, through them, came a rustle: the sound of wind through branches.
Arkady knew at once what was happening. Limpy had learnt his Native Cat couplets for walking pace, at four miles an hour, and we were travelling at twenty-five.
Arkady shifted into bottom gear, and we crawled along no faster than a walker. Instantly, Limpy matched his tempo to the new speed. He was smiling. His head swayed to and fro. The sound became a lovely melodious swishing; and you knew that, as far as he was concerned, he _was_ the Native Cat....
Such anecdotes make vividly evident the felt correspondence between the oral language and the landscape, an alliance so thorough that the speaker must pace his stories or songs to match the speed with which he moves through the terrain. It is as though specific loci in the land release specific stories or stanzas in those Aboriginal persons who travel by them. Or as though, at such times, it is not the native person who speaks, but rather the land that speaks _through_ him as he journeys across it.
This correspondence between the speaking voice and the animate landscape is an intensely felt affinity, a linkage of immense import for the survival of the people. In a land as dry as the Australian outback, where rainfall is always uncertain, the ability to _move_ in response to climatic changes is indispensable. An oral Dreaming cycle, practically considered, is a detailed set of instructions for moving through the country, a safe way through the arid landscape. Anthropologist Helen Payne has analyzed a continuous series of significant Dreaming sites along a single songline, and found that each of the sites contained either a source of water, a potential shelter, a high vantage point from which to view the surrounding terrain, or a cluster of several such characteristics. Indeed, these Dreaming sites were the _only_ places with such assets in an otherwise arid desert.
Payne found as well that geographic sites of particular abundance were commonly crossed by more than one Dreaming—having figured in the adventures of more than one Dreamtime Ancestor—and were thus sacred to several totemic clans. The number and complexity of the rituals associated with any particular Dreaming site varied in direct proportion to the abundance of food, water, and/or shelter to be found at that place.
Each person, by borrowing or trading for the right to sing distant stretches of her own or another's Dreaming tracks, may continually expand her knowledge of potential routes through the countryside along which she may travel in lean times. And since every Aboriginal band is comprised of individuals from different totemic clans, or Dreamings, it will usually have access to multiple songlines, multiple ways to move whenever lack of water or food necessitates such a move.
The Dreaming songs, in other words, provide an auditory _mnemonic_ (or memory tool)—an oral means of recalling viable routes through an often harsh terrain.
Yet there is another mnemonic structure at work in the Dreaming. The two anecdotes cited above—both of them occurring in moving automobiles—indicate that the telling of specific stories or the chanting of particular songs is itself prompted by the sensible encounter with specific sites. Just as the song structure carries the memory of how to orient in the land, so the sight of particular features in the land activates the memory of specific songs and stories. The landscape itself, then, provides a _visual_ mnemonic, a set of visual cues for remembering the Dreamtime stories.
The importance of this second mnemonic relation becomes apparent as soon as we acknowledge that the songs and stories carry much _more_ than a set of instructions for moving through the terrain. While the topographic function of the songs is obviously of immense importance, the songs and stories also provide the codes of behavior for the community; they suggest, through multiple examples, how to act, or how _not_ to act, in particular situations. The Dreamtime Ancestors depicted in the stories are neither more nor less moral than their human progeny in the contemporary world, yet the situations in which the Ancestors variously find themselves, and the often difficult results that follow from particular actions, offer a ready set of guidelines for proper behavior on the part of those who sing or hear those stories today. Social taboos, customs, interspecies etiquette—the right way to hunt particular animals or gather particular foods and medicines—all are contained in the Dreamtime songs and stories. And it is the land itself that is the most potent reminder of these teachings, since each feature in the landscape activates the memory of a particular story or cluster of stories.
We earlier encountered a similar correspondence among the Western Apache, for whom the auditory memory of particular teaching stories was triggered by contact with the specific sites where those stories unfolded. One of the strong claims of this book is that the synaesthetic association of visible topology with auditory recall—the intertwining of earthly place with linguistic memory—is common to almost all indigenous, oral cultures. It is, we may suspect, a spontaneous propensity of the human organism—one that is radically transformed, yet not eradicated, by alphabetic writing.
Indeed, even within European culture there is a celebrated example of this propensity, albeit in a thoroughly altered form. In her justly famous book, _The Art of Memory_ , Frances Yates describes the mnemonic technique utilized by the classical orators of Greece and Rome to remember their long speeches (a technique regularly practiced by rhetoricians up until the spread of typographic texts during the late Renaissance). The orator would imagine an elaborate palace, filled with diverse halls and rooms and intricate structural details. He would then envision himself walking through this palace, and would deposit at various places within the rooms a sequence of imagined objects associated with the different parts of his planned speech. Thereafter, to recall the entire speech in its correct sequence and detail, the orator had only to envision himself once again walking the same route through the halls and rooms of the memory palace: each locus encountered on his walk would remind him of the specific phrase to be spoken or the particular topic to be addressed at that point within the discourse. Rather than striving to memorize the composed speech on its own, the orator found it much easier, and certainly much safer, to correlate the diverse parts of the speech to diverse _places_ within an imaginary structure, within an envisioned topology through which he could imaginatively stroll. Yet while the classical orators had to construct and move through such topological matrices in their private imaginations, the native peoples of Australia found themselves corporeally immersed in just such a linguistic-topological field, walking through a material landscape _whose every feature was already resonant with speech and song!_
In aboriginal Australia, then, we can discern two basic mnemonic relationships between the Dreamtime stories and the earthly landscape. First, the spoken or sung Dreamings provide a way of recalling viable routes through an often difficult terrain. Second, the continual encounter with various features of the surrounding landscape stirs the memory of the spoken Dreamings that pertain to those sites. While the sung stories provide an auditory mnemonic for orienting within the land, the land itself provides a visual mnemonic for recalling the Dreamtime stories. Thus, for Aboriginal peoples the Dreamtime stories and the encompassing terrain are _reciprocally_ mnemonic, experientially coupled in a process of mutual invocation. The land and the language—insofar as the language is primarily embodied in the ancestral Dreamings—are inseparable.
Given this radical interdependence between the spoken stories and the sensible landscape, the ethnographic practice of writing down oral stories, and subsequently disseminating them in published form, must be seen as a peculiar form of violence, wherein the stories are torn from the visible landforms and topographic features that materially embody and _provoke_ those stories. For example, _The Speaking Land_ , Ronald and Catherine Berndt's published compendium of Aboriginal stories gathered over the course of four decades of research, is an honorable and meticulous piece of scholarship, yet it cannot help but disappoint those readers who hope to find therein a collection of stirring adventures and vital narratives. The printed stories seem curious at best, and very poorly plotted at worst; something seems missing, some key that would unlock the abstruse logic of these tales. And that key is nothing other than the living land itself, the expressive physiognomy of the local earth. What is missing is the silent topography, the sensuous hillsides and streambeds that pose the place-specific questions to which these stories all reply. The narratives respond directly to the land, as the land responds directly to the spoken or sung stories; here, cut off from that sensuous reference, transposed onto the flat and featureless terrain of the page, the ancient stories begin to lose their Dreaming power.
IN THIS CHAPTER WE HAVE PONDERED A FEW OF THE WAYS IN WHICH the spoken discourse of traditionally oral, tribal cultures remains bound to the expressive sounds, shapes, and gestures of an animate earth. In the absence of formal writing systems, human discourse simply cannot isolate itself from the larger field of expressive meanings in which it participates. Hence, the linguistic patterns of an oral culture remain uniquely responsive, and responsible, to the more-than-human life-world, or bioregion, in which that culture is embedded.
It should be easy, now, to understand the destitution of indigenous, oral persons who have been forcibly displaced from their traditional lands. The local earth is, for them, the very matrix of discursive meaning; to force them from their native ecology (for whatever political or economic purpose) is to render them speechless—or to render their speech meaningless— _to dislodge them from the very ground of coherence_. It is, quite simply, to force them out of their mind. The massive "relocation" or "transmigration" projects underway in numerous parts of the world today in the name of "progress" (for example, the forced "relocation" of oral peoples in Indonesia and Malaysia in order to make way for the commercial clearcutting of their forests) must be understood, in this light, as instances of cultural genocide.
Yet while such civilizational "progress" rumbles forward, a mounting resistance is beginning to emerge within technological civilization itself, fired in part by a new respect for oral modes of sensibility and awareness. The kinds of studies drawn upon in this chapter—studies that document the intimate dependence of oral peoples and their lifeways upon the particularities of the lands that they inhabit—are today being utilized with increasing effectiveness to halt, on _legal_ grounds, the industrial exploitation of native lands. Keith Basso's documentation of the close relation between Western Apache teaching stories and the perceivable landscape has already been used successfully in litigation to protect Western Apache land and water rights. Meanwhile, documentation of the Aboriginal Dreaming tracks is increasingly utilized in Australian courts of law to protect vital or sacred sites from further "development."
For the Amahuaca, the Koyukon, the Western Apache, and the diverse Aboriginal peoples of Australia—as for numerous indigenous, oral cultures—the coherence of human language is inseparable from the coherence of the surrounding ecology, from the expressive vitality of the more-than-human terrain. It is the animate earth that speaks; human speech is but a part of that vaster discourse.
# **6**
# **_Time, Space, and the Eclipse of the Earth_**
We must stand apart from the conventions of history, even while using the record of the past, for the idea of history is itself a western invention whose central theme is the rejection of habitat. It formulates experience outside of nature and tends to reduce place to only a stage upon which the human drama is enacted. History conceives the past mainly in terms of biography and nations. It seeks causality in the conscious, spiritual, ambitious character of men and memorializes them in writing.
—PAUL SHEPARD
I wonder if the Ground has anything to say? I wonder if the ground is listening to what is said?
—YOUNG CHIEF, of the Cayuses tribe (upon signing over their lands to the U.S. government, in 1855)
## PART I: ABSTRACTION
STORIES HOLD, IN THEIR NARRATIVE LAYERS, THE SEDIMENTED knowledge accumulated by our progenitors. To hear a story told and retold in one's childhood, and to recount that tale in turn when one has earned the right to do so (now inflected by the patterns of one's own experience and the rhythms of one's own voice), is to actively preserve the coherence of one's culture. The practical knowledge, the moral patterns and social taboos, and indeed the very language or manner of speech of any nonwriting culture maintain themselves primarily through narrative chants, myths, legends, and trickster tales—that is, through the telling of stories.
Yet the stories told within an oral culture are often, as we have seen, deeply bound to the earthly landscape inhabited by that culture. The stories, that is, are profoundly and indissolubly place-specific. The Distant Time stories of the Koyukon, the ′ _agodzaahi_ tales of the Western Apache, and the Dreaming stories of the Pintupi and Pitjantjatjara present three very different ways whereby tribal stories weave the people who tell them into their particular ecologies. Or, still more precisely, three ways in which earthly locales may _speak through_ the human persons that inhabit them. For meaningful speech is not—in an oral culture—experienced as an exclusively human capacity, but as a power of the enveloping earth itself, in which humans participate.
The stories of such cultures give evidence, then, of the unique power of particular bioregions, the unique ways in which different ecologies call upon the human community. Yet these stories often provide evidence, as well, about specific sites _within_ those larger regions. In the oral, indigenous world, to tell certain stories without saying precisely where those events occurred (or, if one is recounting a vision or dream, to neglect to say where one was when "granted" the vision), may alone render the telling powerless or ineffective.
The singular magic of a place is evident from what happens there, from what befalls oneself or others when in its vicinity. To tell of such events is implicitly to tell of the particular power of that site, and indeed to participate in its expressive potency. The songs proper to a specific site will share a common style, a rhythm that matches the pulse of the place, attuned to the way things happen there—to the sharpness of the shadows or the rippling speech of water bubbling up from the ground. In traditional Ireland, a country person might journey to one distant spring in order to cure her insomnia, to another for strengthening her ailing eyesight, and to yet another to receive insight and protection from thieves. For each spring has its own powers, its own blessings, and its own curses. Different gods dwell in different places, and different demons. Each place has its own dynamism, its own patterns of movement, and these patterns engage the senses and relate them in particular ways, instilling particular moods and modes of awareness, so that unlettered, oral people will rightly say that each place has its own mind, its own personality, its own intelligence.
### _The Abstraction of Space and Time_
As the technology of writing encounters and spreads through a previously oral culture, the felt power and personality of particular places begins to fade. For the stories that express and embody that power are gradually recorded in writing. Writing down oral stories renders them separable, for the first time, from the actual places where the events in those stories occurred. The tales can now be carried elsewhere; they can be read in distant cities or even on alien continents. The stories, soon, come to seem independent of any specific locale.
Previously, the power of spoken tales was rooted in the potency of the particular places where their events unfolded. While the recounting of certain stories might be provoked by specific social situations, their instructive value and moral efficacy was often dependent (as we saw with the Western Apache) upon one's visible or sensible contact with the actual sites where those stories took place. Other stories might be provoked by a direct encounter with the species of bird or animal whose exploits figure prominently in the tales, or with a particular plant just beginning to flower, or by local weather patterns and seasonal changes. In such cases, contact with the regional landscape—and the diverse sites or places within that landscape—was the primary mnemonic trigger of the oral stories, and was thus integral to the preservation of those stories, and of the culture itself.
Once the stories are written down, however, _the visible text becomes the primary mnemonic activator of the spoken stories_ —the inked traces left by the pen as it traverses the page replacing the earthly traces left by the animals, and by one's ancestors, in their interactions with the local land. The places themselves are no longer necessary to the remembrance of the stories, and often come to seem wholly incidental to the tales, the arbitrary backdrops for human events that might just as easily have happened elsewhere. The transhuman, ecological determinants of the originally oral stories are no longer emphasized, and often are written out of the tales entirely. In this manner the stories and myths, as they lose their oral, performative character, forfeit as well their intimate links to the more-than-human earth. And the land itself, stripped of the particularizing stories that once sprouted from every cave and streambed and cluster of trees on its surface, begins to lose its multiplicitous power. The human senses, intercepted by the written word, are no longer gripped and fascinated by the expressive shapes and sounds of particular places. The spirits fall silent. Gradually, the felt primacy of place is forgotten, superseded by a new, abstract notion of "space" as a homogeneous and placeless void.
Of course, many factors other than, but linked to, writing, contributed to the loss of a full and differentiated sense of place. The development of writing in the Middle East, as in China and Mesoamerica, was accompanied by a large increase in the scale of human settlements, as well as by a concomitant growth in the human ability, or willingness, to manipulate and cultivate the earth. Although the earliest shifts from hunting and foraging lifestyles to more sedentary, agricultural modes of subsistence are very ancient, and may have been prompted by climatic changes at the end of the last ice age, once the agricultural revolution began to accelerate, writing began to play an important role in the stabilization and subsequent spread of the new, sedentary economies. The ability to precisely measure and inventory agricultural surpluses, itself made possible by numerical and linguistic notation, enabled the new, highly centralized cities to survive and perpetuate themselves—especially through times of climatic extremity—and ultimately enabled the commercial trading of surpluses, and the rise of nation-states. The new concentration of persons within permanent towns and cities, and the increased dependence upon the regulation and manipulation of spontaneous natural processes, could only intensify the growing estrangement of the human senses from the wild, animate diversity in which those senses had evolved. But my concern in this work is neither with agriculture nor urbanization—the enormous influences of which have been elucidated in numerous volumes—but rather with the curious question of _writing;_ that is, with the influence of writing upon the human senses and upon our direct sensorial experience of the earth around us.
We have seen that alphabetic writing functions to undermine the embedded, place-specific character of oral cultures in two distinct but related ways, one basically perceptual, the other primarily linguistic. First, reading and writing, as a highly concentrated form of participation, displaces the older participation between the human senses and the earthly terrain (effectively freeing human intention from the direct dictates of the land). Second, writing down the ancestral stories disengages them from particular places. This double retreat, of the senses and of spoken stories, from the diverse places that had once gripped them, cleared the way for the notion of a pure and featureless "space"—an abstract conception that has nevertheless come to seem, today, more primordial and _real_ than the earthly places in which we remain corporeally embedded.
BUT IF ALPHABETIC WRITING WAS AN IMPORTANT FACTOR IN THE emergence of abstract, homogeneous "space," it was no less central to the emergence of abstract, linear "time." To indigenous, oral cultures, the ceaseless flux that we call "time" is overwhelmingly cyclical in character. The senses of an oral people are still attuned to the land around them, still conversant with the expressive speech of the winds and the forest birds, still participant with the sensuous cosmos. Time, in such a world, is not separable from the circular life of the sun and the moon, from the cycling of the seasons, the death and rebirth of the animals—from the eternal return of the greening earth. According to anthropologist Åke Hultkrantz:
Western time concepts include a beginning and an end; American Indians understand time as an eternally recurring cycle of events and years. Some Indian languages lack terms for the past and the future; everything is resting in the present.
Today it is easy for most of us, living amid the ever-changing constructions of literate, technological civilization, to conceive and _even feel_ , behind all the seasonal recurrences in the sensuous terrain, the inexorable thrust of a linear and irreversible time. But for cultures without writing there is simply no separate vantage point from which to view and take note of the subtle mutations and variations in the endless cycles of nature. Those changes that are noticed are often assumed to be part of other, larger cycles. For the overall trajectory of the visible, tangible world—the world disclosed to humankind by our unaided senses—is circular. Thus, in the words of Hehaka Sapa, or Black Elk, of the Oglala Sioux:
Everything the Power of the World does is done in a circle.... The Wind, in its greatest power, whirls. Birds make their nests in circles, for theirs is the same religion as ours. The sun comes forth and goes down again in a circle. The moon does the same, and both are round.... Even the seasons form a great circle in their changing, and always come back again to where they were. The life of a man is a circle from childhood to childhood and so it is in everything where power moves....
The curvature of time in oral cultures is very difficult to articulate on the page, for it defies the linearity of the printed line. Yet to fully engage, sensorially, with one's earthly surroundings is to find oneself in a world of cycles within cycles within cycles. The ancestral stories of an oral culture are recounted again and again—only thus can they be preserved—and this regular, often periodic repetition serves to bind the human community to the ceaseless round dance of the cosmos. The mythic creation stories of these cultures are not, like Western biblical accounts of the world's creation, descriptions of events assumed to have happened only once in the far-off past. Rather, the very telling of these stories actively participates in a creative process that is felt to be _happening right now_ , an ongoing emergence whose periodic renewal actually _requires_ such participation. Mircea Eliade, in his important and enigmatic work _Cosmos and History: The Myth of the Eternal Return_ , has shown as well as any scholar the extent to which indigenous peoples inhabit a cyclical time periodically regenerated through the ritual repetition of mythic events. Within "archaic" cultures (Eliade's term), every effective activity—from hunting, fishing, and gathering plants, to winning a sexual partner, constructing a home, or giving birth—is the recurrence of an archetypal event enacted by ancestral or totemic powers in the mythic times.
The myths preserve and transmit the paradigms, the exemplary models, for all the responsible activities in which men engage. By virtue of these paradigmatic models revealed to men in mythical times, the Cosmos and society are periodically regenerated.
By performing such activities with care, employing the very phrases and gestures disclosed in the Mythic Time, one actually becomes the ancestral being, and thus rejuvenates the emergent order of the world (just as the Pintupi tribesman on Walkabout, walking in the footsteps of his totem ancestor, is singing the world itself back into existence).
Even highly unusual, extraordinary events are spontaneously assimilated to recurrent mythic prototypes. Thus, Cortes's arrival on the shores of Mexico is interpreted by the Aztecs as the return of the minor god Quetzalcoatl to his kingdom (an interpretation instantly encouraged and exploited by the sly Cortés himself); similarly Captain Cook's arrival in Hawaii is construed by Native Hawaiians as the return of the deity Lono. To oral cultures, and even to a partially literate society like the Aztec (whose largely pictorial writing remained perceptually bound to the visible forms of surrounding nature), human events take on meaning only to the extent that they can be located within a storied universe that continually retells itself; unprecedented events, singular encounters that have no place among the cycling stories, can have no place, either, among the turning seasons or the cycles of earth and sky. The multiple ritual enactments, the initiatory ceremonies, the annual songs and dances of the hunt and the harvest—all are ways whereby indigenous peoples-of-place actively engage the rhythms of the more-than-human cosmos, and thus embed their own rhythms within those of the vaster round.
THE ALPHABET ALTERS ALL THIS. IN ORDER TO READ PHONETICALLY, we must disengage the synaesthetic participation between our senses and the encompassing earth. The letters of the alphabet, each referring to a particular sound or sound-gesture of the human mouth, begin to function as mirrors reflecting us back upon ourselves. They thus establish a new reflexivity between the human organism and its own signs, short-circuiting the sensory reciprocity between that organism and the land (the "reflective intellect" is precisely this new reflexive loop, this new "reflection" between ourselves and our written signs). Human encounters and events begin to become interesting in their own right, independent of their relation to natural cycles.
Recording mythic events in writing establishes, as well, a new experience of the permanence, fixity, and unrepeatable quality of those events. Once fixed on the written surface, mythic events are no longer able to shift their form to fit current situations. Current happenings are thus robbed of their mythic, storied resonance; when the myths are written down, contemporary events acquire a naked specificity and uniqueness hitherto unknown. As some of these naked occurrences come to be de-scribed or written down, they, too, are thereby fixed in their particularity, and so assume their singular place within the slowly accreting sequence of recorded events. Thus does oral story gradually give way to written history. The cyclical shape of earthly time gradually fades behind the new awareness of an irreversible and rectilinear progression of itemizable events. And historical, linear time becomes apparent.
But now let us step back for a moment. For by discussing in this somewhat cursory manner the influence of alphabetic writing upon the emergence of homogeneous "space" and linear "time," I have perhaps left the impression that space and time were always—for oral peoples as for ourselves—distinguishable dimensions of experience, and that the literate revolution simply altered the experiential character of these two, already distinct, phenomena. In truth, however, the very differentiation of "space" from "time" was itself born of the same perceptual and linguistic changes that we are discussing. For a time that is cyclical, or circular, is just as much _spatial_ as it is _temporal_.
### The Indistinction of Space and Time in the Oral Universe
We touch here upon one of the most intransigent barriers preventing genuine understanding between the modern, alphabetized West and indigenous, oral cultures. Unlike linear time, time conceived as cyclical cannot be readily abstracted from the spatial phenomena that exemplify it—from, for instance, the circular trajectories of the sun, the moon, and the stars. Unlike a straight line, moreover, a circle demarcates and encloses a spatial field. Indeed, the visible space in which we commonly find ourselves when we step outdoors is itself encompassed by the circular enigma that we have come to call "the horizon." The precise contour of the horizon varies considerably in different terrains, yet whenever we climb to a prominent vantage point, the circular character of the visible world becomes explicit. Thus cyclical time, the experiential time of an oral culture, has the same shape as perceivable space. And the two circles are, in truth, one:
The Lakota define the year as a circle around the border of the world. The circle is a symbol of both the earth (with its encircling horizons) and time. The changes of sunup and sundown around the horizon during the course of the year delineate the contours of time, time as a part of space.
On high plateaus in the Rocky Mountains, where the visible horizon is especially vast and wide, are circular arrangements of stones arrayed around a central hub. It is known that such "medicine wheels," still used by various North American tribes, once served a calendrical function. Or, rather, they enabled a person to orient herself within a dimension that was neither purely spatial nor purely temporal—the large stone that is precisely aligned with the place of the sun's northernmost emergence, marks a place that is as much in time (the summer solstice) as in space. A similar unity—of that which _to us_ are two different dimensions, the spatial and the temporal—existed among the Aztecs at the time of the conquest, according to Diego Duran, a Spanish monk who arrived in Mexico in the first half of the sixteenth century:
Duran reports that among the Aztecs, who distribute their years into cycles according to the cardinal points, "the years most feared by the people were those of the North and of the West, since they remembered that the most unhappy events had taken place under those signs."
So a cyclical mode of time does not readily distinguish itself from the spatial field in which oral persons find themselves experientially immersed. We must remember, however, that this experiential space is itself very different from the static, homogeneous void that alphabetic civilization has come to call "space." As we saw above, space, for an oral culture, is directly experienced as _place_ , or as _places_ —as a differentiated realm containing diverse sites, each of which has its own power, its own way of organizing our senses and influencing our awareness. Unlike the abstraction of an infinite and homogeneous "space," place is from the first a qualitative matrix, a pulsing or potentized field of experience, able to move us even in its stillness. It is a mode of space, then, that is always already temporal, and we should not be surprised that oral peoples speak of what to us are purely spatial phenomena as animate, emergent processes, and of space itself as a kind of dynamism, a continual unfolding. For instance, a recent, book-length analysis of spatial concepts among the Diné, or Navajo, concludes that for them
[s]pace, like the entities or objects within it, is dynamic. That is, all "entities," "objects," or similar units of action and perception must be considered as units that are engaged in continuous processes. In the same way, spatial units and spatial relationships are "qualitative" in this same sense and cannot be considered to be clearly defined, readily quantifiable and static in essence.
The authors assert, therefore, that a complex notion of space-time (or, in their words, "time-space") would likely be a more relevant translation of Navajo experience "than clearly distinct concepts of one-dimensional time and three-dimensional space."
A similar situation was discovered by the American linguist Benjamin Lee Whorf in his extensive analyses of the Hopi language during the 1930s and early 1940s. Whorf found no analog, in the Hopi language, to the linear, sequential, uniformly flowing time that Western civilization takes for granted. Indeed, Whorf found no reference to any independent temporal dimension of reality, and no terms or expressions that "refer to space in such a way as to exclude that element of extension or existence that we call time, and so by implication leave a residue that could be referred to as time." What we call _time_ , in other words, could not be isolated from the Hopi experience of _space:_
In this Hopi view, [that which we call] time disappears and [that which we call] space is altered, so that it is no longer the homogeneous and instantaneous timeless space of our supposed intuition or of classical Newtonian mechanics.
Whorf's fascinating disclosures were often taken simplistically, by researchers in other disciplines, to mean, among other things, that the Hopi people have no temporal awareness whatsoever, or that the Hopi language is utterly static, and has no way of distinguishing between earlier and later events, or between occurrences more or less distant from the speaker in what _we_ would call time. Such misreadings, doubtless encouraged by Whorf's occasional propensity for vigorous overstatement, have led various linguists in recent years to decry Whorf's findings. Several researchers, working closely with the Hopi language, claim to have refuted Whorf's conclusions entirely. Such refutations, however, are themselves dependent upon an oversimplified reading of Whorf's conclusions, upon a crusading refusal to discern that Whorf was not asserting an absence of temporal awareness among the Hopi, but rather an absence, in their discourse, of any _metaphysical_ concept of time that could be isolated from their dynamic awareness of spatiality.
While Whorf did not find separable notions of space and time among the Hopi, he did discern, in the Hopi language, a distinction between two basic modalities of existence, which he terms the "manifested" and the "manifesting." The "manifested" corresponds roughly to our notion of "objective" existence, and it comprises "all that is or has been accessible to the senses... with no attempt to distinguish between present and past, but excluding everything that we call future." The "manifesting," on the other hand,
comprises all that we call future, _but not merely this;_ it includes equally and indistinguishably all that we call mental—everything that appears or exists in the mind, or, as the Hopi would prefer to say, in the _heart_ , not only the heart of man, but the heart of animals, plants, and things, and behind and within all the forms and appearances of nature, in the heart of nature [itself]....
The "manifested," in other words, is that aspect of phenomena already evident to our senses, while the "manifesting" is that which is not yet explicit, not yet present to the senses, but which is assumed to be psychologically gathering itself toward manifestation within the depths of all sensible phenomena. One's own feeling, thinking, and desiring are a part of, and hence participant with, this collective desiring and preparing implicit in all things—from the emergence and fruition of the corn, to the formation of clouds and the bestowal of rain. Indeed, human intention, especially when concentrated by communal ceremony and prayer, contributes directly to the becoming-manifested of such phenomena.
WHILE THE LANGUAGE OF THE HOPI BELONGS TO THE UTO-AZTECAN family of languages, the neighboring Diné, or Navajo, speak an Athapaskan language—like the Koyukon and other tribes of the far Northwest, from whence the ancestors of the Apache and the Navajo first headed south many centuries ago. (The nomadic Navajo first came into contact with the Pueblo peoples of the Rio Grande valley around six hundred years ago, and ultimately adopted a range in the Arizona desert less than two hundred years ago.) Nevertheless, the Navajo language also seems to maintain a broad notion of the influence of human desire and imagination upon a continually emergent world, a notion very analogous to that found by Whorf among the Hopi. In the 1983 study of Navajo semantics alluded to earlier, the authors claim that "existence," for the Navajo, "should be understood as a continuous manifestation... [as] a series of events, rather than states or situational persistences through time." They then go on to suggest that what Western people call "the future" is experienced by the Navajo to be
like a stock of possibilities, of incompletely realized events and circumstances. They [these circumstances] are still most of all 'becoming' (rather than being) and involved in a process of 'manifesting' themselves. A human being can, through his thought and desire, exert an influence on these 'possibles.'
Thus, in place of any clear distinction between space and time, we find, in examples of both the Uto-Aztecan and the Athapaskan language groups, a subtle differentiation between manifest and unmanifest spatiality—that is, a sense of space as a continual emergence from implicit to explicit existence, and of human intention as participant with this encompassing emergence.
The indistinction of space and time was also evident in our discussion, in the last chapter, of Aboriginal Australian notions of the _Alcheringa_ , or Dreamtime. Like the Distant Time of the Koyukon, the Dreamtime does not refer to the past in any literal sense (to a time that is finished and done with), but rather to the temporal and psychological latency of the enveloping landscape. Different paths through the present terrain resonate with different stories from the Dreamtime, and indeed every water hole, every forest, every cluster of boulders or dry creekbed has its own Dreaming, its own implicit life. The vitality of each place, moreover, is rejuvenated by the human enactment, and en-chant-ment, of the storied events that crouch within it. The Dreamtime, then, is integral to the spatial surroundings. It is not a set of accomplished events located in some finished past, but is the very depth of the experiential present—the earthly sleep, or dream, out of which the visible landscape continually comes to presence. And once again human dreaming, human intention, human action and chanting participate vividly in this coming-to-presence.
Numerous other examples could be cited. These few instances, from opposite sides of the earth, should suffice at least to demonstrate that separable "time" and "space" are not absolute givens in all human experience. It is likely that without a formal system of numerical and linguistic notation it is not possible to entirely abstract a uniform sense of progressive "time" from the direct experience of the animate, emergent environment—or, what amounts to the same thing, to freeze the dynamic experience of earthly place into the intuition of a static, homogeneous "space." If this is the case, then writing must be recognized as a necessary condition for the belief in an entirely distinct space and time.
### _Exiled in the word_
According to Mircea Eliade, the ancient Hebrews were the first people to "discover" a linear, nonrepeating mode of time:
[F]or the first time, the prophets placed a value on history, succeeded in transcending the traditional vision of the cycle (the conception that ensures all things will be repeated forever), and discovered a one-way time. This discovery was not to be immediately and fully accepted by the consciousness of the entire Jewish people, and the ancient conceptions were still long to survive.
To the ancient Hebrews, or what we know of them through the lens of the Hebrew Bible, the cyclical return of seasonal events commanded far less attention than those happenings that were unique and without precedent (natural catastrophes, sieges, battles, and the like), for it was these nonrepeating events that signaled the will of YHWH, or God, in relation to the Hebrew people. In Eliade's terms, these unique occurrences, whose consequences were often devastating (either to the Hebrews or to their enemies), were interpreted by the prophets as "negative theophanies," as expressions of YHWH's wrath. Thus interpreted, these discordant and nonrepeating events acquired a coherence previously unknown, and so began to stand out from the cyclical unfolding of natural phenomena. And the Hebrew nation came to comprehend itself in relation to this new, nonrepeating modality of time—that is, in relation to history.
[F]or the first time, we find affirmed, and increasingly accepted, the idea that historical events have a value in themselves, insofar as they are determined by the will of God.
Yet it is crucial to recognize what Eliade does _not_ mention in his discussion—that the Hebrews are, as well, the first truly alphabetic culture that we know of, the first "People of the Book." Indeed, at the founding event of the Jewish nation—the great theophany atop Mount Sinai—Moses _inscribes_ the commandments dictated by YHWH (the most sacred of God's names) upon two stone tablets, presumably in an alphabetic script. (Contemporary scholars place the exodus from Egypt sometime around 1250 B.C.E.; it is just at this time that the twenty-two-letter, consonantal _aleph-beth_ was coming into use in the area of Canaan, or Palestine.)
In truth, the new recognition of a nonmythological, nonrepeating time by the Hebrew scribes can only be comprehended with reference to alphabetic writing itself. Recording cultural stories in writing, as we have seen, fixes the storied events in their particularity, providing them with a new and unchanging permanence while inscribing them in a steadily accreting sequence of similarly unique occurrences. A new sense of time as a nonrepeating sequence begins to make itself felt over and against the ceaseless cycling of the cosmos. The variously scribed layers of the Hebrew Bible are the first sustained record of this new sensibility.
As we have also discerned, the ancient _aleph-beth_ , as the first thoroughly _phonetic_ writing system, prioritized the human voice. The increasingly literate Israelites found themselves caught up in a vital relationship not with the expressive natural forms around them, nor with the static images or idols common to pictographic or ideographic cultures, but with an all-powerful human voice. It was a voice that clearly preceded, and outlasted, every individual life—the voice, it would seem, of eternity itself—but which nevertheless addressed the Hebrew nation directly, speaking, first and foremost, through the written letters.
While the visible landscape provides an oral, tribal culture with a necessary mnemonic, or memory trigger, for remembering its ancestral stories, alphabetic writing enabled the Hebrew tribes to preserve their cultural stories intact even when the people were cut off, for many generations, from the actual lands where those stories had taken place. By carrying on its lettered surface the vital stories earlier carried by the terrain itself, _the written text became a kind of portable homeland for the Hebrew people_. And indeed it is only thus, by virtue of this portable _ground_ , that the Jewish people have been able to preserve their singular culture, and thus themselves, while in an almost perpetual state of exile from the actual lands where their ancestral stories unfolded.
Yet many of the written narratives in the Bible are already stories of displacement, of exile. The most ancient stratum of the Hebrew Bible is structured, from the first, by the motif of exile—from the expulsion of Adam and Eve from the garden of Eden, to the long wandering of the Israelites in the desert. The Jewish sense of exile was never merely a state of separation from a specific locale, from a particular ground; it was (and is) also a sense of separation from the very _possibility_ of being placed, from the very possibility of being entirely at home. This deeper sense of displacement, this sense of always _already_ being in exile, is inseparable, I suggest, from alphabetic literacy, this great and difficult magic of which the Hebrews were the first real caretakers. Alphabetic writing can engage the human senses only to the extent that those senses sever, at least provisionally, their spontaneous participation with the animate earth. To begin to read, alphabetically, is thus already to be dis-placed, cut off from the sensory nourishment of a more-than-human field of forms. It is also, however, to feel the still-lingering savor of that nourishment, and so to yearn, to hope, that such contact and conviviality may someday return. _"Because being Jewish,"_ as Edmond Jabes has written, _"means exiling yourself in the word and, at the same time, weeping for your exile."_ 22
The pain, the sadness of this exile, is precisely the trace of what has been lost, the intimation of a forgotten intimacy. The narratives in Genesis remain deeply attuned to the animistic power of places, and it is this lingering power that lends such poignancy to the motifs of exodus and exile. The stories of the patriarchs are filled with sacred place-names, and many of these narratives seem structured so as to tell how particular places came to have their specific names. While these sacred sites never seem to have an entirely autonomous power (many, for instance, take their sacredness from the fact that YHWH there speaks or otherwise reveals Himself to one of the protagonists), earthly place nevertheless remains a structuring element of biblical space.
Moreover, the trajectory of time, for the ancient Hebrews, was by no means entirely linear. The holy days described in the Bible are closely bound to the intertwined cycles of the sun and the moon. Further, the nonrepeating, historical time alluded to by Eliade seems to correlate with the sense of existential separation and exile. It is thus that, in Hebrew tradition, the expulsion from the eternity of Eden (and, later, the destruction of the Temple) is mirrored, at the other end of sequential history, by the promised return from exile, the coming of the Messiah, and an end to separated time. The forward trajectory of time, that is, will at last open outward, flowing back into the spacious eternity of living place (the "promised land"), and so into a golden age of peace between all nations. Eternity lies not in a separated heaven (the ancient Hebrews knew of no such realm) but in the promise of a future reconciliation on the earth.
Time and space are still profoundly influenced by one another in the Hebrew Bible. They are never _entirely_ distinguishable, for they are still informed, however distantly, by a participatory experience of _place_.
IT REMAINED FOR THE ANCIENT GREEKS, POSSESSED OF THEIR OWN version of the alphabet, to derive an entirely placeless notion of eternity—a strictly intelligible, nonmaterial realm of pure Ideas resting entirely outside of the sensible world. It is obvious that the Greek alphabet contributed to a kind of theoretical abstraction very different from that engaged in by the Hebrew prophets and scribes. In part, this may be attributed to the very different historical trajectories of the Hebrew and the Greek peoples, to the obvious contrasts between desert-dwelling peoples and seafaring peoples, and to a host of other influences upon Greek culture arriving, like the alphabet, from abroad. But it is also the consequence of a simple but profound structural change introduced into the alphabet by the Greek scribes when they adapted this writing system from its earlier, Semitic incarnation. We must leave to the next chapter a careful discussion of this structural change and its experiential ramifications. Here we need only observe that Greek thinkers were the first to begin to objectify space and time as entirely distinct and separable dimensions.
Yet this was a sporadic and fragmentary process, resulting from the overlapping descriptive, analytic, and speculative writings of diverse individuals and schools of thought. The earliest historians, like Hecataeus of Miletos (c. 550–489 B.C.E.), Herodotus (c. 480–425 B.C.E.), and Thucydides (c. 460–400 B.C.E.) pioneered the use of written prose, rather than poetry, to record past events. They practiced a new skepticism regarding the storied gods and goddesses of the animate environment, and by separating past events from the tradition-bound rhythms of verse and chanted story, they loosened time itself from the recurrent cycling of the sensuous earth, opening the prospect of a nonrepeating, historical time extending indefinitely into the past.
A century later Aristotle (384–322 B.C.E.) sought to _define_ the dimension of time as it makes itself evident in our experience. He concluded that "time is just this: the number of a motion with respect to the prior and the posterior." Time, in other words, is what is counted whenever we measure a movement between earlier and later moments of its unfolding. Time is thus inseparable from number and sequence; it appears in Aristotle's writings as a continuous linear series of points, each a punctiform "now" dividing the past from the future.
Shortly thereafter, in his remarkably influential text _Elements_ , the Greek geometrician Euclid (c. 300 B.C.E.) implied by his various definitions and postulates that space itself could be conceived as an entirely homogeneous and limitless three-dimensional continuum. The homogeneous character of Euclidian space was indicated, in particular, by his assertion that parallel straight lines, no matter how far they are extended in either direction, will never meet. While this postulate holds true for a perfectly flat and featureless ideal space, the experienced world that we bodily inhabit is not so regular. Indeed, we now know that the sphericality of the earth itself—this very surface on which we dwell—confounds Euclid's parallel postulate: two straightest-possible lines that start out parallel to each other on the curved surface of a sphere will eventually converge and'cross, like meridians at the North Pole. That we still commonly envision the curved surface of the earth, with all of its local irregularities (its mountains and river valleys), to be embedded within a three-dimensional space lacking any curvature of its own, is exquisite testimony to the lasting influence of Euclidean conceptions. Euclid's assumptions provided the classical basis for Western, scientific notions of space, from the Renaissance until the work of Albert Einstein, and even today our supposedly "commonsense" experience remains profoundly under the influence of such assumptions.
While evolving techniques of numerical notation and measurement obviously played an explicit role in the development of these early descriptions, the spread of alphabetic literacy was at work behind the scenes, altering the perceptual relations between the Greeks and the sensible world around them, and thus gradually disclosing the new, apparently independent dimensions of space and time to which the numbers and measurements were then applied.
### Absolute Space and Absolute Time
Yet a thorough description of homogeneous "space" and sequential "time," as objectively existing entities, had to wait until the invention of the printing press. For it was the dissemination of printed texts (texts that until then had been meticulously copied by hand and preserved, like treasures, in monastic libraries and universities) into the wider community of persons, and the subsequent rise of vernacular literatures, that effectively sealed the ascendancy of alphabetic modes of thought over the oral, participatory experience of nature. The thorough differentiation of "time" from "space" was impossible as long as large portions of the community still experienced the surrounding terrain as animate and alive, as long as material (spatial) phenomena were still perceived by many as having their own inherent spontaneity and (temporal) dynamism. The burning alive of tens of thousands of women (most of them herbalists and midwives from peasant backgrounds) as "witches" during the sixteenth and seventeenth centuries may usefully be understood as the attempted, and nearly successful, extermination of the last orally preserved traditions of Europe—the last traditions rooted in the direct, participatory experience of plants, animals, and elements—in order to clear the way for the dominion of alphabetic reason over a natural world increasingly construed as a passive and mechanical set of objects.
It was Isaac Newton, in his great _Principia Mathematica_ of 1687, who finally gave an absolute formulation to separable "time" and "space" as the necessary frame for his clockwork universe:
Absolute, true and mathematical time, of itself and from its own nature, flows equably without regard to anything external....
Absolute space, in its own nature, without regard to anything external, remains always similar and immovable....
By these formulations Newton meant to distinguish "absolute time" from that "relative time" which is simply the order of succession of perceivable events, and to distinguish "absolute space" from that "relative space" which is the order of coexistence between perceivable things. While "relative time" is merely a relationship between material events, and so has no existence apart from those events, "absolute, true and mathematical time" is, for Newton, an independent reality that we cannot perceive directly, but which underlies all material events and their relations. Similarly "absolute, true, and mathematical space" subsists independent of all perceivable things. In itself it is empty—a void. Like absolute time, it is infinite in extent; it can neither be created nor destroyed, and no part of it can be distinguished from any other part.
By assuming the existence of this empty and "immovable" space—this space that is at rest relative to any and all motion—Newton was then able to calculate the motion of the moon or the earth relative to this absolute space; it was only by assuming these absolute references that he was able to derive his theory of universal attraction, or "gravity." After the publication of his _Principia_ , Newton's assumptions regarding space and time were challenged by numerous philosophers, and he found himself in extended debates with such illustrious thinkers as Leibniz and Berkeley over the question of whether one could rationally distinguish absolute from relative space, or absolute from relative time. However, although they challenged the absolute character of Newton's space and time, none of these thinkers challenged the assumption of an absolute difference _between_ space and time—the by now commonplace assumption that space and time were entirely distinct dimensions of experience.
In 1781, Immanuel Kant, in his _Critique of Pure Reason_ , capped the debates regarding the absolute or relative nature of time and space. He agreed with Newton that space and time were absolute, that they were independent of particular things and events. For Kant, however, these distinct dimensions did not belong to the surrounding world as it exists in itself, but were necessary forms of human awareness, the two forms by which the human mind inevitably structures the things it perceives. Thus, while he denied that space and time necessarily exist apart from human experience, Kant's work seemed to establish more forcefully than ever that, at least as far as humans were concerned, "space" and "time" were distinct and inescapable dimensions.
Needless to say, Kant's writings could not be translated into Navajo or Pintupi.
## PART II: THE LIVING PRESENT
When I returned to North America from my travels among traditional peoples in Indonesia and Nepal, I quickly found myself perplexed and confused by many aspects of my own culture. Assumptions that I had previously taken for granted, or that I had since childhood accepted as obvious and unshakable truths, now made little sense to me. The belief, for instance, in an autonomous "past" and "future." Where _were_ these invisible realms that had so much power over the lives of my family and friends? Everybody that I knew seemed to be expending a great deal of effort thinking about and trying to hold onto the past—obsessively photographing and videotaping events, and continually projecting and fretting about the future—ceaselessly sending out insurance premiums for their homes, for their cars, even for their own bodies. As a result of all these past and future concerns, everyone appeared (to me in my raw and newly returned state) to be strangely unaware of happenings unfolding all around them _in the present_. They seemed utterly oblivious to all those phenomena to which I had had to sensitize myself in order to communicate with indigenous magicians in the course of my fieldwork: the lives of other animals, the minute gestures of insects and plants, the speech of birds, the tastes in the wind, the flux of sounds and smells.... My family and my old friends all seemed so oblivious to the sensuous presence of the world. The present, for them, seemed nothing more than a point, an infinitesimal now separating "the past" from "the future." And indeed, the more I entered into conversation with my family and friends, the more readily I, too, felt my consciousness cut off, as though by a sheet of reflective glass, from the life of the land....
There is a useful exercise that I devised back then to keep myself from falling completely into the civilized oblivion of linear time. You are welcome to try it the next time you are out of doors. I locate myself in a relatively open space—a low hill is particularly good, or a wide field. I relax a bit, take a few breaths, gaze around. Then I close my eyes, and let myself begin to feel the whole bulk of my past—the whole mass of events leading up to this very moment. And I call into awareness, as well, my whole future—all those projects and possibilities that lie waiting to be realized. I imagine this past and this future as two vast balloons of time, separated from each other like the bulbs of an hourglass, yet linked together at the single moment where I stand pondering them. And then, very slowly, I allow both of these immense bulbs of time to begin leaking their substance into this minute moment between them, into the present. Slowly, imperceptibly at first, the present moment begins to grow. Nourished by the leakage from the past and the future, the present moment swells in proportion as those other dimensions shrink. Soon it is very large; and the past and future have dwindled down to mere knots on the edge of this huge expanse. At this point I let the past and the future dissolve entirely. And I open my eyes....
_I FIND MYSELF STANDING IN THE MIDST OF AN ETERNITY, A VAST and inexhaustible present. The whole world rests within itself—the trees at the field's edge, the hum of crickets in the grass, cirrocumulus clouds rippling like waves across the sky, from horizon to horizon. In the distance I notice the curving dirt road and my rusty car parked at its edge—these, too, seem to have their place in this open moment of vision, this eternal present. And smells—the air is rich with faint whiffs from the forest, the heather, the soil underfoot—so many messages mingling between different elements in the encircling land. The jagged snag of a single withered oak tree standing alone in the field does not, in this eternity, seem really dead. It is surrounded by an admiring clump of low bushes, and a large boulder reposes at the edge of these bushes, dialoguing with the old tree about shadows and sunlight_.
_Stepping closer, I see that the crumbling bark around the oak's trunk is crossed by two lines of ants, one moving up the trunk and the other heading down into the soil. From this closer vantage I see, too, that the shadows on the boulder are not really shadows at all, but patches of lichen spreading outward from various points on the rock's surface, in diverse textures and hues—dull blacks and crinkly grays and powdery, deep reds—as though through them the rock was expressing its inner moods. I scratch my leg. Strangely, the vividness of this world does not dissipate. I stomp on the ground, spin around, even stand on my head. But the open present does not disperse. Several jet black crows race out of the woods, chasing each other in swoops and sudden dives; one of them lands on the crumbling snag_. "Kahhr!... Kahr! Kahr!" _Now it glides down to the ground just in front of me_ —"Kahr!"— _and stands there looking at me, sideways, through a purple eye. The lids blink swiftly, like shutters. It hops around me and the big beak opens_. "Kawhhr!" _I try to reply_ , "Cawr!" _and the bird steps forward. Crow does not hop, I see, but walks, clumsily, on this ground. I can see the tiny feathers covering the nostrils on its beak as the breeze picks it up off the ground, feel myself swoop through the swirling breeze toward the forest edge...._
_Things are different in this world without "the past" and "the future," my body quivering in this space like an animal. I know well that, in some time out of this time, I must return to my house and my books. But here, too, is home. For my body is at home, in this open present, with its mind. And this is no mere illusion, no hallucination, this eternity—there is something too persistent, too stable, too unshakable about this experience for it to be merely a mirage...._
THE UNSHAKABLE SOLIDITY OF THIS EXPERIENCE IS CURIOUS indeed. It seems to have something to do with the remarkable affinity between this temporal notion that we term "the present" and the spatial landscape in which we are embedded. When I allow the past and the future to dissolve, imaginatively, into the immediacy of the present moment, then the "present" itself expands to become an enveloping field of _presence_. And this presence, vibrant and alive, spontaneously assumes the precise shape and contour of the enveloping sensory landscape, as though this were its native shape! It is this remarkable fit between temporal concept (the "present") and spatial percept (the enveloping presence of the land) that accounts, I believe, for the relatively stable and solid nature of this experience, and that prompts me to wonder whether "time" and "space" are really as distinct as I was taught to believe. There is no aspect of this realm that is strictly temporal—for it is composed of spatial things that have density and weight, and is spatially extended around me on all sides, from the near trees to the distant clouds. And yet there is no aspect, either, that is strictly spatial or static—for every perceivable being, from the stones to the breeze to my car in the distance, seems to vibrate with life and sensation. In this open present, I am unable to isolate space from time, or vice versa. I am immersed in the world.
IN 1905, ALBERT EINSTEIN CHALLENGED THE NEWTONIAN VIEW OF absolute time and absolute space with his "special theory of relativity." Einstein's equations in this, and later in the "general theory of relativity," did not treat of time and space; they assumed, instead, the existence of a unitary continuum that Einstein termed "space-time." Space-time, however, was a highly abstract concept unthinkable apart from the complex mathematics of relativity theory. Einstein's mathematical revelations, in other words, did little to challenge the Kantian assumption that separable space and time were necessary and unavoidable forms in all ordinary perception. While space-time held sway within the _conceptual_ order of relativity physics, our direct, _perceptual_ experience was still assumed to be structured according to the separable dimensions of time and space.
It thus fell to the tradition of phenomenology to call into question the distinction between space and time at the level of our direct, preconceptual experience. Of course, phenomenology did not set out to undermine this distinction—only to attend, as closely as possible, to the way phenomena present themselves in our immediate, lived experience. Indeed, phenomenologists tended to assume, at the outset, a clear distinction between space and time. It was only toward the end of his investigations regarding the phenomenology of "time consciousness" that Edmund Husserl was led to suggest that the experience of time is rooted in a deeper dimension of experience that is not, in itself, strictly temporal.
Husserl's assistant, the German phenomenologist Martin Heidegger, returned again and again to the analysis of temporal experience. In his massive and influential work _Being and Time_ , Heidegger disclosed, underneath the commonplace Aristotelian idea of time as an infinite sequence of "now points," a forgotten sense of time as the very mystery of Being, as that strange power—essentially resistant to all objectification or representation—that nevertheless structures and makes possible all our relations to each other and to the world. This mystery cannot be represented, precisely because it is never identical to itself; primordial time, for Heidegger, is from the first outside-of-itself, or "ecstatic." Indeed, the past, the present, and the future are here described by Heidegger as the three "ecstasies" of time, the three ways in which the irreducible dynamism of existence opens us to what is outside ourselves, to that which is _other_.
Yet Heidegger gradually came to suspect that this implicit, preconceptual sense of time could not be held apart from our preconceptual experience of space. Hence, in an important essay written late in his career, Heidegger alludes to a still more primordial dimension, which he calls "time-space"—a realm neither wholly temporal nor wholly spatial, from whence "time" and "space" have been artificially derived by a process of abstraction.
Meanwhile, Maurice Merleau-Ponty, continually deepening his own investigations of perceptual experience, also came, in his final work, to assert an experiential realm more originary than space and time, from which these two dimensions have been derived. In the working notes to _The Visible and the Invisible_ , Merleau-Ponty writes of "this very time that is space, this very space that is time, which I will have rediscovered by my analysis of the visible and the flesh." Yet this analysis was cut short by his sudden death in 1961.
So all three phenomenologists—Husserl, Heidegger, and Merleau-Ponty—came independently, in the course of their separate investigations, to suspect that the conventional distinction between space and time was untenable from the standpoint of direct, preconceptual experience. Heidegger and Merleau-Ponty were both striving, toward the end of their lives, to articulate a more immediate modality of awareness, a more primordial dimension whose characteristics are neither strictly spatial nor strictly temporal, but are rather—somehow—both at once.
We have seen that such a mode of experience is commonplace for indigenous, oral peoples, for whom time and space have never been sundered. The tradition of phenomenology, it would seem, has been striving to recover such an experience from within literate awareness itself—straining to remember, in the very depths of reflective thought, the silent reciprocity wherein such reflection is born. No single one of these thinkers was entirely successful in reconciling time and space. Yet their later writings provide tantalizing clues, talismans for those who are struggling today to bring their minds and their bodies back together, and so to regain a full-blooded awareness of the present.
### The Earthly Topology of Time
_I remain standing on this hill under rippled clouds, my skin tingling with sensations. The expansiveness of the present holds my body enthralled. My animal senses are all awake—my ears attuned to a multiplicity of minute sounds, the tiny hairs on my face registering every lull and shift in the breeze. I am embedded in this open moment, my muscles stretching and bending with the grass. This present seems endless, inexhaustible. What, then, has become of the past and the future?_
I found my way into this living expanse by dissolving past and future into the sensorial present that envelops me; did I thereby do away with them entirely? I think not. I simply did away with these dimensions as they are conventionally conceived—as autonomous realms existing apart from the sensuous present. By letting past and future dissolve into the present moment, I have opened the way for their gradual rediscovery—no longer as autonomous, mental realms, but now as aspects of the corporeal present, of this capacious terrain that bodily enfolds me. And so now I crouch in the midst of this eternity, my naked toes hugging the soil and my eyes drinking the distances, trying to discern where, in this living landscape, the past and the future might reside.
Merleau-Ponty, in one of the notes found on his desk after his death, addressed the same conundrum:
In what sense the visible landscape under my eyes is not exterior to... other moments of time and the past, but has them really _behind itself_ in simultaneity, inside itself, and not it and they side by side "in" time.
And so we are faced with this puzzle: Where, within the visible landscape, can we locate the past and the future? Where is their place in the sensuous world?
Of course, we may say that we perceive the past all around us, in great trees grown from seeds that germinated long ago, in the eroded banks of a meandering stream, or the widening cracks in an old road. And, too, that we are peering into the future wherever we look—watching a storm cloud emerge from the horizon, or a spiderweb slowly taking shape before our eyes—since all that we perceive is already, in a sense, pregnant with the future. But how, then, can we _distinguish_ these two temporal realms? We certainly have a sense that the past and the future are not the same; nevertheless, they are strangely commingled within all that we perceive. How, then, do they distinguish themselves perceptually? If we say that "the past" is where all that we see comes from and "the future" is where it is all going, we simply beg the question, naming two allegedly obvious domains that we remain unable to locate within the perceivable landscape—as though past and future are, indeed, pure intuitions of the mind, existing in some incorporeal dimension outside of the sensory world. This, presumably, is what prompts many scientists and philosophers to assert that other animals have no real awareness of time—no sense of a past or a future—since they lack any intellect that could apprehend this non-sensuous dimension.
As an animal myself, I remain suspicious of all these dodges, all these ways whereby my species lays claim to a source of truth that supposedly lies outside of the bodily world wherein plants, stones, and streams have their being, outside of this earthly terrain that we share with the other animals. And yet, as a philosopher, I feel pressed to account for these mysteries, for these "times" that are somehow _not present_ , for these other "whens." And so now let us bring the human animal and the philosopher in ourselves together, and try to locate the "past" and the "future" within the sensory landscape.
FIRST, WE SHOULD TAKE SOME METHODOLOGICAL GUIDANCE FROM Merleau-Ponty, who in 1960 was already struggling to give voice to "this very time that is space, this very space that is time." In his last work Merleau-Ponty describes the relation between the perceptual world and the world of our supposedly incorporeal ideals and thoughts: "it is by borrowing from the world's structure that the universe of truth and of thought is constructed for us." These words assert the primacy of the bodily world relative to the universe of ideas; they suggest that the structures of our apparently incorporeal ideas are lifted, as it were, from the structures of the perceptual world. If we read Merleau-Ponty's words carefully, and accept their guidance, we discern that what we are here hunting for, in our deepening quest, are specific aspects of the perceivable landscape that have lent their particular character, or shape, to these two persistent ideas, "the past" and "the future." We are searching, that is, for a structural correspondence—an isomorphism, or match—between the conceptual structure of "the past" and "the future" and the perceptual structure of the surrounding sensory world.
If we have taken a kind of method from Merleau-Ponty, it is to Martin Heidegger that we should turn for a careful structural description of "the past" and "the future." Throughout his life, from his first to his final writings, Heidegger gave special attention to the phenomenon of time, and it is he, more than any other thinker, who developed a phenomenology of time's dimensions. In the middle of a late essay entitled "Time and Being," Heidegger asks the very question we ourselves have posed: "Where is time? _Is_ time at all and does it have a place?" He then goes on to distinguish that time into which he is inquiring from the common _idea_ of time as a linear sequence of "nows":
Obviously, time is not nothing. Accordingly, we maintain caution and say: there is time. We become still more cautious, and look carefully at that which shows itself as time, by looking ahead to Being in the sense of presence, the present. However, the present in the sense of presence differs so vastly from the present in the sense of the now.... [T]he present as presence and everything which belongs to such a present would have to be called real time, even though there is nothing immediately about it of time as time is usually represented in the sense of a succession of a calculable sequence of nows.
Heidegger's philosophical move, here, to disclose behind the present considered as "now" a deeper sense of the present as "presence," approximates our own experiential move to expand the punctiform "now" by dissolving the "past" and the "future" as conventionally experienced, thereby locating ourselves in a vast and open present—which we, too, have called "the present as presence." According to Heidegger, it is only from within this experience of the present as presence that "real time" (which, later in the essay, he will call "time-space") can begin to make itself evident. In our case the present has determined itself as presence only by taking on the precise contours of the visible landscape that enfolds us. We are now free to look around us, in this vast terrain, for the place of the past and of the future.
And Heidegger offers us a helpful clue. In _Being and Time_ , he writes of past, present, and future as the three "ecstasies" of time, suggesting that the past, the present, and the future all draw us outside of ourselves. Time is ecstatic in that it opens us outward. Toward what? The three ecstasies of time, according to Heidegger, "are not simply raptures in which one gets carried away. Rather, there belongs to each ecstasy a 'whither' to which one is carried...." Each of time's ecstasies carries us, Heidegger says, toward a particular "horizon."
As soon as we pay heed to this curious description, we notice an obvious correspondence between the conceptual structure of time, as described by Heidegger, and the perceptual structure of the enveloping landscape. The horizon itself! Heidegger uses the term "horizon" as a structural metaphor, a way of expressing the ecstatic nature of time. Just as the power of time seems to ensure that the perceivable present is always open, always already unfolding beyond itself, so the distant horizon seems to hold open the perceivable landscape, binding it always to that which lies beyond it.
The visible horizon, that is, a kind of gateway or threshold, joining the presence of the surrounding terrain to that which exceeds this open presence, to that which is hidden _beyond_ the horizon. The horizon carries the promise of something more, something _other_. Here we have made our first discovery: the way that other places—places not explicitly present within the perceivable landscape—are nevertheless joined to the present landscape by the visible horizon. And so let us ask: is it possible that the realms we are looking for, the place of _the past_ and that of _the future_ , are precisely beyond the horizon?
Certainly this is a useful first step. For clearly, neither the past nor the future are entirely out in the open of the perceivable present, _and yet they seem everywhere implied_. Since the horizon effectively implicates all that lies beyond the horizon within the present landscape that it bounds, it seems plausible to suppose that both the past and the future reside beyond the horizon.
Yet this leaves me somewhat confused, for I am unable, then, to account for the _difference_ between the past and the future. The horizon of the perceivable landscape is provided, I know, by the relation of my body to the vast and spherical Body of the earth. This is not merely something that I have read, or learned in school. It has become evident and true for me in the course of many journeys across the land, watching the horizon continually recede as I move toward it, watching it disgorge unexpected vistas that expand and envelop me even as the horizon itself maintains its distance. And yet if I glance behind me as I journey, I see that this enigmatic edge is also following me, keeping its distance behind me as well as in front, gradually swallowing those terrains that I walk, drive, or pedal away from. May I then conclude that _the future_ is beyond that part of the horizon toward which I am facing, while _the past_ is beyond that part of the horizon that lies behind me? Then I would need only to turn around in order for my past to become my future, and vice versa. But this does not seem quite right. If I journey toward the horizon—toward any part of that horizon—I will indeed disclose new things and places that were previously in my future, beyond the horizon. Certainly I can attempt the reverse, as when I journey back toward that distant town where I used to dwell. But in this I am never quite successful. For that town, when I arrive, is no longer as it was. The old schoolhouse now stands half-collapsed in a field overgrown with wildflowers and thistles; the marsh where each spring I used to await the arrival of herons has vanished beneath a huge shopping mall.... The land has changed. I cannot, it seems, journey toward the past in the same way that I can journey toward the future. For the past does not _remain_ past beyond the horizon; it does not wait for me there like the future.
It is this strange asymmetry of past and future in relation to the present that Heidegger describes in his late essay "Time and Being." While in _Being and Time_ Heidegger wrote of the centrifugal, ecstatic character of time—of time as that which draws us outside of ourselves, opening us to what is other—in this later essay he stresses the centripetal, inward-extending nature of time, describing time as a mystery that continually approaches us from beyond, extending and offering the gift of presence while nevertheless withdrawing behind the event of this offering. Such descriptions may sound strange, even uncanny, to our ears, and yet we should listen to them closely. For as Heidegger's thought matured, he increasingly sought to loosen human awareness from the bondage of outworn assumptions, precisely by wielding common words in highly unusual ways, shaking terms free from their conventional usages. Thus _past_ and _future_ are here articulated as hidden powers that approach us, offering and opening the present while nevertheless remaining withdrawn, concealed from the very present that they make possible. In Heidegger's description, both the past and the future remain hidden from the open presence that they mutually bring about. And yet the way the future conceals itself in its offering is quite different from the manner in which the past is concealed in its giving. Specifically, the future, or that which is to come, _withholds_ its presence, while the past, or that which has been, _refuses_ its presence. The future withholds, while the past refuses. In his most complete description of the vicissitudes of time, Heidegger puts the matter thus:
What has been, which, by refusing the present, lets that become present which is no longer present; and the coming toward us of what is to come, which, by withholding the present, lets that be present which is not yet present—both [make] manifest the manner of an extending opening up which gives all presencing into the open.
The strange character of Heidegger's language here is part of his project: he is trying to avoid the use of nouns, of nominative forms that would freeze the temporal flux. It is precisely this strangeness that enables his words to approach, and to open us onto, the silent structuration of this mystery we call time. If we ponder these words from within the open presence of the land around us, we are led to ask: Where can we perceive this _withholding_ and this _refusal_ of which Heidegger speaks? Where can we glimpse this refusal and this withholding that open and make possible the sensuous presence of the world around us?
We have already noticed the magic by which the horizon encloses and yet holds open the visible landscape: precisely by concealing, or better, _withholding_ , that which lies beyond it. Thus, the horizon may indeed be felt as a withholding. But it is hardly a refusal. The horizon's lips of earth and sky may touch one another, but they are never sealed; and we know that if we journey toward that horizon, it will gradually disclose to us that which it now withholds.
Where, then, can we locate the refusal to which Heidegger alludes? Do we perceive such a refusal anywhere around us? More important: how do we even know what we are looking for? Here again, Heidegger provides a clue. In "Time and Being," he writes of the past and of the future as _absences_ that by their very absence concern us, and so make themselves felt within the present. This description aids us a great deal. Now at least we can say what we are searching for in our attempt to locate, or place, the past and the future. We are hunting for modes of absence which, by their very way of being absent, make themselves felt within the sensuous presence of the open landscape. Or in Merleau-Ponty's terminology (the terminology of _The Visible and the Invisible_ ) we could say we are searching for certain _invisible_ aspects of the visible environment, certain unseen regions whose very hiddenness somehow enables or makes possible the open visibility of the land around us. The _beyond-the-horizon_ is just such an absent or unseen realm.
And so we must now ask: Is there _another_ unseen aspect, another absent region whose very concealment is somehow necessary to the open presence of the landscape?
Of course, there are those facets that I cannot see of the things or bodies that surround me—the sides of the trees that are facing away from me, or the other side of that lichen-covered rock. Yet these concealments are all analogous, in a sense, to that which lies hidden beyond the horizon. The other side of that rock, for instance, is withheld from my gaze, but it is not refused, for I can disclose it by walking over there, just as I can disclose what lies beyond the horizon by making a longer journey.
What of my own body? Well, most of my body is present to my awareness, and visible to my gaze. I can see my limbs, my torso, and even my nose, although my back, of course, is hidden beyond the horizon of my shoulders. The back of my body is inaccessible to my vision, and yet I know that it exists, that it is visible to the crows perched behind me in the trees, as I know that the fields and forests hidden beyond the horizon are yet visible and present to those who dwell there.
Yet while pondering the unseen aspect of my body, I soon notice another unseen region: that of the whole _inside_ of my body. The inside of my body is not, of course, entirely absent; but it is hidden from visibility in a manner very different from the concealment of my back, or of that which lies beyond the horizon. It is an instance, I suddenly realize, of a vast mode of absence or invisibility entirely proper to the present landscape—an absence I had almost entirely forgotten. It is the absence of what is _under the ground_.
LIKE THE BEYOND-THE-HORIZON, THE ABSENCE OF THE UNDER-THE-GROUND is an absence so familiar, and so necessary to the open presence of the world around us, that we take it entirely for granted, and so it has been very difficult for me to bring it into awareness. But once I have done so, the recognition of this hidden realm begins to clarify and balance the enigmatic power of that other unseen region beyond the horizon.
For these would seem to be the two primary dimensions from whence things enter the open presence of the landscape, and into which they depart. Sensible phenomena are continually appearing out of, and continually vanishing into, these two very different realms of concealment or invisibility. One trajectory is a passage out toward, or inward from, a vast openness. The other is a descent into, or a sprouting up from, a packed density. While the open horizon withholds the visibility of that which lies beyond it, the ground is much more resolute in its concealment of what lies beneath it. It is this resoluteness, this _refusal_ of access to what lies beneath the ground, that enables the ground to solidly support all those phenomena that move or dwell upon its surface. Thus, although the absence of the beyond-the-horizon and that of the under-the-ground reciprocate one another, they contrast markedly in their relation to the perceivable present. We may describe this reciprocity and this contrast thus: _The beyond-the-horizon, by withholding its presence, holds open the perceived landscape, while the under-the-ground, by refusing its presence, supports the perceived landscape_. The reciprocity and asymmetry between these two realms bear an uncanny resemblance to the reciprocity and contrast between the _future_ (or "what is to come") and the _past_ (or "what has been") in Martin Heidegger's description above—the one _withholding_ presence, the other _refusing_ presence; both of them thus making possible the open presence of the present. Dare we suspect that these two descriptions describe one and the same phenomenon? I believe that we can, for the isomorphism is complete.
BY READING MERLEAU-PONTY AND HEIDEGGER TOGETHER, AND by setting their words in relation to our own experience, we have begun to realize that the past and the future—these curious dimensions—may be just as much spatial as they are temporal. Indeed, we have begun to _place_ these dimensions, to discern their location within the sensuous world. The conceptual abstraction that we commonly term "the future" would seem to be born from our bodily awareness of that which is hidden beyond the horizon—of that which exceeds, and thus holds open, the living present. What we commonly term "the past" would seem to be rooted in our carnal sense of that which is hidden under the ground—of that which resists, and thus supports, the living present. As ground and horizon, these dimensions are no more temporal than they are spatial, no more mental than they are bodily and sensorial.
We can now discern just how close Merleau-Ponty was to this discovery by reading his aforementioned note of November 1960 in the light of our disclosures:
In what sense the visible landscape under my eyes is not exterior to, and bound systematically to... other moments of time and the past, but has them really _behind itself_ in simultaneity, _inside itself_ and not it and they side by side "in" time.
For we can now understand this _behind_ and this _inside_ in a remarkably precise manner. The visible landscape has the other moments of time "behind itself," precisely in that the future waits beyond the horizon, as well as _behind_ every entity that I see, as the unseen "other side" of the many visibles that surround me. And the visible landscape has the other moments of time "inside itself," precisely in that the past preserves itself under the ground, as well as _inside_ every entity that I perceive. The sensorial landscape, in other words, not only opens onto that distant future waiting beyond the horizon but also onto a near future, onto an immanent field of possibilities waiting behind each tree, behind each stone, behind each leaf from whence a spider may at any moment come crawling into our awareness. And this living terrain is supported not only by that more settled or sedimented past under the ground, but by an immanent past resting inside each tree, within each blade of grass, within the very muscles and cells of our own bodies.
It is thus that ecologists and environmental scientists may study the recent past of a particular place by "coring" several of the standing trees, in order to count their interior rings and to interpret the varying width of those rings (an extra-wide layer, fourteen rings in from the cambium, suggests a season of abundant rain fourteen years into the depth of the past, while an extra-thin layer tells of a year without rainfall). The deeper past may be pondered by digging a "soil pit" to expose the sedimented layers of the soil, and to interpret the composition and structure of those layers (a layer of charcoal, for instance, bespeaks a forest fire at that depth of the past). Meanwhile, archaeologists, paleontologists, and geologists dig still deeper beneath the ground of the present in order to unearth traces of ancient epochs and eons.
That which has been and that which is to come are not elsewhere—they are not autonomous dimensions independent of the encompassing present in which we dwell. They are, rather, the very depths of this living place—the hidden depth of its distances and the concealed depth on which we stand.
PROMPTED BY THE PLACE-CENTERED DISCOURSE OF ORAL, INDIGENOUS peoples—which seems to lack any absolute distinction between "space" and "time"—and prompted as well by our analysis of writing and its perceptual effects, we have been searching for a possible reconciliation between time and space. If the distinction between these dimensions is not a necessary distinction, then we should be able to demonstrate the possibility of another way of construing events, one in which spatial and temporal aspects are not distinguishable.
And we have succeeded in demonstrating that there is at least one way to unify the experience of time and of space, that it is indeed possible to perceptually reconcile the temporal and the spatial in a manner that accounts for the apparent openness of what we have come to call the "future" and the apparent closedness of what we have come to call the "past." Heretofore, such a perceptual reconciliation was thought to be impossible, usually because space—even perceived space—was assumed to be essentially homogeneous, and so to lack any structural asymmetry that might correspond with the evident asymmetry of time. It is evident, however, that when our awareness of time is joined with our awareness of space, space itself is transformed. Space is no longer experienced as a homogeneous void, but reveals itself as this vast and richly textured field in which we are corporeally immersed, this vibrant expanse structured by both a ground and a horizon. It is precisely the ground and the horizon that transform abstract space into space-time. And these characteristics—the ground and the horizon— _are granted to us only by the earth_. Thus, when we let time and space blend into a unified space-time, we rediscover the enveloping earth.
It would seem, then, that the conceptual separation of time and space—the literate distinction between a linear, progressive time and a homogeneous, featureless space—functions to _eclipse_ the enveloping earth from human awareness. As long as we structure our lives according to assumed parameters of a static space and a rectilinear time, we will be able to ignore, or overlook, our thorough dependence upon the earth around us. Only when space and time are reconciled into a single, unified field of phenomena does the encompassing earth become evident, once again, in all its power and its depth, as the very ground and horizon of all our knowing.
### In the Depths of the Sensuous
The importance that our analysis has led us to place on such taken for granted phenomena as the _ground_ and the _horizon_ will seem strange to most readers, indeed to all of us raised in a culture that asks us to distrust our immediate sensory experience and to orient ourselves instead on the basis of an abstract, "objective" reality known only through quantitative measurement, technological instrumentation, and other exclusively human involvements. But for those indigenous cultures still participant with the more-than-human life-world, for those peoples that have not yet shifted their synaesthetic focus from the animate earth to a purely human set of signs, the riddles of the under-the-ground and the beyond-the-horizon (the inside of things and the other side of things) are felt as vast and powerful mysteries, the principal realms from whence beings enter the animate world, and into which they depart.
For instance, among most native tribes of the American Southwest, where I live—including, among others, the Hopi, the Zuñi, the Tewa, the Tiwa, the Keresan, and the Navajo nations—the people believe that they came into the world from under the ground. According to the Zuñi emergence story, all the _people_ (humans and all other animals) originally lived in the fourth dark underworld within the earth. They were summoned forth from there by Sun, who, along with Moon, inhabited the bright world above Earth's surface. And so the animal-people gathered all their sacred bundles for making rain, and for coaxing seeds to grow, and climbed upward along a reed through the four underworlds—through the soot world, the sulfur-smell world, the fog world, and the feather-wing world—until, finally, they emerged into _this_ world. From the _sipapu_ , or place of emergence, the people then spread out and began to settle the land.
The Emergence is one of the most sacred and widely held beliefs among native North Americans today, although it is particularly evident in the Southwest. In its structure the story of the people's emergence from under the ground, usually climbing up a reed or a tree, mimics the emergence from the soil of the corn and other plants harvested by the horticultural tribes of the Southwest. The people who climb up from those depths in search of sunlight and rain are like corn growing up through the soil.
But the Emergence is also akin to the process by which all mammals, including humans, are born into this world, emerging from the darkness of their mother's womb into the spaciousness of the open earth. "When we came up on this earth, it was just like a child being born from its mother." In fact, earlier tellings of the Zuñi Emergence, recorded in the last century, relate that long before the existence of the people, the Sun cohabited with the Earth, and it is thus that life was conceived within the deepest, fourth womb of the Earth. Hence, the Emergence may be understood as the collective _birth_ of all peoples—of all animals and plants—after a prolonged period of gestation in the dark depths of the ground.
The most sacred ceremonies of the pueblo-dwelling tribes take place in the _kivas_ , the underground or partially underground chambers also called "wombs" by many of the pueblo people. One enters a kiva by climbing down a ladder through a hole in the roof, and after the ceremony one leaves the kiva by climbing up through the same opening, the same _sipapu_ , reexperiencing—and renewing—the primordial emergence from the underworld. In fact, all sorts of earthly openings—holes, caves, canyons, small depressions in the ground and even in stones—are considered _sipapu_ by the Pueblo peoples, and so remind them of their origin under the ground that now supports them.
The individual experience of birth is thus related to the collective emergence of life from under the ground. Similarly, human death, for oral peoples, is not just a personal event but also a transformation in the land, a process whereby one's individual sensibility opens outward to rejoin the encompassing, more-than-human field of sensations. In an old Pawnee tale, a dead man returns as a ghost, saying, "I am in everything; in the grass, the water." The dead do not _leave_ the sensuous world, forsaking it for an immaterial heaven. Rather, the vitality of one who dies is often thought to journey just _beyond the visible horizon_ , to a nearby land where all of the ancestors traditionally gather, and from whence they still influence events within the land of the living. Among the above-mentioned Pueblo peoples, for instance, the dead are thought to travel to the village of the _kachinas_ , which for the Zuñi is located under a lake several days journey to the west. The kachinas, the godlike ancestors, regularly return to the various pueblos for the seasonal ceremonies at which they are impersonated, or made visible, by masked dancers. But the kachinas also visit the pueblos, whenever they wish, as rain-bearing clouds that approach from _beyond the horizon_ , carrying the life-giving moisture so necessary to the corn and the other plants upon which these horticultural peoples depend:
the Hopis—like the other Pueblos—believe their ancestors to be fertilizing clouds, bringers of rain who will nourish the crops upon which the living subsist. The necessity of death... becomes even more accentuated, therefore.... Death brings into existence the ancestors, who turn into clouds and kachinas that bring rain; moisture feeds the corn and other foods that in turn nourish the Hopi people themselves, and in the eternal cycle, death feeds life.
Among nonhorticultural tribes as well, the dead are often thought to journey to a land beyond the horizon, from whence they may return among the living in the guise of animals and other natural elements. Indeed, for many hunting peoples, the realm beyond the mountains, or beyond the ocean, was where various animal species resided when they were not evident in the present landscape, a realm where the deer or the salmon were thought to remove their animal guises and to live in quasi-human form. To cite a single example, the Skagit Indians of northwestern North America held that the salmon, when they are not spawning in the rivers, live beyond the horizon in human form. Hence, in the nineteenth century, when several of these Indians traveled to the eastern coast of North America and saw the abundance of pale-skinned people living there, they reported back that they had been to salmon country and had seen salmon walking around as human beings.
For the American Plains tribes, at least in the nineteenth century, the home of the dead beyond the horizon was commonly believed to be a land always abundant in edible plants and wild game—the "happy hunting ground" of popular legend. While some such indigenous notion of a fertile and abundant terrain where the ancestors dwell was likely the archaic source of even the Christian belief in a heavenly paradise, it is important to realize that for oral peoples such realms were never wholly cut off from the sensuous world of the living present. They were not projected entirely outside of the experienced world, but were felt as the mystery and hidden depth of the sensuous world itself.
If we pay close attention to the life and activity of the great celestial powers—the sun, the moon, and the clustered stars—we will see that even these entities, so commonly associated with height and vertical transcendence, seem to emerge from, and return to, the lands beyond the horizon. Hence, if the Shoshoni Indians, for example, assert that a dead person "follows the Milky Way" to the land of the dead, this need not indicate, as some anthropologists have claimed, that the Shoshoni believe in a celestial heaven. For the Milky Way is but a visible trail or "way" followed by the spirits of the dead, and this trail—as we can readily see—leads precisely beyond-the-horizon.
Yet here we must acknowledge a strange ambiguity. The beyond-the-horizon is that realm where the sun goes when it leaves us, and the realm from which it emerges at dawn; it is where the moon goes to and returns from. But we could just as well say the sun sinks into the under-the-ground and the moon emerges from under-the-ground. For when we attend closely to our direct, sensory experience of the rising and the setting, we see that the moon's journey beyond the horizon is also experienced as a movement down into the ground, and indeed that the sun's rise each morning is as much an emergence from under the ground as is the emergence of a groundhog at the end of winter! Hence, for example, these words by Kiowa author N. Scott Momaday:
"Where does the sun live?"... [T]o the Indian child who asks the question, the parent replies, "The sun lives in the earth." The sun-watcher among the Rio Grande Pueblos, whose sacred task it is to observe, each day, the very point of the sun's emergence on the skyline, knows in the depths of his being that the sun is alive and that it is indivisible with the earth, and he refers to the farthest eastern mesa as "the sun's house."... Should someone say to the sun, "Where are you going?" the sun would surely answer, "I am going home," and it is understood at once that home is the earth. All things are alive in this profound unity in which are all elements, all animals, all things.... [M]y father remembered that, as a boy, he had watched with wonder and something like fear the old man Koi-khan-hole, "Dragonfly," stand in the first light, his arms outstretched and his painted face fixed on the east, "praying the sun out of the ground."
Phenomenologically considered, it is as though the luminous orb of the sun journeys into the ground each evening, moving all night through the density underfoot, to emerge, at dawn, at the opposite side of the visible world. For some indigenous cultures, it is precisely during this journey through the ground that the sun impregnates the earth with its fiery life, giving rise to the myriad living things—human and nonhuman—that blossom forth on earth's surface.
So the journey beyond-the-horizon can lead under-the-ground, and vice versa. We begin to glimpse here the secret identity, for oral peoples, of those topological regions that we have come to call "the past" and "the future"—the curious manner in which these two very different modes of absence can nevertheless transmute into each other, blur into one another, like moods. It is thus that many indigenous cultures have but a single term to designate the very deep past and the far distant future. Among the Inuit of Baffin Island, for example, the term _uvatiarru_ may be translated both as "long ago" and "in the future." The cyclical metamorphosis of the distant past into the distant future, or of that-which-has-been into that-which-is-to-come, would seem to take place continually, in the depths far below the visible present, in that place where the unseen lands beyond the horizon seem to fold into the invisible density beneath our feet.
MARTIN HEIDEGGER, WHOSE CAREFUL DESCRIPTIONS OF THE PAST and the future have helped us to recognize these realms as actual dimensions of the perceptual field, did not write of only two temporal dimensions, however, but of _three_ , including that of the present. In _Being and Time_ , Heidegger asserts that the present has its own ecstasy, its own proper transcendence, its own " 'whither' to which one is carried away." The implication is that phenomena can be hidden not just within the past or the future, but also within the very thickness of the present, itself—that there is an enigmatic, hidden dimension at the very heart of the sensible present, into which phenomena may withdraw and out of which they continually emerge. Thus in "Time and Being," Heidegger writes that "even in the present itself, there always plays a kind of approach and bringing about, that is, a kind of presencing." As though, paradoxically, there is a modality of absence entirely native to the present, out of which the present, itself, comes to presence: "In the present, too, presencing is given."
Is there, then, yet another mode of absence or invisibility entirely endemic to the open landscape? I have already noticed, here within the perceivable present, the hidden nature of what lies behind the tree trunks and stones that surround me, which corresponds to the unseen character of that which lies on the other side of these nearby hills, and ultimately to those lands entirely beyond the horizon of the perceivable present, from whence numerous entities enter the visible terrain and into which various phenomena withdraw, recede, and finally vanish from view. I have acknowledged as well the concealed character of that which rests _inside_ the trunks of these trees, inside the stones and the hills, which corresponds, ultimately, to the unseen nature of the under-the-ground, from whence beings sprout and unfurl, and into which they also crumble, decompose, and are submerged. Is there some other obvious style of absence, in the very thickness of the present, that is unique to itself, and not a mere modification of the under-the-ground or the beyond-the-horizon? Some mode of concealment that is, paradoxically, already out in the open, from whence the visible landscape itself continually comes to presence?
Perhaps I am pushing my method too far, here, in trying to place not only the _withholding_ of presence by the future and the _refusal_ of presence by the past, but also this concealment of presence _from within the present itself_. For now, more than ever, I feel confused,—unable to grasp, or to conceive of, what it is that I am searching for. Even as I gaze out across the wooded hills, my mind seems muddled by these questions, by ideas and associations that keep me from directly sensing and responding to the animate earth around me. I try to relax, and so begin to breathe more deeply, enjoying the coolness of the breeze as it floods in at my nostrils, feeling my chest and abdomen slowly expand and contract. My thinking begins to ease, the internal chatter gradually taking on the rhythm of the in-breath and the out-breath, the words themselves beginning to dissolve, flowing out with each exhalation to merge with the silent breathing of the land. The interior monologue dissipates, slowly, into the rustle of pine needles and the stately gait of the clouds.
A butterfly glides by, golden wings navigating delicate air currents with a few momentary flutters before they settle on a white flower. The seedstalks of the grasses bounce in the breeze, while clustered wildflowers tremble on their stems, awaiting the humming insects that motor haphazardly from one to the other. Fragrant whiffs from new blossoms in the overgrown orchard by the creek stir not only the winged beings, but my own flaring nostrils as they reach me from afar, drifting like spiderwebs on the faint winds. My sensing body now vividly awake to the world, I gradually become conscious of a third mode of invisibility, of an unseen dimension in which I am so thoroughly and deeply immersed that even now I can hardly bring it to full awareness....
It is the invisibility of the air.
# **7**
# **_The Forgetting and Remembering of the Air_**
Let's sit down here... on the open prairie, where we can't see a highway or a fence. Let's have no blankets to sit on, but feel the ground with our bodies, the earth, the yielding shrubs. Let's have the grass for a mattress, experiencing its sharpness and its softness. Let us become like stones, plants, and trees. Let us be animals, think and feel like animals. Listen to the air. You can hear it, feel it, smell it, taste it. _Woniya wakan_ —the holy air—which renews all by its breath. _Woniya, woniya wakan_ —spirit, life, breath, renewal—it means all that. _Woniya_ —we sit together, don't touch, but something is there; we feel it between us, as a presence. A good way to start thinking about nature, talk about it. Rather talk to it, talk to the rivers, to the lakes, to the winds as to our relatives.
—JOHN FIRE LAME DEER
WHAT A MYSTERY IS THE AIR, WHAT AN ENIGMA TO THESE human senses! On the one hand, the air is the most pervasive presence I can name, enveloping, embracing, and caressing me both inside and out, moving in ripples along my skin, flowing between my fingers, swirling around my arms and thighs, rolling in eddies along the roof of my mouth, slipping ceaselessly through throat and trachea to fill the lungs, to feed my blood, my heart, my self. I cannot act, cannot speak, cannot think a single thought without the participation of this fluid element. I am immersed in its depths as surely as fish are immersed in the sea.
Yet the air, on the other hand, is the most outrageous absence known to this body. For it is utterly invisible. I know very well that there is something there—I can feel it moving against my face and can taste it and smell it, can even hear it as it swirls within my ears and along the bark of trees, but still, I cannot see it. I can see the steady movement it induces in the shapeshifting clouds, the way it bends the branches of the cottonwoods, and sends ripples along the surface of a stream. The fluttering wing feathers of a condor soaring overheard; the spiraling trajectory of a leaf as it falls; a spider web billowing like a sail; the slow drift of a seed through space—all make evident, to my eyes, the sensuous presence of the air. Yet these eyes cannot see the air itself.
Unlike the hidden character of what lies beyond the horizon, and unlike the unseen nature of that which resides under the ground, the air is invisible _in principle_. That which today lies beyond the horizon can at least partly be disclosed by journeying into that future, as that which waits under the ground can be somewhat unearthed by excavations into the past. But the air can never be opened for our eyes, never made manifest. Itself invisible, it is the medium through which we see all else in the present terrain.
And this unseen enigma is the very mystery that enables life to live. It unites our breathing bodies not only with the under-the-ground (with the rich microbial life of the soil, with fossil and mineral deposits deep in the bedrock), and not only with the beyond-the-horizon (with distant forests and oceans), but also with the interior life of all that we perceive in the open field of the living present—the grasses and the aspen leaves, the ravens, the buzzing insects and the drifting clouds. What the plants are quietly breathing out, we animals are breathing in; what we breathe out, the plants are breathing in. The air, we might say, is the soul of the visible landscape, the secret realm from whence all beings draw their nourishment. As the very mystery of the living present, it is that most intimate absence from whence the present presences, and thus a key to the forgotten presence of the earth.
NOTHING IS MORE COMMON TO THE DIVERSE INDIGENOUS CULTURE of the earth than a recognition of the air, the wind, and the breath, as aspects of a singularly sacred power. By virtue of its pervading presence, its utter invisibility, and its manifest influence on all manner of visible phenomena, the air, for oral peoples, is the archetype of all that is ineffable, unknowable, yet undeniably real and efficacious. Its obvious ties to speech—the sense that spoken words are structured breath (try speaking a word without exhaling at the same time), and indeed that spoken phrases take their communicative power from this invisible medium that moves between us—lends the air a deep association with linguistic meaning and with thought. Indeed, the ineffability of the air seems akin to the ineffability of awareness itself, and we should not be surprised that many indigenous peoples construe awareness, or "mind," not as a power that resides inside their heads, but rather as a quality that they themselves _are inside of_ , along with the other animals and the plants, the mountains and the clouds.
According to Robert Lawlor, a researcher who has lived and studied among the indigenous cultures of Australia, Aboriginal peoples tend to consider the visible entities around them—rocks, persons, leaves—as crystallizations of conscious awareness, while the invisible medium _between_ such entities is experienced as what Westerners would call "the unconscious," the creative but unseen realm from which such conscious forms arise. Thus, the _Alcheringa_ , or Dreamtime—that implicit realm of dreamlike happenings from whence the visible present is continually emerging—resides not just within the hills and landforms of the surrounding terrain, but also in the invisible depths of the air itself, in the thickness of the very medium that flows within us and all around us. This leads Aboriginal Australians to accord awesome significance to various atmospheric phenomena. Flashes of lightning are experienced as violent discharges from the depths of the Dreaming. Birds, who wing their way through the invisible, are often experienced as messengers of the unconscious, while the rainbow (the Rainbow Snake, who arcs upward across the sky and then dives back into the earth) is felt to personify all the most implacable, dangerous, and yet life-giving forces in the land. For the rainbow is perceived as the very _edge_ of the Dreaming, as that place where the invisible, unconscious potentials begin to become visible.
### Wind and Spirit on the Great Plains
The omnipresent and yet invisible nature of the air ensures that the indigenous beliefs and teachings regarding this elemental mystery are among the most sacred and secret of oral traditions. Native teachings regarding the wind or the breath are exceedingly difficult to track or to record, for to give voice to them unnecessarily may violate the mystery and holiness of this enveloping power, this enigmatic presence (or absence) so obviously essential to one's life and the life of the land.
We do know that the air was an uncommonly sacred power for most of the native peoples of North America. Among the Creek Indians of the Southeast, for instance, the creator god—the only divinity equal to or exceeding the Earth and the Sun in its power—is called Hesakitumesee, the Master of Breath; it is this being who sends fog, wind, and other weather across the land, affecting the destiny of the people.
For the Lakota Nation, the most sacred or _wakan_ aspect of Wakan Tanka, the Great Mysterious, is Taku Škanškan, the Enveloping Sky. Known to the shamans simply as Škan, Taku Škanskan is felt to be everywhere, the omnipresent spirit that imparts life, motion, and thought to all things, yet is visible to us only as the blue of the sky. (It is this deity that contemporary Lakota persons sometimes address, in English, as the Great Spirit.) Tate (pronounced "Tah-day")— _Wind_ —is created by Škan out of his own substance, to be a companion for Škan and to carry his wishes and messages throughout the world. (Škan and Tate—Sky and Wind—are thus sometimes spoken of as the same entity by the Lakota shamans.) And it was Tate who mated with Ite ("Ee-day"), a beautiful woman of the Buffalo people; from this union Ite gave birth to the North Wind, the East Wind, the South Wind, and the West Wind (as well as to Yum, the little whirlwind or dust-devil). These four Winds structure, and lend their particular magics, to every Lakota ritual practiced today.
Meanwhile, the peace pipe is the most _wakan_ of all possessions for the Lakota. Carved from dark red pipestone found only in the northern plains—a stone considered to be the petrified blood of their ancestors—the sacred pipe is smoked in ritual fashion during all of the diverse Lakota ceremonies, from the sweat lodge to the Sun Dance. The pipe smoke makes the invisible breath visible, and as it rises from the pipe, it makes visible the flows and currents in the air itself, makes visible the unseen connections between those who smoke the pipe in offering and all other entities that dwell within the world: the winged peoples, the other walking and crawling peoples, and the multiple rooted beings—trees, grasses, shrubs, mosses. Further, the rising smoke carries the prayers of the Lakota people to the sky beings—to the sun and the moon, to the stars, to the thunder beings and the clouds, to all those powers embraced by _woniya wakan_ , the holy air.
_Woniya wakan_ —the holy air—which renews all by its breath. _Woniya, woniya wakan_ —spirit, life, breath, renewal—it means all that. _Woniya_ —we sit together, don't touch, but something is there; we feel it between us, as a presence.
At the opening of any ceremony, a Lakota medicine person fills and lights the sacred pipe, and then, before smoking it himself, offers the mouthpiece to the West Wind so that the wind itself may partake of the smoke. Then, circling, he offers the pipe to be smoked by the North Wind, then to the East Wind, and finally to the South Wind. As the messengers of the gods, the winds are the first powers to be addressed in any ceremony.
The winds of the four directions are also deeply associated with the cyclical, spatial sense of time. An old Lakota shaman, named Sword, interviewed early in the twentieth century, related that in any ceremony, after offering the mouthpiece of the lit pipe to each of the four winds,
the shaman should move the pipe in the same manner until the mouthpiece again points toward the west, and say: "Circling, I complete the four quarters and the time." He should do this because the four winds are the four quarters of the circle and mankind knows not where they may be or whence they may come and the pipe should be offered directly toward them. The four quarters embrace all that are in the world and all that are in the sky. Therefore, by circling the pipe, the offering is made to all the gods. The circle is the symbol of time, for the day time, the night time, and the moon time are circles above the world, and the year time is a circle around the border of the world. Therefore the lighted pipe moved in a complete circle is an offering to all the times.
After completing the circle, the shaman points the mouthpiece of the pipe toward the sky, and offers it to Wind, Tate, the father of the four Winds. Finally, then, "the shaman should smoke the pipe and while doing so should say: 'I smoke with the Great Spirit. Let us have a blue day.' "
### Air and Awareness Among the Diné, or Navajo
While the air is held sacred _throughout_ native North America, the most extensively documented interpretation of the air is probably the Diné, or Navajo, concept of _nilch'i_ —the Holy Wind. Long misunderstood by anthropologists, the Navajo term _nilch'i_ refers to the whole body of the air or the atmosphere, including the air when in motion, as well as the air that swirls within us as we breathe. According to James Kale McNeley, in his meticulously documented book _Holy Wind in Navajo Philosophy, nilch'i_ , "meaning Wind, Air, or Atmosphere," suffuses all of nature, and is that which grants life, movement, speech, and awareness to all beings. Moreover, the Holy Wind serves as the means of communication _between_ all beings and elements of the animate world. _Nilch'i_ is thus utterly central to the Diné, or Navajo worldview.
Although _nilch'i_ is conceived by the Navajo as a single, unified phenomenon, the Wind in its totality is also assumed to be comprised of many diverse aspects, a plurality of partial Winds, each of which have their own name in the Navajo language. One of these— _nilch'i hwii'siziinii_ , or "the Wind within one"—refers to that part of the overall Wind that circulates within each individual. This notion was mistaken by early missionaries, and by the important missionary/ethnologist Father Berard Haile, to be a phenomenon akin to the personal soul of Christian belief. Thus, "the Wind within one" was interpreted, until recently, to be an immaterial spirit or soul, a thoroughly autonomous entity that enters the individual at birth, acts as the internal source of his or her life and behavior, and then departs from the individual at death. Only recently have anthropologists like McNeley been able to break out of the interpretive blinders imposed by the Christian worldview in order to recognize that the powers attributed by Western culture to a purely internal soul or mind are experienced by the Navajo as attributes of the enveloping Wind or Atmosphere as a whole. The "Wind within one" is in no way autonomous, for it is in a continual process of interchange with the various winds that surround one, and indeed is entirely a part of the Holy Wind itself.
WE MAY BRING OURSELVES CLOSE TO THE ORAL EXPERIENCE OF THE air by consulting the words of the Navajo elders themselves, and by pondering the preeminent influence of Wind, or Air, within the Navajo universe.
Wind existed first, as a person, and when the Earth began its existence Wind took care of it. We started existing where Darknesses, lying on one another, occurred. Here, the one that had lain on top became Dawn, whitening across. What used to be lying on one another back then, this is Wind. It (Wind) was Darkness. That is why when Darkness settles over you at night it breezes beautifully. It is this, it is a person, they say. From there when it dawns, when it dawns beautifully becoming white-streaked through the dawn, it usually breezes. Wind exists beautifully, they say. Back there in the underworlds, this was a person it seems.
Already in the underworlds, in those times or realms beneath the ground, prior to the emergence of the Holy People into the world of the present, Wind existed and provided both breath and guidance to the other Holy Ones, such as First Man, First Woman, Talking God, and Calling God. When these Holy People emerged from the ground into this world on Earth's surface, they were accompanied by Wind. Already differentiated as the Winds of Darkness and of Dawn, Wind now differentiated itself further into the Blue Wind of noon and the Yellow Wind of twilight. These four Winds spread out from the emergence place and were then placed by the Earth in the four directions, _along the horizon of the world_ —Dawn Woman in the east, Horizontal Blue Girl in the south, Horizontal Yellow Boy (or evening twilight) in the west, and Darkness Man (or night) in the north (the precise names of these Winds vary from one chant to another; often they are simply spoken of as White Wind, Blue Wind, Yellow Wind, and Dark Wind). These four Winds—or four _Words_ , as they are also called—are said to be the means of breath of the four sacred mountains that visibly rise at the edge of the Navajo cosmos, one in each of the cardinal directions. "They [the Winds] stand within the mountains, these [mountains] from then on being, by them, our sacred ones to the end of time." Similarly, the Sun and the Moon have their own Winds, which are their means of life and breath. Other Winds surround and move between these great powers, as their means of communication with each other and with other phenomena. From its sacred home in each of the four directions, the Holy Wind is said to approach and enter into the various natural phenomena of the world, and so to provide the means of life, movement, thought, and speech to the plants, to the animals, and to all the other Earth Surface People, including the Navajo people themselves.
Wind is believed by the Diné to be present within a person from the very moment of conception, when two Winds, one from the bodily fluids of the father and one from those of the mother, form a single Wind within the embryo. It is the motion of this Wind that produces the movement and growth of the developing fetus. When the baby is born, the Navajo say that the Wind within it "unfolds him" and it is then, when the infant commences breathing, that another, _surrounding_ Wind enters into the child. This Wind may be sent from one of the four directions along the horizon, or from the Sun, or the Moon, or from the Ground itself—indeed from _any_ natural phenomenon. Of course, the particular Wind that enters with the first breath will have a powerful influence upon the whole course of that person's life. Yet other Winds will enter at later moments in the development of the child, so that, as McNeley writes, "the growing child is believed to be continually subject to the influence of Winds existing around him."
Although invisible, the Holy Wind can be recognized by the swirling and spiraling traces that it continually leaves in the visible world. The Winds that enter a human being leave their trace, according to the Navajo, in the vortices or swirling patterns to be seen on our fingertips and the tips of our toes, and in the spiraling pattern made by the hairs as they emerge from our heads. As one elder explains:
There are whorls here at the tips of our fingers. Winds stick out here. It is the same way on the toes of our feet, and Winds exist on us here where soft spots are, where there are spirals. At the tops of our heads some children have two spirals, some have only one, you see. I am saying that those (who have two) live by means of two Winds. These (Winds sticking out of the) whorls at the tips of our toes hold us to the Earth. Those at our fingertips hold us to the Sky. Because of these, we do not fall when we move about.
Further, it is Wind that enables us to speak. We have already noted that the four Winds of the cardinal directions are also called the "four Words." Since we speak only by means of the breath, Wind itself—the collective breath—is said to hold the power of language: "It is only by means of Wind that we talk. It exists at the tip of our tongues."
Summing up these various conceptions, McNeley writes:
[A]ccording to the Navajo conception, then, Winds exist all around and within the individual, entering and departing through respiratory organs and whorls on the body's surface. _That which is within and that which surrounds one is all the same and it is holy_.
Finally, and most profoundly, this invisible medium, in which we are bodily immersed, is what provides us with the capacity for conscious thought. It was mentioned above that the sacred Mountains in the four directions have various Winds that move between them as their means of communication with each other and with other entities. The invisible Wind that swirls within and around each individual person is assumed to consist, in part, of such messenger Winds from the four directions. Two of these Winds, often spoken of as Little Winds or Wind's Children, are believed to be the individual's "means of knowing." These two Little Winds linger within the spiraling folds of our two ears, and it is from there that they offer guidance to us, alerting us to near and distant difficulties, helping us to plan and to make choices. When a Navajo person finds himself thinking in words, this is said to be the voice of one or both of these two Little Winds speaking into his ears. Of course these Wind's Children are simply little currents or vortices within the vast body of _nilch'i_ —the Holy Wind—which exists everywhere. In the words of one elder; "The one called Wind's Child, this is just like living in water"—that is, Wind's Child is inseparable from the swirling body of air in which we are thoroughly immersed.
Such Little Winds from the four directions dwell not only in human ears, but in the ears or earlike aspects of all living things, providing their means of hearing, knowing, and communicating with others. It is thus that other animals, for instance, know what we humans are thinking about them: "When we are thinking well of them—horses, cattle, goats, and everything that we live by—they know about it by means of Wind. They know our thinking." Some elders say that nowadays Little Winds from the four directions no longer advise or speak to the Navajo as clearly as they once did, but that such Little Winds still speak clearly into the ears of other animals, telling them of what is happening in the world; and animals like Coyote and Owl often communicate such knowledge to the Navajo, warning the humans of dangerous situations by specific sounds and behaviors.
Now, when referring to the multiple and diverse Winds such as Dawn Man or Dawn Woman, Sky Blue Woman, Twilight Man, Dark Wind, Wind's Child, Revolving Wind, Glossy Wind, Rolling Darkness Wind, and others, the Navajo are not speaking of abstract or ideal entities but of palpable phenomena—of the actual gusts, breezes, whirlwinds, eddies, stormfronts, crosscurrents, gales, whiffs, blasts, and breaths that they perceive in the fluid medium that surrounds and flows through their bodies. The profound belief in the overall unity of _nilch'i_ , the Navajo conviction that all of these subsidiary Winds are internal expressions of a single, inexhaustible mystery, is obviously born of the observation that the multiple atmospheric vortices made by their own breathing—or by the heat rising in waves from the sun-baked cliff, or the branches of trees as they divide the surging air, or the minute trembling of a rattlesnake's tail—that all these evident currents and eddies swirling around and even inside them are not entirely autonomous forces, but rather are momentary articulations within the vast and fathomless body of Air itself.
It is clear, however, that there is a kind of _provisional_ autonomy or identity to the various winds that are part of the overall atmosphere—the warm and sluggish air lingering in the sandy arroyo every afternoon is obviously different from the cool breeze blowing through the cottonwoods along the river. To the Navajo there are unpredictable Winds as well as steady Winds, helpful Winds and Harmful Winds. Certain dangerous Winds, for example, can alter the character of the good Winds within a living person, or can bring difficulty and harm to the community, or to the land. Each person must navigate through this world of diverse invisible influences with great care, strengthening her contact with the various good Winds by respecting the land itself, striving to bring her life into harmony, or _hozho_ , with the four directions, into reciprocity with the Ground and the Sky, with the Sun, the Moon, and the Stars.
Like the mountains of the four directions, and like the other animals and the plants, humans are themselves one of the Wind's dwelling places, one of its multiple centers, and just as we are nourished and influenced by the Air at large, so do our actions and thoughts affect the Air in turn. The individual, that is, is not passive with respect to the Holy Wind; rather she participates _in_ it, as one of its organs. Her own desire and intent (her own interior Wind) participates directly in the life of the invisible Wind all around her, and hence can engage and subtly influence events in the surrounding terrain—even, in some measure, the becoming-abundant of rain clouds, the gestation of seeds, and the seasonal procreation of animals.
Hence the emphasis among the Navajo, and indeed among so many native peoples, upon concentrated thought and prayer in order to influence and aid the continual emergence of such earthly occurrences from unmanifest (implicit, invisible) to manifest (visible) existence.
It is through the ritual power of speech and song that the Navajo are enabled most powerfully to affect and alter events in the enveloping cosmos. According to Gary Witherspoon, in his landmark study of _Language and Art in the Navajo Universe_ , the Navajo consider the act of speech to be an externalization of thought, "an imposition of form upon the external world" in which the surrounding Air is transformed. And because the Air or Wind is the very medium in which the other natural forces live and act, by transforming the Air through song, the singer is able to affect and subtly influence the activity of the great natural powers themselves.
When a Navajo person wishes to renew or reestablish, in the world, the harmonious condition of well-being and beauty expressed by the Navajo word _hozho_ he must first strive, through ritual, to create this harmony and peacefulness within his own being. Having established such _hozho_ within himself, he can then actively impart this state of well-being to the enveloping cosmos, through the transforming power of song or prayer. Finally, according to Witherspoon,
[a]fter a person has projected _hozho_ into the air through ritual form, he then, at the conclusion of the ritual, breathes that _hozho_ back into himself and makes himself a part of the order, harmony, and beauty he has projected into the world through the ritual mediums of speech and song.
This brief quote from Witherspoon makes especially evident the reciprocal, even circular character of the relation between the Navajo people and the animate cosmos that enfolds and includes them. They are not passive with respect to the other powers of this world, or rather they are both passive _and_ active, inhaling _and_ exhaling, receiving the nourishment of the diverse beings and actively nourishing them in turn. As it is spoken in the Blessingway ceremony:
_With everything having life, with everything having the power of speech, with everything having the power to breathe, with everything having the power to teach and guide, with that in blessing we will live._ 31
For the Navajo, then, the Air—particularly in its capacity to provide awareness, thought, and speech—has properties that European, alphabetic civilization has traditionally ascribed to an interior, individual human "mind" or "psyche." Yet by attributing these powers to the Air, and by insisting that the "Winds within us" are thoroughly continuous with the Wind at large—with the invisible medium in which we are immersed—the Navajo elders suggest that that which we call the "mind" _is not ours_ , is not a human possession. Rather, mind as Wind is a property of the encompassing world, in which humans—like all other beings—participate. One's individual awareness, the sense of a relatively personal self or psyche, is simply that part of the enveloping Air that circulates within, through, and around one's particular body; hence, one's own intelligence is assumed, from the start, to be entirely participant with the swirling psyche of the land. Any undue harm that befalls the land is readily felt within the awareness of all who dwell within that land. And thus the health, balance, and well-being of each person is inseparable from the health and well-being of the enveloping earthly terrain.
THE NAVAJO IDENTIFICATION OF AWARENESS WITH THE AIR—THEIR intuition that the psyche is not an immaterial power that resides inside us, but is rather the invisible yet thoroughly palpable medium in which we (along with the trees, the squirrels, and the clouds) are immersed—must seem at first bizarre, even outrageous, to persons of European ancestry. Yet a few moments' etymological research will reveal that this identification is not nearly so alien to European civilization as one might assume. Indeed, our English term "psyche"—together with all its modern offspring like "psychology," "psychiatry," and "psychotherapy"—is derived from the ancient Greek word _psychê_ , which signified not merely the "soul," or the "mind," but also a "breath," or a "gust of wind." The Greek noun was itself derived from the verb _psychein_ , which meant "to breathe," or "to blow." Meanwhile, another ancient Greek word for "air, wind, and breath"—the term _pneuma_ , from which we derive such terms as "pneumatic" and "pneumonia"—also and at the same time signified that vital principle which in English we call "spirit."
Of course, the word "spirit" itself, despite all of its incorporeal and non-sensuous connotations, is directly related to the very bodily term "respiration" through their common root in the Latin word _spiritus_ , which signified both "breath" and "wind." Similarly, the Latin word for "soul," _anima_ —from whence have evolved such English terms as "animal," "animation," "animism," and "unanimous" (being of one mind, or one soul), also signified "air" and "breath." Moreover, these were not separate meanings; it is clear that _anima_ , like _psychê_ , originally named an elemental phenomenon that somehow comprised both what we now call "the air" and what we now term "the soul." The more specific Latin word _animus_ , which signified "that which thinks in us," was derived from the same airy root, _anima_ , itself derived from the older Greek term _anemos_ , meaning "wind."
We find an identical association of the "mind" with the "wind" and the "breath" in innumerable ancient languages. Even such an objective, scientifically respectable word as "atmosphere" displays its ancestral ties to the Sanskrit word _atman_ , which signified "soul" as well as the "air" and the "breath." Thus, a great many terms that now refer to the air as a purely passive and insensate medium are clearly derived from words that once identified the air with life and awareness! And words that now seem to designate a strictly immaterial mind, or spirit, are derived from terms that once named the breath as the very substance of that mystery.
It is difficult to avoid the conclusion that, for ancient Mediterranean cultures no less than for the Lakota and the Navajo, the air was once a singularly sacred presence. As the experiential source of both psyche and spirit, it would seem that the air was once felt to be the very matter of awareness, the subtle body of the mind. _And hence that awareness, far from being experienced as a quality that distinguishes humans from the rest of nature, was originally felt as that which invisibly_ **_joined_** _human beings to the other animals and to the plants, to the forests and to the mountains_. For it was the unseen but common medium of their existence.
But how, then, did the air come to lose its psychological quality? How did the psyche withdraw so thoroughly from the world around us, leaving the cedar trees, the spiders, the stones, and the storm clouds without that psychological depth in which they used to dwell (without, indeed, any psychological resonance or even relevance)? How did the psyche, the spirit, or the _mind_ retreat so thoroughly into the human skull, leaving the air itself a thin and taken-for-granted presence, commonly equated, today, with mere empty space? Read on.
### Wind, Breath, and Speech
Like so many ancient and tribal languages, Hebrew has a single word for both "spirit" and "wind"—the word _ruach_. What is remarkable here is the evident centrality of _ruach_ , the spiritual wind, to early Hebraic religiosity. The primordiality of _ruach_ , and its close association with the divine, is manifest in the very first sentence of the Hebrew Bible:
When God began to create heaven and earth—the earth being unformed and void, with darkness over the surface of the deep and a wind [ _ruach_ ] from God sweeping over the water...
At the very beginning of creation, before even the existence of the earth or the sky, God is present as a wind moving over the waters. Remember the similar primordiality of the wind in the Navajo telling: "Wind existed first... and when the Earth began its existence Wind took care of it." And breath, as we learn in the next section of Genesis, is the most intimate and elemental bond linking humans to the divine; it is that which flows most directly between God and man. For after God forms an earthling _(adam)_ , from the dust of the earth _(adamah)_ , he blows into the earthling's nostrils the breath of life, and the human awakens. Although _ruach_ may be used to refer to the breath, the Hebrew term used here is _neshamah_ , which denotes both the breath and the soul. While _ruach_ generally refers to the wind, or spirit, at large, _neshamah_ commonly signifies the more personal, individualized aspect of wind, the wind or breath of a particular body—like the "Wind within one" of a Navajo person. In this sense, _neshamah_ is also used to signify conscious awareness.
We moderns tend to view ancient Hebraic culture through the intervening lens of Greek and Christian thought; even Jewish scholarship, and much contemporary Jewish self-understanding, has been subtly influenced and informed by centuries of Hellenic and Christian interpretation. It is only thus that many persons today associate the ancient Hebrews with such anachronistic notions as the belief in an otherworldly heaven and hell, or a faith in the immateriality and immortality of the personal soul. Yet such dualistic notions have no real place in the Hebrew Bible. Careful attention to the evidence suggests that ancient Hebraic religiosity was far more corporeal, and far more responsive to the sensuous earth, than we commonly assume.
Of course, the ancient Hebrews were, as we have seen, among the first communities to make sustained use of phonetic writing, the first bearers of an alphabet. Moreover, unlike the other Semitic peoples, they did not restrict their use of the alphabet to economic and political record-keeping, but used it to record ancestral stories, traditions, and laws. They were perhaps the first nation to so thoroughly shift their sensory participation away from the forms of surrounding nature to a purely phonetic set of signs, and so to experience the profound epistemological independence from the natural environment that was made possible by this potent new technology. To actively participate with the visible forms of nature came to be considered _idolatry_ by the ancient Hebrews; _it was not the land but the written letters that now carried the ancestral wisdom_.
Yet although the Hebrews renounced all animistic engagement with the _visible_ forms of the natural world (whether with the moon, or the sun, or those animals—like the bull—sacred to other peoples of the Middle East), they nevertheless retained a participatory relationship with the invisible medium of that world—with the wind and the breath.
The power of this relationship may be directly inferred from the very structure of the Hebrew writing system, the _aleph-beth_. This ancient alphabet, in contrast to its European derivatives, had no letters for what we have come to call "the vowels." The twenty-two letters of the Hebrew _aleph-beth_ were all consonants. Thus, in order to read a text written in traditional Hebrew, one had to infer the appropriate vowel sounds from the consonantal context, and add them when sounding out the written syllables.
This lack of written vowels is only partly explained by the morphological structure of the Semitic languages, in which words with the same combination of consonants (usually grouped in clusters of three) tend to have a related meaning. This morphology ensured that a person fluent in the Hebrew language could, with effort, correctly decipher a Hebrew text without the aid of written vowels. Nevertheless, additional letters for vowels would have greatly facilitated the reading of ancient Hebrew. The fact that some later Hebrew scribes, taking their lead from a standard practice of the Aramaeans, occasionally used the consonants _H, W_ , and _Y_ to suggest specific vowel sounds, is evidence that the lack of written vowels was indeed felt as a difficulty. When, in the seventh century C.E., vowel indicators in the form of little dots and dashes inserted below and above the letters were finally introduced into Hebrew texts, the usefulness of those marks made them a standard component of many Hebrew texts thereafter.
Another, perhaps more significant, reason for the absence of written vowels in the traditional _aleph-beth_ has to do with the nature of the vowel sounds themselves. While consonants are those shapes made by the lips, teeth, tongue, palate, or throat, that momentarily obstruct the flow of breath and so give form to our words and phrases, the vowels are those sounds that are made by the unimpeded breath itself. _The vowels, that is to say, are nothing other than sounded breath_. And the breath, for the ancient Semites, was the very mystery of life and awareness, a mystery inseparable from the invisible _ruach_ —the holy wind or spirit. The breath, as we have noted, was the vital substance blown into Adam's nostrils by God himself, who thereby granted life and consciousness to humankind. It is possible, then, that the Hebrew scribes refrained from creating distinct letters for the vowel-sounds in order to avoid making a visible representation of the invisible. To fashion a visible representation of the vowels, of the sounded breath, would have been to concretize the ineffable, _to make a visible likeness of the divine_. It would have been to make a visible representation of a mystery whose very essence was to be invisible and hence unknowable—the sacred breath, the holy wind. And thus it was not done.
Of course, we do not know if the thought of imaging the vowels, or the sounded breath, even occurred to the ancient Semitic scribes; it is entirely possible that their reverent relation to the wind and the air—their sense of the sacredness of this element that lends its communicative magic to all spoken utterances—simply precluded such a notion from even arising. In any case, whether the avoidance of vowel notation was conscious or inadvertent, the absence of written vowels marks a profound difference between the ancient Semitic _aleph-beth_ and the subsequent European alphabets.
For example, unlike texts written with the Greek or the Roman alphabets, a Hebrew text simply could not be experienced as a double—a stand-in, or substitute—for the sensuous, corporeal world. The Hebrew letters and texts were not sufficient unto themselves; in order to be read, they had to be added to, enspirited by the reader's breath. The invisible air, the same mystery that animates the visible terrain, was also needed to animate the visible letters, to make them come alive and to speak. The letters themselves thus remained overtly dependent upon the elemental, corporeal life-world—they were activated by the very breath of that world, and could not be cut off from that world without losing all of their power. In this manner the absence of written vowels ensured that Hebrew language and tradition remained open to the power of that which exceeds the strictly human community—it ensured that the Hebraic sensibility would remain rooted, however tenuously, in the animate earth. (While the Hebrew Bible would become, as we have seen, a kind of portable homeland for the Jewish people, it could never entirely take the place of the breathing land itself, upon which the text manifestly depends. Hence the persistent themes of exile and longed-for return that reverberate through Jewish history down to the present day.)
The absence of written vowels in ancient Hebrew entailed that the reader of a traditional Hebrew text had to actively _choose_ the appropriate breath sounds or vowels, yet different vowels would often vary the meaning of the written consonants (much as the meaning of the consonantal cluster "RD," in English, will vary according to whether we insert a long _o_ sound between those consonants, "RoaD"; or a long _i_ sound, "RiDe"; a short _e_ sound, "ReD"; or a long _e_ sound, "ReaD"). The reader of a traditional Hebrew text must actively choose one pronunciation over another, according to the fit of that meaning within the written context, yet the precise meaning of that context would itself have been determined by the particular vowels already chosen by that reader.
The traditional Hebrew text, in other words, overtly demanded the reader's conscious participation. The text was never complete in itself; it had to be actively engaged by a reader who, by this engagement, gave rise to a particular reading. Only in relation—only by being taken up and actively interpreted by a particular reader—did the text become meaningful. And there was no single, definitive meaning; the ambiguity entailed by the lack of written vowels ensured that diverse readings, diverse shades of meaning, were always possible.
Some form of active participation, as we have seen, is necessary to _all_ acts of phonetic reading, whether of Greek, or Latin, or English texts such as this one. But the purely consonantal structure of the Hebrew writing system rendered this participation—the creative interaction between the reader and the text—particularly conscious and overt. It simply could not be taken for granted, or forgotten. Indeed, the willful engagement with the text that was necessitated by the absence of written vowels lent a deeply _interactive_ or _interpretive_ character to the Jewish community's understanding of its own most sacred teachings. The scholar Barry Holtz alludes to this understanding in his introduction to a book on the sacred texts of Judaism:
We tend usually to think of reading as a passive occupation, but for the Jewish textual tradition, it was anything but that. Reading was a passionate and active grappling with God's living word. It held the challenge of uncovering secret meanings, unheard-of explanations, matters of great weight and significance. An active, indeed interactive, reading was their method of approaching the sacred text called Torah and through that reading process of finding something at once new and very old....
By "interactive" I mean to suggest that for the rabbis of the tradition, Torah called for a living and dynamic response. The great texts in turn are the record of that response, and each text in turn becomes the occasion for later commentary and interaction. The Torah remains unendingly alive because the readers of each subsequent generation saw it as such, taking the holiness of Torah seriously, and adding their own contribution to the story. For the tradition, Torah _demands_ interpretation.
The reader, that is, must actively respond to the Torah, must bring his own individual creativity into dialogue with the teachings in order to reveal new and unsuspected nuances. The Jewish people must enter into dialogue with the received teachings of their ancestors, questioning them, struggling with them. The Hebrew Bible is not a set of finished stories and unchanging laws; it is not a static body of dogmatic truths but a living enigma that must be questioned, grappled with, and interpreted afresh in every generation. For, as it is said, the guidance that the Torah can offer in one generation is very different from that which it waits to offer in another.
This ongoing tradition of textual interpretation and commentary, and of commentary upon earlier commentary, has given rise to the numerous postbiblical texts of the Jewish tradition, from the Mishnah, the Talmud, and the collections of midrash, to the Zohar and other Kabbalistic works. Collectively, all these texts are known as the "Oral Torah," since they all originated in oral discussion and commentary upon the "Written Torah," upon the teachings ostensibly revealed to Moses, the first Jewish scribe, atop Mount Sinai. The process of writing down oral commentaries and interpretations, with the intent of preserving them, began in the second or third century C.E.
The first of such compilations, the Talmud, is today printed with the primary layer of text, the Mishnah, in the center of each page, and with subsequent commentaries upon that text arrayed around it—in successive layers, as it were. Thus, in its visible arrangement the Talmud displays a sense of the written text not as a definitive and finished object but as an organic, open-ended process to be entered into, an evolving being to be confronted and engaged.
### _The Power of Letters_
Yet this sense of the written text as an animate, living mystery is nowhere more explicit than in the Kabbalah, the esoteric tradition of Jewish mysticism. For here it is not just the text as a whole but the very _letters_ that are thought to be alive! Each letter of the _aleph-beth_ is assumed by the Kabbalists to have its own personality, its own profound magic, its own way of organizing the whole of existence around itself. Because the written commandments were ostensibly dictated to Moses directly by God on Mount Sinai, so the written letters comprising that first Hebrew text—the twenty-two letters of the _aleph-beth_ —are assumed to be the visible traces of divine utterance. Indeed, some Kabbalists claimed that it was by first generating the twenty-two letters, and then combining them into such utterances as "Let there be light," that God spoke the visible universe itself into existence. The letters, that is, are sensible concretions of the very powers of creation.
By meditating, when reading, not upon the written phrases, or even upon the words, but upon the individual _letters_ that gaze out at him from the surface of the page, the Jewish mystic could enter into direct contact with the divine energies. By combining and permutating the letters of particular phrases and words until the words themselves lost all evident meaning and only the letters stood forth in all their naked intensity, the Kabbalist was able to bring himself into increasingly exalted states of consciousness, awakening creative powers that previously lay dormant within his body. Sometimes, when the practitioner was reading in this concentrated and magical fashion, "the letters sprang to life of their own accord," and began "speaking" directly to the mystic. At least one practitioner was alarmed to see the written letters expanding "to the size of mountains" before his eyes. Others reported that, after combining and recombining the letters, they saw the letters suddenly take wing and fly forth from the surface of the page
A close acquaintance with the living letters, and a working knowledge of their individual energies, was assumed to give the Kabbalist magical abilities with which to ease suffering, illness, and discord in the world about him. The Kabbalists, in other words, considered the _aleph-beth_ to be a highly concentrated and divine form of magic; therefore, they consciously cultivated their synaesthetic participation with the written letters.
Since the letters of the _aleph-beth_ also at times served as numbers for the Hebrew people (with the first letter, _aleph_ , signifying the number 1, the second letter, _beth_ , the number 2, on up through 10, and with other letters signifying 20, 30, 40, etc., and still others signifying 100, 200, 300, and 400), written words and phrases could also be compared by calculating the total numerical value of the letters that comprise them—a Kabbalistic technique called _gematria_. Through both permutating the letters and calculating their numerical values, mystics were able to demonstrate hidden equivalences and correspondences between various words and names contained in the scripture. _Elohim_ , for instance, one of the most sacred names of God in the Hebrew Bible, could be shown to have the same numerical value as the Hebrew word for nature, _hateva_ —evidence of the hidden unity of God and nature. (Such pantheistic notions equating God with nature—common to many practitioners of Kabbalah—would startle the various environmentalists today who charge that Hebraic religion expelled all divinity from the natural world.)
Indeed, all the diverse names of God in the Hebrew Bible, and the letters that comprise them, figure prominently in Kabbalistic theory, providing essential clues for the practitioner who seeks direct experience of the divine. Supreme among these names is the Tetragrammaton, the four-letter name, YHWH. Often written, in non-Hebrew texts, as Yahweh, the true manner of pronouncing this most powerful combination of letters is said to have been forgotten. Nevertheless, some of the most concentrated of Kabbalistic practices involved pronouncing each letter of the Tetragrammaton separately, combining it, in turn, with each of the five possible breath sounds, or vowels. A much more elaborate, and presumably dangerous, practice entailed isolating each letter of the Tetragrammaton and combining it, one at a time, with every other letter of the _aleph-beth_ , pronouncing each one of _these_ combinations, in turn, with each of the various vowel sounds. By carefully reciting this incantation over an earthen form in the shape of a human being, it was said that one could bring the clay figure—a _golem_ —to life. A clue to the sympathetic magic involved in this incantation may be found in the teaching of the great thirteenth-century Kabbalist Abraham Abulafia, who asserted that the spoker vowels and the written consonants are as interdependent "as the soul and the body." To combine the vowels—the sounded breath—with the visible consonants was akin to breathing life into a clump of clay, as YHWH had lent his breath to the earthen Adam.
FINALLY, WE MUST ACKNOWLEDGE THE VAST IMPORTANCE, WITHIN the Jewish mystical tradition, of the breath itself. In the thirteenth-century _Zohar_ , the most important of all Kabbalistic texts, the central figure, Rabbi Shim'on bar Yohai, insists that the union between humans and God is best effected through the medium of the breath. According to Rabbi Shim'on, King Solomon learned from his father, King David, the breathing techniques involved in invoking the holy breath, the inspiration of the divine. "By learning and practicing the secrets inherent in the breath, Solomon could lift nature's physical veil from created things and see the spirit within." In a manner startlingly reminiscent of a Navajo or a Lakota ceremony, Rabbi Shim'on's son, El'azar, begins a prayer session by exhorting "the winds to come from all four directions and fill his breath," and instructs his companions to circulate the air inhaled from all four directions interchangeably within their bodies. Elsewhere in the _Zohar_ , one of Rabbi Shim'on's companions speaks of "the soul-breath" sent from YHWH to enter the body of the righteous person at birth. Much like the "wind within one" of the Navajo people, "the soul-breath that enters at birth directs and trains the human being and initiates him into every straight path. This sense of the breath as medium between the individual and the divine is exemplified in a commentary on prayer by a nineteenth-century Hasidic master (Hasidism was a vibrant wave of Jewish mysticism that swept East European Jewry in the eighteenth and nineteenth centuries):
_If prayer is pure and untainted_ ,
_surely that holy breath_
_that rises from your lips_
_will join with the breath of heaven_
_that is always flowing_
_into you from above...._
_Thus that part of God_
_which is within you_
_is reunited with its source_.
Yet the sacred breath enters not just into human beings (providing awareness and guidance), it also animates and sustains the whole of the sensible world. Like the wind itself, the breath of God permeates all of nature. In a classic text entitled "The Portal of Unity and Faith," the eighteenth-century Hasidic master Schneur Zalman of Ladi describes how the syllables and letters of God's creative utterances, such as "Let there be light," or "Let the waters bring forth swarms of living creatures," gradually generate, through a concatenated series of permutations and numerical substitutions, the exact names, and hence _the exact forms_ , of all natural entities (in Hebrew a single term, _davar_ , means both "word" and "thing"). Yet without the continual outflow of God's breath, which Schneur Zalman calls "the Breath of His Mouth," all of the letters that stand within the things of this world—all the letter combinations embodied in particular animals, plants, and stones—would return to their undifferentiated source in the divine Unity, and the sensible world, along with all sensing beings, would be extinguished. Just as the consonantal letters of a traditional Hebrew text depend, for their communicative power, upon the sounded breath that animates them, so the divine letters and letter combinations that structure the physical universe are dependent upon the divine breath that continually utters them forth. All things vibrate with "the Breath of His Mouth."
And it is by virtue of this continual breath that nature is always new; the world around us is a continual, ongoing utterance! Thus, the activity of speech, like breathing, links humans not just to God but to all that surrounds us, from the stones to the sparrows. This is simply illustrated in another Hasidic commentary on prayer:
_See your prayer as arousing the letters_
_through which heaven and earth_
_and all living things were created_.
_The letters are the life of all;_
_when you pray through them_ ,
_all Creation joins with you in prayer_.
_All that is around you can be uplifted;_
_even the song of a passing bird_
_may enter into such a prayer. 55_
GIVEN THE SUBTLE IMPORTANCE PLACED UPON THE WIND AND THE breath within the Hebrew tradition, we may be tempted to wonder whether, long before the employment of phonetic writing and the _aleph-beth_ , the monotheism of Abraham and his descendants was borne by a new way of experiencing the invisible air, a new sense of the unity of this unseen presence that flows not just within us but between all things, granting us life and speech even as it moves the swaying grasses and the gathering clouds. Is it possible that a volatile power once propitiated as a local storm god came to be generalized, by one tribe of nomadic herders, into the capricious power of the encompassing atmosphere itself? We know that the singular mystery revered by the children of Abraham was an ineffable power that could not be localized in any visible phenomenon, could not be imaged in any idol. Prior to the use of writing by Moses and the later scribes, however, it may be that this power was not intangible, but simply invisible—that it was experienced not as an abstract power entirely _outside of_ sensuous nature, but as the unseen medium, the _ruach_ , the ubiquitous wind or spirit that enlivens the visible world.
It is remarkable that the most holy of God's names, the four-letter Tetragrammaton, is composed of the most breath-like consonants in the Hebrew _aleph-beth_ (the same three letters, Y, H, and W, that were sometimes used by ancient scribes to stand in for particular vowels). The most sacred of God's names would thus seem to be the most breath-like of utterances—a name spoken, as it were, by the wind. Some contemporary students of Kabbalah suggest that the forgotten pronunciation of the name may have entailed forming the first syllable, "Y-H," on the whispered inbreath, and the second syllable, "W-H," on the whispered outbreath—the whole name thus forming a single cycle of the breath. If their suspicion is in any sense correct, then the awesome mystery invoked by the Tetragrammaton may not be separable from the mystery of _breathing_ —this ebb and flow that ceaselessly binds us to the invisible.
Setting all speculations aside, however, it should be clear from the foregoing discussion that the strictly consonantal character of the Hebrew script encouraged a unique relation to the sacred texts, and to the sacred in general. In particular, the absence of written vowels fostered (1) a consciously interactive relation with the text—even, for some, an overtly animistic participation with the written letters themselves, and (2) a continued respect and reverence for the air—for the invisible medium that activates the visible letters even as it animates the visible terrain. While they certainly developed a new, literate distance from the surrounding world of nature, the Hebrews—the first "People of the Book"—nevertheless retained a profoundly oral relation to the invisible medium of that world, to the wind and the breath.
### The Forgetting of the Air
It is precisely this oral awareness of the invisible depths that enfold us—this sense of the unseen air as an awesome mystery joining the human and extrahuman worlds—that was sundered by the Greek scribes.
When they adapted the ancient Semitic _aleph-beth_ for their own use, probably in the eighth century B.C.E., the Greek scribes took on (with modifications) the shapes as well as the names of the early Semitic letters. Yet, as we mentioned in chapter 4, those names had no extraliterate reference for the Greeks, as they did for the Hebrews. Remember that for the Hebrews, _aleph_ (Greek: _alpha_ ) signified not just the first letter but also, and more primordially, "ox," similarly _beth_ ( _beta_ ) meant "house," _gimmel_ ( _gamma_ ) was the word for "camel," etc. But to the Greeks, these words named only the letters themselves; they had no other significance. And as the names of the letters shed their worldly, extraliterate significance in the transfer across the Mediterranean, any pictographic resonance between the written letters and those worldly phenomena (oxen, houses, camels, ****etc.) was forgotten as well. In the journey to Greece, in other words, the letters of the _aleph-beth_ loosened and left behind their vestigial ties to the enveloping life-world; they thereby became a much more abstract set of symbols.
But the Greeks also introduced a strange new element into the alphabet, an innovation that would ultimately increase the abstract capacity of this writing system far more than the above-mentioned factors. For the Greek scribes introduced written _vowels_ into the previously consonantal system of letters.
Actually, many of the new letters were adapted from already existing Semitic letters. Certain characters in the Semitic _aleph-beth_ signified consonants that had no existence in the Greek language, and it was these apparently superfluous letters that were appropriated by the Greek scribes to represent vowel sounds. The letter _aleph_ , for instance, was not a vowel but a consonant in the original Hebrew usage; it signified the opening of the throat prior to all utterance. Since the Greeks had no use for this consonant, they adapted this character, which they called _alpha_ , to signify the vowel sound _A_. Other Hebrew letters were altered to represent the vowels _E_ , _I_ , and _O_. Finally, the Greeks added the letter _upsilon_ , which eventually became the Roman letter _U_.
The resulting alphabet was a very different kind of tool from its earlier, Semitic incarnation—one that would have very different effects upon the senses that engaged it, and upon the various languages that adopted it as their own. For the addition of written vowels enabled a much more thorough transcription of spoken utterance onto the flat surface of the page. A text written with the new alphabet had none of the ambiguity that, as we have seen, was inherent in a traditional Hebrew text. While for any Hebrew text of sufficient length there were various possible pronunciations, or readings, each of which would yield a slightly different set of words and meanings, a comparable Greek text would likely admit of only a single correct reading. It is thus that texts written with the Greek (and later the Roman) alphabet did not invite the kind of active and ever-renewed interpretation that was demanded by the Hebrew texts. The interactive, synaesthetic participation involved in reading—in transforming a series of visible marks into a sequence of sounds—could now become entirely habitual and automatic. For there was no longer any choice in how to sound out the text; all the cues for one's participation were spelled out upon the page. Relative to Semitic texts, then, the Greek texts had a remarkable autonomy—they seemed to stand, and even to speak, on their own.
Yet the apparent precision and efficiency of the new alphabet was obtained at a high price. For by using visible characters to represent the sounded breath, the Greek scribes effectively _desacralized_ the breath and the air. By providing a visible representation of that which was—by its very nature—invisible, they nullified the mysteriousness of the enveloping atmosphere, negating the uncanniness of this element that was both here and yet not here, present to the skin and yet absent to the eyes, immanence and transcendence all at once.
The awesomeness of the air had resided precisely in its ubiquitous and yet unseen nature, its capacity to grant movement and life to visible nature while remaining, in itself, invisible and ungraspable. Hebraic writing had preserved this mystery by refraining from representing the air itself upon the parchment or the page—by refusing to image, or objectify, this unseen flux that sustains both the word and the visible world. By breaking this taboo, _by transposing the invisible into the register of the visible, the Greek scribes effectively dissolved the primordial power of the air_.
The effects of this perceptual dissolution were not, of course, evident all at once. In Greece, as we have seen, the new alphabet met substantial resistance in the form of a well-developed and flourishing oral culture, and so took several centuries to make itself felt within the common discourse. As late as the middle of the sixth century B.C.E., the Milesian philosopher Anaximenes could still assert:
As the _psychê_ , being air, holds a man together and gives him life, so breath and air hold together the entire universe and give it life.
A century and a half later, however, when the alphabet was at last being taught within the educational curriculum and was thereby spreading throughout Greek culture, Plato and Socrates were able to co-opt the term _psychê_ —which for Anaximenes was fully associated with the breath and the air—employing the term now to indicate something not just invisible but utterly intangible. The Platonic _psychê_ was not at all a part of the sensuous world, but was rather of another, utterly non-sensuous dimension. The _psychê_ , that is, was no longer an invisible yet tangible power continually participant, by virtue of the breath, with the enveloping atmosphere, but a thoroughly abstract phenomenon now enclosed within the physical body as in a prison.
We have already seen how the new relation that Plato wrote of, between the immortal _psychê_ and the transcendent realm of eternal "Ideas," was itself dependent upon the new affinity between the literate intellect and the visible letters (and words) of the alphabet. We can now discern that this relation between the _psychê_ and the bodiless Ideas was dependent, as well, upon a gradual forgetting of the air and the breath, itself made possible by the spread of the new technology. For it was only as the unseen air lost its fascination for the human senses that this other, more extreme invisibility came to take its place—the utterly incorporeal realm of pure "Ideas," to which the Platonic, rational _psychê_ was connected much as the earlier, breathlike _psychê_ was joined to the atmosphere.
THOSE WHO SPEAK FACILELY OF A "JUDEO-CHRISTIAN TRADITION" fail to discern the remarkably different approaches that distinguish the ancient Jewish and the Christian faiths, differences rooted partly in the sensorial effects of the very different writing systems employed by these two highly text-centered traditions. Unlike the Hebrew Bible, the Christian New Testament was originally written primarily in the Greek alphabet, and thus the dualistic sensibility promoted by the Greek writing system was early on allied with Christian doctrine. Under the aegis of the Church, the belief in a non-sensuous heaven, and in the fundamentally incorporeal nature of the human soul—itself "imprisoned," as Plato had suggested, in the bodily world—accompanied the alphabet as it spread, first throughout Europe and later throughout the Americas. _And wherever the alphabet advanced, it proceeded by dispelling the air of ghosts and invisible influences—by stripping the air of its_ anima, _its psychic depth_.
In the oral, animistic world of pre-Christian and peasant Europe, all things—animals, forests, rivers, and caves—had the power of expressive speech, and the primary medium of this collective discourse was the air. In the absence of writing, human utterance, whether embodied in songs, stories, or spontaneous sounds, was inseparable from the exhaled breath. The invisible atmosphere was thus the assumed intermediary in all communication, a zone of subtle influences crossing, mingling, and metamorphosing. This invisible yet palpable realm of whiffs and scents, of vegetative emanations and animal exhalations, was also the unseen repository of ancestral voices, the home of stories yet to be spoken, of ghosts and spirited intelligences—a kind of collective field of meaning from whence individual awareness continually emerged and into which it continually receded, with every inbreath and outbreath.
We might say that the air, as the invisible wellspring of the present, yielded an awareness of transformation and transcendence very different from that total transcendence expounded by the Church. The experiential interplay between the _seen_ and the _unseen_ —this duality entirely proper to the sensuous life-world—was far more real, for oral peoples, than an abstract dualism between sensuous reality as a whole and some other, utterly non-sensuous heaven.
Thus it was that the progressive spread of Christianity was largely dependent upon the spread of the alphabet, and, conversely, that Christian missions and missionaries were by far the greatest factor in the advancement of alphabetic literacy in both the medieval and the modern eras. It was not enough to preach the Christian faith: one had to induce the unlettered, tribal peoples to begin to use the technology upon which that faith depended. Only by training the senses to participate with the written word could one hope to break their spontaneous participation with the animate terrain. _Only as the written text began to speak would the voices of the forest, and of the river, begin to fade. And only then would language loosen its ancient association with the invisible breath, the spirit sever itself from the wind, the psyche dissociate itself from the environing air_. The air, once the very medium of expressive interchange, would become an increasingly empty and unnoticed phenomenon, displaced by the strange _new_ medium of the written word.
### _Membranes and Barriers_
The progressive forgetting of the air—the loss of the invisible richness of the present—has been accompanied by a concomitant internalization of human awareness. We have just seen how the ancient Greek _psychê_ , or soul, was transformed from a phenomenon associated with the air and the breath into a wholly immaterial entity trapped, as it were, within the human body. In contact with the written word a new, apparently autonomous, sensibility emerges into experience, a new self that can enter into relation with its own verbal traces, can view and ponder its own statements even as it is formulating them, and can thus reflexively interact with itself in isolation from other persons and from the surrounding, animate earth. This new sensibility seems independent of the body—seems, indeed, of another order entirely—since it is borne by the letters and texts whose changeless quality contrasts vividly with the shifting life of the body and the flux of organic nature. That this new sensibility comes to view itself as an isolated intelligence located "inside" the material body can only be understood in relation to the forgetting of the air, to the forgetting of this sensuous but unseen medium that continually flows in and out of the breathing body, binding the subtle depths within us to the fathomless depths that surround us.
We may better comprehend this curious development—the withdrawal of mind from sensible nature and its progressive incarceration in the human skull—by considering that every human language secretes a kind of perceptual boundary that hovers, like a translucent veil, between those who speak that language and the sensuous terrain that they inhabit. As we grow into a particular culture or language, we implicitly begin to structure our sensory contact with the earth around us in a particular manner, paying attention to certain phenomena while ignoring others, differentiating textures, tastes, and tones in accordance with the verbal contrasts contained in the language. We simply cannot take our place within any community of human speakers without ordering our sensations in a common manner, and without thereby limiting our spontaneous access to the wild world that surrounds us. Any particular language or way of speaking thus holds us within a particular community of human speakers only by invoking an ephemeral border, or boundary, between our sensing bodies and the sensuous earth.
Nevertheless, the perceptual boundary constituted by any language may be exceedingly porous and permeable. Indeed, for many oral, indigenous peoples, the boundaries enacted by their languages are more like permeable membranes binding the peoples to their particular terrains, rather than barriers walling them off from the land. By affirming that the other animals have their own languages, and that even the rustling of leaves in an oak tree or an aspen grove is itself a kind of voice, oral peoples bind their senses to the shifting sounds and gestures of the local earth, and thus ensure that their own ways of speaking remain informed by the life of the land. Still, the membrane enacted by their language is felt, and is acknowledged as a margin of danger and magic, a place where the relations between the human and the more-than-human worlds must be continually negotiated. The shamans common to oral cultures dwell precisely on this margin or edge; the primary role of such magicians, as I suggested at the outset of this book, is to act as intermediaries between the human and more-than-human realms. By regularly shedding the sensory constraints induced by a common language, periodically dissolving the perceptual boundary in order to directly encounter, converse, and bargain with various nonhuman intelligences—with otter, or owl, or eland—and then rejoining the common discourse, the shaman keeps the human discourse from rigidifying, and keeps the perceptual membrane fluid and porous, ensuring the greatest possible attunement between the human community and the animate earth, between the familiar and the fathomless.
The emergence or adoption of a formal writing system significantly solidifies the ephemeral perceptual boundary already established by a common tongue; now the spoken language has a visible counterpart that floats, fixed and immobile, between the human body and the sensuous world. Yet while formal writing thus solidifies the linguistic-perceptual boundary, many ancient writing systems implicitly refer the human senses to that which lies _beyond_ the boundary; their often pictorially derived characters cannot help but remind the reading body of its inherence in a more-than-human field of animate forms. Language is not, here, a purely human possession—it remains tied, however distantly, to the larger field of expressive powers.
The advent of _phonetic_ writing further rigidifies the perceptual boundary enclosing the human community. For the written characters no longer depend, implicitly, upon the larger field of sensuous phenomena; they refer, instead, to a strictly human set of sounds. The letters, as we have said, begin to function as mirrors reflecting the human community back upon itself. Nevertheless, even this mirrored boundary may remain somewhat open to what lies beyond it. We have seen that in the original _aleph-beth_ the vowels, or rather the _absence_ of vowels, provided the pores, the openings in the linguistic membrane through which the invisible wind—the living breath—could still flow between the human and the more-than-human worlds.
It was only with the plugging of these last pores—with the insertion of visible letters for the vowels themselves—that the perceptual boundary established by the common language was effectively sealed, and what had once been a porous membrane became an impenetrable barrier, a hall of mirrors. The Greek scribes, that is, transformed the breathing boundary between human culture and the animate earth into a seamless barrier segregating a pure inside from a pure outside. With the addition of written vowels—with the filling of those gaps, or pores, in the early alphabet—human language became a largely self-referential system closed off from the larger world that once engendered it. And the "I," the speaking self, was hermetically sealed within this new interior.
Today the speaking self looks out at a purely "exterior" nature from a purely "interior" zone, presumably located somewhere inside the physical body or brain. Within alphabetic civilization, virtually _every_ human psyche construes itself as just such an individual "interior," a private "mind" or "consciousness" unrelated to the other "minds" that surround it, or to the environing earth. For there is no longer any common medium, no reciprocity, no _respiration_ between the inside and the outside. There is no longer any flow between the self-reflexive domain of alphabetized awareness and all that exceeds, or subtends, this determinate realm. Between consciousness and the unconscious. Between civilization and the wilderness.
### _Remembering_
In the world of modernity the air has indeed become the most taken-for-granted of phenomena. Although we imbibe it continually, we commonly fail to notice that there is anything there. We refer to the unseen depth between things—between people, or trees, or clouds—as mere empty space. The invisibility of the atmosphere, far from leading us to attend to it more closely, now enables us to neglect it entirely. Although we are wholly dependent upon its nourishment for all of our actions and all our thoughts, the immersing medium has no mystery for us, no conscious influence or meaning. Lacking all sacredness, stripped of all spiritual significance, the air is today little more than a conveniently forgotten dump site for a host of gaseous effluents and industrial pollutants. Our fascination is elsewhere, carried by all these _other_ media—these newspapers, radio broadcasts, television networks, computer bulletin boards—all these fields or channels of _strictly human communication_ that so readily grab our senses and mold our thoughts once our age-old participation with the original, more-than-human medium has been sundered.
As a child, growing up on the outskirts of New York City, I often gazed at great smokestacks billowing dark clouds into the sky. Yet I soon stopped wondering where all that sooty stuff went: since the adults who decided such things saw fit to dispose of wastes in this manner, it must, I concluded, be all right. Later, while learning to drive, I would watch with some alarm as the trucks roaring past me on the highway spewed black smoke from their gleaming exhaust pipes, but I quickly forgave them, remembering that my car, too, offered its hot fumes to the air. Everybody did it. As the vapor trails from the jets soaring overhead seemed to disperse, perfectly, into the limitless blue, so we assumed that these wastes, these multicolored smokes and chemical fumes, would all cancel themselves, somehow, in the invisible emptiness.
It was as though after the demise of the ancestral, pagan gods, Western civilization's burnt offerings had become ever more constant, more extravagant, more acrid—as though we were petitioning some unknown and slumbering power, trying to stir some vast dragon, striving to invoke some unknown or long-forgotten power that, awakening, might call us back into relation with something other than ourselves and our own designs.
Indeed, the outpouring of technological by-products and pollutants since the Industrial Revolution could go on only so long before it would begin to alter the finite structure of the world around us, before its effects would begin to impinge upon our breathing bodies, inexorably drawing us back to our senses and our sensorial contact with the animate earth.
Today the technological media—the newspapers and radios and televisions—are themselves beginning to acknowledge and call attention to the changes underway in the air itself. It is through these secondary media that we recently learned of the massive buildup in the upper atmosphere of manufactured chemical compounds that every year burn an ever-widening hole in the stratospheric ozone layer above Antarctica, while thinning the rest of that protective layer worldwide. From these media we also learn of the drastic increase in atmospheric carbon dioxide since the onset of the Industrial Revolution, and we hear over and again that this surfeit of carbon dioxide, along with other heat-absorbing gases, is already promoting a substantial warming of the earthly climate, a change which in turn endangers the survival of numerous ecosystems, numerous animal and plant species already stressed, many to the edge of extinction, by the ever-burgeoning human population.
Nevertheless, such published and broadcast information, reaching us as it does through these technological channels, all too often remains an abstract cluster of statistics; it does little to alter our intellectual detachment from the sensuous earth until, returning from a journey, we see for ourselves the brown haze that now settles over the town where we live, until we feel the chemical breeze stinging the moist membranes that line our nose, or until we watch, with alarm, as gale-force winds rip the awning off our storefront. Or perhaps, after recovering from our fifth fevered illness in a single winter, we realize that our bodily resistance has been dampened by the increased radiation that daily pours through the exhausted sky, or by airborne fallout from the latest power-plant failure across the continent.
Phenomenologically considered—experientially considered—the changing atmosphere is not just one component of the ecological crisis, to be set alongside the poisoning of the waters, the rapid extinction of animals and plants, the collapse of complex ecosystems, and other human-induced horrors. All of these, to be sure, are interconnected facets of an astonishing dissociation—a monumental forgetting of our human inherence in a more-than-human world. Yet our disregard for the very air that we breathe is in some sense the most profound expression of this oblivion. For it is the air that most directly envelops us; the air, in other words, is that element that we are most intimately _in_. As long as we experience the invisible depths that surround us as empty space, we will be able to deny, or repress, our thorough interdependence with the other animals, the plants, and the living land that sustains us. We may acknowledge, intellectually, our body's reliance upon those plants and animals that we consume as nourishment, yet the civilized mind still feels itself somehow separate, autonomous, independent of the body and of bodily nature in general. Only as we begin to notice and to experience, once again, our immersion in the invisible air do we start to recall what it is to be fully a part of this world.
For the primordial affinity between awareness and the invisible air simply cannot be avoided. As we become conscious of the unseen depths that surround us, the inwardness or interiority that we have come to associate with the personal psyche begins to be encountered in the world at large: we feel ourselves enveloped, immersed, caught up _within_ the sensuous world. This breathing landscape is no longer just a passive backdrop against which human history unfolds, but a potentized field of intelligence in which our actions participate. As the regime of self-reference begins to break down, as we awaken to the air, and to the multiplicitous Others that are implicated, with us, in its generative depths, the shapes around us seem to awaken, to come alive....
# **_Coda:_**
# **_Turning Inside Out_**
Ah, not to be cut off,
not through the slightest partition
shut out from the law of the stars.
The inner—what is it?
if not intensified sky,
hurled through with birds and deep
with the winds of homecoming.
—RAINER MARIA RILKE
N _OT TO BE CUT OFF ,_ AS RILKE SAYS. AND YET WE SEEM, today, so estranged from the stars, so utterly cut off from the world of hawk and otter and stone. This book has traced some of the ways whereby the human mind came to renounce its sensuous bearings, isolating itself from the other animals and the animate earth. By writing these pages I have hoped, as well, to _renew_ some of those bearings, to begin to recall and reestablish the rootedness of human awareness in the larger ecology.
Each chapter has disclosed the subtle dependence of various "interior," mental phenomena upon certain easily overlooked or taken-for-granted aspects of the surrounding sensuous world. Language was disclosed as a profoundly bodily phenomenon, sustained by the gestures and sounds of the animate landscape. The rational intellect so prized in the West was shown to rely upon the external, visible letters of the alphabet. The presumably interior, mental awareness of the "past" and the "future" was shown to be dependent upon our sensory experience of that which is hidden beneath the ground and concealed beyond the horizon. Finally, the experience of awareness itself was related to mysteries of the breath and the air, to the tangible but invisible atmosphere in which we find ourselves immersed.
The human mind is not some otherworldly essence that comes to house itself inside our physiology. Rather, it is instilled and provoked by the sensorial field itself, induced by the tensions and participations between the human body and the animate earth. The invisible shapes of smells, rhythms of cricketsong, and the movement of shadows all, in a sense, provide the subtle body of our thoughts. Our own reflections, we might say, are a part of the play of light and _its_ reflections. "The inner—what is it, if not intensified sky?"
By acknowledging such links between the inner, psychological world and the perceptual terrain that surrounds us, we begin to turn inside-out, loosening the psyche from its confinement within a strictly human sphere, freeing sentience to return to the sensible world that contains us. Intelligence is no longer ours alone but is a property of the earth; we are in it, of it, immersed in its depths. And indeed each terrain, each ecology, seems to have its own particular intelligence, its unique vernacular of soil and leaf and sky.
Each place its own mind, its own psyche. Oak, madrone, Douglas fir, red-tailed hawk, serpentine in the sandstone, a certain scale to the topography, drenching rains in the winter, fog off-shore in the summer, salmon surging in the streams—all these together make up a particular state of mind, a place-specific intelligence shared by all the humans that dwell therein, but also by the coyotes yapping in those valleys, by the bobcats and the ferns and the spiders, by all beings who live and make their way in that zone. Each place its own psyche. Each sky its own blue.
THE SENSE OF BEING IMMERSED IN A SENTIENT WORLD IS preserved in the oral stories and songs of indigenous peoples—in the belief that sensible phenomena are all alive and aware, in the assumption that all things have the capacity of speech. Language, for oral peoples, is not a human invention but a gift of the land itself.
I do not deny that human language has its uniqueness, that from a certain perspective human discourse has little in common with the sounds and signals of other animals, or with the rippling speech of the river. I wish simply to remember that this was not the perspective held by those who first acquired, for us, the gift of speech. Human language evolved in a thoroughly animistic context; it necessarily functioned, for many millennia, not only as a means of communication between humans, but as a way of propitiating, praising, and appeasing the expressive powers of the surrounding terrain. Human language, that is, arose not only as a means of attunement between persons, but also between ourselves and the animate landscape. The belief that meaningful speech is a purely human property was entirely alien to those oral communities that first evolved our various ways of speaking, and by holding to such a belief today we may well be inhibiting the spontaneous activity of language. By denying that birds and other animals have their own styles of speech, by insisting that the river has no real voice and that the ground itself is mute, we stifle our direct experience. We cut ourselves off from the deep meanings in many of our words, severing our language from that which supports and sustains it. We then wonder why we are often unable to communicate even among ourselves.
IN ELUCIDATING THE PROCESS WHEREBY CIVILIZATION HAS TURNED in upon itself, isolating itself from the breathing earth, I have concentrated upon the curious perceptual and linguistic transformations made possible by the advent of formal writing systems, and in particular by the advent of _phonetic_ writing. I do not, however, wish to imply that writing was the sole factor in this process—a complex process that, after all, has been under way for several thousand years. Many other factors could have been chosen. I have hardly alluded, in this work, to the emergence of agriculture at the dawn of the Neolithic era, although the spread of agricultural techniques radically transformed the experienced relation between humans and other species. Nor have I addressed the development of formal numbering systems, and the consequent influence of numerical measurement, and quantification, upon our interactions with the land. And of course I have said little or nothing regarding the countless technologies spawned by alphabetic civilization itself, from telephones to televisions, from automobiles to antibiotics. By concentrating upon the written word, I have wished to demonstrate less a particular thesis than a particular stance, a particular way of pondering and of questioning _any_ factor that one might choose.
It is a way of thinking that strives for rigor without forfeiting our animal kinship with the world around us—an attempt to think in accordance with the senses, to ponder and reflect without severing our sensorial bond with the owls and the wind. It is a style of thinking, then, that associates _truth_ not with static fact, but with a quality of relationship.
Ecologically considered, it is not primarily our verbal statements that are "true" or "false," but rather the kind of relations that we sustain with the the rest of nature. A human community that lives in a mutually beneficial relation with the surrounding earth is a community, we might say, that lives in truth. The ways of speaking common to that community—the claims and beliefs that enable such reciprocity to perpetuate itself—are, in this important sense, _true_. They are in accord with a right relation between these people and their world. Statements and beliefs, meanwhile, that foster violence toward the land, ways of speaking that enable the impairment or ruination of the surrounding field of beings, can be described as _false_ ways of speaking—ways that encourage an unsustainable relation with the encompassing earth. A civilization that relentlessly destroys the living land it inhabits is not well acquainted with _truth_ , regardless of how many supposed facts it has amassed regarding the calculable properties of its world.
Hence I am less concerned with the "literal" truth of the assertions that I have made in this work than I am concerned with the kind of relationships that they make possible. "Literal truth" is entirely an artifact of alphabetic literacy: to be _literally true_ originally meant to be true to "the letter of scripture"—to "the letter of the law." In this work I have tried to reacquaint the reader with a mode of awareness that precedes and underlies the literate intellect, to a way of thinking and speaking that strives to be faithful not to the written record but to the sensuous world itself, and to the other bodies or beings that surround us.
For such an oral awareness, to _explain_ is not to present a set of finished reasons, but to tell a story. That is what I have attempted in these pages. It is an unfinished story, told from various angles, sketchy in some parts, complete with gaps and questions and unrealized characters. But it is a story, nonetheless, not a wholly determinate set of facts.
Of course, not all stories are successful. There are good stories and mediocre stories and downright bad stories. How are they to be judged? If they do not aim at a static or "literal" reality, how can we discern whether one telling of events is any better or more worthy than another? The answer is this: a story must be judged according to whether it _makes sense_. And "making sense" must here be understood in its most direct meaning: to make sense is _to enliven the senses_. A story that makes sense is one that stirs the senses from their slumber, one that opens the eyes and the ears to their real surroundings, tuning the tongue to the actual tastes in the air and sending chills of recognition along the surface of the skin. To _make sense_ is to release the body from the constraints imposed by outworn ways of speaking, and hence to renew and rejuvenate one's felt awareness of the world. It is to make the senses wake up to where they are.
THE APPARENTLY AUTONOMOUS, MENTAL DIMENSION ORIGINALLY opened by the alphabet—the ability to interact with our own signs in utter abstraction from our earthly surroundings—has today blossomed into a vast, cognitive realm, a horizonless expanse of virtual interactions and encounters. Our reflective intellects inhabit a global field of information, pondering the latest scenario for the origin of the universe as we absently fork food into our mouths, composing presentations for the next board meeting while we sip our coffee or cappuccino, clicking on the computer and slipping into cyberspace in order to network with other bodiless minds, exchanging information about gene sequences and military coups, "conferencing" to solve global environmental problems while oblivious to the moon rising above the rooftops. Our nervous system synapsed to the terminal, we do not notice that the chorus of frogs by the nearby stream has dwindled, this year, to a solitary voice, and that the song sparrows no longer return to the trees.
In contrast to the apparently unlimited, global character of the technologically mediated world, the sensuous world—the world of our direct, unmediated interactions—is always local. The sensuous world is the particular ground on which we walk, the air we breathe. For myself as I write this, it is the moist earth of a half-logged island off the northwest coast of North America. It is this dark and stone-rich soil feeding the roots of cedars and spruces, and of the alders that rise in front of the cabin, their last leaves dangling from the branches before being flung into the sky by the early winter storms. And it is the salty air that pours in through the loose windows, spiced with cedar and seaweed, and sometimes a hint of diesel fumes from a boat headed south tugging a giant raft of clear-cut tree trunks. Sometimes, as well, there is the very faint, fishy scent of otter scat. Each day a group of otters slips out of the green waters onto the nearby rocks at high tide, one or two adults and three smaller, sleek bodies, at least one of them dragging a half-alive fish between its teeth. The otters, too, breathe this wild air, and when the storm winds batter the island, they stretch their necks into the invisible surge, drinking large drafts from the tumult.
In the interior of this island, in the depths of the forest, things are quieter. Huge and towering powers stand there, unperturbed by the winds, their crusty bark fissured with splitting seams and crossed by lines of ants, inchworms, and beetles of varied shapes and hues. A single woodpecker is thwacking a trunk somewhere, the percussive rhythm reaching my ears without any echo, absorbed by the mosses and the needles heavy with water drops that have taken hours to slide down the trunks from the upper canopy (each drop lodging itself in successive cracks and crevasses, gathering weight from subsequent drips, then slipping down, past lichens and tiny spiders, to the next protruding ridge or branch). Fallen firs and hemlocks, and an old spruce tree tunneled by termites, lie dank and rotting in the ferns, the jumbled branches of the spruce blocking the faint deer trail that I follow.
The deer on this island have recently molted, forsaking their summer fur for a thicker, winter coat. I watch them in the old orchard at dusk. No longer the warm brown color of sunlight on soil, their fur is now grey against the shadowed trunks and the all-grey sky. These quiet beings seem entirely a part of this breathing terrain, their very texture and color shifting with the local seasons.
Human persons, too, are shaped by the places they inhabit, both individually and collectively. Our bodily rhythms, our moods, cycles of creativity and stillness, and even our thoughts are readily engaged and influenced by shifting patterns in the land. Yet our organic attunement to the local earth is thwarted by our ever-increasing intercourse with our own signs. Transfixed by our technologies, we short-circuit the sensorial reciprocity between our breathing bodies and the bodily terrain. Human awareness folds in upon itself, and the senses—once the crucial site of our engagement with the wild and animate earth—become mere adjuncts of an isolate and abstract mind bent on overcoming an organic reality that now seems disturbingly aloof and arbitrary.
The alphabetized intellect stakes its claim to the earth by _staking it down_ , extends its dominion by drawing a grid of straight lines and right angles across the body of a continent—across North America, across Africa, across Australia—defining states and provinces, counties and countries with scant regard for the oral peoples that already live there, according to a calculative logic utterly oblivious to the life of the land.
If I say that I live in the "United States" or in "Canada," in "British Columbia" or in "New Mexico," I situate myself within a purely human set of coordinates. I say very little or nothing about the earthly place that I inhabit, but simply establish my temporary location within a shifting matrix of political, economic, and civilizational forces struggling to maintain themselves, today, largely at the expense of the animate earth. The great danger is that I, and many other good persons, may come to believe that our breathing bodies really inhabit these abstractions, and that we will lend our lives more to consolidating, defending, or bewailing the fate of these ephemeral entities than to nurturing and defending the actual places that physically sustain us.
The land that includes us has its own articulations, its own contours and rhythms that must be acknowledged if the land is to breathe and to flourish. Such patterns, for instance, are those traced by rivers as they wind their way to the coast, or by a mountain range that rises like a backbone from the plains, its ridges halting the passage of clouds that gather and release their rains on one side of the range, leaving the other slope dry and desertlike. Another such contour is the boundary between two very different kinds of bedrock formed by some cataclysmic event in the story of a continent, or between two different soils, each of which invites a different population of plants and trees to take root. Diverse groups of animals arrange themselves within such subtle boundaries, limiting their movements to the terrain that affords them their needed foods and the necessary shelter from predators. Other, more migratory species follow such patterns as they move with the seasons, articulating routes and regions readily obscured by the current human overlay of nations, states, and their various subdivisions. Only when we slip beneath the exclusively human logic continually imposed upon the earth do we catch sight of this other, older logic at work in the world. Only as we come close to our senses, and begin to trust, once again, the nuanced intelligence of our sensing bodies, do we begin to notice and respond to the subtle logos of the land.
There is an intimate reciprocity to the senses; as we touch the bark of a tree, we feel the tree _touching us;_ as we lend our ears to the local sounds and ally our nose to the seasonal scents, the terrain gradually tunes us in in turn. The senses, that is, are the primary way that the earth has of informing our thoughts and of guiding our actions. Huge centralized programs, global initiatives, and other "top down" solutions will never suffice to restore and protect the health of the animate earth. _For it is only at the scale of our direct, sensory interactions with the land around us that we can appropriately notice and respond to the immediate needs of the living world_.
Yet at the scale of our sensing bodies the earth is astonishingly, irreducibly diverse. It discloses itself to our senses not as a uniform planet inviting global principles and generalizations, but as this forested realm embraced by water, or a windswept prairie, or a desert silence. We can know the needs of any particular region only by participating in its specificity—by becoming familiar with its cycles and styles, awake and attentive to its other inhabitants.
OF COURSE, THE INTENSELY PLACE-CENTERED CHARACTER OF THE older, oral cultures was not without its drawbacks. Exquisitely integrated into their surrounding ecologies, indigenous, oral cultures were often so bound to their specific terrains that other, neighboring ecologies—other patterns of flora, fauna, and climate—could seem utterly incongruous, threatening, even monstrous. While such uncanniness may have helped to limit territorial incursions into neighboring bioregions, and thus may have minimized the potential for intertribal conflict, still there were times when human bands were displaced from their familiar lands—whether by climatic changes, by changes in the migration routes of prey, or simply by accident—and suddenly found themselves in a world where their ritual gestures, their prayers, and their stories seemed to lose all meaning, where the shapes of the landforms lacked coherence, _where nothing seemed to make sense_.
Without a set of stories and songs appropriate to the new surroundings, without an etiquette matched to _this_ land and its specific affordances of food, fuel, and shelter, the displaced and often frightened newcomers could easily disrupt and even destroy a large part of the biotic community. The extinctions of various large animals that occurred immediately after migrating humans first crossed the Bering Strait and spread throughout North and South America may well have been precipitated by just such a situation—by a lack of cultural and linguistic patterns tuned to the diverse ecologies of this continent. A similar wave of extinctions appears to have occurred much earlier, during the first centuries of human incursion into Australia, while other extinctions have marked the arrival of our species in various island ecologies, including New Zealand, Hawaii, and Madagascar. Such events suggest that the deep attunement to place characteristic of so many oral peoples emerges only after several generations in one general terrain.
It is also evident that encounters between human groups from entirely different bioregions could at times precipitate violence—in some cases quite bloody violence—merely as a result of the incommensurability of cultural universes and the consequent terror that each group might induce in the other. Such considerations must lead us to wonder whether the strange sense of human commonality made possible by the spread of formal writing systems is not something very worthy after all. Is there not something terrifically valuable about the modern faith in human equality? Although achieved at the cost of our cultural attunement to the particular places we inhabit, is there not something wondrous about the spreading recognition that we are part of a single, unitary earth?
Perhaps there is. And yet it is a precarious value. For at the very moment that human populations on every continent have come to recognize the planet as a unified whole, we discover that so many other species are rapidly dwindling and vanishing, that the rivers are choking from industrial wastes, that the sky itself is wounded. At the very moment that the idea of human equality has finally spread, via the printed word or the electronic media, into every nation, it becomes apparent that it is indeed nothing more than an idea, that in some of the most "developed" of nations humans are nevertheless destroying each other, physically and emotionally, in unprecedented numbers—whether through warfare, through the callousness of corporate greed, or through a rapidly spreading indifference.
Clearly, something is terribly missing, some essential ingredient has been neglected, some necessary aspect of life has been dangerously overlooked, set aside, or simply forgotten in the rush toward a common world. In order to obtain the astonishing and unifying image of the whole earth whirling in the darkness of space, humans, it would seem, have had to relinquish something just as valuable—the humility and grace that comes from being fully a part of that whirling world. We have forgotten the poise that comes from living in storied relation and reciprocity with the myriad things, the myriad _beings_ , that perceptually surround us.
Only if we can renew that reciprocity—grounding our newfound capacity for literate abstraction in those older, oral forms of experience—only then will the abstract intellect find its real value. It is surely not a matter of "going back," but rather of coming full circle, uniting our capacity for cool reason with those more sensorial and mimetic ways of knowing, letting the vision of a common world root itself in our direct, participatory engagement with the local and the particular. If, however, we simply persist in our reflective cocoon, then all of our abstract ideals and aspirations for a unitary world will prove horribly delusory. If we do not soon remember ourselves to our sensuous surroundings, if we do not reclaim our solidarity with the other sensibilities that inhabit and constitute those surroundings, then the cost of our human commonality may be our common extinction.
Indeed, many persons and communities, both within and outside of the industrialized nations, are already engaged in such a process of remembering. Individuals with the most varied backgrounds and skills—farmers, physicists, poets, professors, herbalists, engineers, mapmakers—have all been drawn toward the practice that some call "reinhabitation." They have begun to apprentice themselves to their particular places, to the ecological regions they inhabit. Many, for instance, have become careful students of the plants and trees that grow in their terrain, learning each plant's nutritive and/or medicinal properties, and its associations with specific insects and animals. Others have taken as teachers the local animals themselves, spending their spare time monitoring migrations, or learning the life cycle and behavior of particular species. They work to restore damaged habitats, and gradually to restore native species that had been locally eradicated by human recklessness. Working together, they shut down the factory that pollutes the estuary, and they woo the salmon back into the streams. In the heart of the city they plant collective gardens with endemic species, and hold equinox feasts with the homeless. At every juncture they strive to discern those modes of human community that are most appropriate to the region, most responsive and responsible to the earthly surroundings.
In North America this spontaneous and quietly growing movement goes by many names. In truth, it is less a movement than a common sensibility shared by persons who have, in Robinson Jeffers's phrase, "fallen in love outward" with the world around them. As their compassion for the land deepens, they choose to resist the contemporary tendency to move always elsewhere for a better job or more affluent lifestyle, and resolve instead to dedicate themselves to the terrain that has claimed them, to meet the generosity of the land with a kind of wild faithfulness. They rejuvenate their senses by entering into reciprocity with the sensuous surroundings. This does not prevent them from engaging in the political realities of counties and countries, from supporting statewide initiatives and voting in national elections. They are aware, however, that political and economic institutions not aligned with earthly realities are not likely to last, that such structures are like ephemeral phantoms to which we must attend without letting them distract us from what is really _here_. Such persons ally themselves not with the ever-expanding human monoculture, nor with the abstract vision of a global economy, but with the far more sustainable prospect of a regionally diverse and interdependent web of largely self-sufficient communities—a multiplicity of technologically sophisticated, vernacular cultures tuned to the structure and pulse of particular places. They know well that if humankind is to flourish without destroying the living world that sustains us, then we must grow out of our adolescent aspiration to encompass and control all that is. Sooner or later, they suspect, our technological ambition must begin to scale itself down, allowing itself to be oriented by the distinct needs of specific bioregions. Sooner or later, that is, technological civilization must accept the invitation of gravity and settle back into the land, its political and economic structures diversifying into the varied contours and rhythms of a more-than-human earth.
YET THE PRACTICE OF REALIGNMENT WITH REALITY CAN HARDLY afford to be Utopian. It cannot base itself upon a vision hatched in our heads and then projected into the future. Any approach to current problems that aims us toward a mentally envisioned future implicitly holds us within the oblivion of linear time. It holds us, that is, within the same illusory dimension that enabled us to neglect and finally to forget the land around us. By projecting the solution somewhere outside of the perceiveable present, it invites our attention away from the sensuous surroundings, induces us to dull our senses, yet again, on behalf of a mental ideal.
A genuinely ecological approach does not work to attain a mentally envisioned future, but strives to enter, ever more deeply, into the sensorial present. It strives to become ever more awake to the other lives, the other forms of sentience and sensibility that surround us in the open field of the present moment. For the other animals and the gathering clouds do not exist in linear time. We meet them only when the thrust of historical time begins to open itself outward, when we walk out of our heads into the cycling life of the land around us. This wild expanse has its own timing, its rhythms of dawning and dusk, its seasons of gestation and bud and blossom. It is here, and not in linear history, that the ravens reside.
Of course, if we live in the thick of the city, or even among the sprawling malls of suburbia, the sensuous world itself seems to surge toward a transcendent future, as high-rise buildings spring up from vacant lots, as wetlands give way to highways and billboard advertisements become 3-D holograms. Yet this restless progression takes place only within the encircling horizon of the breathing earth. New York City remains, first and foremost, an island settlement in the Hudson River estuary, subject to the coastal weather of that geography. For all the international commerce that goes on within its glassy walls, Manhattan could not exist without its grounding amid the waters with their tidal surges. Meanwhile, the inhabitants of Los Angeles awaken, often enough, to the trembling power of their own terrain. To return to our senses is to renew our bond with this wider life, to feel the soil beneath the pavement, to sense—even when indoors—the moon's gaze upon the roof.
BUT WHAT, THEN, OF WRITING? THE PRECEDING PAGES HAVE CALLED attention to some unnoticed and unfortunate side-effects of the alphabet—effects that have structured much of the way we now perceive. Yet it would be a perilous mistake for any reader to conclude from these pages that he or she should simply relinquish the written word. Indeed, the story sketched out herein suggests that the written word carries a pivotal magic—the same magic that once sparkled for us in the eyes of an owl and the glide of an otter.
For those of us who care for an earth not encompassed by machines, a world of textures, tastes, and sounds other than those that we have engineered, there can be no question of simply abandoning literacy, of turning away from all writing. Our task, rather, is that of _taking up_ the written word, with all of its potency, and patiently, carefully, writing language back into the land. Our craft is that of releasing the budded, earthly intelligence of our words, freeing them to respond to the speech of the things themselves—to the green uttering-forth of leaves from the spring branches. It is the practice of spinning stories that have the rhythm and lilt of the local sound-scape, tales for the tongue, tales that want to be told, again and again, sliding off the digital screen and slipping off the lettered page to inhabit these coastal forests, those desert canyons, those whispering grasslands and valleys and swamps. Finding phrases that place us in contact with the trembling neck-muscles of a deer holding its antlers high as it swims toward the mainland, or with the ant dragging a scavenged rice-grain through the grasses. Planting words, like seeds, under rocks and fallen logs—letting language take root, once again, in the earthen silence of shadow and bone and leaf.
AN ALDER LEAF, LOOSENED BY WIND, IS DRIFTING OUT WITH THE tide. As it drifts, it bumps into the slender leg of a great blue heron staring intently through the rippled surface, then drifts on. The heron raises one leg out of the water and replaces it, a single step. As I watch I, too, am drawn into the spread of silence. Slowly, a bank of cloud approaches, slipping its bulged and billowing texture over the earth, folding the heron and the alder trees and my gazing body into the depths of a vast breathing being, enfolding us all within a common flesh, a common story now bursting with rain.
# **_Notes_**
## CHAPTER 1: THE ECOLOGY OF MAGIC
1. This work was done at the Philadelphia Association, a therapeutic community directed by Dr. R. D. Laing and his associates.
2. A simple illustration of this may be found among many of the indigenous peoples of North America, for whom the English term "medicine" commonly translates a word meaning "power"—specifically, the sacred power received by a human person from a particular animal or other nonhuman entity. Thus, a particular _medicine person_ may be renowned for her "badger medicine" or "bear medicine," for his "eagle medicine," "elk medicine," or even "thunder medicine." It is from their direct engagement with these nonhuman powers that medicine persons derive their own abilities, including their ability to cure human ailments.
3. To the Western mind such views are likely to sound like reckless "projections" of human consciousness into inanimate and dumb materials, suitable for poetry perhaps, but having nothing, in fact, to do with those actual birds or that forest. Such is our common view. This text will examine the possibility that it is civilization that has been confused, and not indigenous peoples. It will suggest, and provide evidence, that one perceives a world at all only by projecting oneself into that world, that one makes contact with things and others only by actively participating in them, lending one's sensory imagination to things in order to discover how they alter and transform that imagination, how they reflect us back changed, how they are different from us. It will suggest that perception is _always_ participatory, and hence that modern humanity's denial of awareness in nonhuman nature is borne not by any conceptual or scientific rigor, but rather by an inability, or a refusal, to fully perceive other organisms.
4. The similarity between such animistic worldviews and the emerging perspective of contemporary ecology is not trivial. Atmospheric geochemist James Lovelock, elucidating the well-known Gaia hypothesis—a theory stressing the major role played by organic life in the ceaseless modulation of the earth's atmospheric and climatic conditions—insists that the geological environment is itself constituted by organic life, and by the products of organic metabolism. In his words, we inhabit "a world that is the breath and bones of our ancestors." See, for instance, "Gaia: the World as Living Organism," in the _New Scientist_ , December 18, 1986, as well as _Scientists on Gaia_ , ed. Stephen Schneider and Penelope Boston (Cambridge: M.I.T. Press, 1991).
## CHAPTER 2: PHILOSOPHY ON THE WAY TO ECOLOGY
1. Galileo Galilei, cited in Edwin Jones, _Reading the Book of Nature_ (Athens: Ohio University Press, 1989), p. 22.
2. "Phenomenon," in _Merriam-Webster's Collegiate Dictionary_ , 10th ed., signifies "an object or aspect known through the senses rather than by thought or intuition." It is commonly contrasted with the term "noumenon" (from the Greek _nooumenon:_ "that which is apprehended by thought"—itself derived from the Greek term _nous_ , for "mind").
3. Maurice Merleau-Ponty, _Phenomenology of Perception_ , trans. Colin Smith (London: Routledge & Kegan Paul, 1962), pp. viii–ix.
4. Edmund Husserl, _Cartesian Meditations: An Introduction to Phenomenology_ , trans. Dorion Cairns (The Hague: Martinus Nijhoff Publishers, 1960). (Husserl completed the original text in 1929.)
5. Edmund Husserl, "Epilogue," in _Ideas Pertaining to a Pure Phenomenology II_ , trans. Richard Rozcewicz and André Schuwer, 1989, p. 421. The notion of intersubjectivity did not reach the American popular awareness until the 1960s, when various authors began to describe objective reality as the "consensus reality" of the cultural mainstream.
6. Husserl's notion of the life-world was developed in his last, unfinished book, _The Crisis of European Sciences_ and _Transcendental Phenomenology_ , written from 1934 to 1937, in the shadow of the impending world war. As a German Jew, Husserl was denied any public platform from which to lecture, teach, or publish in his own country; hence, the lectures from which _The Crisis_ grew were presented on journeys to Vienna and to Prague, and the first installments of the book were published in Yugoslavia shortly before Husserl's death in 1938. The "Crisis" of the title, which he wrote of as "the loss of science's meaning for life," was soon to be exemplified in the supreme indifference to life of many of Germany's scientists and medical doctors as they wrote numerous scientific articles on the biological inferiority of particular races, and later, in the objective and technological efficiency of the death factories at Auschwitz, Dachau, Buchenwald, and Treblinka. Although the gas chambers are no more, the same crisis—the same estrangement of a presumably "objective" rationality from living, sensuous reality—continues today in the reckless poisoning of the waters and the winds, and the forced extinction of countless forms of life, by a technological "progress" utterly oblivious to the living world on which it feeds.
7. Husserl, "Foundational Investigations of the Phenomenological Origin of the Spatiality of Nature," trans. Fred Kersten, in Peter McCormick and Frederick A. Elliston, eds., _Husserl: Shorter Works_ (Brighton, Eng.: Harvester Press, 1981).
8. Ibid., p. 227.
9. Ibid., p. 231.
10. See Maurice Merleau-Ponty, _Signs_ , trans. Richard McCleary (Evanston, Ill.: Northwestern University Press, 1964), pp. 180–81.
11. In this chapter I will be intertwining Merleau-Ponty's conclusions with my own experiential illustrations of those conclusions. I am less interested in merely repeating Merleau-Ponty's insights thirty years after his death than I am in demonstrating the remarkable usefulness of those insights for a deeply philosophical (and psychological) ecology. While my explications will at times move beyond the exact content of Merleau-Ponty's writings, they are nonetheless inspired by a close and long-standing acquaintance with those writings, and they remain faithful, I trust, to the unfinished and open-ended character of his thinking.
12. Aristotle, _The Politics of Aristotle_ , trans. E. Barker (Oxford: Oxford University Press, 1946), p. 13 (1254b).
13. Merleau-Ponty, _Phenomenology of Perception_ , p. 214.
14. Ibid.
15. Ibid., p. 317.
16. Ibid., pp. 211–12.
17. Ibid., pp. 320, 322.
18. Lucien Lévy-Bruhl, _How Natives Think_ , (reprint, Princeton: Princeton University Press, 1985), p. 77.
19. Merleau-Ponty, _Phenomenology of Perception_ , p. 227.
20. Ibid., p. 229.
21. Ibid., p. 228.
22. Genuine art, we might say, is simply human creation that does not stifle the nonhuman element but, rather, allows whatever is Other in the materials to continue to live and to breathe. Genuine artistry, in this sense, does not impose a wholly external form upon some ostensibly "inert" matter, but rather allows the form to emerge from the participation and reciprocity between the artist and his materials, whether these materials be stones, or pigments, or spoken words. Thus understood, art is really a cooperative endeavor, a work of cocreation in which the dynamism and power of earth-born materials is honored and respected. In return for this respect, these materials contribute their more-than-human resonances to human culture.
23. Maurice Merleau-Ponty, _The Visible and the Invisible_ , trans. Alphonso Lingis, (Evanston, Ill.: Northwestern University Press, 1968).
24. Ibid., p. 127.
25. Richard K. Nelson, _Make Prayers to the Raven: A Koyukon View of the Northern Forest_ (Chicago: University of Chicago Press, 1983), p. 14.
26. Ibid., p. 241.
27. Kenneth Lincoln, "Native American Literatures," in _Smoothing the Ground: Essays on Native American Oral Literature_ , Brian Swann, ed., (Berkeley: University of California Press, 1983), p. 18.
28. Ibid., p. 22.
## CHAPTER 3: THE FLESH OF LANGUAGE
1. Merleau-Ponty, _Phenomenology of Perception_ , p. 184.
2. Ibid.
3. James M. Edie, introduction to Merleau-Ponty, _Consciousness and the Acquisition of Language_ (Evanston, Ill.: Northwestern University Press, 1973), p. xviii.
4. Giambattista Vico, _The New Science of Giambattista Vico_ , trans. Thomas G. Bergin and Max H. Fisch, 3rd ed. (Garden City, N.Y.: Doubleday & Co., 1961).
5. Jean-Jacques Rousseau, "Essay on the Origin of Languages," trans. John H. Moran; and Johann Gottfried Herder, "Essay on the Origin of Language," trans. Alexander Gode in Rousseau and Herder, _On the Origin of Language_ (Chicago: University of Chicago Press, 1966). Wilhelm von Humboldt later took up and extended Herder's views on language, contesting the mainstream view of language as an objective and determinate system. He insisted that we must think of language primarily as _speech_ , and of speech as a dynamic and creative _activity_ , not as a finished phenomenon—as _energeia_ , not _ergon_. See Charles Taylor, _Human Agency and Language_ (New York: Cambridge University Press, 1985), p. 256.
6. Merleau-Ponty, _Phenomenology of Perception_ , p. 184.
7. Ample evidence for such a view may be found by studying the phonetic texture of particular words. To present a single example: Philosopher Peter Hadreas has sampled the words for "sea" and for "earth" (or "ground") in fifteen European and Asian languages currently in use, and found that the words for "sea" consistently depend upon continuant consonants, while the words for "earth" or "ground" depend upon plosive consonants. (Continuant consonants are those consonants that do not involve a stoppage of air flow. With such consonants— _n, m, ng, s, z, f, v, h, sh_ —the breath is shaped by the vocal organs without being obstructed by them. Plosives, on the other hand, involve a momentary stoppage of the air flow and a subsequent, slightly explosive, release. Such are _t, d, ch, j, p, b, and g_.) Here is Hadreas's chart:
_language_ | _"sea"_ | _"earth"_ or _"ground"_
---|---|---
French | mer | terre
Italian | mare | terra
Spanish | mer | tierra
German | meer | erde
Dutch | zee | aarde
Russian | more | potshva
Polish | morze | gleba
Czech | more | puda
Lithuanian | jura | padas
Latvian | jura | augsne
Turkish | deniz | toprak
Arabic | bahar | trab
Japanese | umi | dai chi
Korean | hoswu | taeji
Chinese | hoi | tati
Hadreas offers this explanation of the findings: "The sea as we move over or through it does not involve an obstruction of movement; whereas the earth or ground, at least insofar as it breaks a fall, always does." Accordingly, the words for "earth" or "ground" all employ plosives, while the words for "sea" employ only continuants. (Even in the apparent exceptions of Turkish and Arabic, the words for "sea" are relatively less plosive than those for "earth.") See Peter J. Hadreas, _In Place of the Flawed Diamond_ (New York: Peter Lang Publishers, 1986); pp. 100–102.
8. Although an almost dogmatic insistence upon the arbitrariness of the relation between linguistic signs and that which they signify has been common among linguists throughout the twentieth century, several major researchers have dared to challenge this profoundly dualistic assumption, and have undertaken careful studies of the implicit significance carried by particular speech sounds, or "phonemes." Among those theorists who have stressed the importance of this layer of meanings immanent in the speech sounds themselves are the German linguist Hans Georg von der Gabelentz (1840–93); the French linguist Maurice Grammont (1866–1946), whose work focused on the evocative significance of the different vowel sounds; the well-known American linguist Edward Sapir (1884–1939); and Sapir's correspondent, the outstanding Danish linguist Otto Jesperson (1860–1943), who accomplished substantial research on the role of onomatopoeias and "sound symbolism" in the ongoing evolution of spoken languages (see, for instance, chap. 20 of Jesperson's book _Language—Its Nature, Development, and Origin_ [New York: Henry Holt, 1922]). Finally, I must acknowledge the great Russian investigator of languages, Roman Jacobson (1896–1982), whose wonderful chapter "The Spell of Speech Sounds," from a late book written with Linda R. Waugh entitled _The Sound Shape of Language_ (Bloomington: Indiana University Press, 1979), was my initial encounter with the first two linguists mentioned above.
9. Merleau-Ponty, _Phenomenology of Perception_ , p. 197.
10. Ferdinand de Saussure, _Course in General Linguistics_ , ed. Charles Bally and Albert Sechehaye, trans. Wade Baskin (New York: McGraw-Hill, 1966).
11. Merleau-Ponty, _Signs_ , p. 39.
12. Ibid., pp. 40, 42.
13. Merleau-Ponty, _The Visible and the Invisible_. See, for instance, the note on "Perception and Language," p. 213.
14. Ibid., p. 125.
15. Ibid., p. 194. Compare these well-known lines from Dogen, the great thirteenth-century Japanese Zen teacher: "That the self advances and realizes the ten thousand things is delusion. That the ten thousand things advance and realize the self is enlightenment" (from the _Genjo Koan_ by Dogen).
16. Merleau-Ponty, _The Visible and the Invisible_ , p. 155.
17. Merleau-Ponty's approach to language and to meaning, disclosing their source in a carnal field of participation that subtends the strictly human universe of instituted and inert meanings, provides a powerful response to the significant challenge posed to Western rationality by the postmodern "deconstructionist" thinkers. While these theorists aim to effect a deconstruction of _all_ philosophical foundations, Merleau-Ponty's work suggests that, underneath all those admittedly shaky foundations, there remains the actual ground that we stand on, the earthly ground of rock and soil that we share with the other animals and the plants. This dark source, to which we can readily point even in the silence, will outlast all our purely human philosophies as it outlasts all the other artificial structures we erect upon it. We would do well, then, to keep our thoughts and our theories close to this nonarbitrary ground that already supports all our cogitations. The density beneath our feet is a depth we cannot fathom, and it spreads out on all sides into the horizon, and beyond. Unlike all the human-made foundations we construct upon its surface, the silent and stony ground itself can never be grasped in a purely human act of comprehension. For it has, from the start, been constituted (or "constructed") by many organic entities besides ourselves.
18. See Marcel Griaule, _Conversations with Ogotemmêli_ (London: Oxford University Press, 1965), pp. 16–21.
19. See, for instance, Howard Norman, "Crow Ducks and Other Wandering Talk," in David M. Guss, ed., _The Language of the Birds_ (San Francisco: North Point Press, 1985), p. 19.
20. Translated by Edward Field, in Jerome and Diane Rothenberg, eds., _Symposium of the Whole_ (Berkeley: University of California Press, 1983), p. 3.
21. Norman, "Crow Ducks," p. 20.
22. See, for instance, Richard Nelson, _Make Prayers to the Raven: A Koyukon View of the Northern Forest_ (Chicago: University of Chicago Press, 1983).
23. Mircea Eliade, _Shamanism: Archaic Techniques of Ecstasy_ , trans. Willard R. Trask (Princeton: Princeton University Press, 1964), pp. 96–98.
24. Brian Swann, ed., _Smoothing the Ground: Essays on Native American Oral Literature_ (Berkeley: University of California Press, 1983), p. 28.
25. Merleau-Ponty, _The Visible and the Invisible_ , p. 213.
26. Edward Sapir, "The Status of Linguistics as a Science," in David G. Mandelbaum, ed., _Selected Writings of Edward Sapir_ (Berkeley: University of California Press, 1949), p. 162.
## CHAPTER 4: ANIMISM AND THE ALPHABET
1. Perhaps the most influential of such analyses has been historian Lynn White Jr.'s much-reprinted essay "The Historical Roots of Our Ecologic Crisis," originally published in _Science_ 155 (1967), pp. 1203–1207.
The Genesis quote is from _Tanakh: The Holy Scriptures_ , translated by the Jewish Publication Society according to the traditional Hebrew text (Philadelphia: Jewish Publication Society, 1985).
2. Jacques Derrida and other theorists have claimed that there is no self-identical author or subject standing behind any text that one reads, legislating its "actual" meanings; the precise meaning of a text, like its real origin, can only be indicated by referring to other texts to which this one responds, and since those, in turn, mark divergences from still other texts, the clear source, or the true meaning, is always deferred, always elsewhere. Since neither the origin nor the precise meaning of a text can ever be made wholly explicit, there can be no real meeting between the reader and the writer, at least not in the traditional sense of a pure coinciding of one's "self" with the exact intention of a supposed "author."
My equation of "meaning" with "meeting" would seem, at first blush, to fall easy prey to this critique. Yet Derrida's critique has bite only if one maintains that the other who writes is an exclusively _human_ Other, only if one assumes that the written text is borne by an exclusively human subjectivity. Here, however, I am asserting a homology between the act of reading and the ancestral, indigenous act of _tracking_. I am suggesting that that which lurks behind all the texts that we read is not a human subject but another animal, another shape of awareness (ultimately the otherness of animate nature itself). The meeting that I speak of, then, is precisely the encounter with a presence that can never wholly coincide with our own, the confrontation with an enigma that cannot be dispelled by thought, an otherness that can never be fully overcome.
3. J. Gernet, quoted in Jacques Derrida, _Of Grammatology_ , trans. Gayatri Spivak (Baltimore: Johns Hopkins University Press, 1976), p. 123.
4. The approximate dates referred to in this paragraph are drawn from several texts, including Albertine Gaur, _A History of Writing_ (New York: British Library/Cross River Press, 1992); J. T. Hooker et al., _Reading the Past: Ancient Writing from Cuneiform to the Alphabet_ (Berkeley: British Museum/University of California Press, 1990); and Jack Goody, _The Interface_ _Between the Written and the Oral_ (Cambridge: Cambridge University Press, 1987).
5. The written characters or glyphs that I have referred to as ideograms are sometimes called logograms (word signs) by contemporary linguists, in order to emphasize that these characters are regularly used to transcribe or invoke particular words. The term "logogram," however, hides or masks the pictorial element that remains subtly operative in many of these written characters, and it is for this reason that I, like many others, have retained the popular terminology. The pictorial, "iconic" nature of many characters within a script inevitably influences the experience of language and linguistic meaning common to those who use that script. In the Mayan languages, for instance, the words for "writing" and "painting" were and are the same—the same artisans practiced both crafts, and the patron deities of both crafts were twin monkey gods. As Dennis Tedlock informs us in his introduction to the Mayan _Popol Vuh_ , "In the books made under the patronage of these twin gods... the writing not only records words but sometimes has elements that picture or point to their meaning without the necessity of a detour through words." Dennis Tedlock, trans., _Popul Vuh: The Mayan Book of the Dawn of Life_ (New York: Simon & Schuster, 1985), p. 30.
6. That the contemporary Chinese word for "writing," as we saw earlier, also applies to the tracks of animals and the marks on a turtle shell may well be attributed to the fact that China has retained a somewhat iconic or pictorially derived mode of writing down to the present day.
7. Jack Goody, _The Interface Between the Written and the Oral_ , pp. 34, 38.
8. It is important to realize that many pictorially derived writing systems commonly assumed by Western thinkers to be largely ideographic—like Egyptian hieroglyphs, the Chinese script, and even the recently deciphered Mayan system—utilize a host of conventional rebuses as phonetic indicators in combination with ideographic signs. These phonetic characters, however, commonly retain pictorial ties to the sensuous world. Although a hasty reader might choose to read these phonetic symbols without giving thought to their pictorial significance, according to Dennis Tedlock "the other meanings were still there for a reader who could see and hear them—even the same reader perhaps, in a different mood." A striking demonstration of the imagistic logic that animates such nonalphabetic writing systems may be found in the chapter entitled "Eyes and Ears to the Book" in Tedlock's remarkable study of Mayan culture, _Breath on the Mirror_ (San Francisco: HarperCollins, 1993, pp. 109–14).
9. Walter J. Ong, _Orality and Literacy: The Technologizing of the Word_ (New York: Methuen, 1982), pp. 87–88.
10. Ibid.
11. J. A. Hawkins, "The Origin and Dissemination of Writing in Western Asia," in P. R. S. Moorey, ed., _Origins of Civilization_ (London: Oxford University Press, 1979), p. 132.
12. Ong, p. 89. See also Hooker et al., pp. 210–11; Gaur, p. 87.
13. However, the _aleph_ in the Hebrew _aleph-beth_ does not represent a vowel sound—rather, it signifies the opening of the throat prior to any sound.
14. Another common version of the early Semitic 'qoph' consisted of a _semicircle_ intersected by a vertical line: . Linguist Geoffrey Sampson writes that "no-one familiar with the look of heavy simian eyebrows ought... to find it difficult to see ['qoph'] as a full-face view of an ape." Likewise, the Semitic letter 'gimel' (which means camel in Hebrew) consisted of a rising and descending line: —Sampson believes that this may be a stylized image of a camel's most prominent feature: its hump. Other letters took their forms from a hand, mouth, a snake. See Geoffrey Sampson, _Writing Systems: A Linguistic Introduction_ (Stanford: Stanford University Press, 1985), pp. 78–81.
These letter shapes are from the original Hebrew _aleph-beth_ , known in the later Jewish tradition as _Ksav Ivri_ (literally: "script of the Hebrews"). These letters were eventually replaced, between the fifth and the third century B.C.E., by the square Hebrew letters used today, themselves borrowed from a late Aramaic version of the _aleph-beth_. See Hooker, et al., pp. 226–27; also Gaur, p. 92.
15. David Diringer, _The Alphabet_ (New York: Philosophical Library, 1948), p. 159.
16. Plato, _Phaedrus_ , trans. R. Hackforth, in _Plato: The Collected Dialogues_ , ed. Edith Hamilton and Huntington Cairns (Princeton: Princeton University Press, 1982), sec. 230d.
17. Homer, _The Odyssey_ , trans. Robert Fitzgerald (Garden City, N.Y.: Doubleday & Co., 1961); and Homer, _The Iliad_ , trans. Robert Fitzgerald (Garden City, N.Y.: Doubleday & Co., 1974).
18. Eric Havelock, _The Muse Learns to Write: Reflections on Orality and Literacy from Antiquity to the Present_ (New Haven: Yale University Press, 1986), pp. 19, 83, 90. See also Havelock's seminal text _Preface to Plato_ (Cambridge: Harvard University Press, 1963).
19. The earliest Greek inscriptions of an alphabetic nature yet to be discovered are from around 740 or 730 B.C.E. (Hooker et al., pp. 230–32). See also Rhys Carpenter, "The Antiquity of the Greek Alphabet," _American Journal of Archaeology_ 37 (1933); Havelock, _Preface to Plato_ , pp. 49–52; Havelock, _The Muse Learns to Write_ , pp. 79–97; Goody, _The Interface Between the Written and the Oral_ , pp. 40–47.
20. The evidence for this resistance is carefully documented by Eric A. Havelock, the most accomplished scholar of the transition from orality to literacy in ancient Greece, particularly in his essay "The Special Theory of Greek Orality," in _The Muse Learns to Write_.
21. Havelock, _The Muse Learns to Write_ , p. 87.
22. There is a linguistic parallel here with the Vedic _sutras_ , so named because they, too, are sewn, or _sutured_ , together.
23. See Adam Parry, ed., _The Making of Homeric Verse: The Collected Papers of Milman Parry_ (Oxford: Clarendon Press, 1971). See also Albert Lord, _The Singer of Tales_ (Cambridge: Harvard University Press, 1960).
24. Ivan Illich and Barry Sanders, _The Alphabetization of the Popular Mind_ (San Francisco: North Point Press, 1988), p. 18.
25. See Ong, p. 35: "Fixed, often rhythmically balanced, expressions of this sort and of other sorts can be found occasionally in print, indeed can be 'looked up' in books of sayings, but in oral cultures they are not occasional. They are incessant. They form the substance of thought itself. Thought in any extended form is impossible without them, for it consists in them."
26. Today these disks are housed in the Parry Collection at Harvard University.
27. See especially "Whole Formulaic Verses in Greek and Southslavic Heroic Song," as well as other essays in Adam Parry, _The Making of Homeric Verse_.
28. Ibid., p. 378.
29. Ong, p. 59. In recent years Milman Parry's conclusion that the Homeric epics originated in a completely oral context has been disputed by Jack Goody, another careful student of oral-literate contrasts. Goody points out that while the Yugoslavian bards recorded by Parry and Lord were themselves nonliterate, the culture in which they sang and improvised their epic poems was not entirely untouched by literacy. Goody himself has worked among the LoDagaa people of northern Ghana—a tribe unacquainted with literacy until quite recently—and he undertook to record their oral myth, "the Bagre," which is ritually recited during the course of a long series of initiatory ceremonies. (Jack Goody, _The Myth of the Bagre_ (Oxford: Clarendon Press, 1972). Along with many obvious similarities, he has found marked differences between the LoDagaa recitation and both the Slavic and the Homeric epics. The epic poems of Yugoslavia and of ancient Greece seem much more formal and tightly composed than their African counterparts (see "Africa, Greece and Oral Poetry," in Goody, _The Interface Between the Written and the Oral_ ). Further, according to Goody, the epic mode of the bardic tales, centered on the legendary acts of a human hero or a group of heroes, is foreign both to the Bagre and to other oral compositions of indigenous Africa (on this, see also Ruth Finnegan, _Oral Literature in Africa_ [London: Oxford University Press, 1970]). Goody's evidence suggests that the epic mode is more proper to the poetry of cultures in the earliest stages of literacy, rather than to that of purely oral peoples. He argues from this that the culture in which the _Iliad_ and the _Odyssey_ took shape should not be considered a pristinely oral culture, since even if the culture was without writing it had nevertheless been influenced (1) by the much earlier existence of nonalphabetic writing systems (Linear A and Linear B, which had been used, for economic and military accounting, by the Minoan and Mycenaean cultures on the island of Crete, until such writing vanished around 1100 B.C.E.), and (2) by the literacy of the neighboring societies of the Near East, societies with which the Greek merchants must have been in frequent contact ("Africa, Greece and Oral Poetry," pp. 98, 107–9). Goody's premise, that pre-Homeric Greece may have been influenced by the limited literacy of its Minoan and Mycenaean forebears, or by the protophonetic literacy of some cultures across the Mediterranean, may help us to understand why the Homeric gods and goddesses are as anthropomorphic as they are, much more human in form than are the deities of most cultures entirely untouched by literacy. We may, however, accept Goody's argument for the _indirect_ influence of literacy without concluding that mainland Greece from 1100 to 750 B.C.E. made any _direct_ use of writing, or had any wish to do so. For a lively debate on the orality of the Homeric epics, see "Becoming Homer: An Exchange," in the _New York Review of Books_ , May 14, 1992.
30. Havelock, _The Muse Learns to Write_ , p. 112.
31. Philip Wheelwright, ed., _The Presocratics_ (New York: Macmillan Publishing Co., 1985), p. 45.
32. Illich and Sanders, pp. 22–23.
33. Plato, _Meno_ , trans. W. K. C. Guthrie, in _Plato: The Collected Dialogues_ , ed. Hamilton and Cairns sec. 72a (Princeton: Princeton University Press, 1982).
34. The reader may object that the alphabet gave a fixed and visible form _not_ to the actual quality we call "justice," but only to the word, to the verbal label that "stands for" that quality. Surely Socrates was asking his discussants to ponder the quality itself, not the mere word. However, the clear distinction assumed by this objection, between words and what they "stand for," is a fairly recent distinction, itself made possible by the spread of phonetic writing. Only after spoken words were fixed in writing could they begin to be thought of as arbitrary "labels." In the Athens of Socrates and Plato, however—a society only emerging into literacy—the word was still directly participant with the phenomenon that it invoked, the phenomenon still participant with the spoken word. If the new technology of writing imparted to the spoken word "virtue" a new sense of autonomy and permanence, it brought a new sense of changelessness to the quality itself.
35. Ernest Fenollosa, cited in Ezra Pound, _ABC of Reading_ (New York: New Directions Press, 1960), pp. 19–22.
36. Jacques Derrida has explored at great length the consequences of this curious vanishing throughout the trajectory of Western (alphabetic) philosophy, a tradition that ceaselessly forgets, or represses, its dependence upon writing. See, for instance, _Of Grammatology_ , trans. Gayatri Spivak (Baltimore: Johns Hopkins University Press, 1976). Derrida, however, does not notice some of the most glaring differences between alphabetic and nonalphabetic modes of thought, differences that make themselves evident in our experienced relation to the animate earth. While Derrida assimilates all language to writing _(l'écriture)_ , my approach has been largely the reverse, to show that all discourse, even written discourse such as this, is implicitly sensorial and bodily, and hence remains bound, like the sensing body, to a world that is never exclusively human.
37. By suggesting that the relation, in Plato's writing, between the immortal psyche and the intelligible Ideas is dependent upon the experienced relation between the new, literate intellect and the visible letters of the alphabet, my intention is not to effect a reduction of transcendent, philosophic notions to banal, mundane experience, but rather to reawaken a sense of the profoundly magical, transcendent activity that reading _is_. In this I am simply practicing the method of wakefulness urged by Merleau-Ponty, whose phrase "the primacy of perception" expressed an intuition that even the most transcendental philosophies remain rooted in, and dependent upon, the very corporeal, sensuous world that they seek to forget.
38. _Phaedrus_ , 275a.
39. Ibid., 275b.
40. Ibid., 277e.
41. Ibid., 278a.
42. Ibid., 230d.
43. Two reputable and accessible firsthand accounts of how visions and "medicine power" were and sometimes still are invoked among the Plains tribes are _Lame Deer, Seeker of Visions_ by John Fire Lame Deer and Richard Erdoes, and _Black Elk Speaks_ , by John Neihardt. Both books exist in numerous editions.
44. _Phaedrus_ , 236e.
45. Ibid., 275b.
46. Ibid., 259a-d.
47. Ibid., 259b-c.
48. Richard Nelson, _Make Prayers to the Raven_ (Chicago: University of Chicago, 1983), p. 17.
49. Jack Goody, in _The Domestication of the Savage Mind_ (Cambridge: Cambridge University Press, 1977), has shown the dependence of such "mental" lists upon visible, written lists. See also Walter Ong: "Primary oral cultures commonly situate their equivalent of lists in narrative, as in the catalogue of the ships and captains in the _Iliad_ (Book 11, lines 461–879).... In the text of the Torah, which set down in writing thought forms still basically oral, the equivalent of geography (establishing the relationship of one place to another) is put into a formulary action narrative (Numbers 33:16ff.): 'Setting out from the desert of Sinai, they camped at Kibroth-hattaavah. Setting out from Kibroth-hattaavah, they camped at Hazeroth. Setting out from Hazeroth, they camped at Rithmah...' and so on for many more verses. Even genealogies out of such orally framed tradition are in effect commonly narrative. Instead of a recitation of names, we find a sequence of 'begats,' of statements of what someone did: 'Irad begat Mahajael, Mahajael begat Methusael, Methusael begat Lamech' (Genesis 4:18)." Ong, p. 99.
50. Walter Ong writes of this as the "agonistic" requirement in oral storytelling. See Ong, pp. 43–45.
51. _Phaedrus_ , 262d.
52. Ibid., 247c.
53. For instance, the research of Milman Parry and Albert Lord (see n. 23 above).
54. Such is the focus of the research undertaken by such diverse scholars as Eric Havelock, Marshall McLuhan, Walter Ong, Jack Goody, and, most recently, Ivan Illich. See Havelock, _Preface to Plato_ and _The Muse Learns to Write: Reflections on Orality and Literacy from Antiquity to the Present;_ Marshall McLuhan, _The Gutenberg Galaxy: The Making of Typographic Man_ (Toronto: University of Toronto Press, 1962); Ong, _Interfaces of the Word_ (London: Cornell University Press, 1977) and _Orality and Literacy: The Technologizing of the Word;_ Goody, _The Interface Between the Written and the Oral_ and _The Domestication of the Savage Mind_ (Cambridge: Cambridge University Press, 1977); Illich and Sanders, _The Alphabetization of the Popular Mind_.
55. This is a special concern in Illich and Sanders, and in Goody, _The Interface Between the Written and the Oral_.
56. Ivan Illich, _In the Vineyard of the Text_ (Chicago: University of Chicago Press, 1993). Also Illich and Sanders, pp. 45–51.
57. This reciprocity, the circular manner in which a nuanced sense of self emerges only through a deepening relation with other beings, is regularly acknowledged in Buddhism as the "dependent co-arising of self and other."
58. Indeed, Merleau-Ponty takes the visual focus to be paradigmatic for synaesthesia in general: "... the senses interact in perception as the two eyes collaborate in vision." _Phenomenology of Perception_ , pp. 233–34.
59. It is important to realize that the focused structure of perception ensures that I am able to participate with any phenomenon only by _not_ participating with other phenomena. I cannot directly perceive a particular entity, in all its synaesthetic depth and otherness, without forfeiting, for the moment, a direct encounter with other entities, which must therefore remain part of the indeterminate background—at least until they themselves succeed in winning the focus of my senses. Thus, among many indigenous, oral peoples, for whom all things are potentially animate, it is nonetheless clear that _not all phenomena are experienced as animate all the time_. Indeed, certain phenomena, certain plants or insects that we ask about, may have little or no overt significance to the tribal community; they may not even have names within the storied language of the culture. Since these phenomena do not solicit the focused attention of the human community, they are rarely, if ever, experienced by them as unique entities with their own intensity and depth. Only those phenomena that regularly engage our synaesthetic attention stand out from the body of the land as autonomous powers in their own right. If there is no focus, no juxtaposition of diverse sensory modalities, then the phenomenon has no chance to move us, no chance to play one part of our experience off another, no chance to teach us. It thus remains flat, without much depth or dynamism, a purely background phenomenon.
60. A Carrier Indian, quoted in Diamond Jenness, _The Carrier Indians of the Bulkley River_ , Bureau of American Ethnology: Bulletin 133 (Washington, D.C.: Smithsonian Institution, 1943), p. 540 (emphasis added).
61. Tzvetan Todorov, _The Conquest of America_ , trans. Richard Howard (New York: Harper & Row, 1984).
62. Ibid., p. 89.
63. Ibid., pp. 61–62.
## CHAPTER 5: IN THE LANDSCAPE OF LANGUAGE
1. _Phaedrus_ , 275d.
2. See especially Elizabeth Eisenstein's two-volume work, _The Printing Press as an Agent of Change: Communications and Cultural Transformations_ _in Early Modern Europe_ (New York: Cambridge University Press, 1979). An older, more iconoclastic source is McLuhan's _The Gutenberg Galaxy: The Making of Typographic Man_.
3. For an auditory example of such tuning, the reader may wish to listen to a compact disc entitled _Voices of the Rainforest_ (Rykodisk, 1991), a compilation of field recordings of the Kaluli people of Papua New Guinea made by ethnomusicologist Steven Feld. The Kaluli people sing with birds, with insects, with tree frogs and tumbling waterfalls, with the rain itself. "And when the Kaluli sing with them, they sing _like_ them. Nature is music to the Kaluli ears. And Kaluli music is naturally part of the surrounding soundscape.... In this rainforest musical ecology, the world really is a tuning fork." The songful language of the Kaluli is rich with onomatopoeic words that echo the speech of animals as well as mimic the diverse swirling, bubbling, and plopping sounds made by water in the rain forest. But like all oral languages, the participatory songs of the Kaluli people are now threatened with extinction, in this case due to the encroachment of oil-drilling operations: the new voices in the forest are those of helicopters and drilling rigs. See also Steven Feld's book, _Sound and Sentiment: Birds, Weeping, Poetics and Song in Kaluli Expression_ , rev. ed. (Philadelphia: Temple University Press, 1991).
4. F. Bruce Lamb, _Wizard of the Upper Amazon: The Story of Manuel Córdova-Rios_ , (Boston: Houghton Mifflin, 1971).
5. Ibid., pp. 63–64.
6. Ibid., p. 51.
7. Ibid., pp. 48–49.
8. See, for instance, the works of Otto Jesperson and Roman Jacobson cited in chap. 3, n. 8. Late in his life, Jacobson claimed that the reluctance of linguists to acknowledge the inner significance of speech sounds arose simply because early attempts to document this significance had failed to dissect the speech sounds into their most basic constituents. Jacobson and Waugh, _The Sound Shape of Language_ (Bloomington: Indiana University Press, 1979), p. 185.
9. Richard Nelson, _Make Prayers to the Raven: A Koyukon View of the Northern Forest_ (Chicago: University of Chicago Press, 1983), p. 2.
10. Ibid., p. 20.
11. Richard Nelson, _The Island Within_ (San Francisco: North Point Press, 1989), p. 110.
12. Nelson, _Make Prayers to the Raven_ , p. 172.
13. Ibid., p. 86.
14. Ibid., p. 87.
15. Ibid.
16. Ibid., p. 109.
17. Ibid., p. 115.
18. Ibid., p. 110.
19. Ibid., p. 116.
20. Ibid., p. 111.
21. Ibid., p. 106.
22. Ibid.
23. Ibid.
24. Ibid., p. 115.
25. Ibid., p. 119.
26. Ibid., p. 16.
27. Nelson, _Make Prayers to the Raven_ , p. 118.
28. John Bierhorst, _The Mythology of North America_ (New York: William Morrow & Co., 1985), pp. 6–7. See also, for instance, Gerald Vizenor, _Anishnabe Adisokan: Tales of the People_ (Minneapolis: Nodin Press, 1970), p. 9.
29. Ibid., p. 127.
30. Ibid., p. 148.
31. Ibid., p. 155.
32. Ibid., p. 176.
33. Ibid., p. 177.
34. Nelson, _The Island Within_ , p. 117.
35. Ibid., p. 69.
36. A Koyukon elder, quoted in Nelson, _Make Prayers to the Raven_ , p. 26.
37. The choice of cultures here is determined both by my intention to present examples from contrasting biotic regions as well as by my wish to suggest, in the short space of a chapter, the wildly variant ways in which oral languages display their earthly dependence.
38. Keith Basso, " 'Stalking with Stories': Names, Places, and Moral Narratives Among the Western Apaches" (henceforth Basso, "Stalking") in Daniel Halpern, ed., _On Nature: Nature, Landscape, and Natural History_ (San Francisco: North Point Press, 1987).
39. Ibid., p. 101.
40. Ibid., pp. 105–6. See also Keith H. Basso, " 'Speaking with Names': Language and Landscape Among the Western Apache" (henceforth Basso, "Speaking"), in _Cultural Anthropology_ , May 1988, p. 111.
41. Basso, "Stalking," p. 95.
42. Basso, "Speaking," pp. 117–18. I have, however, transcribed the story into the form used by Basso in Basso, "Stalking."
43. Basso, "Stalking," p. 107.
44. Ibid., p. 108.
45. Ibid., pp. 111–12.
46. Ibid., p. 112.
47. Ibid., p. 110.
48. Ibid.
49. Nick Thompson, quoted in Basso, "Stalking," p. 112.
50. Basso, "Stalking," p. 113; Basso, "Speaking," pp. 118, 121–22.
51. Nick Thompson, quoted in Basso, "Stalking," p. 96.
52. Wilson Lavender, quoted in Basso, "Stalking," p. 97.
53. Basso, "Stalking," p. 112.
54. Ibid., pp. 112–14.
55. Basso, "Speaking," p. 110.
56. Quoted in Basso, "Stalking," pp. 100–101.
57. Ronald M. Berndt and Catherine H. Berndt, "How Ooldea Soak Was Made," in _The Speaking Land: Myth and Story in Aboriginal Australia_ (London: Penguin Books, 1989), p. 42.
58. Berndt and Berndt, p. 213.
59. "Leech at Mamaraawiri," in Berndt and Berndt, p. 211.
60. Bruce Chatwin, _The Songlines_ (London: Penguin Books, 1987), p. 60.
61. The Pintupi people say that they recognize a song by its smell _(mayu)_ or taste _(ngurru)_ —a remarkable example of synaesthesia.
62. See Gary Snyder, _The Practice of the Wild_ (San Francisco: North Point Press, 1990), p. 83, as well as Basso, "Stalking," p. 101.
63. Chatwin, p. 60. See also T. G. H. Strehlow, _Aranda Traditions_ (Melbourne: Melbourne University Press, 1947), p. 17.
64. Billy Marshall-Stoneking, "Paddy: A Poem for Land Rights," in _Singing the Snake: Poems from the Western Desert_ (Pymble, Austral.: Angus & Robertson, 1990).
65. Chatwin, p. 14.
66. Colin Tatz, ed., _Black Viewpoints: the Aboriginal Experience_ (Sydney: Australia and New Zealand Book Co., 1975), p. 29. On this point, see also an interview with aboriginal writer and educator Eric Willmot in _Omni_ , June 1987.
67. W. E. H. Stanner, "The Dreaming," in Jerome Rothenberg and Diane Rothenberg, eds., _Symposium of the Whole_ (Berkeley: University of California Press, 1983), pp. 201–5. See also Nancy Munn, _Walbiri Iconography: Graphic Representation and Cultural Symbolism in a Central Australian Society_ (Ithaca, N.Y.: Cornell University Press, 1973), pp. 131–33.
68. From Marshall-Stoneking, "Passage," in _Singing the Snake_ , p. 30.
69. Chatwin, pp. 105–6.
70. Helen Payne, "Rites for Sites or Sites for Rites? The Dynamics of Women's Cultural Life in the Musgraves," in Peggy Brock, ed., _Women_ , _Rites, and Sites: Aboriginal Women's Cultural Knowledge_ (North Sydney, Austral.: Allen & Unwin Limited, 1989), p. 56.
71. Chatwin, p. 52.
72. Catherine J. Ellis and Linda Barwick, "Antikirinja Women's Song Knowledge 1963–1972," in _Women, Rites, and Sites_ , pp. 31–32.
73. Ibid., pp. 34–36. While I have spoken of various aboriginal traditions in the present tense, the reader should be aware that many of these traditions are rapidly being lost under the influence of alphabetic civilization.
74. Gary Snyder, _The Practice of the Wild_ , p. 82.
75. Chatwin, pp. 293–94.
76. Payne, "Rites for Sites or Sites for Rites?" in _Women, Rites, and Sites_ , p. 45.
77. Ibid.
78. Basso, "Speaking," pp. 110–13.
79. The gender specificity here is intentional: almost all orators were men.
80. Frances A. Yates, _The Art of Memory_ (Chicago: University of Chicago Press, 1966).
81. Basso, "Stalking," pp. 115–16.
## CHAPTER 6:
TIME, SPACE, AND THE ECLIPSE OF THE EARTH
1. See Charles A. Reed, ed., _Origins of Agriculture_ (The Hague: Mouton & Co., 1977).
2. Åke Hultkrantz, _Native Religions of North America_ (San Francisco: Harper & Row, 1987), pp. 32–33.
3. T. C. McLuhan, _Touch the Earth_ (New York: Outerbridge and Dienstfrey, 1971), p. 42.
4. Mircea Eliade, _The Myth of the Eternal Return_ (New York: Harper & Row, 1959).
5. Ibid., p. vii.
6. See Todorov, pp. 116–19.
7. Marshall Sahlins, _Historical Metaphors and Mythical Realities_ (Ann Arbor: University of Michigan Press, 1981).
8. Hultkrantz, p. 33.
9. Todorov, p. 85.
10. Rik Pinxten, Ingrid Van Doren, and Frank Harvey, _Anthropology of Space: Explorations into the Natural Philosophy and Semantics of the Navajo_ (Philadelphia: University of Pennsylvania Press, 1983), p. 168.
11. Ibid., p. 36.
12. Benjamin Lee Whorf, "An American Indian Model of the Universe," in Dennis Tedlock and Barbara Tedlock, eds., _Teachings from the American Earth_ (New York: Liveright, 1975), p. 122.
13. Ibid.
14. See especially Ekkehart Malotki, _Hopi Time: A Linguistic Analysis of the Temporal Concepts in the Hopi Language_ (New York: Mouton Publishers, 1983).
15. Whorf, "An American Indian Model," p. 124.
16. Ibid.
17. Pinxten et al., p. 18.
18. Ibid., pp. 20–21.
19. Eliade, _The Myth of the Eternal Return_ , p. 104.
20. Ibid.
21. Indeed, the _original_ tablets, smashed by Moses in anger upon seeing the golden calf, were according to the Hebrew Bible inscribed directly "by the finger of God." Exodus 31:18. See also Rabbi Michael L. Munk, _The Wisdom in the Hebrew Alphabet: The Sacred Letters as a Guide to Jewish Deed and Thought_ (Brooklyn: Mesorah Publications, 1983).
22. Edmond Jabes, _Elya_ (Berkeley, Calif.: Tree Books, 1974), p. 72.
23. Aristotle, _Physics_ , trans. Hippocrates G. Apostle (Bloomington: Indiana University Press, 1969), book IV.
24. By the era of the printing press, the mechanical clock was slowly exerting its influence throughout Europe. The presence of alphabetic writing may help explain why the mechanical clock was invented in Europe and had spread throughout European culture long before taking hold in the more ideographic world of the Orient. Actually, a few elaborate clocklike machines had been designed and built for the private use of Chinese emperors as early as the eleventh century, yet these were intended strictly as calendrical devices modeling the movements of the heavens—machines that would allow the emperor to _align_ his intentions and decrees more precisely with astrological events. The order of time remained inseparable from such cosmic, spatial phenomena.
In the West, on the contrary, the mechanical clock functioned to _sever_ the experience of time from the spatial cycles of the sun, moon, and stars, marking out a series of determinate intervals that paid little heed to the heavens or to the shifting lengths of daylight and darkness. Mechanical clocks originated in monasteries (the strongholds of alphabetic literacy throughout the Middle Ages), where they were used to regulate the times for prayer. But by the middle of the fourteenth century, large clocks in the belfries of churches and town halls rang the equal hours for the whole populace, regulating the daily activities of the community according to an artificially determined and unvarying measure. Because the fixed hours of the clock were ultimately independent of the sun, independent of its rising and setting and the length of the daylight (all of which might vary not just in different seasons but in different locations), clock-time could ultimately be used to regulate transactions _between_ different villages and towns, eventually establishing the sense of a wholly objective, quantitative time impervious to the particular rhythms of different locales and seasons. The voice of this objective time was the implacable "tick-tock" of the clock's internal mechanism, which lent auditory force to the Aristotelian sense of time as a countable series of discrete now-points. See Daniel Boorstin, _The Discoverers_ (New York: Random House, 1983), pp. 36–46, 56–78.
25. Quoted in Alexandre Koyré, _From the Closed World to the Infinite Universe_ (New York: Harper & Brothers, 1958), pp. 161, 162.
26. Ibid., pp. 161–62, 245.
27. Ibid., pp. 221–72.
28. Edmund Husserl, _Phenomenology of Internal Time-Consciousness_ , trans. James S. Churchill (Bloomington: Indiana University Press, 1964), pp. 104, 150. See also David Wood, _The Deconstruction of Time_ (Atlantic Highlands, N.J.: Humanities Press, 1989), pp. 106–9.
29. Martin Heidegger, _Being and Time_ , trans. John Macquarrie and Edward Robinson (Oxford: Basil Blackwell, 1967).
30. Martin Heidegger, "Time and Being," in _On Time and Being_ , trans. Joan Stambaugh (New York: Harper & Row, 1972).
31. Merleau-Ponty, _The Visible and the Invisible_ , p. 259. It is worthy of note that these words were written by Merleau-Ponty on June 1, 1960, less than a year before his death, and more than a year and a half before Heidegger's introduction of "time-space" in his January 1962 lecture "Time and Being."
32. Ibid., p. 267.
33. Ibid., p. 259.
34. Ibid., p. 13.
35. Heidegger, "Time and Being," p. 11.
36. Ibid., pp. 11–12.
37. Heidegger, _Being and Time_ , p. 416.
38. In truth, the idea of time is a thoroughly horizon-laden thought for Heidegger; in _Being and Time_ he can hardly mention the phenomenon of time in any capacity without linking it to the horizon metaphor. Thus, when explicating the genesis of our ordinary conception of time as a linear sequence, Heidegger translates Aristotle's definition of time in the following manner: "For this is time: that which is counted in the movement which we encounter within the horizon of the earlier and later" ( _Being and Time_ , p. 473). And indeed, the entire book ends with the question "Does _time_ itself manifest as the horizon of _Being?_ " ( _Being and Time_ , p. 488).
39. Heidegger, "Time and Being," p. 13.
40. Ibid., pp. 16–17.
41. Ibid., p. 17.
42. Ibid., pp. 13, 17.
43. Merleau-Ponty, _The Visible and the Invisible_ , p. 267.
44. John Bierhorst, _The Mythology of North America_ (New York: William Morrow & Co., 1985), pp. 77–92. See also Åke Hultkrantz, _Native Religions of North America_ , pp. 91–94.
45. One finds resonances throughout the Americas: "The people came 'out of the ground' (Nez Percé); 'the people grew up from the soil' (Tarahumara); 'the people came out of the hills' (Tzotzil); 'the first man emerged from the earth' (Toba)." See John Bierhorst, _The Way of the Earth_ (New York: William Morrow & Co., 1994), p. 98.
46. A Jicarilla Apache storyteller, quoted in Bierhorst, _Mythology of North America_ , 1985, p. 82.
47. Hultkrantz, pp. 91–92.
48. George B. Grinell, _Pawnee Hero Stories and Folk-tales_ (1889) (Lincoln: University of Nebraska Press, 1961), pp. 149–50.
49. Christopher Vecsey, _Imagine Ourselves Richly: Mythic Narratives of North American Indians_ (San Francisco: HarperCollins, 1991), p. 45. See also Dennis Tedlock, "An American Indian View of Death" in Dennis Tedlock and Barbara Tedlock, eds., _Teachings from the American Earth: Indian Religion and Philosophy_ (New York: Liveright, 1975), especially pp. 264–70.
50. June McCormick Collins, "The Mythological Basis for Attitudes Towards Animals Among Salish-Speaking Indians," _Journal of American Folklore_ 65, no. 258 (1952), p. 354.
51. Åke Hultkrantz, for instance, asserts that the belief among the Wind River Shoshoni that the dead must follow the Milky Way to "the land of the dead" conflicts with "another belief" according to which the dead dwell beyond the mountains (Hultkrantz, p. 59). Examined phenomenologically, however, the two beliefs are not in conflict at all, since the visible path of the Milky Way leads precisely beyond the mountains.
52. N. Scott Momaday, "Personal Reflections," in Calvin Martin, ed., _The American Indian and the Problem of History_ (New York: Oxford University Press, 1987), pp. 156–61.
53. See, for instance, John James Houston, "Songs in Stone: Animals in Inuit Sculpture," in _Orion Nature Quarterly_ 4, no. 4 (Autumn 1985), p. 8.
54. Heidegger, _Being and Time_ , p. 416.
55. Heidegger, "Time and Being," p. 15.
56. Ibid., p. 13.
## CHAPTER 7:
THE FORGETTING AND REMEMBERING OF THE AIR
1. Robert Lawlor, _Voices of the First Day: Awakening in the Aboriginal Dreamtime_ (Rochester, Vt.: Inner Traditions, 1992), p. 41.
2. See, for instance, Berndt and Berndt, pp. 73–125.
3. Lawlor, p. 42.
4. See Christopher Vecsey, _Imagine Ourselves Richly: Mythic Narratives of North American Indians_ (San Francisco: HarperCollins, 1991), chap. 7.
5. See, for instance, the words of the Lakota medicine man, Finger, recorded by Dr. James R. Walker in Tedlock and Tedlock, pp. 208–13.
6. See D. M. Dooling, ed., _The Sons of the Wind: The Sacred Stories of the Lakota_ (New York: Parabola Books, 1984), for a beautiful and carefully researched telling of the sacred Lakota stories. Precise insights into the nature of the _wakan_ beings may be gleaned from these stories, aided by the very useful glossary at the front of the book. The stories should be supplemented by the words of the old Lakota holy men—Sword, Finger, One-Star, and Tyon—recorded early in the twentieth century by Dr. James R. Walker and excerpted in chap. 13, "Oglala Metaphysics," in Tedlock and Tedlock, pp. 205–18. Dr. Walker's own essential research may be found in J. R. Walker, _The Sun Dance and Other Ceremonies of the Teton Dakota_ , Anthropological Papers of the American Museum of Natural History 16 (1917); and in Elaine Jahner, _Lakota Myth_ (Lincoln: University of Nebraska Press, 1983).
7. The peace pipe was given to the Lakotas by White Buffalo Woman as a gift from the buffalo, whose sacred breath is also visible when it is seen on a cold day. Yet it is not only animals and plants that are assumed to breathe and to partake of the air: in the _inipi_ , or sweat lodge, ceremony, water is poured on the red-hot rocks to release the living breath of the rocks themselves: "You pray to the Great Spirit, to the sacred rocks, the _tunka_ , the _inyan_. They have no mouth, no eyes, no arms or legs, but they exhale the breath of life." From John Fire Lame Deer and Richard Erdoes, _Lame Deer, Seeker of Visions_ (New York: Simon & Schuster, 1972), p. 180.
8. John Fire Lame Deer, in Lame Deer and Erdoes, p. 119.
9. See Tedlock and Tedlock, pp. 217–18.
10. Ibid., p. 218.
11. Ibid., p. 218.
12. James Kale McNeley, _Holy Wind in Navajo Philosophy_ (Tucson: University of Arizona Press, 1981), p. 1. This book is the fruit of twenty years of association with the Navajo. McNeley is married to a Diné woman, and the two of them teach on the Navajo Reservation in Arizona. Although the Navajo commonly refer to themselves as "Diné"—the People—I have mostly used the more familiar term "Navajo," for convenience' sake, in this work.
13. Ibid., p. 2.
14. A Navajo singer and healer quoted in McNeley, pp. 9–10. Most of the elders interviewed by McNeley requested that their identities remain unpublished.
15. Ibid., pp. 16, 21.
16. Ibid., pp. 14–31.
17. Ibid., pp. 23, 33–34.
18. Ibid., pp. 34–35.
19. Ibid., p. 35.
20. Ibid., p. 35.
21. Ibid., p. 35. Emphasis added.
22. Ibid., p. 36.
23. Ibid., pp. 11, 36–37.
24. Ibid., p. 36.
25. Ibid., p. 24.
26. Ibid., p. 24.
27. Ibid., pp. 37–38.
28. Gary Witherspoon, _Language and Art in the Navajo Universe_ (Ann Arbor: University of Michigan Press, 1977).
29. Witherspoon, p. 31; McNeley, p. 57.
30. Witherspoon, p. 61.
31. Leland C. Wyman, _Blessingway_ (Tucson: University of Arizona Press, 1970), p. 616. These words are translated from River Junction Curly's version of the Blessingway.
32. C. T. Onions, ed., _The Oxford Dictionary of English Etymology_ (Oxford: Clarendon Press, 1966), p. 720.
33. Ibid., p. 691.
34. Eric Partridge, _Origins: A Short Etymological Dictionary of Modern English_ (London: Routledge & Kegan Paul, 1958), pp. 651–52.
35. Onions, p. 38; Partridge, p. 18.
36. Here is how the British linguist and historian Owen Barfield addressed these curious evidences embedded in our words:
such a purely material content as "wind"... and... such a purely abstract content as "the principle of life within man or animal" are both _late_ arrivals in human consciousness. Their abstractness and their simplicity are alike evidence of long ages of intellectual evolution. So far from the psychic meaning of "spiritus" having arisen because someone had the idea, "principle of life..." and wanted a word for it, the abstract idea "principle of life" is itself a _product_ of the old concrete meaning of "spiritus," which contained within itself the germs of both later significations. We must, therefore, imagine a time when "spiritus" or "pneuma," or older words from which these had descended, meant neither _breath_ , nor _wind_ , nor _spirit_ , nor yet all three of these things, but when they simply had their own peculiar meaning, which has since, in the course of the evolution of consciousness, crystallized into the three meanings specified....
See Owen Barfield, _Saving the Appearances_ (Middletown, Conn., Wesleyan University Press, 1965), pp. 80–81.
37. _Tanakh: The Holy Scriptures_ (Philadelphia: Jewish Publication Society, 1985), Genesis 1:2. This is the most authoritative English translation of the Hebrew Bible from the traditional Hebrew text. The traditional Hebrew name for the Bible, _Tanakh_ , is an acronym formed by the first letters of the three sections of the Hebrew text: _T_ orah (Instruction), _N_ evi'im (Prophets), and _K_ ethuvim (Writings). Although the relevant phrase in the first sentence of the Torah is commonly translated into English as "the spirit of God," "a wind from God" is actually a more direct rendering of the original Hebrew.
38. McNeley, p. 10.
39. Genesis 2:7. Just as the Hebrew term for human _(adam)_ relates directly to the word for earth _(adamah)_ , so also the English term "human" relates directly to the word "humus"—the earth or soil. Thus, both the Hebrew _adam_ and the English "human" can be precisely translated as "earthling," or "earthborn one."
40. This is not to suggest that all of the ancient Hebrews were able, or even allowed, to read—far from it. Yet to the extent that they took the written commandments as their supreme laws, and to the extent that the story about receiving those scribed commandments, at Mount Sinai, became their foundational story, _every_ Hebrew life was structured in accordance with Scripture—with writing—whether the individual was literate or not.
41. An excellent analysis of the extent to which the lack of vowel letters in the Hebrew writing system can or cannot be thoroughly explained by the structure of the Hebrew language is found in Geoffrey Sampson's masterful text, _Writing Systems: A Linguistic Introduction_ (Stanford: Stanford University Press, 1985), pp. 77–98. Sampson's analysis shows that even a reader fluent in Hebrew encounters a relatively high degree of ambiguity when reading a traditional text without vowel marks; it is this ambiguity that forces the reader of Hebrew to actively grapple with conflicting meanings, conflicting ways of sounding the text.
42. While Hebrew words that share the same group of consonants tend to have a related meaning, meanings can still change drastically with different vowel sounds. For instance, while the word "TSaHaK" means "sexual intercourse," the word "TSaHoK" means "laughter." Or consider the Hebrew words "D R" (a stable), "DāR" (mother-of-pearl), "DōR" (generation), "DūR" (ruin), "DōR" (to dwell).
43. Barry W. Holtz, ed., _Back to the Sources: Reading the Classic Jewish Texts_ (New York: Summit Books, 1984), pp. 16–17.
44. See, for instance, Lawrence Kushner, _The Book of Letters: A Mystical Alef-bait_ (New York: Harper & Row, 1975). Also Rabbi Michael L. Munk, _The Wisdom in the Hebrew Alphabet: The Sacred Letters as a Guide to Jewish Deed and Thought_ (New York: Mesorah Publications, 1983). See also Aryeh Kaplan, _Sefer Yetzirah: The Book of Creation_.
45. Moshe Idel, _Kabbalah: New Perspectives_ (New Haven: Yale University Press, 1988), pp. 234–37.
46. Perle Epstein, _Kaballah: The Way of the Jewish Mystic_ (Boston: Shambhala, 1988), pp. 98–99.
47. The voluminous research on Kabbalah conducted by the great twentieth-century scholar Gershom Scholem has led many to believe that the Kabbalah was something of an anomaly within traditional Judaism, sparked by non-Jewish influences, like Gnosticism, which ostensibly infiltrated Jewish circles early on and combined with other, Neoplatonic influences during the Middle Ages. However, more recent scholarship—particularly the extensive and ongoing research of the brilliant Israeli scholar Moshe Idel—has called into question some of Scholem's assumptions, and has begun to suggest the profoundly endemic relation of Kabbalah to the very core of ancient and medieval Judaism. Idel's carefully reasoned scholarship suggests that many Kabbalistic beliefs and practices were preserved and transmitted orally long before being written down, and that the fragmentary written teachings that first surfaced during the twelfth century were expressions of a coherent tradition of esoteric Jewish praxis that likely extended back to the archaic origins of Judaism itself. See especially Idel, _Kabbalah_ , particularly chaps. 2, 5, and 7. See also Gershom Scholem, _Major Trends in Jewish Mysticism_ (New York: Schocken Books, 1961).
48. Idel, pp. 97–103; Epstein, pp. 93–94.
49. Idel, p. 100; Epstein, p. 88; Gershom Scholem, _Kabbalah_ (New York: New American Library, 1974), pp. 351–55.
50. Quoted in Epstein, pp. 59–60. The _Zohar_ was almost certainly written in the latter half of the thirteenth century by Moses de León of Guadalajara. De León himself, however, ascribed authorship to the second-century sage who figures as the central character in the text, Rabbi Shim'on bar Yohai.
51. Quoted in Epstein, p. 66.
52. Daniel Chanan Matt, ed. and trans., _Zohar, The Book of Enlightenment_ (New York: Paulist Press, 1983), pp. 60–62. In the _Zohar_ the soul-breath, or _neshamah_ , also has intermediary aspects resembling the Messenger Winds of the Navajo, as is evident from this quote: "The _neshamah_ of a human being... leaves him every single night. In the morning she returns to him and dwells in his nostrils." See p. 219n.
53. Arthur Green and Barry W. Holtz, eds., _Your Word Is Fire: The Hasidic Masters on Contemplative Prayer_ (New York: Schocken Books, 1987), p. 48.
54. Shneur Zalman of Ladi, "The Portal of Unity and Faith," in _An Anthology of Jewish Mysticism_ , trans. Raphael Ben Zion (New York: Judaica Press, 1981), pp. 83–128. For a book-length commentary on this important text, see Adin Steinsaltz, _The Sustaining Utterance_ , trans. Yehuda Hanegbi (London: Jason Aronson, 1974).
55. Green and Holtz, p. 43.
56. Even in the written narratives of the Bible, YHWH typically manifests himself in atmospheric phenomena, from the rains that flood the earth for forty days in Genesis, to the tumultuous whirlwind that addresses Job in the later writings. In the pivotal theophany atop Mount Sinai, YHWH displays himself to the assembled tribes as a storm cloud, thundering and lightning, and it is as a cloud that YHWH accompanies the Israelites in their subsequent wanderings through the desert.
57. The rest of Europe inherited these Greek innovations only by way of the Romans, who modified the Greek shapes into the capital letter forms used today throughout Western Europe and the Americas. See David Diringer's excellent, if somewhat dated, overview, _The Alphabet: A Key to the History of Mankind_ (New York: Philosophical Library, 1953).
58. Plato provides a remarkably precise description of this new situation when he has Socrates state, in the _Phaedrus_ , that written words "seem to talk to you as though they were intelligent, but if you ask them anything about what they say, from a desire to be instructed, they go on telling you just the same thing forever" ( _Phaedrus_ , 275d). Socrates' description clearly indicates the apparent autonomy of Greek texts, yet at the same time makes evident the monotonous, almost mechanical efficiency of the new alphabet. A Hebrew reader could never claim that a traditional text "goes on telling you just the same thing forever," for the simple reason that the consonantal text may subtly vary its words, and hence its meanings, each time that the reader engages it!
59. Philip Wheelwright, ed., _The Pre-Socratics_ (New York: Macmillan Publishing Co., 1985), pp. 60, 288. Anaximenes, it is reported, also claimed that the air was the immortal and ever-moving source of all phenomena; that even the gods themselves were born of the air! See Wheelwright, pp. 61–63.
60. _Phaedrus_ , 250c.
61. The explicit fusion of Christian theology with Platonic philosophy was accomplished by the early Church theologians—first by Justin Martyr; later by Clement of Alexandria and Origen; finally, and most profoundly, by Augustine. For an accessible and engaging discussion of Christianity's alliance with Greek philosophy, see Richard Tarnas's sweeping work _The Passion of the Western Mind_ (New York: Ballantine Books, 1991).
62. Thus it was that two decades ago a careful scientific study of the atmosphere, using new, highly sensitive instruments, yielded a new astonishment at the anomalous chemical makeup of the medium. The chemical composition of the earthly atmosphere was very far from any stable equilibrium, and yet, remarkably, this composition seemed to be actively and quite sensitively maintained by some unknown and enigmatic set of processes. This disclosure led several scientists to hypothesize that the composition of the atmosphere was being actively monitored and modulated by all of the earth's organic constituents acting collectively, as a vast, planetary metabolism. The Gaia hypothesis—named for the ageless mother of the gods in the oral mythology of ancient Greece—proposed that the earthly world in which we find ourselves must be reconceptualized as a living entity.
Whatever the scientific fate of the Gaia hypothesis, its emergence provides a striking illustration of the way in which a renewed awareness of the air forces us to recognize, ever more vividly, our interdependence with the countless organisms that surround us, and ultimately encourages us to speak of the encompassing earth in the manner of our oral ancestors, as an animate, living presence. See David Abram, "The Perceptual Implications of Gaia" in _The Ecologist_ (Summer 1985); reprinted in _Dharma Gaia: A Harvest of Essays in Buddhism and Ecology_ , edited by A. H. Badiner (San Francisco: Parallax Press, 1990). Also see Stephen Schneider and Penelope Boston, eds., _Scientists on Gaia_ (Cambridge: M.I.T. Press, 1991).
## CODA: TURNING INSIDE OUT
1. Paul S. Martin, "40,000 Years of Extinction on the 'Planet of Doom,' " in _Paleogeography, Paleoclimatology, Paleoecology_ 82 (1990), pp. 187–201. See also Paul Martin and Richard Klein, eds., _Quaternary Extinctions_ (Tucson: University of Arizona Press, 1984).
2. In contrast to a long-standing tendency of Western social science, this work has not attempted to provide a rational explanation of animistic beliefs and practices. On the contrary, it has presented an animistic or participatory account of rationality. It has suggested that civilized reason is sustained only by a deeply animistic engagement with our own signs. To tell the story in this manner—to provide an animistic account of reason, rather than the other way around—is to imply that animism is the wider and more inclusive term, and that oral, mimetic modes of experience still underlie, and support, all our literate and technological modes of reflection. When reflection's rootedness in such bodily, participatory modes of experience is entirely unacknowledged or unconscious, reflective reason becomes dysfunctional, unintentionally destroying the corporeal, sensuous world that sustains it.
# **_Bibliography_**
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# **_Permissions Acknowledgments_**
#
Chapter One, "The Ecology of Magic," was originally published in somewhat different form in _Orion Magazine_.
_Grateful acknowledgment is made to the following for permission to reprint previously published material:_
_The Ecco Press_ : Excerpt from "From March '79" by Tomas Transtromer and translated by John F. Deane, from _Selected Poems 1954–1986_ , edited by Robert Hass. Copyright © 1987 by Tomas Transtromer and John F. Deane. Published by The Ecco Press in 1987. Excerpt from "Stalking with Stories" by Keith Basso from _Antaeus: On Nature_ , edited by Daniel Halpern, copyright © 1986 by _Antaeus_ , New York. First published by The Ecco Press in 1986. Reprinted by permission of The Ecco Press.
_ETT Imprint_ : Excerpts from "Passage" and "Paddy: A Poem for Land Rights" from _Singing the Snake_ by Billy Marshall-Stoneking (Angus & Robertson, Pymble, Australia, 1990). Reprinted by permission of ETT Imprint.
_Barry W. Holtz_ : Excerpts from _Your Word Is Fire_ , edited and translated by Arthur Green and Barry W. Holtz (Schocken Books, New York, 1987). Reprinted by permission of Barry W. Holtz.
_Elizabeth S. Lamb_ : Excerpt from _Wizard of the Upper Amazon_ by F. Bruce Lamb, copyright © 1971, 1974 by F. Bruce Lamb. Reprinted by permission of Elizabeth S. Lamb, owner of copyright.
_Pantheon Books_ : Excerpt from _No Nature: New and Selected Poems_ by Gary Snyder. Copyright © 1992 by Gary Snyder. Reprinted by permission of Pantheon Books, a division of Random House, Inc.
_Penguin Books Australia Ltd_ : Excerpt from _The Speaking Land by_ Ronald M. Berndt and Catherine H. Berndt (Penguin Books, Ringwood, Australia, 1989). Reprinted by permission of Penguin Books Australia Ltd.
_Routledge_ : Excerpts from _Phenomenology of Perception_ by Maurice Merleau-Ponty, translated by Colin Smith (Routledge & Kegan Paul, London, 1962). Reprinted by permission of Routledge.
_The University of Arizona Press_ : Excerpts from _Holy Wind in Navajo Philosophy_ by James Kale McNeley (The University of Arizona Press, Tucson, 1981). Reprinted by permission of The University of Arizona Press.
_University of California Press_ : Excerpts from _Smoothing the Ground: Essays on Native American Oral Culture_ by Brian Swann, copyright © 1983 by The Regents of the University of California. Reprinted by permission of the University of California Press.
_The University of Chicago Press_ : Excerpts from _Make Prayers to the Raven_ by Richard Nelson (The University of Chicago Press, Chicago, 1983). Reprinted by permission of The University of Chicago Press.
_Viking Penguin and Jonathan Cape_ : Excerpts from _The Songlines_ by Bruce Chatwin, copyright © 1987 by Bruce Chatwin. Rights outside the United States administered on behalf of the Estate of Bruce Chatwin by Jonathan Cape, London. Reprinted by permission of Viking Penguin, a division of Penguin Books USA Inc., and Jonathan Cape.
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,870 |
package EnsEMBL::Web::XS::Test;
use 5.014002;
use strict;
use warnings;
require Exporter;
our @ISA = qw(Exporter);
# Items to export into callers namespace by default. Note: do not export
# names by default without a very good reason. Use EXPORT_OK instead.
# Do not simply export all your public functions/methods/constants.
# This allows declaration use EnsEMBL::Web::XS::Test ':all';
# If you do not need this, moving things directly into @EXPORT or @EXPORT_OK
# will save memory.
our %EXPORT_TAGS = ( 'all' => [ qw(
hello_planet
) ] );
our @EXPORT_OK = ( @{ $EXPORT_TAGS{'all'} } );
our @EXPORT = qw(
);
our $VERSION = '0.01';
require XSLoader;
XSLoader::load('EnsEMBL::Web::XS::Test', $VERSION);
# Preloaded methods go here.
1;
__END__
=head1 NAME
EnsEMBL::Web::XS::Test - Example XS extension
=head1 SYNOPSIS
use EnsEMBL::Web::XS::Test;
hello_planet("World");
hello_planet();
=head1 DESCRIPTION
This is a test XS library for use when debugging XS code and writing new
XS code. Its methods do nothing useful for EnsEMBL beyond assisting
writing other modules and must not be called in production code.
Note that while this is a separate library so that it is not included
in production systems, it is generally a good idea to bundle our own stuff
into a single library to ease deployment and reduce time coding wrappers.
=head2 EXPORT
None by default.
=head1 AUTHOR
EnsEMBL <helpdesk@ensembl.org>
=head1 COPYRIGHT AND LICENSE
Covered by main EnsEMBL licence.
=cut
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,357 |
\section*{Introduction}
The generalization of the results of Bost and Connes \cite{bos-con}
to general number fields has received significant attention for more
than ten years, but was only recently formulated in detail as an
explicit problem, see \cite[Problem 1.1]{cmn1}. We paraphrase here
this formulation for easy reference:
{\em Given an algebraic number field $K$, construct a C$^*$-dynamical
system $(\mathcal A, \sigma)$ such that
\begin{itemize}
\item[(i)] the partition function of the system is the Dedekind zeta
function of $K$; \item[(ii)] the quotient of the idele class group
$C_K$ by the connected component $D_K$ of the identity acts as
symmetries of the system; \item[(iii)] for each inverse temperature
$0<\beta \leq 1$ there is a unique KMS$_\beta$-state; \item[(iv)]
for each $\beta >1$ the action of the symmetry group $C_K/D_K$ on
the extremal KMS$_\beta$-states is free and transitive; \item[(v)]
there is a $K$-subalgebra $\mathcal A_0$ of $\mathcal A$ such that
the values of the extremal KMS$_\infty$-states on elements of
$\mathcal A_0$ are algebraic numbers that generate the maximal
abelian extension $\kab$ of $K$; and \item[(vi)] the Galois action
of $\gal(\kab/K)$ on these values is realized by the action of $C_K
/ D_K$ on the extremal KMS$_\infty$-states via the class field
theory isomorphism $s\colon C_K / D_K \to \gal(\kab/K)$.
\end{itemize}
}
Systems with properties (i)-(iv) have been constructed by several
authors for various classes of number fields, see \cite[Section
1.4]{con-mar} for a discussion of these constructions and an
extensive list of references. However, the last two properties have
proven quite elusive. This should not come as a surprise, since a
system satisfying (v) and (vi) has the potential to shed light onto
Hilbert's $12$th problem about the explicit class field theory of
$K$, although this will ultimately depend on the specific
expressions obtained for the extremal KMS$_\infty$-states and the
generators of the subalgebra~$\mathcal A_0$. Since imaginary
quadratic fields are the only fields beyond $\mathbb Q$ for which explicit
class field theory is completely understood, it is natural that they
should be the first case to be solved, and indeed, Connes, Marcolli,
and Ramachandran have produced a complete solution of the problem
for these fields, see \cite[Theorem 3.1]{cmn1}. It should be noticed
also that properties (v) and (vi) are intrinsically related so that
the `right' Galois symmetries and the `right' arithmetic subalgebra
must match each other for the system to have genuine class field
theory content. This failed for instance in the system constructed
in \cite{lvf}, where it was natural to include certain cyclotomic
elements in the arithmetic subalgebra $\mathcal A_0$, but the Galois
action on the corresponding values of extremal KMS$_\infty$ states
did not match the symmetry action of the idele class group on these
elements, see~\cite[Theorem 4.4]{lvf}.
In this paper we study a generalization of the system from
\cite{cmn1} to all algebraic number fields; this construction is
also isomorphic to a particular case, for Shimura data arising from
a number field, of a general construction due to Ha and Paugam
\cite{ha-pa}.
In Section~\ref{sKMS} we show how to reduce the study of KMS states
of certain restricted groupoid C$^*$-algebras to measures of the
(unrestricted) transformation group $(G,X)$ satisfying a scaling
condition. Similar results are well known when $G$ acts freely,
see~\cite{ren}; the key result in this section allows a certain
degree of non-freeness and is motivated by our considerations
in~\cite{finitecm}. We remark that for our applications in
Section~\ref{sBC} we could use instead the earlier results
from~\cite{lac} on crossed products by lattice semigroups. However,
our present results can be applied to a wider class of systems, e.g.
those studied in \cite{lvf} for fields of class number bigger than
one.
Under further assumptions, in \proref{phasetransition}, we give a
natural parametrization of the extremal KMS${}_\beta$-states in
terms of a specific subset of the space~$X$. In
Proposition~\ref{ground} we prove a similar result for ground
states.
In Section~\ref{sBC} we discuss the dynamical system $(\mathcal A,
\sigma)$. We initially construct the C$^*$-algebra $\mathcal A$
along the lines of \cite{con-mar,ha-pa}, using the restricted
groupoid obtained from an action of the group of fractional ideals
on the cartesian product of $\gal(\kab/K) $ by the finite adeles,
balanced over the integral ideles. Since we choose to incorporate
the class field theory isomorphism in the construction, the symmetry
group of the system is $\gal(\kab/K)$ itself. We also indicate that
$\mathcal A$ is a semigroup crossed product of the type discussed in
\cite{lac}, and that, for totally imaginary fields of class number
one, the resulting system is isomorphic to the one constructed in
\cite{lvf} using Hecke algebras.
Using the results of Section~\ref{sKMS} we show in \thmref{mainthm}
that for an arbitrary number field $K$ the system $(\mathcal A,
\sigma)$ satisfies properties (i) through (iv) above. The
description of the symmetry action and classification of the
KMS${}_\beta$-states generalize the corresponding results of
\cite{cmn1} and complete the initial results of \cite[Section
6]{ha-pa}. The argument goes along familiar lines~\cite{lac,nes},
but we include a complete proof to make the paper self-contained.
Finally, we introduce in Section~\ref{sKlat} a notion of
$n$-dimensional $K$-lattices, which generalizes the $n$-dimensional
$\mathbb Q$-lattices from \cite{con-mar} and the $1$-dimensional
$K$-lattices for imaginary quadratic fields $K$ from \cite{cmn1}.
After discussing some of their basic properties we show in
\corref{Klatscal} how $1$-dimensional $K$-lattices are related to
the systems of Section~\ref{sBC}. Therefore these systems can be
introduced without using any class field theory data. This may turn
out to be useful in verifying properties (v) and (vi), as was the
case for $\mathbb Q$ and imaginary quadratic fields~\cite{con-mar,cmn1}.
\bigskip
\section{KMS states and measures}\label{sKMS}
Throughout this section we suppose that $G$ is a countable discrete
group acting on a second countable, locally compact, Hausdorff
topological space $X$ and that $Y$ is a clopen subset of $X$
satisfying $G Y = X$. The C$^*$-algebra $C_0(X)\rtimes_r G$ is the
reduced C$^*$-algebra of the transformation groupoid $G\times X$.
Consider the subgroupoid
$$
G\boxtimes Y=\{(g,x)\mid x\in Y,\ gx\in Y\}
$$
and denote by $C^*_r(G\boxtimes Y)$ its reduced C$^*$-algebra. In
other words, $C^*_r(G\boxtimes Y)=\31_Y(C_0(X)\rtimes_r G) \31_Y$,
where $\31_Y$ is the characteristic function of $Y$.
We endow $C^*_r(G\boxtimes Y)$ with the dynamics $\sigma$ associated
to a given homomorphism $N\colon G \to (0,+\infty)$, so
$$
\sigma_t(f)(g,x)=N(g)^{it}f(g,x)
$$
for $t\in\mathbb R$ and $f\in C_c(G\boxtimes Y)\subset C^*_r(G\boxtimes
Y)$. Recall that a KMS state for $\sigma$ at inverse temperature
$\beta\in\mathbb R$, or $\sigma$-KMS${}_\beta$-state, is a
$\sigma$-invariant state~$\varphi$ such that $\varphi(ab) =
\varphi(b\sigma_{i\beta}(a))$ for $a$ and $b$ in a set of
$\sigma$-analytic elements with dense linear span.
Denote by $E$ the usual conditional expectation from $C_0(X)
\rtimes_r G$ onto $C_0(X)$: thus with $u_g$ denoting the element in
the multiplier algebra $M(C_0(X)\rtimes_r G)$ corresponding to $g\in
G$, we have $E(fu_g)=f$ if $g=e$ and $0$ otherwise. Observe that the
image under $E$ of the corner $C_r^*(G\boxtimes Y)$ is $C_0(Y)$. By
restriction to $C_0(Y)$, a state~$\varphi$ on $C_r^*(G\boxtimes Y)$
gives rise to a Radon probability measure $\mu$, and conversely, a
Radon probability measure on $Y$ can be extended via the conditional
expectation to a state on $C_r^*(G\boxtimes Y)$. Clearly $(\mu_*
\circ E)|_{C_0(Y)} = \mu_*$, but in general it is not true that
$(\varphi |_{C_0(Y)}) \circ E$ will be the same as~$\varphi$. We
will show that, under certain combined assumptions on $N$ and the
action of $G$ on $X$, the $\sigma$-KMS$_\beta$-states do arise from
their restrictions to $C_0(Y)$, and are thus in one-to-one
correspondence with a class of measures on~$Y$ characterized by a
scaling condition.
\begin{proposition}\label{KMSmeasures}
Under the general assumptions on $G$, $X$, $Y$ and $N$ listed above,
suppose there exist a sequence $\{Y_n\}^\infty_{n=1}$ of Borel
subsets of $Y$ and a sequence $\{g_n\}^\infty_{n=1}$ of elements of
$G$ such that \enu{i} $\cup^\infty_{n=1} Y_n $ contains the set of
points in $Y$ with nontrivial isotropy; \enu{ii} $N(g_n) \neq 1$ for
all $n\ge1$; \enu{iii} $g_n Y_n = Y_n$ for all $n\ge1$.
\smallskip
Then for each $\beta\ne0$ the map $\mu \mapsto \varphi =(\mu_* \circ
E)|_{C_r^*(G\boxtimes Y)}$ is an affine isomorphism between Radon
measures $\mu$ on $X$ satisfying $\mu(Y) = 1$ and the scaling
condition
\begin{equation}\label{normalrescale}
\mu(gZ) = N(g)^{-\beta} \mu(Z)
\end{equation}
for Borel $Z\subset X$,
and $\sigma$-KMS$_\beta$-states $\varphi$ on $C_r^*(G\boxtimes Y)$.
\end{proposition}
\begin{proof}
It is straightforward to check that any measure satisfying the
scaling condition extends via~$E$ to a KMS$_\beta$-state.
Conversely, let $\varphi$ be a KMS$_\beta$-state. Denote by $\mu$
the probability measure on $Y$ defined by $\varphi|_{C_0(Y)}$.
Applying the KMS-condition to elements of the form
$u_gfu_g^*=f(g^{-1}\,\cdot)$, it is easy to see that
(\ref{normalrescale}) is satisfied for Borel $Z\subset Y$ such that
$gZ\subset Y$. In particular, $\mu(Y_n)=0$ by conditions (ii) and
(iii).
To show that $\varphi=\mu_*\circ E$ fix $g\ne e$. Let $f\in C_c(Y)$
be such that $g^{-1}(\operatorname{supp} f)\subset Y$. Then $fu_g\in
C_r^*(G\boxtimes Y)$ and we have to prove that $\varphi(fu_g)=0$.
Denote by $Y_g$ the set of points of $Y$ left invariant by $g$. If
$\operatorname{supp} f\cap Y_g=\emptyset$ then we can write $f$ as a finite sum of
functions $h_1h_2$ such that $g(\operatorname{supp} h_1)\cap\operatorname{supp} h_2=\emptyset$.
By the KMS-condition we have
$$
\varphi(h_1h_2u_g)=\varphi(h_2u_gh_1)=\varphi(h_2h_1(g^{-1}\cdot)u_g)=0.
$$
Therefore $\varphi(fu_g)=0$.
Assume now that $\operatorname{supp} f\cap Y_g\ne\emptyset$. As $\mu(Y_n)=0$, by
condition (i) we get $\mu(Y_g)=0$. Hence there exists a norm-bounded
sequence $\{f_n\}_n\subset C_c(Y)$ such that $\operatorname{supp} f_n \cap
Y_g=\emptyset$, $g^{-1}(\operatorname{supp} f_n)\subset Y$ and $f_n\to f$ in
measure $\mu$. Then $\varphi(f_nu_g)=0$. On the other hand, by the
Cauchy-Schwarz inequality,
$$
|\varphi(fu_g)-\varphi(f_n u_g)|\le\varphi(|f-f_n|)^{1/2}
\varphi(u^*_g|f-f_n|u_g)^{1/2}
\le\|f-f_n\|^{1/2}\varphi(|f-f_n|)^{1/2},
$$
whence $\varphi(fu_g)=0$. Therefore $\varphi=\mu_*\circ E$.
To finish the proof it remains to note that the measure $\mu$
extends uniquely to a measure on $X$, which we still denote by
$\mu$, such that (\ref{normalrescale}) is satisfied for all $g\in G$
and Borel $Z\subset X$. Explicitly, we can write
$$
\mu(Z)=\sum_iN(h_i)^\beta\mu(h_iZ\cap Z_i),
$$
where $h_i\in G$ and $Z_i\subset Y$ are such that $X$ is the
disjoint union of the sets $h^{-1}_iZ_i$, see
\cite[Lemma~2.2]{cmgl2}.
\end{proof}
Our next goal is to classify measures satisfying the scaling
condition. The classification depends on convergence of certain
Dirichlet series. More precisely, when $S$ is a subset of $G$ the
zeta function associated to $S$ is defined to be
\[
\zeta_S(\beta) := \sum_{s\in S} N(s)^{-\beta} .
\]
\begin{proposition}\label{phasetransition}
Assume the hypotheses of \proref{KMSmeasures}. Let $\beta\ne0$, $S$
be a subset of~$G$, and $Y_0 \subset Y$ a nonempty Borel set such
that \enu{i} $gY_0 \cap Y_0 = \emptyset$ for $g\in G \setminus
\{e\}$; \enu{ii} $SY_0 \subset Y$;\enu{iii} if $gY_0\cap
Y\ne\emptyset$ then $g\in S$; \enu{iv} $Y\setminus S
U\subset\cup_nY_n$ for every open set $U$ containing $Y_0$; \enu{v}
$\zeta_S(\beta)<\infty$.
\smallskip
Then \enu{1} the map $\varphi=\mu_*\circ
E\mapsto\zeta_S(\beta)\mu|_{Y_0}$ is an affine isomorphism between
the $\sigma$-KMS$_\beta$-states on $C^*_r(G\boxtimes Y)$ and the
Borel probability measures on $Y_0$; the inverse map is given by
$\nu\mapsto\mu_*\circ E$, where the measure $\mu$ on~$Y$ is defined
by
\begin{equation} \label{eextension}
\mu(Z)=\zeta_S(\beta)^{-1}\sum_{s\in S}N(s)^{-\beta}\nu(s^{-1}Z\cap Y_0);
\end{equation}
\enu{2} if $\mu$ is the measure on $Y$ defined by a probability
measure $\nu$ on $Y_0$ by \eqref{eextension}, and $H_S$ is the
subspace of $L^2(Y, d \mu)$ consisting of functions $f$ such that
$f(sy)=f(y)$ for $y\in Y_0$ and $s\in S$, then for $f\in H_S$ we
have
\begin{equation} \label{enorm}
\|f\|^2_2=\zeta_S(\beta)\int_{Y_0}|f(y)|^2d\mu(y);
\end{equation}
furthermore, the
orthogonal projection $P\colon L^2(Y,d\mu)\to H_S$ is given by
\begin{equation} \label{epro0}
Pf|_{Sy}=\zeta_S(\beta)^{-1}\sum_{s\in S}N(s)^{-\beta}f(sy)\ \
\text{for}\ \ y\in Y_0.
\end{equation}
\end{proposition}
\bp By \proref{KMSmeasures} any KMS$_\beta$-state is determined by a
Radon measure $\mu$ such that $\mu(Y)=1$ and $\mu$ satisfies the
scaling condition \eqref{normalrescale}. By assumptions (i) and
(ii), for such a measure $\mu$ we have
$$
1\ge\mu(SY_0)=\sum_{s\in
S}N(s)^{-\beta}\mu(Y_0)=\zeta_S(\beta)\mu(Y_0).
$$
On the other hand, as $\mu(Y_n)=0$, by assumption (iv) we have
$$
1=\mu(Y)\le\mu(SU)\le\sum_{s\in S}\mu(sU)=\zeta_S(\beta)\mu(U)
$$
for any open set $U$ containing $Y_0$. By regularity of the measure
we conclude that $\zeta_S(\beta)\mu(Y_0)\ge1$, and hence
$\zeta_S(\beta)\mu(Y_0)=1$. It follows that $SY_0$ is a subset of
$Y$ of full measure. Since $\mu$ satisfies the scaling condition, we
conclude that $\mu$ is completely determined by its restriction to
$Y_0$.
To finish the proof of (1) we have to construct the inverse map. Let
$\nu$ be a Borel measure on $Y_0$ with $\nu(Y_0)=1$. Similarly to
the proof of Proposition~\ref{KMSmeasures} define a measure $\mu$ on
$X$ by
$$
\mu(Z)=\zeta_S(\beta)^{-1}\sum_{g\in G}N(g)^\beta\nu(gZ\cap Y_0)\ \
\hbox{for Borel}\ \ Z\subset X.
$$
Then $\zeta_S(\beta)\mu$ extends $\nu$ by assumption (i) and
satisfies \eqref{normalrescale}. Furthermore, by assumptions (ii)
and (iii) we have $gY\cap Y_0\ne\emptyset$ if and only if $g^{-1}\in
S$, and in the latter case $Y_0\subset gY$. It follows that for
$Z\subset Y$ we have \eqref{eextension}. In particular,
$\mu(Y)=\zeta_S(\beta)^{-1}\sum_{s\in S}N(s)^{-\beta}\nu(Y_0)=1$.
\smallskip
Turning to the proof of (2), suppose $\mu$ is the measure on $Y$
defined by a probability measure $\nu$ on $Y_0$ by
\eqref{eextension} and recall that we have already shown that
$SY_0$ is a subset of $Y$ of full $\mu$-measure. Then (2) is a
particular case of \cite[Lemma~2.9]{cmgl2}. For the reader's
convenience we sketch a proof.
Equality \eqref{enorm} follows from the identity
$$
\int_{sY_0}|f|^2d\mu(y)=N(s)^{-\beta}\int_{Y_0}|f(s\,\cdot)|^2d\mu(y),
$$
valid for $f\in L^2(Y,d\mu)$, on summing over $s\in S$. Furthermore,
as
$$
\sum_{s\in S}N(s)^{-\beta}|f(sy)|^2\ge\zeta_S(\beta)
\left|\zeta_S(\beta)^{-1}\sum_{s\in S}N(s)^{-\beta}f(sy)\right|^2
$$
the above identity and \eqref{enorm} show that the operator $T$ on
$L^2(Y,d\mu)$ defined by the right hand side of \eqref{epro0} is a
contraction. Since $Tf=f$ for $f\in H_S$, and the image of~$T$
is~$H_S$, we conclude that $T=P$. \ep
In our applications the set $S$ will be a subsemigroup of $\{g\in
G\mid N(g)\ge1\}$ and $Y_0$ the complement of the union of the sets
$gY$, $g\in S\setminus\{e\}$.
\smallskip
We next give a similar classification of ground states. Recall that
a $\sigma$-invariant state $\varphi$ is called a ground state if the
holomorphic function $z\mapsto\varphi(a\sigma_z(b))$ is bounded on
the upper half-plane for $a$ and $b$ in a set of $\sigma$-analytic
elements spanning a dense subspace. If a state $\varphi$ is a
weak$^*$ limit point of a sequence of states $\{\varphi_n\}_n$ such
that $\varphi_n$ is a $\sigma$-KMS$_{\beta_n}$-state and
$\beta_n\to+\infty$ as $n\to\infty$, then $\varphi$ is a ground
state. Such ground states are called
$\sigma$-KMS$_\infty$-states~\cite{con-mar}.
\begin{proposition}\label{ground}
Under the general assumptions on $G$, $X$, $Y$ and $N$ listed before
Proposition~\ref{KMSmeasures}, define $Y_0=Y\setminus\cup_{\{g\colon
N(g)>1\}}gY$. Assume $Y_0$ has the property that if $gY_0\cap
Y_0\ne\emptyset$ for some $g\in G$ then $g=e$. Then the map
$\mu\mapsto\mu_*\circ E$ is an affine isomorphism between the Borel
probability measures on~$Y$ supported on $Y_0$ and the ground states
on $C^*_r(G\boxtimes Y)$.
\end{proposition}
\bp Assume first that $\mu$ is a probability measure on $Y$
supported on $Y_0$, $\varphi=\mu_*\circ E$. If $a=f_1u_g$ and
$b=f_2u_h$ with $g^{-1}(\operatorname{supp} f_1),h^{-1}(\operatorname{supp} f_2)\subset Y$, then
$E(a\sigma_z(b))$ is nonzero only if $h=g^{-1}$. In the latter case
the function $\varphi(a\sigma_z(b))=N(g)^{-iz}\varphi(ab)$ is
clearly bounded on the upper half-plane if $N(g)\le1$. So assume
$N(g)>1$. As $u_gf_2u_g^{-1}=f_2(g^{-1}\,\cdot)$ is supported on
$gY$, we see that the support of $f_1f_2(g^{-1}\,\cdot)$ is
contained in $Y\setminus Y_0$, whence $\varphi(a\sigma_z(b))=0$.
\smallskip
Conversely, assume $\varphi$ is a ground state. Let $\mu$ be the
probability measure on $Y$ defined by $\varphi|_{C_0(Y)}$. Take an
element $g\in G$ with $N(g)>1$. If $f\in C_c(Y\cap g^{-1}Y)$ is
positive, $a=u_gf^{1/2}$ and $b=f^{1/2}u_{g^{-1}}$, then the
function $z\mapsto\varphi(a\sigma_z(b))$ can be bounded on the upper
half-plane only if it is identically zero. Therefore
$\varphi(f(g^{-1}\,\cdot))=0$. Hence $\mu(gY\cap Y)=0$. Thus $\mu$
is supported on $Y_0$.
It remains to show that $\varphi(fu_g)=0$ for all $g\ne e$ and $f\in
C_c(Y)$ with $g^{-1}(\operatorname{supp} f)\subset Y$. If $x\in\operatorname{supp} f\cap Y_0$
then $g^{-1}x\notin Y_0$ by our assumptions on $Y_0$. Hence there
exists $h\in G$ with $N(h)>1$ such that $g^{-1}x\in hY$. This shows
that the sets $Y\setminus Y_0$ and $ghY$ with $N(h) >1$ form an open
cover of $\operatorname{supp} f$. Using a partition of unit subordinate to this
cover we decompose $f$ into a finite sum of functions with supports
contained in these sets. Therefore we may assume that either $\operatorname{supp}
f\subset Y\setminus Y_0$ or $g^{-1}(\operatorname{supp} f)\subset hY$ for some $h$
with $N(h)>1$. In the first case we have $\varphi(fu_g)=0$ as $\mu$
is supported on $Y_0$. In the second case write $f$ as a
product~$f_1f_2$ of continuous functions with the same support,
letting e.g. $f_1=|f|^{1/2}$ and $f_2=f|f|^{-1/2}$. Consider the
elements $a=f_1u_{gh}$ and $b=f_2(gh\cdot)u_{h^{-1}}$ of
$C^*_r(G\boxtimes Y)$, so that $fu_g=ab$. Since $N(h)>1$, the
function $z\mapsto\varphi(a\sigma_z(b))$ can be bounded on the upper
half-plane only if it is identically zero. Therefore
$\varphi(fu_g)=0$. \ep
\bigskip
\section{Bost-Connes systems for number fields}\label{sBC}
Suppose $K$ is an algebraic number field with subring of integers
$\OO$. Recall some notation. Denote by~$V_K$ the set of places of
$K$, and by $V_{K,f}\subset V_K$ the subset of finite places. For
$v\in V_K$ denote by~$K_v$ the corresponding completion of $K$. If
$v$ is finite, let $\OO_v$ be the closure of $\OO$ in $K_v$. The
ring of finite integral adeles is $\hat\OO=\prod_{v\in
V_{K,f}}\OO_v$, and $\akf=K\otimes_{\OO}\hat\OO$ is the ring of
finite adeles. Denoting by $K_\infty=\prod_{v|\infty}K_v$ the
completion of $K$ at all infinite places, we get the ring
$\ak=K_\infty\times\akf$ of adeles. The idele group is $I_K=\ak^*$.
Consider the topological space $\gal(\kab / K)\times \akf$, where
$\gal(\kab/K)$ is the Galois group of the maximal abelian extension
of $K$. On this space there is an action of the group $\jkf$ of
finite ideles, via the Artin map $s\colon I_K \to \gal(\kab/K)$ on
the first component and via multiplication on the second component:
\[
j(\gamma, m) = (\gamma s(j)^{-1} , jm) \ \ \hbox{for}\ \ j\in \jkf,
\ \ \gamma\in \gal(\kab/K), \ \ m\in \akf.
\]
Following~\cite{cmn1} we consider the quotient space
\[
X := \gal(\kab / K)\times_{\ohs} \akf
\]
in which the direct product is balanced over the compact open
subgroup of integral ideles $\ohs \subset \jkf$, in the sense that
one takes the quotient by the action given by $u (\gamma, m) =
(\gamma s(u)^{-1} , um)$ for $u\in \ohs$. This enables a quotient
action of the quotient group $ \jkf/\ohs$, which is isomorphic to
the (discrete) group $J_K$ of fractional ideals in $K$. We remark
that the space $X$ is isomorphic to the one that arises from the
construction of Ha and Paugam when applied to the Shimura data
associated to a number field, see \cite[Definition 5.5]{ha-pa}.
Finally we restrict to the clopen subset $Y := \gal(\kab /
K)\times_\ohs \hat\OO$ of $X$, and we consider the dynamical system
$(C^*_r(J_K \boxtimes Y), \sigma)$, in which the dynamics $\sigma$
is defined in terms of the absolute norm $N\colon J_K \to
(0,+\infty)$. Denote by $J_K^+\subset J_K$ the subsemigroup of
integral ideals, and recall that the norm of such an ideal $\aaa$ is
given by $|\OO/\aaa|$. Remark that by Theorem 2.1 and Theorem 2.4 of
\cite{minautex} the corner $C^*_r(J_K\boxtimes Y)= \31_Y ( C_0(X)
\rtimes J_K ) \31_Y$ is the semigroup crossed product $C(Y) \rtimes
J_K^+$.
In this situation the zeta function of the semigroup $J_K^+$ is
precisely the Dedekind zeta function $\zeta_K(\beta) = \sum_{\aaa\in
J_K^+} N(\aaa)^{-\beta}$; it converges for $\beta>1$ and diverges
for $\beta\in(0,1]$.
\begin{theorem} \label{mainthm}
For the system $(C(\gal(\kab / K)\times_\ohs \hat\OO)\rtimes
J_K^+,\sigma)$ we have: \enu{i} for $\beta<0$ there are no
KMS$_\beta$-states; \enu{ii} for each $0< \beta \leq 1$ there is a
unique KMS$_\beta$-state; \enu{iii} for each $1< \beta<\infty$ the
extremal KMS$_\beta$-states are indexed by $Y_0 := \gal(\kab /
K)\times_\ohs \ohs \cong \gal(\kab / K)$, with the state
corresponding to $w\in Y_0$ given by
\begin{equation} \label{eKMSext}
\varphi_{\beta,w} (f) = \frac{1}{\zeta_K(\beta) }\sum_{\aaa\in
J_K^+} N(\aaa)^{-\beta} f(\aaa w)\ \ \hbox{for}\ \ f\in C(\gal(\kab
/ K)\times_\ohs \hat\OO);
\end{equation}
\enu{iv} the ground states of the system are KMS$_\infty$-states,
and the extremal ground states are indexed by~$Y_0$, with the state
corresponding to $w\in Y_0$ given by $\varphi_{\infty,w}(f)=f(w)$.
\end{theorem}
\bp We apply Proposition~\ref{KMSmeasures} to $G=J_K$, $X= \gal(\kab
/ K)\times_{\ohs} \akf$ and $Y=\gal(\kab / K)\times_\ohs\hat\OO$. If
the image of a point $(\alpha,a)\in\gal(\kab/K)\times\akf$ in $X$
has nontrivial isotropy then $a_v=0$ for some~$v$, since this is
true already for the action of $J_K=\jkf/\ohs$ on $\akf/\ohs$.
Therefore for the sequence $\{(g_n,Y_n)\}_n$ we can take the pairs
$(\pp_v,Y_v)$ indexed by the finite places $v$, where $\pp_v$ is the
prime ideal of $\OO$ corresponding to $v$ and $Y_v\subset Y$
consists of the images in $X$ of all pairs
$(\alpha,a)\in\gal(\kab/K)\times\hat\OO$ with $a_v=0$. By
Proposition~\ref{KMSmeasures} we conclude that the
KMS$_\beta$-states for $\beta\ne0$ correspond to the measures $\mu$
on $X$ such that $\mu(Y)=1$ and $\mu$ satisfies the scaling
condition \eqref{normalrescale}.
\smallskip
Clearly there are no such measures for $\beta<0$, since otherwise
the inclusion $\aaa Y\subset Y$ would imply $N(\aaa)^{-\beta}\le1$.
This proves (i).
\smallskip
To prove part (iii) notice that $S=J_K^+$ and $Y_0=\gal(\kab /
K)\times_\ohs \ohs\subset Y$ satisfy conditions (i), (ii) and (iii)
of Proposition~\ref{phasetransition}. In order to verify condition
(iv) let $A\subset V_{K,f}$ be a finite set and denote by $\OO_A$
the product of $\OO_v$ over $v\in A$, and by $\hat\OO_A$ the product
of $\OO_v$ over $v\notin A$, so that $\hat\OO=\OO_A\times\hat\OO_A$.
Consider the open subset
$$
W_A=\gal(\kab /
K)\times_{\ohs}\left(\OO_A^*\times\hat\OO_A\right)
$$
of $Y$. The intersection of these sets over all finite $A$ coincides
with $Y_0$. Since $Y$ is compact and the sets $W_A$ are closed, it
follows that any neighborhood of $Y_0$ contains $W_A$ for some $A$.
The complement of $J_K^+W_A$ in $Y$ consists of the images of
points $(\alpha,a)\in\gal(\kab/K)\times\hat\OO$ such that $a_v=0$
for some $v\in A$, so it is covered by the sets $Y_v$, $v\in A$,
introduced above. Thus by Proposition~\ref{phasetransition} for each
$\beta>1$ there is a one-to-one affine correspondence between the
KMS$_\beta$-states and the probability measures on~$Y_0$. In particular,
the extremal KMS$_\beta$-states correspond to points of $Y_0$ via
\eqref{eKMSext}, which is a particular case of \eqref{eextension}.
This finishes the proof of part (iii).
\smallskip
Part (iv) follows from (iii) and Proposition~\ref{ground}.
\smallskip
Turning to (ii), we shall first explicitly construct for each
$\beta\in(0,1]$ a measure $\mu_\beta$ on $X$ such that
$\mu_\beta(Y)=1$ and $\mu_\beta$ satisfies the scaling condition
\eqref{normalrescale}. Define $\mu_\beta$ as the push-forward of the
product measure $\mu_\gal\times\prod_{v\in V_{K,f}}\mu_{\beta,v}$ on
$\gal(\kab/K)\times\akf$, where $\mu_\gal$ is the normalized Haar
measure on $\gal(\kab/K)$ and the measures~$\mu_{\beta,v}$ on $K_v$
are defined as follows. The measure $\mu_{1,v}$ is the additive Haar
measure on $K_v$ normalized by $\mu_{1,v}(\OO_v)=1$. The measure
$\mu_{\beta,v}$ is defined so that it is equivalent to $\mu_{1,v}$
and
$$
\frac{d\mu_{\beta,v}}{d\mu_{1,v}}(a)
=\frac{1-N(\pp_v)^{-\beta}}{1-N(\pp_v)^{-1}}\|a\|_v^{\beta-1},
$$
where $\|\cdot\|_v$ is the normalized valuation in the class $v$, so
$\|\pi\|_v=N(\pp_v)^{-1}$ for any uniformizing parameter $\pi
\in\pp_v$. Equivalently, $\mu_{\beta,v}$ is the unique measure on
$K_v$ such that the restriction of $\mu_{\beta,v}$ to $\OO^*_v$ is
the (multiplicative) Haar measure normalized by
$\mu_{\beta,v}(\OO^*_v)=1-N(\pp_v)^{-\beta}$, and $\mu_{\beta,v}(\pi
Z)=N(\pp_v)^{-\beta}\mu_{\beta,v}(Z)$.
To show that the measure $\mu_{\beta}$ is unique it suffices to show
that the action of $J_K$ on $(X,\mu)$ is ergodic for every measure
$\mu$ on $X$ such that $\mu(Y)=1$ and $\mu$ satisfies the scaling
condition \eqref{normalrescale}. Indeed, the set of such measures is
affine, so if all measures are ergodic the set must consist of one
point.
Equivalently, we have to show that the subspace $H$ of $L^2(Y,d\mu)$
of $J_K^+$-invariant functions consists of scalars. Denote by $P$
the projection onto this space. It is enough to compute how $P$ acts
on the pull-backs of functions on $\gal(\kab/K)\times_\ohs\OO_A$ for
finite $A\subset V_{K,f}$. Denote by $J^+_{K,A}$ the unital
subsemigroup of $J^+_K$ generated by $\pp_v$, $v\in A$. Modulo a set
of measure zero $\gal(\kab/K)\times_\ohs\OO_A$ is the union of the
sets $\aaa(\gal(\kab/K)\times_\ohs\OO_A^*)$, $\aaa\in J^+_{K,A}$.
The compact set $\gal(\kab/K)\times_\ohs\OO_A^*$ is a group
isomorphic to $\gal(\kab/K)/s(\ohs_A)$. Therefore it suffices to
compute $Pf$ for the pull-back $f$ of the function
$$
\gal(\kab/K)\times_\ohs\OO_A\ni a\mapsto
\begin{cases}
\tilde\chi(\aaa^{-1}a), &\ \hbox{if}\ \
a\in\aaa(\gal(\kab/K)\times_\ohs\OO_A^*),\\
0, &\ \hbox{otherwise},
\end{cases}
$$
where $\tilde\chi$ is a character of
$\gal(\kab/K)\times_\ohs\OO_A^*$. The character $\tilde\chi$ is
defined by a Dirichlet character $\chi\mod\mm$ with $\mm\in
J^+_{K,A}$.
For a finite set $B\subset V_{K,f}$ denote by $P_B$ the projection
onto the subspace $H_B\subset L^2(Y,d\mu)$ of $J^+_{K,B}$-invariant
functions. Apply Proposition~\ref{phasetransition}(2) with
$G=J_{K,B}:=(J^+_{K,B})^{-1}J^+_{K,B}$, $S=J^+_{K,B}$ and
$Y_0=W_B=\gal(\kab / K)\times_{\ohs}(\OO_B^*\times\hat\OO_B)$. Note
that $\zeta_{J^+_{K,B}}(\beta)=\prod_{v\in
B}(1-N(\pp_v)^{-\beta})^{-1}$. Furthermore, for $\bb\in J^+_{K,B}$
the set $\bb W_B$ intersects the support of $f$ only if $\aaa|\bb$
and the ideals $\aaa$ and $\bb\aaa^{-1}$ are relatively prime, or
equivalently, $\aaa\in J^+_{K,B}$ and $\bb\in\aaa J^+_{K,B\setminus
A}$. Therefore, assuming $A\subset B$, by \eqref{epro0} we get
\begin{align}
P_Bf|_{J^+_{K,B}a}
&=\prod_{v\in B}(1-N(\pp_v)^{-\beta}) \sum_{\mathfrak{c}\, \in
J^+_{K,B\setminus A}}N(\aaa\mathfrak{c})^{-\beta}\tilde\chi(\mathfrak{c} a)\notag \\
&=N(\aaa)^{-\beta}\tilde\chi(a){\prod_{v\in
B}(1-N(\pp_v)^{-\beta})}\sum_{\mathfrak{c} \, \in
J^+_{K,B\setminus A}}N(\mathfrak{c})^{-\beta}\chi(\mathfrak{c})\notag \\
&=N(\aaa)^{-\beta}\tilde\chi(a)\frac{\prod_{v\in
B}(1-N(\pp_v)^{-\beta})}{\prod_{v\in B\setminus
A}(1-\chi(\pp_v)N(\pp_v)^{-\beta})}\notag
\end{align}
for $a\in W_B$. If $\chi$ is trivial we see that $P_Bf$ is constant,
and hence so is $Pf$. On the other hand, for nontrivial~$\chi$ we
get
$$
\|Pf\|_2=\lim_B\|P_Bf\|_2=N(\aaa)^{-\beta}\lim_B\frac{\prod_{v\in
B}|1-N(\pp_v)^{-\beta}|}{\prod_{v\in B\setminus
A}|1-\chi(\pp_v)N(\pp_v)^{-\beta}|}.
$$
The right hand side divided by $N(\aaa)^{-\beta}$ is an increasing
function in $\beta$ on $(0,+\infty)$. For $\beta>1$ it equals
$|L(\chi,\beta)|/\zeta_K(\beta)$. As $L(\chi,\cdot)$ does not have a
pole at $1$, see e.g. \cite[Lemma 13.3]{neu}, we conclude that the
right hand side is zero for $\beta\in(0,1]$. Therefore in either
case we see that $Pf$ is constant. \ep
\begin{remark}
\mbox{\ } \enu{i} There is an obvious action of the Galois group
$\gal(\kab / K)$ of the maximal abelian extension of~$K$ on $Y$,
given by $\alpha (\gamma, m) = (\alpha\gamma, m)$, and this gives
rise to an action of $\gal(\kab / K)$ as symmetries of
$(C^*_r(J_K\boxtimes Y), \sigma)$. This action is clearly free and
transitive on the set $Y_0$ parametrizing the extreme
KMS$_\beta$-states. \enu{ii} It is known~\cite{ha-pa} and easy to
check that the partition function of our system is the Dedekind zeta
function. More precisely, if $\varphi_{\beta,w}$ is an extremal
KMS$_\beta$-state for some $\beta>1$, and $H_{\beta,w}$ is the
generator of the canonical one-parameter unitary group implementing
$\sigma$ in the GNS-representation of $\varphi_{\beta,w}$, then
$\operatorname{Tr}(e^{-\beta H_{\beta,w}})=\zeta_K(\beta)$. \enu{iii} For totally
imaginary fields of class number one the C$^*$-algebra $C^*_r(J_K
\boxtimes Y)$ described above is isomorphic to the Hecke
C$^*$-algebra $C^*(\Gamma_K; \Gamma_\OO)$ studied in \cite{lvf}. To
see this, observe first that $\gal(\kab/K)\cong
\akf^*/\overline{K^*}\cong\ohs/\overline{\OO^*}$. It follows that $Y
= \gal(\kab/K) \times_\ohs \hat\OO $ can be identified
with~$\hat\OO/{\overline{\OO^*}}$. From \cite[Definition 2.2]{lvf}
and the ensueing discussion, multiplication by an extreme inverse
different transforms this identification into a homeomorphism of the
orbit space $\Omega = \mathcal D^{-1} /{\overline{\OO^*}}$ and $Y$.
It is then easy to check that the multiplicative action of $J_K^+
\cong \OO^\times/\OO^*$ on $C(\Omega) $ described in
\cite[Proposition 2.4]{lvf} corresponds to the action of $J_K^+
\cong \jkf/ \ohs$ inherited by $C(Y)$ from the original
transformation group. By \cite[Theorem 2.5]{lvf} it follows that the
Hecke C$^*$-algebra $C^*(\Gamma_K; \Gamma_\OO)$ is isomorphic to
$C(Y) \rtimes J_K^+ \cong C^*_r(J_K\boxtimes Y)$. The isomorphism
respects the semigroup of isometries and thus the dynamics arising
from the norm, but the Galois group action is changed via the
balancing over $\ohs$, and this resolves the incompatibility pointed
out in \cite[Theorem 4.4]{lvf}.
For higher class numbers the Hecke C$^*$-algebra constructed in
\cite{lvf} is a semigroup crossed product by the semigroup of
principal ideals so it is essentially different from the one studied
here.
\end{remark}
\bigskip
\section{$K$-lattices} \label{sKlat}
In this section we define $n$-dimensional $K$-lattices and interpret
the BC-systems for number fields in terms of $K$-lattices.
Recall the following definition given by Connes and
Marcolli~\cite{con-mar}. An $n$-dimensional $\mathbb Q$-lattice is a pair
$(L,\varphi)$, where $L\subset\mathbb R^n$ is a lattice and
$\varphi\colon\mathbb Q^n/\mathbb Z^n\to\mathbb Q L/L$ is a homomorphism. The notion of a
$1$-dimensional $K$-lattice for an imaginary quadratic field $K$ is
analyzed in \cite{cmn1}. In what follows we generalize $K$-lattices
to arbitrary number fields and dimensions. We refer to \cite{KCoMa}
for a related discussion of the function fields case, see
also~\cite{J}.
Recall that we denote by $K_\infty$ the completion of $K$ at all
infinite places, so $K_\infty\cong\mathbb R^{[K\colon\mathbb Q]}$ as a topological
group under addition. By an $n$-dimensional $\OO$-lattice we mean a
lattice $L$ in $K_\infty^n$ such that $\OO L=L$.
\begin{definition}
An $n$-dimensional $K$-lattice is a pair $(L,\varphi)$, where
$L\subset K_\infty^n$ is an $n$-dimensional $\OO$-lattice and
$\varphi\colon K^n/\OO^n\to KL/L$ is an $\OO$-module map.
\end{definition}
The simplest example of an $n$-dimensional $\OO$-lattice is $\OO^n$.
Since $K^n=\mathbb Q\OO^n$, any two finitely generated $\OO$-submodules of
$K^n$ of rank $n$ are commensurable, in particular, any such module
is an $\OO$-lattice. Furthermore, a submodule of $K^n$ of rank $m<n$
is an abelian group of rank $m[K\colon\mathbb Q]$, so it cannot be a
lattice in $K^n_\infty$. Thus for submodules of $K^n$ we get the
usual definition of an $\OO$-lattice: an $\OO$-submodule $M\subset
K^n$ is an $n$-dimensional $\OO$-lattice if and only if it is
finitely generated and has rank $n$.
We now want to give a parametrization of the set of $n$-dimensional
$\OO$-lattices. For this recall that there exists a one-to-one
correspondence between finitely generated $\OO$-submodules of~$K^n$
of rank $n$ and $\hat\OO$-submodules $\LL=\prod_{v\in
V_{K,f}}L_v\subset\akf^n$ such that $L_v$ is a compact open
$\OO_v$-submodule of $K_v^n$ with $L_v=\OO_v^n$ for all but a finite
number of places $v$. Namely, starting from an $\OO$-lattice
define~$\LL$ as its closure. The inverse map is
$\LL\mapsto\cap_v(L_v\cap K^n)$. Hence, given an element
$s=(s_\infty,s_f)\in\gln(\ak)=\gln(K_\infty)\times\gln(\akf)$, we
get an $\OO$-lattice $s_f\hat\OO^n\cap K^n$ in $K^n$, and then an
$\OO$-lattice $s^{-1}_\infty (s_f\hat\OO^n\cap K^n)$ in
$K^n_\infty$.
\begin{lemma} \label{olat}
The map $\gln(\ak)\ni s\mapsto s^{-1}_\infty (s_f\hat\OO^n\cap K^n)$
induces a bijection between
$$
\gln(K)\backslash\gln(\ak)/\gln(\hat\OO)
$$
and the set of $n$-dimensional $\OO$-lattices.
\end{lemma}
\bp It is easy to see that the map from
$\gln(K)\backslash\gln(\ak)/\gln(\hat\OO)$ to $\OO$-lattices is
well-defined. To see that it is injective, assume $r^{-1}_\infty
(r_f\hat\OO^n\cap K^n)=s^{-1}_\infty (s_f\hat\OO^n\cap K^n)$ for
some $r,s\in\gln(\ak)$. Multiplying by $K$ we get $r^{-1}_\infty
K^n=s^{-1}_\infty K^n$, so $g:=s_\infty r^{-1}_\infty\in\gln(K)$.
Taking the closure we get from $g(r_f\hat\OO^n\cap
K^n)=s_f\hat\OO^n\cap K^n$ that $gr_f\hat\OO^n=s_f\hat\OO^n$. Hence
$gr_fu=s_f$ for some $u\in\gln(\hat\OO)$. Since also
$gr_\infty=s_\infty$, this means that $s$ is in a
$\gln(K)$-$\gln(\hat\OO)$-orbit of $r$, so the map is injective.
To prove surjectivity, take an $\OO$-lattice $L\subset K^n_\infty$.
We have $KL=\mathbb Q L\cong\mathbb Q^{n[K\colon\mathbb Q]}$, so $\dim_K KL=n$. In
particular, $L$ is a finitely generated $\OO$-module of rank $n$.
Therefore it suffices to show that there exists $g\in\gln(K_\infty)$
such that $gL\subset K^n$. Let $e_1,\dots,e_n$ be a basis of $KL$
over $K$. Since $KL=\mathbb Q L$ is dense in $K_\infty^n$, the image of
$KL$ under the projection $K^n_\infty\to K^n_v$ is dense in $K^n_v$
for any infinite place $v$. It follows that the images of
$e_1,\dots,e_n$ are linearly independent over $K_v$. So there exists
$g_v\in\gln(K_v)$ which maps these images onto the standard basis of
$K_v^n$. Then $g=(g_v)_{v|\infty}$ is an element in $\gln(K_\infty)$
mapping $e_1,\dots,e_n$ onto the standard basis of $K^n_\infty$, so
that $gKL=K^n$. \ep
For $s\in\gln(\ak)$ and $t\in\mn(\hat\OO)$ consider the
$\OO$-lattice $L=s_f\hat\OO^n\cap K^n$. The map
$s_ft\colon\akf^n\to\akf^n$ maps $\hat\OO^n$ into $s_f\hat\OO^n$,
hence induces an $\hat\OO$-module map $\akf^n/\hat\OO^n\to
\akf^n/s_f\hat\OO^n$. Then there exists a unique $\OO$-module map
$\varphi\colon K^n/\OO^n\to KL/L$ such that the diagram
$$
\xymatrix{\akf^n/\hat\OO^n\ar[r]^{s_ft} & \akf^n/s_f\hat\OO^n\\
K^n/\OO^n\ar[u] \ar[r]_{\varphi} & KL/L\ar[u]}
$$
commutes, where the vertical arrows are the canonical isomorphisms
defined by the inclusions $K^n\subset\akf^n$, $KL\subset\akf^n$. We
shall also denote $\varphi$ by $[s_ft]$. Thus $(L,\varphi)$ is a
$K$-lattice. Therefore $(s_\infty^{-1}L,s_\infty^{-1}\varphi)$ is
also a $K$-lattice, which we denote by $[(s,t)]$.
\begin{lemma} \label{Klattice}
The map $\gln(\ak)\times\mn(\hat\OO)\ni
(s,t)\to[(s,t)]=(s_\infty^{-1}(s_f\hat\OO^n\cap
K^n),s_\infty^{-1}[s_ft])$ induces a bijection between
$$
\gln(K)\backslash\gln(\ak)\times_{\gln(\hat\OO)}\mn(\hat\OO)
$$
and the set of $n$-dimensional $K$-lattices.
\end{lemma}
\bp By Lemma~\ref{olat} we only need to check that any $\OO$-module
map $\akf^n/\hat\OO^n\to\akf^n/s_f\hat\OO^n$, where
$s_f\in\gln(\akf)$, is defined by the matrix $s_ft$ for a unique
$t\in\mn(\hat\OO)$. It suffices to consider $s_f=1$. The problem
then reduces to showing that any $\OO$-module map $K_v/\OO_v\to
K_v/\OO_v$ is given by multiplication by a unique element of
$\OO_v$. If $\pi$ is a uniformizing parameter in $\OO_v$, then any
$\OO$-module map $\OO_v\pi^{-m}/\OO_v\to K_v/\OO_v$ is determined by
the image of $\pi^{-m}$, so it is given by multiplication by an
element in $\OO_v$ which is uniquely determined modulo $\OO_v\pi^m$.
Since $\OO_v$ is complete in the $(\pi)$-adic topology, this gives
the result. \ep
Notice that we have shown in particular that for any $K$-lattice
$(L,\varphi)$ with $L\subset K^n$ the homomorphism $\varphi$ lifts
to a unique $\akf$-module map $\tilde\varphi\colon\akf^n\to\akf^n$.
\begin{definition} \label{defComm}
Two $n$-dimensional $K$-lattices $(L_1,\varphi_1)$ and
$(L_2,\varphi_2)$ are called commensurable if the lattices $L_1$ and
$L_2$ are commensurable and $\varphi_1=\varphi_2$ modulo $L_1+L_2$.
\end{definition}
If $L_1$ and $L_2$ are commensurable then $KL_1=\mathbb Q L_1=\mathbb Q L_2=KL_2$.
In particular, if $L_1\subset K^n$ then also $L_2\subset K^n$. It is
clear that then the lifting of the composition of the homomorphisms
$\varphi_1\colon K^n/\OO^n\to KL_1/L_1$ and $KL_1/L_1\to
K(L_1+L_2)/(L_1+L_2)$ coincides with $\tilde\varphi_1$. Therefore
two $K$-lattices $(L_1,\varphi_1)$ and $(L_2,\varphi_2)$ with
$L_1,L_2\subset K^n$ are commensurable if and only if
$\tilde\varphi_1=\tilde\varphi_2$. This implies that
commensurability is an equivalence relation.
Denote the equivalence relation of commensurability of
$n$-dimensional $K$-lattices by ${\mathcal R}_{K,n}$. Consider now the action
of $\gln(\akf)$ on $\gln(K)\backslash\gln(\ak)\times\mn(\akf)$
defined by
$$
g(s,t)=(sg^{-1},gt).
$$
Define a subgroupoid
$$
\gln(\akf)\boxtimes(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))
=\{(g,s,t)\mid t\in\mn(\hat\OO),\ gt\in\mn(\hat\OO)\}
$$
of the transformation groupoid
$\gln(\akf)\times(\gln(K)\backslash\gln(\ak)\times\mn(\akf))$. We
have a groupoid homomorphism
$$
\gln(\akf)\boxtimes(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))\to
{\mathcal R}_{K,n}
$$
defined by
\begin{equation} \label{ehomo}
(g,s,t)\mapsto ([(sg^{-1},gt)],[(s,t)]).
\end{equation}
To see that $[(s,t)]$ and $[(sg^{-1},gt)]$ are indeed commensurable
recall that by definition we have
$[(s,t)]=(s_\infty^{-1}(s_f\hat\OO^n\cap K^n),s_\infty^{-1}[s_ft])$
and $[(sg^{-1},gt)]=(s_\infty^{-1}(s_fg^{-1}\hat\OO^n\cap
K^n),s_\infty^{-1}[s_ft])$.
By Lemma~\ref{Klattice} to make the above homomorphism injective we
have to factor out the action of $\gln(\hat\OO)$. Consider the
action of $\gln(\hat\OO)\times\gln(\hat\OO)$ on
$\gln(\akf)\boxtimes(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))$
defined by
$$
(u_1,u_2)(g,s,t)=(u_1gu_2^{-1},su_2^{-1},u_2t),
$$
and denote by
$$
\gln(\hat\OO)\backslash\gln(\akf)\boxtimes_{\gln(\hat\OO)}
(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))
$$
the quotient space.
\begin{proposition}
The map (\ref{ehomo}) induces a bijection between
$$
\gln(\hat\OO)\backslash\gln(\akf)\boxtimes_{\gln(\hat\OO)}
(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))
$$
and ${\mathcal R}_{K,n}$.
\end{proposition}
\bp By Lemma~\ref{Klattice} the map
$$
\gln(\hat\OO)\backslash\gln(\akf)\boxtimes_{\gln(\hat\OO)}
(\gln(K)\backslash\gln(\ak)\times\mn(\hat\OO))\to{\mathcal R}_{K,n}
$$
is well-defined and injective. To prove surjectivity we have to show
that if $(L,\varphi)=[(s,t)]$ is a $K$-lattice then any
commensurable $K$-lattice is of the form $[(sg^{-1},gt)]$ for some
$g\in\gln(\akf)$. We may assume that $L\subset K^n$ and then that
$s_\infty=1$. Then by Lemma~\ref{Klattice} and the discussion
following Definition~\ref{defComm} any commensurable $K$-lattice is
of the form $[(q,r)]$ with $q_\infty=1$ and $q_fr=s_ft$. Letting
$g=q_f^{-1}s_f$ we get $(q,r)=(sg^{-1},gt)$. \ep
\begin{remark}
In the case $K=\mathbb Q$, or more generally for fields with class number
one, there is a better description due to the fact that any
$\mathbb Z$-lattice is free. Indeed, by freeness we have
$\gln(\af)=\glnq\gln(\Zhat)$, where $\glnq$ is the group of rational
matrices with positive determinant. It follows that any
$\gln(\Zhat)\times\gln(\Zhat)$-orbit in
$\gln(\af)\times(\gln(\A)\times\mn(\Zhat))$ has a representative in
$\glnq\times((\gln(\mathbb R)\times\glnq)\times\mn(\Zhat))$. Furthermore,
the map
$$
\glnr\times\glnq\to\glnr,\ \ (g,h)\mapsto h^{-1}g,
$$
induces a bijection between $\glnq\backslash(\glnr\times\glnq)$ onto
$\glnr$. One may then conclude that ${\mathcal R}_{\mathbb Q,n}$ can be identified
with
$$
\slnz\backslash\glnq\boxtimes_{\slnz}(\glnr\times\mn(\Zhat)),
$$
where the action of $\slnz\times\slnz$ on
$\glnq\times\glnr\times\mn(\Zhat)$ is given by
$$
(\gamma_1,\gamma_2)(g,h,m)=(\gamma_1g\gamma_2^{-1},\gamma_2h,\gamma_2m).
$$
\end{remark}
Consider now the case $n=1$ (and $K$ arbitrary). Then we conclude
that there is a bijection between ${\mathcal R}_{K,1}$ and the subgroupoid
$$
(\akf^*/\ohs)\boxtimes((\ak^*/K^*)\times_\ohs\hat\OO)
$$
of the transformation groupoid
$(\akf^*/\ohs)\times((\ak^*/K^*)\times_\ohs\akf)$. We have an
action, called the scaling action, of $K^*_\infty$ on $K$-lattices:
if $(L,\varphi)$ is a $K$-lattice and $k\in K_\infty^*$ then
$k(L,\varphi)=(kL,k\varphi)$. It defines an action of $K^*_\infty$
on ${\mathcal R}_{K,1}$. In our transformation groupoid picture of
${\mathcal R}_{K,1}$ it corresponds to the action of $K^*_\infty$ by
multiplication on $\ak^*/K^*$. Denote by $(K^*_\infty)^\circ$ the
connected component of the identity in $K^*_\infty$. Then we get the
following result.
\begin{corollary} \label{Klatscal}
The quotient of the equivalence relation ${\mathcal R}_{K,1}$ of
commensurability of $1$-dimensional $K$-lattices by the scaling
action of the connected component of the identity in $K^*_\infty$ is
a groupoid that is isomorphic to
$$
(\akf^*/\ohs)\boxtimes((\ak^*/K^*(K^*_\infty)^\circ)
\times_\ohs\hat\OO).
$$
\end{corollary}
Recalling that $\akf^*/\ohs\cong J_K$ and
$\ak^*/\overline{K^*(K^*_\infty)^\circ} \cong \gal(\kab/K)$ by class
field theory, we see that the above groupoid is almost the same that
we used to define the BC-system. The small nuance is that when we
put $\gal(\kab/K)$ in our topological groupoid in Section~\ref{sBC}
we were effectively taking the quotient of $\ak^*$ by the {\em
closure} of $K^*(K_\infty^*)^\circ$. In terms of $K$-lattices this
means that given a $K$-lattice $(L,\varphi)$ we would have to
identify not only all $K$-lattices $(kL,k\varphi)$ with $k\in
(K^*_\infty)^\circ$, but also all $K$-lattices of the form
$(kL,k\psi)$, where $\psi$ is a limit point of the maps $u\varphi$
with $u\in\OO^*\cap(K_\infty^*)^\circ$ in the topology of pointwise
convergence.
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,080 |
using System;
namespace ZXing.OneD.RSS
{
/// <summary>
/// Encapsulates a since character value in an RSS barcode, including its checksum information.
/// </summary>
public class DataCharacter
{
/// <summary>
/// Gets the value.
/// </summary>
public int Value { get; private set; }
/// <summary>
/// Gets the checksum portion.
/// </summary>
public int ChecksumPortion { get; private set; }
/// <summary>
/// Initializes a new instance of the <see cref="DataCharacter"/> class.
/// </summary>
/// <param name="value">The value.</param>
/// <param name="checksumPortion">The checksum portion.</param>
public DataCharacter(int value, int checksumPortion)
{
Value = value;
ChecksumPortion = checksumPortion;
}
/// <summary>
/// Returns a <see cref="System.String"/> that represents this instance.
/// </summary>
/// <returns>
/// A <see cref="System.String"/> that represents this instance.
/// </returns>
override public String ToString()
{
return Value + "(" + ChecksumPortion + ')';
}
/// <summary>
/// Determines whether the specified <see cref="System.Object"/> is equal to this instance.
/// </summary>
/// <param name="o">The <see cref="System.Object"/> to compare with this instance.</param>
/// <returns>
/// <c>true</c> if the specified <see cref="System.Object"/> is equal to this instance; otherwise, <c>false</c>.
/// </returns>
override public bool Equals(Object o)
{
if (!(o is DataCharacter))
{
return false;
}
DataCharacter that = (DataCharacter)o;
return Value == that.Value && ChecksumPortion == that.ChecksumPortion;
}
/// <summary>
/// Returns a hash code for this instance.
/// </summary>
/// <returns>
/// A hash code for this instance, suitable for use in hashing algorithms and data structures like a hash table.
/// </returns>
override public int GetHashCode()
{
return Value ^ ChecksumPortion;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,247 |
Q: Do-while loop in Clojure? So I want to first execute a bunch of code, and then ask the user if he wants to do that again. I thought the most convenient way to do this would be a do-while loop like in C++, and since I couldn't seem to find any do-while functions in Clojure, I wrote the following:
(defmacro do-while
"Executes body before testing for truth expression"
[test & body]
`(do (do ~@body) (while ~test ~@body)))
Would there be a better (as in more idiomatic Clojure-ish) way of writing this macro, or perhaps a better way of doing what I want without going through the do-while route?
A: Here is a slightly changed version of Clojure's while macro, where the test is done after evaluating the body:
(defmacro do-while
[test & body]
`(loop []
~@body
(when ~test
(recur))))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,101 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/calculus\/calculus-3rd-edition\/chapter-7-exponential-functions-chapter-review-exercises-page-387\/63","text":"## Calculus (3rd Edition)\n\n$$\\int \\frac{dx}{\\sqrt{e^{2x}-1}} =\\cos^{-1}e^{-x}+c.$$\nSince $u=e^{-x}$, then $du=- e^{-x}dx$ and hence $$\\int \\frac{dx}{\\sqrt{e^{2x}-1}}=-\\int \\frac{du}{u\\sqrt{(1\/u^2)-1}} =-\\int \\frac{du}{ \\sqrt{1-u^2}} \\\\ =\\cos^{-1 }u+c=\\cos^{-1}e^{-x}+c.$$ (We could also have integrated to inverse sine by excluding the negative sign.)","date":"2021-01-21 01:54:37","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\": 0.9893949627876282, \"perplexity\": 1018.962367223732}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703522150.18\/warc\/CC-MAIN-20210121004224-20210121034224-00374.warc.gz\"}"} | null | null |
(12932) Conedera (1999 TC12) – planetoida z grupy pasa głównego asteroid okrążająca Słońce w ciągu 3,78 lat w średniej odległości 2,42 j.a. Odkryta 10 października 1999 roku.
Zobacz też
lista planetoid 12001–13000
lista planetoid
Linki zewnętrzne
Planetoidy pasa głównego
Nazwane planetoidy
Obiekty astronomiczne odkryte w 1999 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,921 |
{"url":"https:\/\/tug.org\/pipermail\/xetex\/2011-December\/022385.html","text":"# [XeTeX] XeTeX, XeTeXpicfile, and counter-intuitive behaviour\n\nThu Dec 1 20:13:12 CET 2011\n\nOn Thu, Dec 01, 2011 at 06:06:13PM +0000, Philip TAYLOR wrote:\n\n> Heiko Oberdiek wrote:\n>\n> >Because they are solving those problems for you and others now\n> >and in the future. That is one of the main reasons for a package,\n> >providing an easier interface for tricky low level stuff.\n>\n> I don't \/want\/ \"an easier interface\", Heiko : I \/hate\/ LaTeX,\n\nThen use a ConTeXt module or whatever.\n\nAnd the syntax of graphicx' \\includegraphics:\n\n\\includegraphics[<key value list>]{<image file>}\n\nis not very special that this can only be used in LaTeX.\n\n> with its \"Nanny knows best\" mentality, and I want to \/understand\/\n> \"the tricky low level\" stuff because it is (a) the most interesting,\n> (b) the most powerful, and (c) because there is nothing between\n> me and it to stop me from making it do exactly what I want.\n\nThen, please, don't surprise me, that you are suprised that TeX\nhas surprised you. ;-)\n\n> >Back to the problems:\n> >\n> >* non-effective \\vfill\\eject: documented in \"The TeXbook\":\n> >\n> > | Whenever TeX is moving an item from the top of the\n> > | \"recent contributions\" to the bottom of the \"current page,\"\n> > | it discards a discardable item (glue, kern, or penalty)\n> > | if the current page does not contain any boxes.\n> >\n> > Remark: This should be fixed:\n> > any boxes. -> any boxes or rules.\n>\n> The problem is, there is still (as far as I can see) no formal\n> definition of what sort of <thing> \\XeTeXp{df|ic}file inserts.\n> If there were such a definition, tracking down such bugs would\n> be very much simpler.\n\nYes, I don't know a complete, comprehensive documentation of XeTeX.\nI am aware of:\n* Mark A. Wicks: Dvipdfm User's Manual (dvipdfm.pdf)\n* Jin-Hwan Cho: DVI specials for PDF generation;\nTUGboat, Volume 30 (2009), No. 1 (dvipdfmx-special.pdf)\n* Will Robertson: The XeTeX reference guide (XeTeX-reference.pdf)\n\n> >* \\XeTeXpicfile cannot load PDF files: documentd in\n> >\n> > \"The XeTeX reference guide\" by Will Robertson:\n> > | 6 Graphics\n> > | ...\n> > | \\XeTeXpicfile ...\n> > | Insert an image. ...\n> > |\n> > | \\XeTeXpdffile ...\n> > | Insert (pages of) a PDF. ...\n> >\n> > Remark: It could be improved by saying\n> > * that both commands insert a whatsit and\n> > * that they should be used inside a \\hbox to avoid trouble\n> > with discarded items at the top of a page.\n>\n> That would indeed be a great help.\n\nCan you rephrase this in understandable words\nand in excellent English of course :-)\nand send the result to Will that he can update\nthe reference guide?\n\nYours sincerely\nHeiko Oberdiek","date":"2023-03-24 16:03:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8553851246833801, \"perplexity\": 11319.367089504778}, \"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-2023-14\/segments\/1679296945287.43\/warc\/CC-MAIN-20230324144746-20230324174746-00466.warc.gz\"}"} | null | null |
{"url":"https:\/\/yu-group.github.io\/simChef\/reference\/generate_data.html","text":"Generate data from each DGP in the Experiment.\n\nUsage\n\ngenerate_data(experiment, n_reps = 1, ...)\n\nArguments\n\nexperiment\n\nAn Experiment object.\n\nn_reps\n\nThe number of datasets to generate per DGP.\n\n...\n\nNot used.\n\nValue\n\nA list of length equal to the number of DGPs in the Experiment. If the Experiment does not have a vary_across component, then each element in the list is a list of n_reps datasets generated by the given DGP. If the Experiment does have a vary_across component, then each element in the outermost list is a list of lists. The second layer of lists corresponds to a specific parameter setting within the vary_across scheme, and the innermost layer of lists is of length n_reps with the dataset replicates, generated by the DGP.","date":"2023-01-29 09:48:36","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\": 0.4334513545036316, \"perplexity\": 1245.9708057299276}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499710.49\/warc\/CC-MAIN-20230129080341-20230129110341-00489.warc.gz\"}"} | null | null |
Buddhist Kingdom of Bhutan's King and Queen Expecting Second Child
His majesty Jigme Khesar Namgyel Wangchuck and her majesty the queen consort Ashi Jetsun Pema From drukasia.com
Their majesties the king and queen of the Himalayan kingdom of Bhutan have announced that they are expecting their second child. His majesty Jigme Khesar Namgyel Wangchuck, the Druk Gyalpo or "Dragon King" of Bhutan, made the announcement in a royal address in December at Changlimithang Stadium in the capital Thimphu before a capacity crowd, who had gathered to celebrate Bhutan's 112th National Day.
Upon hearing the news, the 30,000 people assembled in the stadium reportedly responded with an ecstatic cheer of approval. The Bhutanese newspaper Kuensel reported that the announcement created such excitement that its journalists had difficultly interviewing people on any other issue. No details were announced about the expected birth date.
Well-wishers described the news of a second royal child as a gift from the throne and an auspicious sign for the years to come. "It was the best National Day gift that we could expect," said Jangchub Choden from the eastern town of Trashiyangtse. (Kuensel)
Remote, landlocked, and perched in the rarified air of the eastern Himalaya, sandwiched between two political and economic heavy hitters India and China, Bhutan is the world's last remaining Vajrayana Buddhist country. The ancient spiritual tradition is embedded in the very consciousness and culture of this remote land, where it has flourished with an unbroken history that dates back to its introduction from Tibet by Padmasambhava, also known as Guru Rinpoche, in the eighth century.
Following the announcement of the royal pregnancy, Buddhist prayers were held for the health of Her Majesty the Queen Consort (Druk Gyaltsuen or "Dragon Queen") Ashi Jetsun Pema. The king said that the royal couple, who were married in 2011, felt extremely fortunate and were "excited beyond words" about the impending arrival of the Royal child. (Kuensel)
The royal couple's first child, the Crown Prince Jigme Namgyel Wangchuck, was born on 5 February 2016. In March 2016, the people of Bhutan planted 108,000 trees In honor of his birth.* Prince Jigme is expected to become the sixth Druk Gyalpo of Bhutan.
112th National Day: His Majesty The King announced that Their Majesties are expecting a second Royal Child. #HisMajesty #KingJigmeKhesar #KingofBhutan #HerMajesty #QueenJetsunPema #QueenofBhutan #Gyalsey #JigmeNamgyel #Bhutan #RoyalBaby #happiness
A post shared by Her Majesty Queen Jetsun Pema (@queenjetsunpema) on Dec 17, 2019 at 7:22pm PST
Bhutan is regularly ranked among the happiest countries in the world. With a population of just 770,000, according to government data for 2017, it is also one of the world's smallest and least industrialized countries, yet it has significant experience in maintaining the delicate balance of managing economic growth in a sustainable manner, famously encapsulated in its conservative "Gross National Happiness" (GNH) approach to economic development.
Bhutan's philosophy of GNH was introduced in the late 1970s by the country's fourth king, Jigme Singye Wangchuk, drawing inspiration from the kingdom's traditional Buddhist culture. An alternative to traditional metrics for measuring national development, such as gross national product (GNP) or gross domestic product (GDP), GNH is founded on four underlying principles or "pillars:" good governance, sustainable socio-economic development, the preservation and promotion of culture, and environmental conservation.
While not opposed to material development or economic progress, GNH rejects the pursuit of economic growth as the ultimate good, instead seeking to cultivate a more holistic approach to balanced development and societal well-being, translating cultural and social priorities into developmental goals to create a happier, more equitable society.
Almost 75 per cent of Bhutan's population of some 770,000 people identify as Buddhists, according to data for 2010 from the Washington, DC-based Pew Research Center, with Hinduism accounting for the majority of the remaining 25 per cent. Most of Bhutan's Buddhists follow either the Drukpa Kagyu or the Nyingma schools of Vajrayana Buddhism. Bhutan held its first elections as a constitutional monarchy in 2008.
Their Majesties are expecting their second child in Spring 2020! Thank you Your Majesties! #KingJigmeKhesar #QueenJetsunPema #TheRoyalFamilyofBhutan #The112thNationalDay2019
A post shared by The Royal Family of Bhutannn (@theroyalfamilyofbhutannn) on Dec 17, 2019 at 12:31am PST
* Bhutan Plants 108,000 Trees to Celebrate Birth of Crown Prince (Buddhistdoor Global)
Bhutan Celebrates News of Second Royal Child (Kuensel)
Bhutan Wins Recognition for Tiger Conservation Efforts
Former Bhutanese Education Minister Makes the Case for "Green Schools"
In Landmark Move, Buddhist Bhutan Gives Teachers, Medics Highest Pay Grade for Civil Servants
BBC Names Dr. Tashi Zangmo of the Bhutan Nuns Foundation among 100 Most Influential Women of 2018
Bhutan, the World's Only Carbon-negative Nation, Sets an Example of Environmental Stewardship for a Planet Grappling with Climate Change
My Story: Walking the Path of a Female Monastic in Bhutan
Changing Mindsets: Tashi Zangmo and the Bhutan Nuns Foundation
Happiness Before Profit: Bhutan Seeks to Redefine Business Using Buddhist Values
A Manifestation of Profound Wisdom and Compassion—Remembering Dilgo Khyentse Rinpoche
An Agent of Change: Empowering Bhutanese Nuns
Ashi Jetsun Pema
Changlimithang Stadium
Druk Gyalpo
Druk Gyaltsuen
drukpa kagyu
GNH
Jigme Namgyel Wangchuck
Jigme Singye Wangchuk
South Asia Tibet / Himalayas / Mongolia People and Personalities | {
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Q: How to acess NavHostController from nested composable Jetpack Compose Assume that my application looks like this
@Composable
fun AppNavigation() {
val navController = rememberNavController()
NavHost(navController, startDestination = Route.Home.route) {
/// other composes
composable("Home") { HomeCompose(navController) }
}
}
@Composable
fun HomeCompose(navController: NavHostController) {
ChildCompose(navController)
}
@Composable
fun ChildCompose(navController: NavHostController) {
navController.navigate("")
}
I want to access navController in nested composable to navigate
but I dont want to pass navController from parent composable to child compsable as above
Is there anyway to access navController from anywhere inside NavHost without passing it through composable hierarchy
Edit: for now, I can use CompositionLocalProvider to access navController in nested compose as below
val AppNavController = compositionLocalOf<NavHostController>() { error("NavHostController error") }
@Composable
fun AppNavigation() {
val navController = rememberNavController()
CompositionLocalProvider(
AppNavController provides navController
) {
NavHost(navController, startDestination = Route.Home.route) {
/// other composes
composable("Home") { HomeCompose() }
}
}
}
@Composable
fun HomeCompose() {
ChildCompose()
}
@Composable
fun ChildCompose(navController: NavHostController) {
val navController = AppNavController.current
Column(modifier = Modifier.clickable {
navController.navigate("Content")
}) {
...
}
}
A: With compose 1.0.0-beta04 and navigation 1.0.0-alpha10 as suggested by the official doc
*
*Pass lambdas that should be triggered by the composable to navigate, rather than the NavController itself.
@Composable
fun ChildCompose(
navigateTo: () -> Unit
) {
//...
navigateTo
}
and to use it:
ChildCompose(navigateTo = {
navController.navigate("...")
})
In this way ChildCompose composable works independently from Navigation
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Royal Air Maroc: Flights to fly in five continents
Do you know the destinations to which Royal Air Maroc flies?
As an African state, it also assigns itself as one of its objectives the achievement of African unity.
Aware of the need to register its action within the framework of international organizations, of which it is an active and dynamic member, the Kingdom of Morocco subscribes to the principles, rights and obligations arising from the Charters of said organizations and reaffirms its attachment to the rights of the 'Man as they are universally recognized.
Likewise, the Kingdom of Morocco reaffirms its determination to work for the maintenance of peace and security in the world.
– Geography
Morocco is located in the northwest of Africa. It is bounded on the north by the Strait of Gibraltar and the Mediterranean Sea, on the south by Mauritania, on the east by Algeria and on the west by the Atlantic Ocean. The Moroccan coast stretches for 3,500 km.
– Demography
The first stage of the exploitation of the General Population and Housing Census of 2004, carried out in Morocco between September 1 and 20, 2004, made it possible to determine the legal population of the country and to study its territorial distribution, both at the level of the two urban and rural areas of residence and at that of the regions, provinces and municipalities. It also made it possible to determine the demographic growth of these territorial entities during the intercensal period 1994-2004.
– Regions
The Kingdom is made up of sixteen regions. The most important region is that of Casablanca. With more than 3.6 million inhabitants, of which only less than 150,000 are rural, the region of Grand Casablanca has 8 prefectures (Casablanca-Anfa, Aïn Sebaa-Hay Mohammadi, Aïn Chok-Hay Hassani, Ben Msik-Sidi Othmane, Al Fida- Derb Soltane-Al Mechouar from Casablanca, Sidi Bernoussi-Zenata and Mohammedia).
Hundreds of destinations are served by RAM
The airline Royal Air Maroc serves destinations all over the world, connecting the interior cities of Morocco to the rest of the world.
With its fleet of fifty-nine planes, Royal Air Maroc regularly serves flights connecting nineteen cities of the kingdom. These are Agadir, Al Hoceima, Beni-Mellal, Casablanca, Dakhla, Essaouira, Fés, Guelmim, Laâyoune, Marrakech, Nador, Ouarzazate, Oujda, Rabat, Tan Tan, Tangier, Tetouan, Zagora and Bouarfa.
Royal air maroc takes you to tempting destinations around the world
Royal Air Maroc serves 94 destinations from its hub at Mohammed V International Airport in Casablanca. In addition to domestic flights through Morocco, the company offers connections to several European destinations, such as Athens, Barcelona, Berlin, Brussels, Copenhagen, Frankfurt, Istanbul, Lisbon, London, Madrid, Milan, Paris, Rome, Venice , Vienna and Zurich. In France alone, there are 11 destinations. The company also serves the Middle East, Africa, the United States and Canada, in addition to the flight to São Paulo.
– In Europe: Amsterdam, Athens, Barcelona, Berlin, Bologna, Bordeaux, Brussels, Copenhagen, Dusseldorf, Frankfurt, Geneva, Gibraltar, Istanbul, Las Palmas, Lisbon, London, Lyon, Madrid, Malaga, Manchester, Marseilles, Milan, Montpellier, Moscow, Munich, Nantes, Naples, Nice, Paris, Port, Rome, Rotterdam, Stockholm, Strasbourg, Tenerife, Toulouse, Turin, Valence, Venice, Vienna and Zurich.
– In Africa: Abidjan, Accra, Alger, Bamako, Bangui, Banjul, Bissau, Brazzaville, Conakry, Cotonou, Dakar, Douala, Freetown, Kinshasa, Lagos, Cairo, Libreville, Lome, Luanda,, Malabo, Monrovia, Niamey, N'jemena, Nouakchott, Ouagadougou, Point-Noire, Praia, Tunis, Yaounde.
– Asia: Amman Beirut Doha Jeddah Beijing Riyadh.
– North America: Boston, Miami, Montreal, New York and Washington.
– South America: Sao Paulo and Rio De Janeiro
One Reply to "Royal Air Maroc: Flights to fly in five continents"
Cheryl Walrath
Please explain cancellation policy if I would like to cancel flight. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,640 |
@class GameCenterGameBriefInfo, GameCenterMyGameView, GiftCenterData, NSString, RankSummaryData, RecentPlayCommModule, RecentPlayRecommendTopic;
@protocol GameCenterMyGameViewDelegate <NSObject>
- (void)onMyGameBannerClick:(NSString *)arg1 AppID:(NSString *)arg2;
- (void)onMyGameCommonModuleClick:(RecentPlayCommModule *)arg1 AppID:(NSString *)arg2;
- (void)onMyGameRankClick:(RankSummaryData *)arg1 AppID:(NSString *)arg2;
- (void)onMyGameGameCircleClick:(RecentPlayRecommendTopic *)arg1 AppID:(NSString *)arg2;
- (void)onMyGameGiftCenterClick:(GiftCenterData *)arg1 AppID:(NSString *)arg2;
- (void)onMyGame:(GameCenterMyGameView *)arg1 downloadGameClick:(GameCenterGameBriefInfo *)arg2;
- (void)onMyGame:(GameCenterMyGameView *)arg1 gameRowClick:(GameCenterGameBriefInfo *)arg2 Index:(long long)arg3;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 80 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,459 |
Cannabis News Journal
Senate Majority Leader Mitch McConnell's Hemp Farming Act of 2018
In a recent breakout session at the American Farm Bureau National Convention in Nashville hemp supporters discussed legislation to remove the crop from the DEA's schedule one substance list. Hemp is only legal in states with certified industrial hemp pilot programs like Kentucky. The federal government currently classifies hemp as an illegal substance due to its similarities to cannabis.
The U.S. Hemp Roundtable, the industry association that joins the nation's leading hemp companies and all of its major grassroots organizations, yesterday lauded U.S. Senate Majority Leader Mitch McConnell upon his announcement of the pending introduction of "The Hemp Farming Act of 2018." Leader McConnell's bill, which is co-sponsored by U.S. Senators Ron Wyden (D-OR) and Rand Paul (R-KY), would permanently remove hemp from regulation as a controlled substance and treat it as an agricultural commodity. Similar legislation, H.R. 3530, the "Industrial Hemp Farming Act of 2017," was introduced last year by Rep. James Comer (R-KY), and has been co-sponsored by 43 of his colleagues, from both sides of the aisle.
"The hemp industry is very grateful to Leader McConnell for his strong leadership over the years on behalf of providing Kentucky farmers – and the whole U.S. agricultural commodity – this exciting new economic opportunity," stated Roundtable President Brian Furnish, an 8th generation tobacco farmer from Cynthiana(KY), who credits the Leader with empowering his transition from tobacco to hemp. "Leader McConnell's persistence and commitment has gotten us to this point – through his work on the 2014 Farm Bill and subsequent legislation that created today's hemp pilot programs. There's no better person to help get us across the finish line."
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U.S. Senate Majority Leader Mitch McConnell (R-KY) and Kentucky Commissioner of Agriculture Ryan Quarles (R-KY) announced on March 26th the impending introduction of legislation in the United States Senate to support Kentucky's hemp industry. The Hemp Farming Act of 2018 will legalize hemp as an agricultural commodity and remove it from the list of controlled substances.
Senator McConnell took the first step to support hemp in 2014 by using his leadership position in the Senate to spearhead a provision to legalize hemp pilot programs in the Farm Bill. Since then, the research has shown the potential of hemp as an agricultural commodity.
"Hemp has played a foundational role in Kentucky's agricultural heritage, and I believe that it can be an important part of our future," Senator McConnell said. "I am grateful to join our Agriculture Commissioner Ryan Quarles in this effort. He and his predecessor, Jamie Comer, have been real champions for the research and development of industrial hemp in the Commonwealth. The work of Commissioner Quarles here in Kentucky has become a nationwide example for the right way to cultivate hemp. I am proud to stand here with him today, because I believe that we are ready to take the next step and build upon the successes we've seen with Kentucky's hemp pilot program."
"Here in Kentucky, we have built the best Industrial Hemp Research Pilot Program in the country and have established a model for how other states can do the same with buy-in from growers, processors, and law enforcement," Commissioner Quarles said. "I want to thank Leader McConnell for introducing this legislation which allows us to harness the economic viability of this crop and presents the best opportunity to put hemp on a path to commercialization."
The Hemp Farming Act of 2018 will help Kentucky enhance its position as the leading state on hemp production. It builds upon the success we have seen through the hemp pilot programs by allowing states to be the primary regulators of hemp, if the U.S. Department of Agriculture approves their implementation plan. This legislation also will remove the federal barriers in place that have stifled the industry, which will help expand the domestic production of hemp. It will also give hemp researchers the chance to apply for competitive federal grants from the U.S. Department of Agriculture – allowing them to continue their impressive work with the support of federal research dollars.
Senator McConnell plans to introduce the bill in the Senate, with Senator Rand Paul and a bipartisan group of members, following this state work period. McConnell acknowledged there was "some queasiness" about hemp in 2014 when federal lawmakers cleared the way for states to regulate it for research and pilot programs. There's much broader understanding now that hemp is a "totally different" plant than its illicit cousin, he said to WBKO News 13 in Kentucky.
"I think we've worked our way through the education process of making sure everybody understands this is really a different plant," the Republican leader said.
McConnell said he plans to have those discussions with Attorney General Jeff Sessions to emphasize the differences between the plants. The Trump administration has taken a tougher stance on all things cannabis.
McConnell said his bill will attract a bipartisan group of co-sponsors and that language of the legislation will be similar to the Hemp Farming Act of 2017. He said the measure would allow states to have primary regulatory oversight of hemp production if they submit plans to federal agriculture officials outlining how they would monitor production.
At least 34 states passed legislation related to industrial hemp. State policymakers have taken action to address various policy issues — the definition of hemp, licensure of growers, regulation and certification of seeds, state-wide commissions and legal protection of growers. Some states establishing these programs require a change in federal laws or a waiver from the DEA prior to implementation.
Thirty eight states and Puerto Rico considered legislation related to industrial hemp in 2017. These bills ranged from clarifying existing laws to establishing new licensing requirements and programs. At least 15 states enacted legislation in 2017 — Arkansas, Colorado, Florida, Hawaii, North Dakota, Nevada, New York, Oregon, South Carolina, Tennessee, Virginia, Washington, West Virginia, Wisconsin and Wyoming. Florida, Wisconsin and Nevada authorized new research or pilot programs. The governors of Arizona and New Mexico vetoed legislation, which would have established new research programs.
(New Mexico, Senate Bill 6, Industrial Hemp Research -Latest Details)
On Tuesday, January 2, 2018, New Mexico's Supreme Court blocked District Judge Sarah Singleton's decision that determined that Governor Martinez did not legally veto 10 bills passed during the 2017 legislative session. The January 2 ruling puts a hold on the promulgation of all 10 bills until the court has a 'full and fair opportunity' to consider the case.
The 10 bills, including two bills legalizing hemp research, were passed by legislators during the 2017 legislative session and then vetoed by Governor Martinez. The Governor's veto was challenged in District Court with Judge Sarah Singleton ruling that the 10 bills were not vetoed through a legal process, and therefore should be promulgated. The Governor's office then challenged Judge Singleton's ruling in the New Mexico Supreme Court. The recent Supreme Court decision puts the 10 bills, including hemp, back in legal limbo until the court address the issue at a later date. A full story can be found at the following link.
While hemp and cannabis products both come from the cannabis plant, hemp is typically distinguished by its use, physical appearance and lower concentration of tetrahydrocannabinol (THC). Hemp producers often grow the plant for the one or more parts — seeds, flowers and stalk. The plant is cultivated to grow taller, denser and with a single stalk.
State statutes, with the exception of West Virginia, define industrial hemp as a variety of cannabis with a THC concentration of not more than 0.3 percent. West Virginia defines hemp as cannabis with a THC concentration of less than 1 percent.
Many state definitions for industrial hemp specify that THC concentration is on a dry weight basis and can be measured from any part of the plant. Some states also require the plant to be possessed by a licensed grower for it to be considered under the definition of industrial hemp.
Map of State Laws Related to Industrial Hemp (Click Here)
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{"url":"https:\/\/www.researcher-app.com\/paper\/1950926","text":"3 years ago\n\n# Age, metallicity and star formation history of spheroidal galaxies in cluster at z~1.2.\n\nP. Saracco, F. La Barbera, A. Gargiulo, F. Mannucci, D. Marchesini, M. Nonino, P. Ciliegi\n\nWe present the analysis, based on spectra collected at the Large Binocular Telescope, of the stellar populations in seven spheroidal galaxies in the cluster XLSSJ0223 at $z$\\sim$1.22. The aim is to constrain the epoch of their formation and their star formation history. Using absorption line strenghts and full spectral fitting, we derive for the stellar populations of the seven spheroids a median age <Age>=2.4$\\pm$0.6 Gyr, corresponding to a median formation redshift %CONTENT%lt;$z_f$>$\\sim2.6_{-0.5}^{+0.7}$(lookback time = 11$_{-1.0}^{+0.6}$Gyr). We find a significant scatter in age, showing that massive spheroids, at least in our targeted cluster, are not coeval. The median metallicity is [Z\/H]=0.09$\\pm$0.16, as for early-types in clusters at 0%CONTENT%lt;$z%CONTENT%lt;0.9. This lack of evolution of [Z\/H] over the range 0%CONTENT%lt;$z$<$1.3, corresponding to the last 9 billions years, suggests that no significant additional star formation and chemical enrichment are required for cluster spheroids to reach the present-day population. We do not detect significant correlation between age and velocity dispersion$\\sigma_e$, or dynamical mass M$_{dyn}$, or effective stellar mass density$\\Sigma_e$. On the contrary, the metallicity [Z\/H] of the seven spheroids is correlated to their dynamical mass M$_{dyn}$, according to a relation similar to the one for local spheroids. [Z\/H] is also anticorrelated to stellar mass density$\\Sigma_e$because of the anticorrelation between M$_{dyn}$and$\\Sigma_e$. Therefore, the basic trends observed in the local universe were already established at$z\\sim1.3\\$, i.e. more massive spheroids are more metal rich, have lower stellar mass density and tend to be older than lower-mass galaxies.\n\nPublisher URL: http:\/\/arxiv.org\/abs\/1901.01595\n\nDOI: arXiv:1901.01595v2\n\nYou might also like\nDiscover & Discuss Important Research\n\nKeeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.\n\nResearcher displays publicly available abstracts and doesn\u2019t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.","date":"2022-08-09 08:11:32","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\": 0.31291520595550537, \"perplexity\": 5547.823392674478}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570913.16\/warc\/CC-MAIN-20220809064307-20220809094307-00542.warc.gz\"}"} | null | null |
\section{Introduction}
In the study of porous media, composite materials and other physical and engineering systems, one is led to the initial or boundary value problems with periodic structures (see, e.g., \cite{All02,CP99,OSY92}). The process of passing from a microscopic description to a macroscopic description of the behaviors of such systems is called \emph{homogenization}. At present, numerous publications can be found on the mathematical aspects of the homogenization theory (see \cite{BJ16,BFF17,BLM19}).
The goal of this paper is to use a \emph{probabilistic approach} to study the limit behavior, as $\epsilon\to 0$, of the solution $u^\epsilon:\R^d\to\R$ of the following \emph{nonlocal} partial differential equation (PDE) of parabolic-type with rapidly oscillating periodic and singular coefficients,
\begin{equation}\label{ue}
\begin{cases}
\frac{\partial u^\epsilon}{\partial t}(t,x) = \L_\e^\alpha u^\epsilon(t,x)+\left( \frac{1}{\epsilon^{\alpha-1}} e\left(\frac{x}{\epsilon}\right)+ g\left(\frac{x}{\epsilon}\right) \right)u^\e(t,x), & t>0,x\in\R^d, \\
u^\epsilon(0,x) = u_0(x), & x\in\R^d,
\end{cases}
\end{equation}
where $1<\alpha<2$ and the linear operator $\L_\e^\alpha$ is a nonlocal integro-differential operator of L\'evy-type given by
\begin{equation*}
\begin{split}
\L_\e^\alpha f(x) :=&\ \int_{\Ro d} \left[ f\left( x+\sigma\left(\frac{x}{\e},y\right)\right)-f(x)- \sigma^i\left(\frac{x}{\e},y\right) \partial_i f(x) \ind_{B}(y) \right] \nu^\alpha(dy) \\
& + \left[\frac{1}{\e^{\alpha-1}} b^i\left(\frac{x}{\e}\right) + c^i\left(\frac{x}{\e}\right) \right] \partial_i f(x),\quad x\in\R^d.
\end{split}
\end{equation*}
Here $B$ is the unit open ball in $\R^d$ centering at the origin, and $\nu^\alpha(dy):=\frac{dy}{|y|^{d+\alpha}}$ is the isotropy $\alpha$-stable L\'evy measure.
In this paper, we use Einstein's convention that the repeated indices in a product will be summed automatically.
For notational simplicity, we introduce the linear operator $\A^{\sigma,\alpha}$ defined by
\begin{equation}\label{A}
\A^{\sigma,\nu^\alpha} f(x) := \int_{\Ro d} \big[ f( x+\sigma(x,y))-f(x) - \sigma^i(x,y) \partial_i f(x)\ind_B(y) \big] \nu^\alpha(dy),\quad x\in\R^d.
\end{equation}
For a function $f$ on $\R^d$ (or $F$ on $\R^d\times\R^d$), we denote $f_\epsilon(x):=f\left(\frac{x}{\e}\right)$ (or $F_\epsilon(x,y):=F\left(\frac{x}{\e},y\right)$). Then
\begin{equation}\label{rewrite_Le}
\L^\alpha_\e=\A^{\sigma_\e,\nu^\alpha}+\left( \frac{1}{\e^{\alpha-1}} b_\e +c_\e\right)\cdot\nabla.
\end{equation}
The main result of this paper is the following theorem. Assumptions \ref{coef1}--\ref{center} are made for the coefficients and will be listed in the sequel. We will prove it at the end of Section \ref{Homogenization}.
\begin{theorem}\label{HomoPDE}
Under Assumptions \ref{coef1}--\ref{center}, the nonlocal PDE \eqref{ue} has a unique mild solution $u^\e$ for each $\e>0$. Moreover, for each $t\ge 0,x\in\R^d$,
\begin{equation}\label{u_conv}
u^\epsilon(t,x)\to u(t,x), \quad\epsilon\to 0,
\end{equation}
where $u$ satisfies the limit nonlocal PDE,
\begin{equation*}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) = \A^{\id_y,\Pi}u(t,x)+\bar C\cdot\nabla u(t,x)+\bar Eu(t,x), & t>0,x\in\R^d, \\
u(0,x) = u_0(x), & x\in\R^d,
\end{cases}
\end{equation*}
where
\begin{equation*}
\A^{\id_y,\Pi} f(x) := \int_{\Ro d} \left[ f( x+y)-f(x)- y^i \partial_i f(x)\ind_B(y) \right] \Pi(dy),
\end{equation*}
and the constant coefficients $\bar C,\bar E$ and measure $\Pi$ are given by \eqref{HomoCoef1}, \eqref{HomoCoef2} and \eqref{homo_nu} respectively. The solution is given by
\begin{equation*}
u(t,x)=\E[u_0(x+\bar Ct+L_t)]e^{\bar Et},
\end{equation*}
where $\{L_t\}_{t\ge0}$ is a symmetric $\alpha$-stable L\'evy processes with jump intensity measure $\Pi$.
\end{theorem}
The original probabilistic approach to the homogenization of \emph{local} linear second order parabolic partial differential operators is presented in \cite[Chapter 3]{BLP78}, which is based on the ergodic theorem, the Feynman-Kac formula and the functional central limit theorem. By now, there are lots of literature concerning the homogenization of second order local PDEs, i.e., the case of replacing the operator $\A^{\sigma_\e,\nu^\alpha}$ in \eqref{rewrite_Le} by a second order partial differential operator with singular coefficients. Two different scales of spatial variables involved in the coefficients have been considered in \cite{BP07}, by using the nonlinear Feynman-Kac formula in the context of backward stochastic differential equations (SDEs). In \cite{PP12}, the authors allowed the singular coefficients to be time-dependent and rapidly oscillating in time with a different scale in contrast to the spatial variable. The paper \cite{HP08} dealt with the case when the second order coefficient matrix can be degenerate, using the existence of a spectral gap and Malliavin's calculus.
There are also some literature for the homogenization of nonlocal PDEs or SDEs with jumps involved. We refer the reader to \cite{PZ17,Sch10} for the periodic homogenization results of some kinds of nonlocal operators involving stable-like terms or convolution type kernels. The methods used in these papers are all analytic. The probabilistic study of homogenization of periodic stable-like processes in pure jump or jump-diffusion case can be found in \cite{Fra06,San16}. The homogenization in random medium is slightly different from the periodic case, referring to \cite{RS09} for related results for jump-diffusion processes in random medium.
The homogenization of a kind of one-dimensional pure jump Markov processes with the following form of generators has been investigated in the paper \cite{HIT77},
\begin{equation*}
\A_\e f(x) = \int_{\R\setminus\{0\}} \left[ f(x+z)-f(x)-zf'(x) \right] a\left(\DivEps x,\DivEps z\right) \frac{dz}{|z|^{1+\alpha}} + \frac{1}{\e^{\alpha-1}} b\left(\frac{x}{\e}\right)f'(x).
\end{equation*}
See \cite{Tom92} for a generalization for multi-dimensional case and with diffusion terms involved. In their context, the jump function $h$ is oscillating both in the spatial variable $x$ and in the noise variable $y$, and they are in the same scale. This means the noise comes from the underlying space of periodic medium. But when we do homogenization for systems with fluctuations, the noise usually comes from the external environment, so that the jump function is no longer oscillating in the noise variable, and there happens to be two different scales. As we will see in Section \ref{Pre&Ass}, the change of variables allows us to write
\begin{equation*}
\begin{split}
\L_\e^\alpha f(x) :=&\ \int_{\Ro d} \left[ f( x+z)-f(x)- z^i \partial_i f(x) \ind_{B}(y) \right] h\left(\DivEps x,z\right)\frac{dz}{|z|^{d+\alpha}} \\
& + \left[\frac{1}{\e^{\alpha-1}} b^i\left(\frac{x}{\e}\right) + c^i\left(\frac{x}{\e}\right) \right] \partial_i f(x),
\end{split}
\end{equation*}
for some function $h$. Note that the main difference is that we do not involve oscillations for $h$ in its noise variable, while \cite{HIT77,Tom92} involve. Meanwhile, the coefficients of drift and the zeroth order term $b,c,e,g$ has two different scales.
In paper \cite{Fra07}, the author considered the homogenization of SDEs driven by multiplicative stable processes, where the noise intensity coefficient $\sigma$ is linear in the noise variable in the sense that $\sigma(x,y)=\sigma_0(x)y$, with $\sigma_0$ three-times continuously differentiable. In the present paper, we generalize his results to the general multiplicative case. That is, the intensity function $\sigma$ need \emph{not to be linear} for the noise component. This is also more realistic in applications (see, e.g., \cite{SY04,BHR15}). For some typical forms of $\sigma$ that are nonlinear in $y$, see Example \ref{exmp}. In addition, the coefficients only need to possess some H\"older or Lipschitz continuity in our context (see Remark \ref{sigma-reg} for the comparison of the regularity assumptions for $\sigma$). This will give rise to several difficulties both in analytic and probabilistic aspects. We also use the homogenization results of SDEs to study the homogenization of the nonlocal PDEs with singular coefficients involved, by utilizing Feynman-Kac formula. The trick we use to remove the singular drift in \eqref{ue} is now known as \emph{Zvonkin's transform}, which appeared originally in \cite{Zvo74}.
The work in this paper is highly motivated by these two considerations. That is, the noise in our homogenization problem comes from the external environment instead of the underlying periodic medium, and is not necessarily linear. Both are more suitable from the practical point of view than in earlier papers. Under these considerations, we further weaken the regularity assumptions for all coefficients in a compatible way.
We denote by $\C^k$ ($\C_b^k$) with integer $k\ge0$ the space of (bounded) continuous functions possessing (bounded) derivatives of orders not greater than $k$.
We shall explicitly write out the domain if necessary. Denote by $\C_b(\R^d):=\C_b^0(\R^d)$, it is a Banach space with the supremum norm $\|f\|_0=\sup_{x\in\R^d}|f(x)|$. The space $\C_b^k(\R^d)$ is a Banach space endowed with the norm $\|f\|_k=\|f\|_0+\sum_{j=1}^k \|\nabla^{\otimes j}f\|$. We also denote by $\C^{1-}$ the class of all Lipschitz continuous functions. For a non-integer $\gamma>0$, the H\"older spaces $\C^\gamma$ ($\C^\gamma_b$) are defined as the subspaces of $\C^{\lfloor\gamma\rfloor}$ ($\C^{\lfloor\gamma\rfloor}_b$) consisting of functions whose $\lfloor\gamma\rfloor$-th order partial derivatives are locally H\"older continuous (uniformly H\"older continuous) with exponent $\gamma-\lfloor\gamma\rfloor$. These two spaces $\C^\gamma$ and $\C^\gamma_b$ obviously coincide when the underlying domain is compact. The space $\C^\gamma_b(\R^d)$ is a Banach space endowed with the norm $\|f\|_\gamma=\|f\|_{\lfloor\gamma\rfloor}+[\nabla^{\lfloor\gamma\rfloor}f]_{\gamma-\lfloor\gamma\rfloor}$, where the seminorm $[\cdot]_{\gamma'}$ with $0<\gamma'<1$ is defined as $[f]_{\gamma'}:=\sup_{x,y\in\R^d,x\ne y}\frac{|f(x)-f(y)|}{|x-y|^{\gamma'}}$. In the sequel, the torus $\T^d:=\R^d/\Z^d$ will be used frequently. Denote by $\D:=\D(\R_+;\T^d)$ the space of all $\T^d$-valued c\`adl\`ag functions on $\R_+$, equipped with the Skorokhod topology. We shall always identify the periodic function on $\R^d$ of period 1 with its restriction on the torus $\T^d=\R^d/\Z^d$. This allows us to regard the space $\C^k(\T^d)$ ($\C^\gamma(\T^d)$) as a sub-Banach space of $\C_b^k(\R^d)$ ($\C_b^\gamma(\R^d)$).
By $B_r$ we means the open ball in $\R^d$ centering at the origin with radius $r>0$, we shall omit the subscript when the radius is one. The capital letter $C$ denotes a finite positive constant whose value may vary from line to line. We also use the notation $C(\cdots)$ to emphasize the dependence on the quantities appearing in the parentheses.
The remainder of the paper is organized as follows. In Section \ref{Pre&Ass}, we present some general assumptions and preliminary results. In Section \ref{NonlocalPoisson}, we study the well-posedness of the nonlocal Poisson equation and the Feller properties of the semigroup associated with $\L^\alpha$. Section \ref{SDEs} is devoted to the strong well-posedness and exponential ergodicity of the L\'evy driven SDE with generator $\L^\alpha$. As a consequence, we obtained the Feynman-Kac representation for the nonlocal PDE \eqref{ue}. Lastly, Section \ref{Homogenization} contains the homogenization results of SDEs and nonlocal PDEs, utilizing the ergodicity and the Feynman-Kac representation.
\section{Preliminaries and general assumptions}\label{Pre&Ass}
Let $(\Omega,\F,\P,\{\F_t\}_{t\ge0})$ be a filtered probability space endowed with a Poisson random measure $N^\alpha$ on $(\Ro d)\times\R_+$ with jump intensity measure $\nu^\alpha(dy)=\frac{dy}{|y|^{d+\alpha}}$, where $1<\alpha<2$. Denote by $\tilde N$ the associated
compensated Poisson random measure, that is, $\tilde N^\alpha(dy,ds):=N^\alpha(dy,ds)-\nu^\alpha(dy)ds$. We assume that the filtration $\{\F_t\}_{t\ge0}$ satisfies the usual conditions. Let $L^\alpha=\{L_t^\alpha\}_{t\ge 0}$ be a $d$-dimensional isotropic $\alpha$-stable L\'evy process given by
\begin{equation*}
L_t^\alpha=\int_0^t\int_{\Bo}y\tilde N^\alpha(dy,ds)+ \int_0^t\int_{B^c}y N^\alpha(dy,ds).
\end{equation*}
Given $\epsilon> 0, x\in\R^d$, consider the following:
\begin{equation}\label{Xe}
dX_t^{x,\epsilon} = \left(\frac{1}{\epsilon^{\alpha-1}} b\left(\frac{X_{t}^{x,\epsilon}}{\epsilon}\right)+ c\left(\frac{X_{t}^{x,\epsilon}}{\epsilon}\right)\right)dt + \sigma\left(\frac{X_{t-}^{x,\epsilon}}{\epsilon},dL_t^{\alpha}\right), \quad X_0^{x,\epsilon}=x,
\end{equation}
or more precisely,
\begin{equation*}
\begin{split}
X_t^{x,\epsilon} = &\ x+\int_0^t \left(\frac{1}{\epsilon^{\alpha-1}} b\left(\frac{X_{s}^{x,\epsilon}}{\epsilon}\right)+ c\left(\frac{X_{s}^{x,\epsilon}}{\epsilon}\right)\right)ds \\
&\ +\int_0^t\int_{\Bo}\sigma\left( \frac{X_{s-}^{x,\epsilon}}{\epsilon},y\right) \tilde N^\alpha(dy,ds)+ \int_0^t\int_{B^c}\sigma \left(\frac{X_{s-}^{x,\epsilon}}{\epsilon},y\right) N^\alpha(dy,ds),
\end{split}
\end{equation*}
where the coefficients $b,c,\sigma(\cdot,y)$ are periodic, for each $y\in\R^d$, of periodic one in each component. The shorthand notation for the stochastic differential term in \eqref{Xe} is due to \cite{Mas07}.
Define $\tilde X_t^{x,\epsilon} :=\frac{1}{\epsilon}X^{x,\epsilon} _{\epsilon^\alpha t}$. It is easy to check that
\begin{equation}\label{Xe_tilde}
d\tilde X_t^{x,\epsilon} = \left( b(\tilde X_{t}^{x,\epsilon})+ \epsilon^{\alpha-1} c(\tilde X_{t}^{x,\epsilon})\right)dt+ \DivEps 1
\sigma\left(\tilde X_{t-}^{x,\epsilon},\e d\tilde L_t^{\alpha}\right), \quad \tilde X_0^{x,\epsilon}=\frac{x}{\epsilon},
\end{equation}
where $\{\tilde L_t^\alpha\}:=\{\frac{1}{\epsilon}L^\alpha_{\epsilon^\alpha t}\}\stackrel{\mathtt d}{=}\{L_t^\alpha\}$ by virtue of the selfsimilarity. We shall also consider the ``limit'' equation, namely
\begin{equation}\label{X_tilde}
d\tilde X_t^x = b(\tilde X^x_{t})dt+
\sigma\left(\tilde X^x_{t-},d\tilde L_t^{\alpha}\right), \quad \tilde X_0^x=x.
\end{equation}
For notational simplicity, we shall allow the parameter $\e$ to be zero in $\tilde X^{x,\e}$ to include $\tilde X^x$, i.e., $\tilde X^{x,0}:=\tilde X^x$.
In the sequel, we will regard the solutions $\tilde X^{x,\e},\tilde X^x$ of \eqref{Xe_tilde} and \eqref{X_tilde} as $\T^d$-valued processes, by mapping all trajectories of the processes on $\R^d$ to the torus $\T^d$, via the canonical quotient map $\pi:\R^d\to\R^d/\Z^d$. Then the periodicity of the coefficients implies that $\tilde X^{x,\e}$ and $\tilde X$ are well-defined stochastic processes on $\T^d$ (cf. \cite[Section 3.3.2]{BLP78}).
Now we list some general assumptions for the nonlocal PDE \eqref{ue} and the SDE \eqref{Xe}. All these assumptions are assumed to hold in the sequel unless otherwise specified.
\begin{assumption}\label{coef1}
The functions $b,c,e,g,u_0$ are all periodic of period 1 in each component. For every $y\in\R^d$, the function $x\to\sigma(x,y)$ is periodic of period 1 in each component.
\end{assumption}
\begin{assumption}\label{regular}
The functions $b,c,e$ are of class $\C_b^\beta$ with exponent $\beta$ satisfying
\begin{equation*}
1-\frac{\alpha}{2}<\beta<1.
\end{equation*}
The functions $g$ and $u_0$ are both continuous.
\end{assumption}
\begin{assumption}\label{sigma}
The function $\sigma:\R^d\times\R^d\to\R^d$ satisfies the following conditions.
(1). \emph{Regularity}. For every $x\in\R^d$, the function $y\to\sigma(x,y)$ is of class $\C^2$. There exists a constant $C>0$, such that for any $x_1,x_2,y\in\R^d$,
\begin{equation*}
|\sigma(x_1,y)-\sigma(x_2,y)|\le C|x_1-x_2| |y|.
\end{equation*}
(2). \emph{Oddness}. For all $x,y\in\R^d$, $\sigma(x,-y)=-\sigma(x,y)$.
(3). \emph{Bounded inverse Jacobian}. The Jacobian matrix with respect to the second variable $\nabla_y\sigma(x,y)$ is non-degenerate for all $x,y\in\R^d$, and there exists a constant $C>0$ such that $|(\nabla_y\sigma(x,y))^{-1}| \le C$ for all $x,y\in\R^d$, where $|\cdot|$ is the operator norm on $\mathscr L(\R^d,\R^d)$.
(4). \emph{Growth condition}. There exists a positive bounded measurable function $\phi:\R^d\to\R_+$, such that for all $x,y\in\R^d$,
\begin{equation*}
\phi(x)^{-1}|y| \le |\sigma(x,y)| \le \phi(x)|y|.
\end{equation*}
\end{assumption}
\begin{remark}\label{rem_sigma}
Some comments on our assumptions will be helpful:
(1). As mentioned in the end of the introduction, $b,c,e,g,u_0$ and the function $x\to\sigma(x,y)$, for every $y\in\R^d$, can be regarded as functions on $\T^d$, and we have $b,c,e\in\C^\beta(\T^d)$, $g,u_0\in\C(\T^d)$, under Assumptions \ref{coef1} and \ref{regular}.
(2). Both the oddness and the growth condition in Assumption \ref{sigma} imply that $\sigma(\cdot,0)\equiv0$.
(3). The bounded inverse Jacobian condition implies that $|\nabla_y\sigma| \ge C^{-1}$. Since by Hadamard's inequality (see, for instance, \cite{Sch70}),
\begin{equation}\label{bij}
|(\nabla_y\sigma)^{-1}| \le C \Rightarrow |\det((\nabla_y\sigma)^{-1})| \le C^d \Leftrightarrow |\det(\nabla_y\sigma)| \ge C^{-d} \Rightarrow |\nabla_y\sigma| \ge C^{-1}.
\end{equation}
(4). The growth condition implies that for any $\gamma>\alpha$, we have
\begin{equation}\label{moment}
\sup_{x\in\R^d}\int_{\Bo} |\sigma(x,y)|^{\gamma} \nu^\alpha(dy)<\infty.
\end{equation}
This ensures that we can apply It\^o's formula to $f(\tilde X^x_t)$ (or $f(\tilde X^{x,\e}_t)$, $f(X^{x,\e}_t)$), for any $f\in\C_b^{\gamma}(\R^d)$ with $\gamma>\alpha$ (cf. \cite[Lemma 4.2]{Pri12}).
(5). By virtue of the oddness condition in Assumption \ref{sigma} and the symmetry of the jump intensity measure $\nu^\alpha$, for any $x \in \R^d$,
\begin{equation}\label{trans}
\text{P.V.}\int_{\sigma(x,\cdot)^{-1}B\setminus B}\sigma^i(x,y)\nu^\alpha(dy)=\text{P.V.}\int_{B\setminus\sigma(x,\cdot)^{-1} B}\sigma^i(x,y)\nu^\alpha(dy)=0.
\end{equation}
Consequently we can rewrite the operator $\A^{\sigma,\nu^\alpha}$ in \eqref{A} as
\begin{equation}\label{A_rewrite}
\A^{\sigma,\nu^\alpha} f(x) = \int_{\Ro d} \left[ f( x+z)-f(x)- z^i \partial_i f(x)\ind_B(z) \right] \nu^{\sigma,\alpha}(x,dz),
\end{equation}
where the kernel $\{\nu^{\sigma,\alpha}(x,\cdot)|x\in\R^d\}$ is given by
\begin{equation}\label{kernel}
\nu^{\sigma,\alpha}(x,A):=\int_{\Ro d}\ind_A(\sigma(x,y))\nu^\alpha(dy), \quad A\in\B(\Ro d).
\end{equation}
Moreover, for any $\gamma>\alpha$, the growth condition in Assumption \ref{sigma} implies that
\begin{equation}\label{Levy-kernel}
\begin{split}
&\ \sup_{x\in\R^d}\int_{\Ro d} (|z|^{\gamma}\wedge1) \nu^{\sigma,\alpha}(x,dz) = \sup_{x\in\R^d}\int_{\Ro d} (|\sigma(x,y)|^{\gamma}\wedge1) \nu^\alpha(dy) \\
\le&\ \sup_{x\in\R^d}\left(\int_{|y|\le\phi(x)} (\phi(x)|y|)^{\gamma} \nu^\alpha(dy) + \int_{|y|\ge\phi(x)^{-1}}\nu^\alpha(dy) \right) \\
\le&\ \frac{1}{\gamma-\alpha} \|\phi\|_{L^\infty}^{2\gamma-\alpha} + \frac{1}{\alpha} \|\phi\|_{L^\infty}^{\alpha} < \infty.
\end{split}
\end{equation}
\qed
\end{remark}
\begin{remark}\label{sigma-reg}
The special case $\sigma(x,y)=\sigma_0(x)y$ with certain $\sigma_0$ is considered in \cite{Fra07}, where the author assumed the function $\sigma_0:\R^d\to \text{GL}(\R^d)$ is periodic and of class $\C^3$. In our context, Assumption \ref{sigma} amounts to saying that
\begin{equation}\label{sigma_0}
\sigma_0:\R^d\to \text{GL}(\R^d) \text{ is periodic and Lipschitz}.
\end{equation}
Since these imply the regularity condition immediately,
the bounded inverse Jacobian and growth conditions are fulfilled by continuity and periodicity, together with the observation $\sup_{x\in\R^d} \|\sigma_0(x)\| \vee \|\sigma_0(x)^{-1}\| < \infty$. The oddness condition is trivial in this case. \qed
\end{remark}
In practice, the noise is not always linear. Here we give some nontrivial examples for $\sigma$, that is, nonlinear in $y$.
\begin{example}\label{exmp}
Suppose $\sigma_0$ to satisfy \eqref{sigma_0}.
(i). The dependence of the noise is a small perturbation of the linear case, namely, $\sigma(x,y)=\sigma_0(x)y+\delta(x,y)$, where the function $\delta$ satisfies the same properties as $\sigma$ but has much smaller scale than $\sigma$.
(ii). Another case is that the function $\sigma$ is separable but not linear in $y$. To be precise, let $\eta:\R^d\to\R^d$ be an odd function of class $\C^2$, satisfying that $\nabla\eta(y)$ is non-degenerate for all $y\in\R^d$, and there exist some constants $C_1,C_2>0$, such that $|\nabla\eta|\ge C_1$ and $C_2^{-1} |y|\le|\eta(y)|\le C_2 |y|$. Now let $\sigma(x,y)=\sigma_0(x)\eta(y)$. Then $\sigma$ satisfies Assumption \ref{sigma}.
(iii). Combining the above two example together, one can obtain a more general example, that is, $\sigma(x,y) = \sigma_0(x)\eta(y)+\delta(x,y)$.
\end{example}
We will need some regularities for the 'partial' inverse of $\sigma$. For a function $F:\R^d\times\R^d\ni(x,y)\to F(x,y)\in\R$, we say $F\in L^\infty_2(\R^d;\C^{1-}_1(\R^d;\R^d))$, if there exists a constant $C>0$ such that for all $x,y \in\R^d$, $|F(x,y)|\le C$, and for all $x_1,x_2,y \in\R^d$, $|F(x_1,y)-F(x_2,y)|\le C|x_1-x_2|$. Then the regularity and growth conditions in Assumption \ref{sigma} imply that the function $(x,y)\to\sigma(x,y)/|y|$ is of class $L^\infty_2(\R^d;\C^{1-}_1(\R^d;\R^d))$.
\begin{lemma}\label{inverse}
Under Assumption \ref{sigma}, for every $x\in\R^d$, the function $y\to\sigma(x,y)$ is a $\C^2$-diffeomorphism. Denote the inverse by $\tau(x,z):=\sigma(x,\cdot)^{-1}(z)$, then for every $z\in\R^d$, the function $x\to\tau(x,z)$ is periodic of period one in each component. Moreover, the function $(x,z)\to\tau(x,z)/|z|$ is of class $L^\infty_2(\R^d;\C^{1-}_1(\R^d;\R^d))$.
\end{lemma}
\begin{proof}
Fix $x\in\R^d$. Since the function $y\to\sigma(x,y)$ is of class $\C^2$, by the bounded inverse Jacobian condition in Assumption \ref{sigma}, together with Hadamard's global inverse function theorem (see \cite[Theorem 6.2.4]{KP12}), $\sigma(x,\cdot)$ is a $\C^2$-diffeomorphism. The periodicity is obvious.
Now using the bounded inverse Jacobian condition, the Jacobian matrix of $\tau(x,z)$ with respect to $z$ satisfies $|\nabla_z\tau(x,z)|\le C$, for all $x,z\in\R^d$. Then by the growth condition and regularity condition, the second assertion follows from the following derivation,
\begin{equation*}
\sup_z\frac{|\tau(x,z)|}{|z|}\le\phi(x),
\end{equation*}
\begin{equation*}
\begin{split}
&\ \sup_z\frac{|\tau(x_1,z)-\tau(x_2,z)|}{|z|} = \sup_y\frac{|\tau(x_1,\sigma(x_1,y))- \tau(x_2,\sigma(x_1,y))|}{|\sigma(x_1,y)|} \\
=&\ \sup_y\frac{|\tau(x_2,\sigma(x_2,y))- \tau(x_2,\sigma(x_1,y))|}{|\sigma(x_1,y)|} \le \|\phi\|_{L^\infty} \|\nabla_z\tau\|_{L^\infty} \sup_y\frac{|\sigma(x_2,y)-\sigma(x_1,y)|}{|y|} \\
\le&\ C \|\phi\|_{L^\infty} \|\nabla_z\tau\|_{L^\infty} |x_1-x_2|.
\end{split}
\end{equation*}
\end{proof}
\begin{assumption}\label{Jacobian}
$\det(\nabla_z\tau) \in L^\infty_2(\R^d;\C^{1-}_1(\R^d;\R))$.
\end{assumption}
This assumption is rather mild, as shown in the following remark.
\begin{remark}
In the case $\sigma(x,y)=\sigma_0(x)y$, the Jacobian of $\tau(x,z)$ with respect to $z$ is $\nabla_z\tau(x,z)\equiv\sigma_0(x)^{-1}$. Then Assumption \ref{Jacobian} reduces to that the function $\det(\sigma_0)^{-1}:\R^d\to\R$ is Lipschitz, which is a direct consequence of \eqref{sigma_0}. When $\sigma(x,y) = \sigma_0(x)\eta(y)$ as in Example \ref{exmp}, $\tau(x,z) = \eta^{-1}(\sigma_0(x)^{-1}z)$. Then it is easy to deduce that Assumption \ref{Jacobian} is implied by the bounded inverse Jacobian condition in Assumption \ref{sigma}, using a similar argument as \eqref{bij}. \qed
\end{remark}
If we let
\begin{equation}\label{h}
h(x,z)=|\det \nabla_z\tau(x,z)| \frac{|z|^{d+\alpha}}{|\tau(x,z)|^{d+\alpha}},
\end{equation}
then by \eqref{kernel}, $\nu^{\sigma,\alpha}(x,dz)=h(x,z)\frac{dz}{|z|^{d+\alpha}}$. Using the growth condition, we also find that for all $x,z\in\R^d$,
\begin{equation}\label{upper_lower_bound}
\|\phi\|_{L^\infty}^{-1}\le\frac{|\tau(x,z)|}{|z|}\le \|\phi\|_{L^\infty}.
\end{equation}
Combining \eqref{h}, \eqref{upper_lower_bound}, Lemma \ref{inverse} and Assumption \ref{Jacobian}, together with the fact that if $f,g\in\C^\gamma$ and $\inf |g|>0$, then $f/g\in\C^\gamma$, we conclude that
\begin{lemma}\label{regularity_h}
Under Assumptions \ref{sigma} and \ref{Jacobian}, $h\in L^\infty_2(\R^d;\C^{1-}_1(\R^d;\R^d))$, namely, there exists a constant $h_0>0$ such that $|h(x_1,z)-h(x_2,z)|\le h_0|x_1-x_2|$ for all $x_1,x_2,z\in\R^d$. Moreover, there also exists a constant $h_1>1$ such that $h_1^{-1}\le h(x,z)\le h_1$ for all $x,z\in\R^d$.
\end{lemma}
In particular, the kernel $\nu^{\sigma,\alpha}$ is comparable to the jump intensity measure of an isotropic $\alpha$-stable process.
\begin{remark}\label{useful-results}
Thanks to Lemma \ref{regularity_h}, the general assumptions in \cite{Bas09,Jin17} are satisfied. Thus, the regularity results and heat kernel estimates therein are available in our context. Actually, these two papers only need that $h\in L^\infty_2(\R^d;\C^\gamma_1(\R^d;\R^d))$ for some $0<\gamma<1$, this is the case by virtue of the natural embedding $\C^{1-}\subset\C^\gamma$. Note that \cite{Bas09} also needs $\alpha+\beta$ not to be an integer, this can be fulfilled by choosing an appropriate $\beta$. \qed
\end{remark}
\section{Nonlocal Poisson equation with zeroth-order term}\label{NonlocalPoisson}
As mentioned in the introduction, we will apply Zvonkin's transform to study the homogenization of SDEs and nonlocal PDEs. Before that, we shall investigate the strong well-posedness of the SDEs presented in the previous section, and Zvonkin's transform will also play an important role in this step (see next section). The key is to consider the following nonlocal Poisson equation with zeroth-order term,
\begin{equation}\label{resovent}
\kappa u-\L^\alpha u = f,
\end{equation}
where $\kappa>0$, and $\L^\alpha$ is the linear integro-partial differential operator given by
\begin{equation}\label{L}
\L^\alpha:=\A^{\sigma,\nu^\alpha}+b\cdot\nabla,
\end{equation}
which may be regarded as the infinitesimal generator of the solution process $\tilde X$ of \eqref{X_tilde} once we prove its well-posedness in the next section.
\subsection{Well-posedness of nonlocal Possoin equation}
We first revisit the maximum principle and the solvability of Poisson equations with zeroth-order term studied in \cite{Pri12}. In this subsection we always assume that Assumptions \ref{regular}, \ref{sigma} and \ref{Jacobian} are in force.
\begin{lemma}\label{MP}
If $u\in\C_b^{1+\gamma}(\R^d), 1+\gamma>\alpha$, is a solution to $\kappa u-\L^\alpha u = f$ with $\kappa>0$ and $f\in\C_b(\R^d)$, then
\begin{equation*}
\kappa\|u\|_0\le \|f\|_0.
\end{equation*}
\end{lemma}
\begin{proof}
Note that the nonlocal operator $\A^{\sigma,\nu^\alpha}$ can be rewritten in the form \eqref{A_rewrite}. For $u\in\C_b^{1+\gamma}(\R^d)$, we have
\begin{equation}\label{difference-estimate-1}
|u(x+z)-u(x)- z\cdot \nabla u(x)| \le |z| \int_0^1 |\nabla u(x+rz)-\nabla u(x)|dr \le \frac{[\nabla u]_\gamma}{1+\gamma}|z|^{1+\gamma}.
\end{equation}
Then by \eqref{Levy-kernel}, there exists a constant $C>0$ such that
\begin{equation*}
\begin{split}
|\L^\alpha u(x)|\le &\ \int_{\Bo} |u(x+z)-u(x)- z\cdot \nabla u(x)| \nu^{\sigma,\alpha}(x,dz) \\
&\ + \int_{B^c} | u(x+z)-u(x)| \nu^{\sigma,\alpha}(x,dz) + |b(x)\cdot\nabla u(x)| \\
\le&\ 2\|u\|_{1+\gamma}\left( \int_{\Ro d} (|z|^{1+\gamma}\wedge1) \nu^{\sigma,\alpha}(x,dz)+\|b\|_0 \right) \\
\le&\ C\|u\|_{1+\gamma}.
\end{split}
\end{equation*}
Based on this estimate, the rest of the proof is exactly the same as that of \cite[Proposition 3.2]{Pri12}, even though it is set up with $\sigma(\cdot,y)\equiv y$ there.
\end{proof}
Now we investigate the solvability of the Poisson equation with a zeroth-order term involved. The results generalize the Schauder estimates in \cite{Pri12} to the anisotropic nonlocal case.
\begin{proposition}\label{res_prob}
For any $\kappa>0$ and $f\in\C_b^\beta(\R^d)$, where $\beta$ is the exponent in Assumptions \ref{regular}, the nonlocal Poisson equation \eqref{resovent} has a unique solution $u=u_\kappa\in\C_b^{\alpha+\beta}(\R^d)$. In addition, there exists a positive constant $C=C(\kappa,\|b\|_\beta)$ such that
\begin{equation}\label{energy1}
\|u_\kappa\|_{\alpha+\beta} \le C (\|u_\kappa\|_0+\|f\|_\beta).
\end{equation}
\end{proposition}
\begin{proof}
The a priori estimate \eqref{energy1} is from \citep[Theorem 7.1, Theorem 7.2]{Bas09}. We thus need to show that the equation \eqref{resovent} has a unique solution $u_\kappa\in\C_b^{\alpha+\beta}(\R^d)$.
Now we prove the existence and uniqueness of solution in $\C_b^{\alpha+\beta}(\R^d)$. It is shown in \cite[Theorem 3.4]{Pri12} that when $\sigma(\cdot,y)\equiv\id_y(y):=y$, the existence and uniqueness hold in $\C_b^{\alpha+\beta}(\R^d)$.
For the general $\sigma$, we apply the method of continuity (see \cite[Section 5.2]{GT01}).
Define a family of linear operators by $\L_\theta:=\theta\A^{\sigma,\nu^\alpha}+ (1-\theta)\A^{\id_y,\nu^\alpha}+b\cdot\nabla$. We consider the family of equations:
\begin{equation}\label{L_theta}
\kappa u-\L_\theta u=f.
\end{equation}
We can also rewrite the nonlocal term in $\L_\theta$ into the form \eqref{A_rewrite}, with the kernel given by $\nu_\theta:=\theta\nu^{\sigma,\alpha}+(1-\theta)\nu^\alpha$. Then the a priori estimate \eqref{energy1} also holds for $u_\theta$ (cf. Remark \ref{useful-results}). As a result, the operator $\L_\theta$ can be considered as a bounded linear operator from the Banach space $\C_b^{\alpha+\beta}(\R^d)$ into the Banach space $\C_b^{\beta}(\R^d)$.
Note that $\L_0=\A^{\id_y,\nu^\alpha}+b\cdot\nabla$, which is the case considered in \cite{Pri12}, and $\L_1=\L^\alpha$.
The solvability of the equation \eqref{resovent} for any $f\in\C_b^\beta(\R^d)$ is then equivalent to the invertibility of the operator $\L_\theta$. We can see from the proof of Lemma \ref{MP} that $\|u_\theta\|_0 \le C \|f\|_0$. Then together with the estimate \eqref{energy1} for $u_\theta$, we have the bound
\begin{equation*}
\|u_\theta\|_{\alpha+\beta} \le C \|f\|_\beta,
\end{equation*}
with the constant $C$ independent of $\theta$. Since, as discussed in \cite{Pri12}, the operator $\L_0=\A^{\id_y,\nu^\alpha}+b\cdot\nabla$ maps $\C_b^{\alpha+\beta}(\R^d)$ onto $\C_b^{\beta}(\R^d)$, the method of continuity is applicable and the result follows.
\end{proof}
\begin{remark}
If we take the periodicity assumption \ref{coef1} into account, then we can slightly strengthen the conclusions in Proposition \ref{res_prob}. That is, if $f\in\C^\beta(\T^d)$, then the unique solution of \eqref{resovent} is of class $\C^{\alpha+\beta}(\T^d)$. \qed
\end{remark}
\subsection{Feller property}
In this subsection, we will study further the operator $\L^\alpha$. It turns out that it is the generator of a Feller semigroup. As a corollary, the solution of equation \eqref{resovent} can be represented in terms of a semigroup, and satisfies a finer estimate. All these results will be used in the next section.
\begin{lemma}\label{Feller}
The linear operator $(\L^\alpha,D(\L^\alpha))$, $D(\L^\alpha)=\C^{\alpha+\beta}(\T^d)$, defined on the Banach space $(\C(\T^d),\|\cdot\|_0)$, is closable and dissipative, its closure generates a Feller semigroup $\{P_t\}_{t\ge0}$ on $\C(\T^d)$.
\end{lemma}
\begin{proof}
Using \eqref{difference-estimate-1} with $\gamma=\alpha+\beta-1$, one can find that for $u\in\C^{\alpha+\beta}(\T^d)$,
\begin{equation*}
\big|u( x+\sigma(x,y))-u(x)- \sigma(x,y)\cdot \nabla u(x)\big| \le \frac{[\nabla u]_{\alpha+\beta-1}}{\alpha+\beta} |\sigma(x,y)|^{\alpha+\beta}.
\end{equation*}
Combining this with \eqref{moment}, a straightforward application of the dominated convergence theorem yields that $\lim_{y\to x}\L^\alpha u(y)=\L^\alpha u(x)$ for any $u\in\C^{\alpha+\beta}(\T^d)$ and $x\in\T^d$. This amounts to saying that $\L^\alpha(\C^{\alpha+\beta}(\T^d))\subset\C(\T^d)$. Therefore, the operator
\begin{equation*}
\L^\alpha:\C(\T^d)\supset\C^{\alpha+\beta}(\T^d)\to\C(\T^d)
\end{equation*}
is a densely defined unbounded operator on $\C(\T^d)$.
Now Lemma \ref{MP} implies that for any $\kappa>0$ and $u\in\C^{\alpha+\beta}(\T^d)$, $\|(\kappa-\L^\alpha)u\|_0\ge\kappa\|u\|_0$, that is, $\L^\alpha$ is dissipative. By Proposition \ref{res_prob}, we have $\C^{\beta}(\T^d)\subset (\kappa-\L^\alpha)(\C^{\alpha+\beta}(\T^d))$ for any $\kappa>0$, which yields that the operator $\kappa-\L^\alpha$ has dense range in $\C(\T^d)$. In addition, $\L^\alpha$ satisfies the positive maximum principle, due to the equivalent form \eqref{A_rewrite} of $\A^{\sigma,\nu^\alpha}$ and Courr\`ege's theorem (see \cite[Corollary 4.5.14]{Jac01}). Now the final assertion follows form the celebrate Hille-Yosida-Ray Theorem (see, for instance, \cite[Theorem 4.2.2]{EK09}).
\end{proof}
Let us recall the notion of martingale problem (see \cite[Section 4.3]{EK09}). First recall that $\D=\D(\R_+;\T^d)$ is the space of all $\T^d$-valued c\`adl\`ag functions on $\R_+$, equipped with the Skorokhod topology. Let $w_t(\omega)=\omega(t),\omega\in\D$, be the coordinate process on $(\D,\B(\D))$, and $\{\F^w_t\}_{t\ge0}:=\sigma(w_s:0\le s\le t)$ be the canonical filtration. Given a probability measure $\nu$ on $\T^d$, we say that a probability measure $\P^\nu$ on $(\D,\B(\D))$ is a solution of the martingale problem for $(\L^\alpha,\nu)$, if $\P^\nu\circ w_0^{-1}=\nu$ and the process
\begin{equation*}
M^f(t):=f(w_t)-f(w_0)-\int_0^t \L^\alpha f(w_s) ds
\end{equation*}
is a $(\D,\B(\D),\{\F^w_t\}_{t\ge0},\P^\nu)$-martingale, for any $f\in D(\L^\alpha)=\C^{\alpha+\beta}(\T^d)$. We denote by $\delta_x$ the Dirac measure, or equivalently, the Dirac function as distribution, focusing on $x\in\R^d$.
\begin{lemma}\label{martingale-prob}
For every $x\in\T^d$, the martingale problem for $(\L^\alpha,\delta_x)$ has a unique solution $\P^x$. Moreover, the coordinate process $\{w_t\}_{t\ge0}$ is a Feller process with generator the closure of $(\L^\alpha,\C^{\alpha+\beta}(\T^d))$, and has a jointly continuous transition probability density $p(t;x,y)$, i.e., $\P^x(w_t\in A)=\int_A p(t;x,y)dy$, $A\in\B(\T^d)$, which satisfies for each $T>0$,
\begin{equation*}
C_1^{-1}\sum_{j\in\Z^d}\left( \frac{t}{|x-y+j|^{d+\alpha}}\wedge t^{-\frac{d}{\alpha}} \right) \le p(t;x,y)\le C_1\sum_{j\in\Z^d}\left( \frac{t}{|x-y+j|^{d+\alpha}}\wedge t^{-\frac{d}{\alpha}} \right),
\end{equation*}
\begin{equation*}
|\nabla_x p(t;x,y)| \le C_2 t^{-\frac{1}{\alpha}}\sum_{j\in\Z^d}\left( \frac{t}{|x-y+j|^{d+\alpha}}\wedge t^{-\frac{d}{\alpha}} \right),
\end{equation*}
for all $x,y\in\R^d$ and $t\in(0,T]$, where $C_1>1, C_2>0$ are two constants depending on $d,\alpha, \|b\|_0, h_0,h_1$. The constants $h_0,h_1$ are related to the function $h$ as in Lemma \ref{regularity_h}.
\end{lemma}
\begin{proof}
The existence of solution of the martingale problem is in \cite[Proposition 3]{MP14}. Taking Lemma \ref{Feller} into account, the uniqueness and the Feller property follow from \cite[Theorem 4.4.1]{EK09}. The existence of transition density and the two estimates can be found in \cite[Theorem 1.4]{Jin17}.
\end{proof}
\begin{remark}\label{rep-Feller}
(1). Combining Lemma \ref{Feller} and Lemma \ref{martingale-prob}, we see that the Feller semigroup $\{P_t\}_{t\ge0}$ generated by the closure of $\L^\alpha$ has the representation
\begin{equation*}
P_t f(x)=\int_{\T^d} f(y) p(t;x,y) dy, \quad f\in \C(\T^d),
\end{equation*}
and the following gradient estimate holds
\begin{equation}\label{gradient-estimate}
\begin{split}
|\nabla P_tf(x)| \le &\ C_2 \|f\|_0 t^{-\frac{1}{\alpha}} \int_{\T^d} \sum_{j\in\Z^d}\left( \frac{t}{|y+j|^{d+\alpha}}\wedge t^{-\frac{d}{\alpha}} \right) dy \\
= &\ C_2 \|f\|_0 t^{-\frac{1}{\alpha}} \int_{\R^d} \left( \frac{t}{|y|^{d+\alpha}}\wedge t^{-\frac{d}{\alpha}} \right) dy \\
\le & \ C_2 \left(1+\frac{1}{\alpha}\right) \|f\|_0 t^{-\frac{1}{\alpha}}.
\end{split}
\end{equation}
(2). Denote the formal generator of $\tilde X^{x,\e}$ by $\tilde\L^\alpha_\e$, i.e.,
\begin{equation*}
\begin{split}
\tilde\L_\e^\alpha f(x) :=&\ \int_{\Ro d} \left[ f\left( x+\DivEps 1\sigma\left(x,\e y\right)\right)-f(x)- \DivEps 1 \sigma^i\left(x,\e y\right) \partial_i f(x) \ind_{B}(y) \right] \nu^\alpha(dy) \\
& + \left[ b^i\left(x\right) + \e^{\alpha-1} c^i\left(x\right) \right] \partial_i f(x),\quad x\in\R^d.
\end{split}
\end{equation*}
Then Lemma \ref{Feller} and \ref{martingale-prob} still hold true with $\tilde\L^\alpha_\e$ in place of $\L^\alpha$.
\qed
\end{remark}
\begin{corollary}\label{precise-estimates}
For any $\kappa>0$ and $f\in\C^\beta(\T^d)$, the unique solution $u_\kappa$ of equation \eqref{resovent} admits the representation
\begin{equation}\label{rep_u_lambda}
u_\kappa(x)=\int_0^\infty e^{-\kappa t} P_t f(x) dt,
\end{equation}
where $\{P_t\}_{t\ge0}$ is the Feller semigroup generated by the closure of $\L^\alpha$, and the integral on the right hand side converges. Moreover, there exists a constant $C>0$ independent of $u,f,b,\kappa$ such that
\begin{equation}\label{energy3}
\kappa\|u_\kappa\|_0+\kappa^{\frac{\alpha+\beta-1}{\alpha}} \|\nabla u_\kappa\|_0 + [\nabla u_\kappa]_{\alpha+\beta-1}\le C\|f\|_\beta.
\end{equation}
\end{corollary}
\begin{proof}
Proposition \ref{res_prob} tells that the interval $(0,+\infty)$ is contained in the resolvent set of $\L^\alpha$. Then by the integral representation of the resolvent (see \cite[Theorem II.1.10.(ii)]{EN00}), we arrive at
\begin{equation*}
u_\kappa=(\kappa-\L^\alpha)^{-1}f = \lim_{t\to\infty} \int_0^t e^{-\kappa s}P_s fds,
\end{equation*}
where the limit is taken in $(\C(\T^d),\|\cdot\|_0)$. The representation \eqref{rep_u_lambda} then follows. Now thanks to the gradient estimate \eqref{gradient-estimate} and representation \eqref{rep_u_lambda}, the estimate \eqref{energy3} is then obtained by the same argument as the proof of \cite[Theorem 3.3, Part I]{Pri12}.
\end{proof}
In the next section, we will remove the large jumps from the SDEs and study their well-posedness by Zvonkin's transform. Thus we consider the following operator, which is a ``flat'' version of $\L^{\alpha}$:
\begin{equation}\label{L-flat}
\L^{\alpha,\flat}f(x) = \int_{\Bo} \big[ f( x+\sigma(x,y))-f(x) - \sigma^i(x,y) \partial_i f(x) \big] \nu^\alpha(dy) + b^i(x)\partial_i f(x).
\end{equation}
We have the following regularity result for $\L^{\alpha,\flat}$.
\begin{corollary}\label{no-large-jump}
There exists a constant $\kappa_*>0$ such that for any $\kappa>\kappa_*$ and $f\in\C^\beta(\T^d)$, there exists a unique solution $u=u^\flat_\kappa\in\C^{\alpha+\beta}(\T^d)$ to the equation
\begin{equation}\label{small-jump}
\kappa u-\L^{\alpha,\flat} u = f.
\end{equation}
In addition, there exists a constant $C>0$ independent of $u,f,b,\kappa$, such that for any $\kappa>\kappa_*$,
\begin{equation}\label{energy4}
(\kappa-\kappa_*)\|u^\flat_\kappa\|_0+(\kappa-\kappa_*)^{\frac{\alpha+\beta-1}{\alpha}} \|\nabla u^\flat_\kappa\|_0 + [\nabla u^\flat_\kappa]_{\alpha+\beta-1}\le C\|f\|_\beta.
\end{equation}
\end{corollary}
\begin{proof}
To obtain the a priori estimate \eqref{energy4}, we rewrite the equation \eqref{small-jump} in the form
\begin{equation*}
\kappa u-\L^{\alpha} u = f - \int_{B^c}[u(x+\sigma(x,y))-u(x)]\nu^\alpha(dy).
\end{equation*}
The estimate \eqref{energy3} implies that
\begin{equation*}
\kappa\|u\|_0+\kappa^{\frac{\alpha+\beta-1}{\alpha}} \|\nabla u\|_0 + [\nabla u]_{\alpha+\beta-1}\le C(\|f\|_\beta+2\nu^\alpha(B^c)\|u\|_\beta).
\end{equation*}
It is easy to see that there exists $\delta>0$ such that
\begin{equation*}
\sup_{|x-y|<\delta} \frac{|u(x)-u(y)|}{|x-y|} \le 2\|\nabla u\|_0,
\end{equation*}
and then
\begin{equation*}
\|u\|_\beta \le \sup_{|x-y|<\delta} \frac{|u(x)-u(y)|}{|x-y|}|x-y|^{1-\beta} + \sup_{|x-y|\ge\delta} \frac{|u(x)-u(y)|}{|x-y|^\beta} \le 2\delta^{1-\beta}\|\nabla u\|_0 + 2\delta^{-\beta}\|u\|_0.
\end{equation*}
Combining these together, we get
\begin{equation*}
\big(\kappa-4C\delta^{-\beta}\nu^\alpha(B^c)\big)\|u\|_0 +\big(\kappa^{\frac{\alpha+\beta-1}{\alpha}} - 4C\delta^{1-\beta}\nu^\alpha(B^c) \big)\|\nabla u\|_0 + [\nabla u]_{\alpha+\beta-1}\le C\|f\|_\beta.
\end{equation*}
Then \eqref{energy4} follows by choosing $\kappa_*=4C\delta^{-\beta}\nu^\alpha(B^c) \vee \big( 4C\delta^{1-\beta}\nu^\alpha(B^c) \big)^\frac{\alpha}{\alpha+\beta-1}$.
Now define a family of operators by
\begin{equation*}
\L_\theta^\flat=\L^{\alpha,\flat}+\theta \int_{B^c}[u(x+\sigma(x,y))-u(x)]\nu^\alpha(dy).
\end{equation*}
Then $\L_1^\flat=\L^\alpha, \L_0^\flat=\L^{\alpha,\flat}$. The well-posedness of equation \eqref{small-jump} follows from the method of continuity and the a priori estimate \eqref{energy4}, just as in the proof of Proposition \ref{res_prob}.
\end{proof}
\section{SDEs with multiplicative stable L\'evy noise}\label{SDEs}
The goal of this section is to study the strong well-posedness of SDEs \eqref{Xe_tilde} and \eqref{X_tilde}, as well as the ergodic properties of the solution processes $\tilde X^{x,\epsilon}$ for each $\epsilon>0$. As corollaries, we also obtain the Feynman-Kac formula and the well-posedness of nonlocal Poisson equation without zeroth-order term, which will be used to study homogenization in the next two sections.
\subsection{Strong well-posedness of SDEs}
We only consider the strong well-posedness for SDE \eqref{X_tilde} since \eqref{Xe_tilde} has the same form. As we have seen in Lemma \ref{martingale-prob}, the existence and uniqueness hold for the martingale problem for $(\L^\alpha,\delta_x)$. Meanwhile, it is known that the martingale solution for $(\L^\alpha,\delta_x)$ is
equivalent to the weak solution of SDE \eqref{X_tilde}, see \cite[Theorem 2.3, Corollary 2.5]{Kur11}. Thus, the existence and uniqueness of weak solution hold for SDE \eqref{X_tilde}.
Moreover, utilizing the fact shown in \cite[Theorem 1.2]{BLP15} that the weak existence and pathwise uniqueness for SDE \eqref{X_tilde} imply strong existence, we only need to prove the pathwise uniqueness. The key is to reduce the SDE \eqref{X_tilde}, whose coefficients have low regularity, to an SDE with Lipschitz coefficients by using Zvonkin's transform.
For $\kappa>\kappa_*$, let $\hat b_\kappa\in\C^{\alpha+\beta}(\T^d)$ be the solution of
\begin{equation*}
\kappa \hat b_\kappa-\L^{\alpha,\flat} \hat b_\kappa = b,
\end{equation*}
where $\L^{\alpha,\flat}$ is the operator in \eqref{L-flat}. The existence and uniqueness of solution $\hat b_\kappa$ is ensured by Corollary \ref{no-large-jump}. Define a map $\Phi_\kappa:\R^d\to\R^d$ by
\begin{equation*}
\Phi_\kappa(x)=x+\hat b_\kappa(x).
\end{equation*}
Then $\Phi_\kappa$ is of class $\C^{\alpha+\beta}$. Moreover, we have
\begin{lemma}\label{Phi}
For $\kappa>0$ large enough, the map $\Phi_\kappa:\R^d\to\R^d$ is a $C^1$-diffeomorphism and its inverse $\Phi_\kappa^{-1}$ is also of class $\C^{\alpha+\beta}$.
\end{lemma}
\begin{proof}
By the estimate in Corollary \ref{no-large-jump}, we have
\begin{equation*}
\kappa^{\frac{\alpha+\beta-1}{\alpha}} \|\nabla\hat b_\kappa \|_0 \le C \|b\|_\beta, \quad \kappa>\kappa_*.
\end{equation*}
Now by choosing $\kappa>\kappa_*\vee(2C \|b\|_\beta)^{\frac{\alpha}{\alpha+\beta-1}}$, we get that $\|\nabla\hat b_\kappa\|_0\le\frac{1}{2}$. Thus
\begin{equation*}
\frac{1}{2}|x_1-x_2|\le \big|\Phi_\kappa(x_1)-\Phi_\kappa(x_2)\big| \le\frac{3}{2}|x_1-x_2|,
\end{equation*}
i.e., $\Phi_\kappa$ is bi-Lipschitz. In particular, $\Phi_\kappa$ is a $\C^1$-diffeomorphism. Moreover,
\begin{equation*}
\nabla(\Phi_\kappa^{-1})= \text{Inv}\circ\nabla\Phi_\kappa\circ\Phi_\kappa^{-1},
\end{equation*}
where the matrix inverse map $\text{Inv}:\text{GL}(\R^d) \to\text{GL}(\R^d)$ is of class $\C^\infty$. Note that $\nabla\Phi_\kappa$ is of class $\C^{\alpha+\beta-1}$, $\Phi_\kappa^{-1}$ is of class $\C^1$. It is easy to see that $\nabla(\Phi_\kappa^{-1})$ is of class $\C^{\alpha+\beta-1}$. The second conclusion of the lemma follows.
\end{proof}
To solve SDE \eqref{X_tilde}, by a standard interlacing technique (cf. \cite[Section 6.5]{App09} or \cite[Theorem IV. 9.1]{IW14}), it suffices to solve the following SDE with no jumps greater than one:
\begin{equation*}
\tilde X^{x,\flat}_t = x+\int_0^t b(\tilde X^{x,\flat}_{s})ds + \int_0^t\int_B \sigma(\tilde X^{x,\flat}_{s-},y) \tilde N^\alpha(dy,ds).
\end{equation*}
Now fix $\kappa>0$ large enough such that the conclusions in Lemma \ref{Phi} hold. We introduce Zvonkin's transform
\begin{equation*}
\tilde X^{*}_t=\Phi_\kappa(\tilde X^{x,\flat}_t).
\end{equation*}
Then by applying It\^o's formula, we have
\begin{equation}\label{Zvonkin}
\tilde X^{*}_t = \Phi_\kappa(x) + \int_0^t b^{*}(\tilde X^{*}_{s}) ds + \int_0^t\int_B \sigma^{*}(\tilde X^{*}_{s-},y) \tilde N^\alpha(dy,ds),
\end{equation}
where
\begin{equation*}
b^{*}(x)=\kappa\hat b_\kappa(\Phi_\kappa^{-1}(x)),
\end{equation*}
\begin{equation*}
\sigma^{*}(x,y)=\hat b_\kappa(\Phi_\kappa^{-1}(x)+ \sigma(\Phi_\kappa^{-1}(x),y)) - \hat b_\kappa(\Phi_\kappa^{-1}(x)) + \sigma(\Phi_\kappa^{-1}(x),y).
\end{equation*}
\begin{proposition}
For each $x\in\R^d$, there is a unique strong solution $\tilde X^x=\{\tilde X_t^x\}_{t\ge0}$ to SDE \eqref{X_tilde}.
\end{proposition}
\begin{proof}
By the above argument, we only need to prove the pathwise uniqueness for SDE \eqref{Zvonkin}. First of all, we have, for any $x,x_1,x_2\in\R^d$,
\begin{equation}\label{b*}
\big|b^{*}(x_1)-b^{*}(x_2)\big| \le C(\|\hat b_\kappa\|_1,\|\Phi_\kappa^{-1}\|_1)|x_1-x_2|,
\end{equation}
Note that for $\gamma\in(0,1)$, $f\in\C_b^{1+\gamma}(\R^d)$, $x,u,v\in\R^d$, there exists a constant $C>0$ such that
\begin{equation*}
|f(u+x)-f(u)-f(v+x)-f(v)| \le C \|f\|_{1+\gamma} |u-v||x|^\gamma,
\end{equation*}
the proof can be found in \cite[Theorem 5.1.(c)]{Bas09}. Then for any $x_1,x_2$,
\begin{equation}\label{sigma*-Lip}
\begin{split}
&\ \left|\sigma^{*}(x_1,y)- \sigma^{*}(x_2,y)\right| \\
\le&\ \big|\hat b_\kappa(\Phi_\kappa^{-1}(x_1)+ \sigma(\Phi_\kappa^{-1}(x_1),y)) - b_\kappa(\Phi_\kappa^{-1}(x_1)) \\
&\ - \hat b_\kappa(\Phi_\kappa^{-1}(x_2)+ \sigma(\Phi_\kappa^{-1}(x_1),y)) + b_\kappa(\Phi_\kappa^{-1}(x_2))\big| \\
&\ + \big|\hat b_\kappa(\Phi_\kappa^{-1}(x_2)+ \sigma(\Phi_\kappa^{-1}(x_1),y)) - \hat b_\kappa(\Phi_\kappa^{-1}(x_2)+ \sigma(\Phi_\kappa^{-1}(x_2),y))\big| \\
&\ + \big|\sigma(\Phi_\kappa^{-1}(x_1),y)- \sigma(\Phi_\kappa^{-1}(x_2),y)\big| \\
\le&\ C \|\hat b_\kappa\|_{\alpha+\beta} \left| \Phi_\kappa^{-1}(x_1)-\Phi_\kappa^{-1}(x_2) \right| \left| \sigma(\Phi_\kappa^{-1}(x_1),y) \right|^{\alpha+\beta-1} \\
&\ +(\|\nabla \hat b_\kappa\|_0+1)\left| \sigma(\Phi_\kappa^{-1}(x_1),y)- \sigma(\Phi_\kappa^{-1}(x_2),y) \right| \\
\le&\ C \left(\|\hat b_\kappa\|_{\alpha+\beta}, \|\Phi_\kappa^{-1}\|_1, \|\phi\|_{L^\infty} \right) |x_1-x_2|(|y|^{\alpha+\beta-1}+|y|).
\end{split}
\end{equation}
where we have used the regularity condition for $\sigma$ in Assumption \ref{sigma}, and $\phi$ is the positive bounded function in the growth condition in that assumption. Noting that $2(\alpha+\beta-1)>\alpha$ by Assumption \ref{regular}, we arrive at
\begin{equation}\label{sigma*}
\int_B \left|\sigma^{*}(x_1,y)- \sigma^{*}(x_2,y)\right|^2 \nu_\alpha(dy) \le C\left(\|\hat b_\kappa\|_{\alpha+\beta}, \|\Phi_\kappa^{-1}\|_1, \|\phi\|_{L^\infty} \right)|x_1-x_2|^2.
\end{equation}
The pathwise uniqueness of SDE \eqref{Zvonkin} follows from \eqref{b*}, \eqref{sigma*} and the classical result \cite[Theorem 4.9.1]{IW14}. The proof is complete.
\end{proof}
\begin{corollary}
The solution process $\tilde X^x$
is a Feller process with generator the closure of $(\L^\alpha,\C^{\alpha+\beta}(\T^d))$.
In particular, $\tilde X^x$ is a strong Markov process.
\end{corollary}
\begin{proof}
By applying It\^o's formula, it is easy to see that for any $f\in D(\L^\alpha)=\C^{\alpha+\beta}(\T^d)$, the following process is a $(\Omega,\F,\P,\{\F_t\}_{t\ge0})$-martingale
\begin{equation*}
\tilde M^f(t):=f(\tilde X^x_t)-f(\tilde X^x_0)-\int_0^t \L^\alpha f(\tilde X^x_s) ds.
\end{equation*}
It is easy to see that $\tilde X^x$ has c\`adl\`ag paths almost surely. Let $\P_{\tilde X^x}:=\P\circ\tilde X^x$ be the pushforward probability measure of $\tilde X^x$ on $(\D,\B(\D))$, then $\P_{\tilde X^x}$ is a solution of martingale problem for $(\L^\alpha,\delta_x)$. By the uniqueness of solutions to the martingale problem obtained in Lemma \ref{martingale-prob}, we find that $\P_{\tilde X^x}=\P^x$, the Feller property follows. The strong Markov property follows from \cite[Theorem III.3.1]{RY13}.
\end{proof}
\begin{remark}
The Feller semigroup $\{P_t\}_{t\ge0}$ in Lemma \ref{Feller} is the semigroup associated with the solution process $\tilde X^x$, that is,
\begin{equation*}
P_tf(x)=\E(f(\tilde X^x_t)), \quad f\in \C(\T^d).
\end{equation*}
\qed
\end{remark}
As a consequence of the Feller property, we can obtain the well-posedness of the parabolic nonlocal PDE and the corresponding Feynman-Kac representation. See \cite{PR14} for the classical version for second order PDE.
\begin{proposition}\label{wellposed-parabolic}
The parabolic nonlocal PDE
\begin{equation*}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) = \L^\alpha u(t,x) +g(x) u(t,x), & t>0,x\in\R^d, \\
u(0,x) = u_0(x), & x\in\R^d,
\end{cases}
\end{equation*}
admits a unique mild solution in the sense that $\int_0^t u(s) ds\in D(\L^\alpha)$ for all $t\ge0$ and
\begin{equation*}
u(t) = u_0 +\L^\alpha \int_0^t u(s) ds + g\int_0^t u(s) ds.
\end{equation*}
Moreover, the unique solution has the following Feynman-Kac representation
\begin{equation*}
u(t,x)=\E\left[u_0(\tilde X^x_t)\exp\left(\int_0^t g(\tilde X^x_s) ds\right)\right].
\end{equation*}
\end{proposition}
\begin{proof}
Choose $G>0$ large enough such that $\|g\|_0<G$. Define
\begin{equation*}
P_t^g f(x)=\E\left[f(\tilde X^x_t)\exp\left(\int_0^t g(\tilde X^x_s) ds- Gt\right)\right], \quad f\in \C(\T^d).
\end{equation*}
Then by an argument similar to that used in \cite[Section 6.7.2]{App09}, one can show that $\{P_t^g\}_{t\ge0}$ is a Feller semigroup with generator the closure of $(\L^\alpha+g-G,\C^{\alpha+\beta}(\T^d))$. This yields that $\{e^{Gt}P_t^g\}_{t\ge0}$ is a $C_0$-semigroup on $\C(\T^d)$ with generator the closure of $(\L^\alpha+g,\C^{\alpha+\beta}(\T^d))$. Note that $u_0\in\C(\T^d)$. Now applying the classic result \cite[Proposition II.6.4]{EN00} in the theory of $C_0$-semigroups, we conclude that the parabolic nonlocal PDE admits a unique mild solution, which can be given by the orbit map $u(t)=e^{Gt}P_t^g u_0$. The desired conclusions follow immediately.
\end{proof}
\subsection{Ergodicity}
Now we deal with the ergodicity of SDEs. To this end, we need the following scaling assumption for the coefficient $\sigma$.
\begin{assumption}\label{scaling}
$\sigma(x,ry) = r\sigma(x,y)$ for all $r>0$ and $x,y\in\R^d$.
\end{assumption}
\begin{remark}\label{sigma-conv}
In the case $\sigma(x,y)=\sigma_0(x)\eta(y)$, this assumption reduces to $\eta(ry)=r\eta(y)$ for all $r>0$ and $y\in\R^d$. That is, the function $\eta$ is positively homogeneous of degree 1. In particular, if $\sigma(x,y)=\sigma_0(x)y$, then this assumption holds automatically. \qed
\end{remark}
By the discussion in previous subsection, for every $\e>0$, SDE \eqref{Xe_tilde} also admits a unique strong solution $\tilde X^{x,\e}$ which is a $\T^d$-valued Feller process. Denote by $p^\epsilon(t;x,y)$ the transition probability density of $\tilde X^{x,\epsilon}$, by $\{P^\epsilon_t\}_{t\ge 0}$ the associated Feller semigroup. Note that under Assumption \ref{scaling}, SDE \eqref{Xe_tilde} becomes
\begin{equation*}
d\tilde X_t^{x,\epsilon} = \left( b(\tilde X_{t}^{x,\epsilon})+ \epsilon^{\alpha-1} c(\tilde X_{t}^{x,\epsilon})\right)dt+
\sigma\left(\tilde X_{t-}^{x,\epsilon},d\tilde L_t^{\alpha}\right), \quad \tilde X_0^{x,\epsilon}=\frac{x}{\epsilon}.
\end{equation*}
The associated generator is $\tilde\L^\alpha_\e=\A^{\sigma,\nu^\alpha}+ (b+\e^{\alpha-1}c)\cdot\nabla$.
\begin{lemma}\label{inv}
For each $0\le\epsilon\le1$, the process $\tilde X^{x,\epsilon}$ possesses a unique invariant distribution $\mu_\epsilon$ on $\T^d$. Moreover, there exist positive constants $C$ and $\rho$, depending only on $d,\alpha, \|b\|_0,\|c\|_0, h_0,h_1$, such that for any periodic bounded Borel function $f$ on $\R^d$ (i.e., $f$ is Borel bounded on $\T^d$),
\begin{equation*}
\sup_{x\in\T^d} \left| P^\e_t f(x)-\int_{\T^d}f(y)\mu_\epsilon(dy) \right| \le C\|f\|_0e^{-\rho t}
\end{equation*}
for every $t\ge0$.
\end{lemma}
\begin{proof}
One can find a version of Doeblin-type result in \cite[Theorem 3.3.1, 3.3.2]{BLP78}, which states that for a Markov process with transition probability densities bounded from below by a positive constant, it has a unique invariant probability measure and the associated semigroup converges exponentially fast. Therefore, it is enough to ensure that the transition probability density $p^\e(1;x,y)$ is bounded from below by a positive constant, which follows immediately from the density estimates in Lemma \ref{martingale-prob}. Moreover, the two constants $C$ and $\rho$ are related to the lower bound of $p^\e(1;x,y)$. Since the generator of each semigroup $\{P^\e_t\}$ is $\tilde\L^\alpha_\e=\A^{\sigma,\nu^\alpha}+ (b+\e^{\alpha-1}c)\cdot\nabla$, the constant $C_1$ associated to $p^\e(t;x,y)$ in Lemma \ref{martingale-prob} are related to $d,\alpha, \|b+\e^{\alpha-1}c\|_0, h_0,h_1$. Hence, for $\e\in[0,1]$, constants $C$ and $\rho$ can be chosen to depend only on $d,\alpha, \|b\|_0,\|c\|_0, h_0,h_1$.
\end{proof}
Denote by $\mu=\mu_0$ the unique invariant probability measure for the limit process $\tilde X^x_t$ in \eqref{X_tilde}. Then we can prove the following lemma.
\begin{lemma}\label{conv_mu}
As $\epsilon\to0$, we have $\mu_\epsilon\to\mu$ weakly.
\end{lemma}
\begin{proof}
Using the same argument as the proof of \cite[Lemma 2.4]{HP08}, and noting that the tightness of the family $\{\mu_\e\}_{\e>0}$ is automatic due to the compactness of $\T^d$, it suffices to prove that $P_t^\e f\to P_tf$ in $\C(\T^d)$ as $\e\to 0$ for any $f\in\C(\T^d)$ and $t\ge0$. By Lemma \ref{martingale-prob} and Remark \ref{rep-Feller}.(2), we know that $\C^{\alpha+\beta}(\T^d)$ is a core for $\L^\alpha$ and each $\tilde\L^\alpha_\e$, $\e>0$. Fix an arbitrary $f\in\C^{\alpha+\beta}(\T^d)$,
\begin{equation*}
|\tilde\L^\alpha_\e f(x)-\L^\alpha(x)| \le \e^{\alpha-1} \left| c(x)\cdot\nabla f(x) \right| \le \e^{\alpha-1}\|c\|_0 \|f\|_1,
\end{equation*}
it converges to zero as $\e\to0$, uniformly in $x$, by the dominated convergence and Assumption \ref{scaling}. Using the Trotter-Kato approximation theorem (see \cite[Theorem III.4.8]{EN00}), $P_t^\e f\to P_tf$ in $\C(\T^d)$ as $\e\to 0$ for all $f\in\C(\T^d)$, uniformly for $t$ in compact intervals.
\end{proof}
Now we combine Lemma \ref{inv} and Lemma \ref{conv_mu} to get the following ergodic theorem.
\begin{proposition}\label{ergodic_thm}
Let $f$ be a bounded Borel function on $\T^d$. Then for any $t>0$,
\begin{equation}
\int_0^t \left|f\left( \frac{X_s^{x,\epsilon}}{\epsilon} \right)- \int_{\T^d}f(x)\mu(dx) \right| ds \to 0
\end{equation}
in probability, as $\epsilon\to 0$.
\end{proposition}
\begin{proof}
We follow the lines of \cite[Proposition 2.4]{Par99}. For $\e>0$, $0\le s<t$, let $\bar f$ be a bounded measurable function on $\T^d$ satisfying $\int_{\T^d}\bar f(x)\mu_\e(dx)=0$. By Lemma \ref{conv_mu}, it suffices to prove that $\int_0^t |\bar f(X^\e_s/\e)| ds \to 0$ in $L^2(\Omega,\P)$. Using Lemma \ref{inv}, we have
\begin{equation*}
\E \left[|\bar f(\tilde X^{x,\e}_t)|\Big| \tilde X^{x,\e}_s\right] = \int_{\T^d} |\bar f(y)|\left[p^\e(t-s,\tilde X_s^{x,\e},y)dy-\mu_\e(dy)\right] \le C\|\bar f\|_0 e^{-\rho(t-s)}.
\end{equation*}
By the Markov property,
\begin{equation*}
\E|\bar f(\tilde X^\e_s) \bar f(\tilde X^\e_t)| = \E\left[|\bar f(\tilde X^\e_s)|\E \left(|\bar f(\tilde X^{\e}_t)|\Big| \tilde X^{\e}_s\right)\right] \le C\|\bar f\|_0^2 e^{-\rho(t-s)}.
\end{equation*}
Hence,
\begin{equation*}
\begin{split}
\E\left[\left( \int_0^t |\bar f(X^\e_s/\e)| ds \right)^2\right] &= \e^{2\alpha}\int_0^{\e^{-\alpha}t} \int_0^{r} \E|\bar f(\tilde X^\e_s) \bar f(\tilde X^\e_r)| ds dr \\
&\le 2C\e^{2\alpha} \|f\|_0^2 \int_0^{\e^{-\alpha}t} \int_0^{r} e^{-\rho(r-s)} dsdr \\
&= 2C\e^{2\alpha} \|f\|_0^2 \rho^{-2} (-1+\rho\e^{-\alpha}t + e^{-\rho\e^{-\alpha}t}) \\
&\to 0,
\end{split}
\end{equation*}
as $\e\to0$. The results follow.
\end{proof}
For every $\gamma>0$, denote by $\C_\mu^\gamma(\T^d)$ the class of all $f\in\C^\gamma(\T^d)$ which are \emph{centered} with respect to the invariant measure $\mu$ in the sense that $\int_{\T^d}f(x)\mu(dx)=0$. It is easy to check that $\C_\mu^\gamma(\T^d)$ is closed, and hence a sub-Banach space of $\C^\gamma(\T^d)$ under the norm $\|\cdot\|_\gamma$.
Thanks to Proposition \ref{res_prob} and Lemma \ref{inv}, we can use the Fredholm alternative to obtain the solvability of the following Poisson equation without zeroth-order term in the smaller space $\C_\mu^{\alpha+\beta}(\T^d)$,
\begin{equation}\label{Poisson}
\L^\alpha u + f = 0,
\end{equation}
for $f\in\C_\mu^\beta(\T^d)$. Before that, we need some lemmas.
\begin{lemma}
The restrictions $\{P_t^\mu := P_t|_{\C_\mu(\T^d)}\}_{t\ge0}$ form a $C_0$-semigroup on the Banach space $(\C_\mu(\T^d),\|\cdot\|_0)$, with generator given by $\L^\alpha_\mu f:= \L^\alpha f$, $D(\L^\alpha_\mu) := \C_\mu^{\alpha+\beta}(\T^d)$.
\end{lemma}
\begin{proof}
Since $\mu$ is invariant with respect to $\{P_t\}_{t\ge0}$, for any $f\in\C_\mu(\T^d)$ and $t\ge0$, we have
\begin{equation*}
\int_{\T^d} P_t f(x) \mu(dx)=\int_{\T^d} f(x) \mu(dx)=0.
\end{equation*}
That is, $\C_\mu(\T^d)$ is $\{P_t\}_{t\ge0}$-invariant, in the sense that $P_t(\C_\mu(\T^d))\subset\C_\mu(\T^d)$ for all $t\ge0$. The lemma then follows from the corollary in \cite[Subsection II.2.3]{EN00}.
\end{proof}
\begin{lemma}\label{res_prob-center}
If $f\in\C_\mu^\beta(\T^d)$, then the unique solution $u_\kappa$ of \eqref{resovent} is of class $\C_\mu^{\alpha+\beta}(\T^d)$, for any $\kappa>0$.
\end{lemma}
\begin{proof}
Since $f$ is centered with respect to $\mu$, by Lemma \ref{inv} we have
\begin{equation}\label{SG_est}
\|P_t f\|_0 \le C\|f\|_0e^{-\rho t}.
\end{equation}
Note the fact that $\mu$ is invariant with respect to $\{P_t\}_{t\ge0}$. Then combining \eqref{SG_est} and the representation \eqref{rep_u_lambda}, a straightforward application of Fubini's theorem implies that
\begin{equation*}
\begin{split}
\int_{\T^d} u_\kappa(x) \mu(dx) & = \int_{\T^d} \int_0^\infty e^{-\kappa t} P_t f(x) dt \mu(dx) = \int_0^\infty e^{-\kappa t} \left(\int_{\T^d} P_t f(x) \mu(dx)\right) dt \\
& = \int_0^\infty e^{-\kappa t} \left(\int_{\T^d} f(x) \mu(dx)\right) dt =0.
\end{split}
\end{equation*}
That is, $u_\kappa$ is also centered with respect to $\mu$.
\end{proof}
The following theorem will solve the well-posedness of equation \eqref{Poisson}, which is more general than the results in \cite[Proposition 3]{Fra07}. We formulate it as follows, referring to \cite[Theorem 1]{PV01} for the classical version for second order partial differential operators.
\begin{proposition}\label{wellposed-Poisson}
For any $f\in\C_\mu^\beta(\T^d)$, there exists a unique solution in $\C_\mu^{\alpha+\beta}(\T^d)$ to the equation \eqref{Poisson}, which satisfies the estimate
\begin{equation}\label{energy2}
\|u\|_{\alpha+\beta} \le C (\|u\|_0+\|f\|_\beta),
\end{equation}
where $C=C(\|b\|_\beta)$ is a positive constant. Moreover, the unique solution admits the representation
\begin{equation}\label{rep_u}
u(x)=\int_0^\infty P_t f(x) dt.
\end{equation}
\end{proposition}
\begin{proof}
The a priori estimate \eqref{energy2} is also from \citep[Theorem 7.1]{Bas09}.
First, we show that if the equation has a solution $u\in\C_\mu(\T^d)$ for $f\in\C_\mu^\beta(\T^d)$, then $u$ must have the representation \eqref{rep_u}, this also implies the uniqueness. By the exponential ergodicity result in Lemma \ref{inv}, we have $\|P_t^\mu f\|_0\le C\|f\|_0 e^{-\rho t}$ for any $f\in\C_\mu(\T^d)$ and $t\ge0$. This yields that, using \cite[Theorem II.1.10.(ii)]{EN00} as in the proof of Corollary \ref{precise-estimates}, the set $\{z\in\Cp|\text{Re}z>-\rho\}$ is contained in the resolvent set of $\L^\alpha_\mu$. Noting that $u=(0-\L^\alpha_\mu)^{-1}f$, the representation and uniqueness follow.
Now we prove the existence. Let $\kappa_0$ be a fixed positive constant. Thanks to Lemma \ref{res_prob-center}, the linear map $\kappa_0-\L^\alpha:\C_\mu^{\alpha+\beta}(\T^d)\to \C_\mu^{\beta}(\T^d)$ is invertible. Furthermore, by virtue of Lemma \ref{MP} and the energy estimate \eqref{energy1}, together with the compact embedding $\C_\mu^{\alpha+\beta}(\T^d)\subset\C_\mu^{\beta}(\T^d)$ (see, for instance, \cite[Lemma 6.36]{GT01}), the resolvent $\mathcal R_{\kappa_0}:=(\kappa_0-\L^\alpha)^{-1}$ is compact from $\C_\mu^{\beta}(\T^d)$ to $\C_\mu^{\beta}(\T^d)$. Consider then the equation
\begin{equation}\label{Fred}
u-\kappa_0\mathcal R_{\kappa_0}u=\mathcal R_{\kappa_0}f, \quad f\in\C_\mu^{\beta}(\T^d),
\end{equation}
Then the Fredholm alternative (see \cite[Section 5.3]{GT01}) implies that the equation \eqref{Fred} always has a unique solution $u\in\C_\mu^{\beta}(\T^d)$ provided the homogeneous equation $u-\kappa_0\mathcal R_{\kappa_0}u=0$ has only the trivial solution $u=0$.
To rephrase these statements in terms of the Poisson equation \eqref{Poisson}, we observe first that since $\mathcal R_{\kappa_0}$ maps $\C_\mu^{\beta}(\T^d)$ onto $\C_\mu^{\alpha+\beta}(\T^d)$, any solution $u\in\C_\mu^{\beta}(\T^d)$ of \eqref{Fred} must also belong to $\C_\mu^{\alpha+\beta}(\T^d)$. Hence, operating on \eqref{Fred} with $\kappa_0-\L^\alpha$ we obtain
\begin{equation*}
-\L^\alpha u= (\kappa_0-\L^\alpha)(u-\kappa_0\mathcal R_{\kappa_0}u)=f.
\end{equation*}
Thus, the solutions of \eqref{Fred} are in one-to-one correspondence with the solutions of the Poisson equation \eqref{Poisson}. Consequently, \eqref{Fred} has a unique solution in $\C_\mu^{\alpha+\beta}(\T^d)$ if we can show that the homogeneous equation $\L^\alpha u=0$ has only the zero solution, while the latter follows from the representation \eqref{rep_u}.
\end{proof}
\begin{remark}
The assumption that $f$ is centered with respect to $\mu$ in Proposition \ref{wellposed-Poisson} is necessary. To see this informally, let's recall the Riesz-Schauder theory for compact operators (cf. \cite[Theorem X.5.3]{Yos95}). The equation \eqref{Fred} admits a solution $u\in\C(\T^d)$ if and only if $\mathcal R_\kappa f\in\text{Ker}(I^*-\kappa\mathcal R_\kappa^*)^\bot$, where the superscript $*$ denotes the adjoint of operators. This is equivalent to say that the equation \eqref{Poisson} admits a solution $u\in\C(\T^d)$ if and only if $f\in\text{Ker}(L^{\alpha,*})^\bot$. On the other hand, we have $\mu\in\text{Ker}(L^{\alpha,*})$ since $\mu$ is the invariant measure with respect to $\{P_t\}_{t\ge0}$. Thus a necessary condition for the existence of \eqref{Poisson} is $\langle\mu,f\rangle=0$, regarding $\mu$ as an element in the dual space of $\C(\T^d)$.
\end{remark}
\section{Homogenization results}\label{Homogenization}
\subsection{Homogenization of SDEs}
The aim of this subsection is to show the homogenization result of the solutions $X^{x,\e}$ of SDEs \eqref{Xe}. It is quite natural to get rid of the drift term involving $\frac{1}{\e^{\alpha-1}}$ in \eqref{Xe}. For this purpose, we again use Zvonkin's transform,
\begin{equation}\label{Xe_hat}
\hat X_t^{x,\epsilon}:=X_t^{x,\epsilon}+\epsilon\left(\hat b\left(\frac{X_t^{x,\epsilon}}{\epsilon}\right)-\hat b\left(\frac{x}{\epsilon}\right)\right),
\end{equation}
where $\hat b$ is the solution of the Poisson equation
\begin{equation}\label{b_hat}
\L^\alpha \hat b + b = 0,
\end{equation}
with the linear operator $\L^\alpha$ given by \eqref{L}. Note that the transform here is slightly different from that used in Section \ref{SDEs}. Due to Proposition \ref{wellposed-Poisson}, $\hat b\in\C_\mu^{\alpha+\beta}(\T^d)$ is uniquely determined under the following assumption.
\begin{assumption}\label{center}
The functions $b$ and $e$ satisfy the \emph{centering condition},
\begin{equation*}
\int_{\T^d}b(x)\mu(dx)=0, \qquad \int_{\T^d}e(x)\mu(dx)=0.
\end{equation*}
\end{assumption}
Note that this assumption is quite natural in the homogenization problems and the reader can also find it in \cite{BLP78,HP08,Par99}. We will let Assumptions \ref{scaling} and \ref{center} hold true in this and next subsection. Now we are in a position to study the homogenization of SDEs with multiplicative stable noise.
\begin{proposition}\label{HomoSDE}
In the sense of weak convergence on the space $\D$, we have that,
\begin{equation}\label{conv_X}
X^{x,\epsilon}\ \Rightarrow X^x, \quad \text{where } X^x_t:=x+\bar Ct+L_t,
\end{equation}
as $\e\to0$. The homogenized coefficient $\bar C$ is given by
\begin{equation}\label{HomoCoef1}
\bar C=\int_{\T^d}(I+\nabla \hat b)c(x)\mu(dx),
\end{equation}
and $\{L_t\}_{t\ge0}$ is a symmetric $\alpha$-stable L\'evy processes with jump intensity measure
\begin{equation}\label{homo_nu}
\Pi(A)=\int_{\Ro d}\int_{\T^d}\ind_A(\sigma(x,y))\mu(dx)\nu^\alpha(dy), \quad A\in\B(\Ro d).
\end{equation}
\end{proposition}
\begin{proof}
Since $\hat b$ is bounded, the theorem will follow if we prove that $\hat X^{x,\epsilon}\Rightarrow X^x$, as $\epsilon\to 0$.
Note that $\nu^\alpha(\epsilon A)=\epsilon^{-\alpha}\nu^\alpha(A)$, $A\in\B(\Ro d)$. By Assumption \ref{scaling} the oddness condition in Assumption \ref{sigma},
\begin{equation}\label{est-1}
\begin{split}
\frac{1}{\epsilon^{\alpha-1}}\A^{\sigma,\nu^\alpha} \hat b \left(\DivEps x\right) &= \int_{\Ro d} \e \left[ \hat b\left( \DivEps x + \sigma\left(\DivEps x,\DivEps y\right) \right) - \hat b\left( \DivEps x \right) - \sigma^i\left(\DivEps x,\DivEps y\right)\partial_i\hat b\left(\DivEps x\right)\ind_B(y) \right]\nu^\alpha(dy) \\
&= \epsilon \A^{\sigma_\e,\nu^\alpha} \hat b_\e(x).
\end{split}
\end{equation}
Then by applying It\^o's formula, and note that $\hat b\in\C^{\alpha+\beta}(\T^d)$ is the solution of Poisson equation \eqref{b_hat},
\begin{equation*}
\begin{split}
\hat X_t^{x,\epsilon} = &\ x+\int_0^t(I+\nabla \hat b)c\left(\frac{X_{s}^{x,\epsilon}}{\epsilon}\right)ds +\int_0^t\int_{\Ro d}\epsilon\left[\hat b_\epsilon\left(X_{s-}^{x,\epsilon}+ \sigma_\epsilon \left(X_{s-}^{x,\epsilon},y\right) \right)-\hat b_\epsilon\left(X_{s-}^{x,\epsilon}\right) \right] \tilde N^\alpha(dy,ds) \\
& +\int_0^t\int_{\Bo} \sigma_\epsilon\left(X_{s-}^{x,\epsilon},y \right)\tilde N^\alpha(dy,ds) +\int_0^t\int_{B^c} \sigma_\epsilon\left(X_{s-}^{x,\epsilon},y \right) N^\alpha(dy,ds) \\
=: &\ x+ \Lambda_1^\e(c)_t + \Lambda_2^\e(\hat b, \tilde N^\alpha)_t +\Lambda_3^\e(\sigma, \tilde N^\alpha)_t+\Lambda_4^\e(\sigma, N^\alpha)_t.
\end{split}
\end{equation*}
where $\hat b_\epsilon(x):=\hat b\left(\frac{x}{\e}\right)$, $\sigma_\epsilon(x,y):=\sigma\left(\frac{x}{\e},y\right)$.
For the last three stochastic integral terms, we figure out the characteristics of them as semimartingales (cf. \cite[Proposition IX.5.3]{JS13}). Choose the truncation function $h_1(x)=x\ind_B(x)$. Denote by $\Xi^\e(s,y):=\epsilon[\hat b_\epsilon(X_{s-}^{x,\epsilon}+ \sigma_\epsilon (X_{s-}^{x,\epsilon},y))-\hat b_\epsilon(X_{s-}^{x,\epsilon})]$. Note that $\Xi^\e(\cdot,0)\equiv0$ by virtue of $\sigma(\cdot,0)\equiv0$ as mentioned in Remark \ref{rem_sigma} (2). Then the characteristics of $\Lambda_2^\e(\hat b, \tilde N^\alpha)$ associated with $h_1$ is given by
\begin{equation*}\left\{
\begin{aligned}
B_2^\e(t) &= -\int_0^t\int_{\Ro d} \Xi^\e(s,y)\ind_{B^c}(\Xi^\e(s,y))\nu^\alpha(dy)ds, \\
C_2^\e & \equiv 0, \\
\nu_2^\e(A\times[0,t]) &= \int_0^t\int_{\Ro d} \ind_A(\Xi^\e(s,y))\nu^\alpha(dy)ds, \quad A\in\B(\Ro d).
\end{aligned} \right.
\end{equation*}
The characteristics of $\Lambda_3^\e(\sigma, \tilde N^\alpha)+\Lambda_4^\e(\sigma, N^\alpha)$ is given by
\begin{equation*}\left\{
\begin{aligned}
B_{3+4}^\e(t) &= \int_0^t\int_{\Ro d} \sigma_\epsilon\left(X_{s-}^{x,\epsilon},y \right)\left[\ind_B\left(\sigma_\epsilon\left(X_{s-}^{x,\epsilon},y \right)\right)-\ind_B(y)\right]\nu^\alpha(dy)ds, \\
C_{3+4}^\e & \equiv 0, \\
\nu_{3+4}^\e(A\times[0,t]) &= \int_0^t\int_{\Ro d} \ind_A\left(\sigma_\epsilon\left(X_{s-}^{x,\epsilon},y \right)\right)\nu^\alpha(dy)ds, \quad A\in\B(\Ro d).
\end{aligned} \right.
\end{equation*}
By the same argument as in \eqref{trans}, we have $B_{3+4}^\e\equiv0$.
Then the theorem is a consequence of the functional central limit theorem in \cite[Theorem VIII.2.17]{JS13} and the following lemma.
\end{proof}
\begin{lemma}\label{FCLT}
For any $t\in\R_+$, and any bounded continuous function $f:\R^d\to\R$ which vanishes in a neighbourhood of the origin, the following convergences hold in probability $\P$ when $\e\to0$:
\begin{enumerate}[(i)]
\item $\sup_{0\le s\le t} \left| \Lambda_1^\e(c)_s- \bar Cs \right| \to 0$;
\item $\sup_{0\le s\le t} |B_2^\e(s)| \to 0$;
\item $\int_0^t\int_{\Ro d}f(x)\nu_2^\e(dx,ds) \to 0$;
\item $\int_0^t\int_{\Ro d}f(x)\nu_{3+4}^\e(dx,ds) \to t\int_{\Ro d}f(x)\Pi(dx)$;
\end{enumerate}
where $\bar C$ and $\Pi$ are defined in \eqref{HomoCoef1} and \eqref{homo_nu}, respectively.
\end{lemma}
\begin{proof}
(i). By Proposition \ref{ergodic_thm}, the convergence in probability of the first integral is immediate,
\begin{equation*}
\sup_{0\le s\le t} \left| \Lambda_1^\e(c)_s- \bar Cs \right| \le \int_0^t \left|(I+\nabla \hat b)c\left(\frac{X_{s}^{x,\epsilon}}{\epsilon}\right)-\bar C\right|ds \to 0, \quad \e\to 0.
\end{equation*}
(ii) and (iii). Since $\Xi^\e$ is integrable with respect to $\tilde N^\alpha$,
the third characteristic of $\Lambda_2^\e$ satisfies that $\int_0^t\int_{\Ro d}(|x|^2\wedge1)\nu_2^\e(dx,ds)<\infty$ for each $\e>0$ and $t\in\R_+$ (cf. \cite[Proposition II.2.9]{JS13}). By the hypothesis, there exist $\rho>0$ and $M>0$ such that $|f|\le M$ on $B_\rho^c$ and $f=0$ on $B_\rho$. Then for any $t\in\R_+$,
\begin{equation*}
\int_0^t\int_{\Ro d}f(x)\nu_2^\e(dx,ds) \le M\int_0^t\int_{\Ro d}\ind_{B_\rho^c}(x) \nu_2^\e(dx,ds),
\end{equation*}
which goes to zero almost surely as $\e\to 0$ by the boundness of $\hat b$ and the dominated convergence theorem, and (iv) follows.
For $B_2^\e$, we have the estimate
\begin{equation*}
\begin{split}
\sup_{0\le s\le t} |B_2^\e(s)| & \le \left[\int_0^t\int_{\Ro d}|x|^2\nu_2^\e(dx,ds) \right]^{\frac{1}{2}} \left[\int_0^t\int_{\Ro d} \ind_{B^c}(x) \nu_2^\e(dx,ds)\right]^{\frac{1}{2}} \\
& =: \sqrt{J^\e_1}\cdot \sqrt{J^\e_2}.
\end{split}
\end{equation*}
By (iv) and a usual approximation procedure, $J^\e_2$ goes to zero surely as $\e\to 0$. For $J^\e_1$,
\begin{equation*}
\begin{split}
J^\e_1 &= \int_0^t\int_{\Ro d} \left|\epsilon\left[\hat b_\epsilon\left(X_{s-}^{x,\epsilon}+ \sigma_\epsilon \left(X_{s-}^{x,\epsilon},y\right) \right)-\hat b_\epsilon\left(X_{s-}^{x,\epsilon}\right) \right] \right|^2 \nu^\alpha(dy) ds \\
&= \int_0^t\left( \int_{B_\e^c}+\int_{B_\e\setminus\{0\}} \right) |\cdots|^2 \nu^\alpha(dy) ds \\
&\le \frac{4t\|\hat b\|_0^2\lambda(\S^{d-1})}{\alpha} \e^{2-\alpha} + \|\hat b\|_1^2\int_{B_\e\setminus\{0\}}\int_0^t \left|\sigma\left( \DivEps{X_{s-}},y \right)\right|^2 ds\nu^\alpha(dy).
\end{split}
\end{equation*}
By the growth condition in Assumption \ref{sigma},
\begin{equation*}
\int_{B_\e\setminus\{0\}}\int_0^t \left|\sigma\left( \DivEps{X_{s-}},y \right)\right|^2 ds\nu^\alpha(dy)
\le \frac{t\lambda(\S^{d-1}) \e^{2-\alpha}}{2-\alpha} \int_0^t \left|\phi\left( \DivEps{X_{s-}}\right)\right|^2 ds.
\end{equation*}
Then (iii) follows from these estimates and Proposition \ref{ergodic_thm}.
(iv). It follows from Proposition \ref{ergodic_thm} that,
\begin{equation*}
\begin{split}
\int_0^t\int_{\Ro d}f(y)\nu^{3+4}_\e(dy,ds) =&\ \int_{\Ro d}\int_0^t f\left(\sigma\left(\DivEps{X^{x,\e}_{s-}},y\right)\right) ds\nu^\alpha(dy) \\
\to &\ t\int_{\Ro d}\int_{\T^d} f(\sigma(x,y))\mu(dx)\nu^\alpha(dy) \\
=&\ t\int_{\Ro d}f(y)\Pi(dy),\quad \e\to 0,
\end{split}
\end{equation*}
where the convergence is in probability.
\end{proof}
\subsection{Homogenization of linear nonlocal PDEs}
Define
\begin{equation}\label{Ye}
Y_t^\epsilon := \int_0^t \left(\frac{1}{\epsilon^{\alpha-1}} e\left(\frac{X_s^{x,\epsilon}}{\epsilon}\right)+ g\left(\frac{X_s^{x,\epsilon}}{\epsilon}\right)\right)ds.
\end{equation}
Thanks to Proposition \ref{wellposed-parabolic}, the nonlocal PDE \eqref{ue} has a unique mild solution, which is given by the Feynman-Kac formula,
\begin{equation}\label{FK}
u^\epsilon(t,x)= \E\left[u_0(X_t^{x,\epsilon})\exp(Y_t^\epsilon)\right].
\end{equation}
Similar to $\hat X^{x,\e}$, we define
\begin{equation*}
\hat Y_t^\epsilon:=Y_t^\epsilon+\epsilon\left(\hat e\left(\frac{Y_t^\epsilon}{\epsilon}\right)-\hat e\left(\frac{x}{\epsilon}\right)\right).
\end{equation*}
Here $\hat e\in\C_\mu^{\alpha+\beta}(\T^d)$, thanks to Proposition \ref{wellposed-Poisson} and Assumption \ref{center}, is the unique solution of the Poisson equation
\begin{equation*}
\L^\alpha \hat e + e = 0,
\end{equation*}
with $\L^\alpha$ given by \eqref{L}. In a similar fashion as \eqref{est-1}, we know that $\frac{1}{\epsilon^{\alpha-1}}\A^{\sigma,\nu^\alpha} \hat e \left(\DivEps x\right) = \epsilon \A^{\sigma_\e,\nu^\alpha} \hat e_\e(x)$. Again using It\^o's formula,
\begin{equation*}
\begin{split}
\hat Y_t^\epsilon =&\ \int_0^t(g+\nabla \hat ec)\left(\frac{X_{s}^{x,\epsilon}}{\epsilon}\right)ds +\int_0^t\int_{\Ro{d}}\epsilon\left[\hat e_\epsilon\left(X_{s-}^{x,\epsilon}+ \sigma_\epsilon \left(X_{s-}^{x,\epsilon},y\right) \right)-\hat e_\epsilon\left(X_{s-}^{x,\epsilon}\right) \right] \tilde N^\alpha(dy,ds) \\
=:&\ \Lambda_1^\e(c,g)_t + \Lambda_2^\e(\hat e, \tilde N^\alpha)_t.
\end{split}
\end{equation*}
Then in the same way as the proof of Proposition \ref{HomoSDE}, we have the convergence of $Y^\e$.
\begin{lemma}
In the sense of weak convergence on the space $\D$, both $Y^\e$ and $\hat Y^\e$ converge in distribution to a deterministic path $y(t)=\bar Et$ as $\e\to0$, where the homogenized coefficient $\bar E$ is given by
\begin{equation}\label{HomoCoef2}
\bar E:=\int_{\T^d}(g+\nabla \hat ec)(x)\mu(dx).
\end{equation}
\end{lemma}
Now we are in the position to prove the main result of this section. Since $\hat b$ and $\hat e$ are bounded on $\R^d$, $u^\e$ has the same limit behavior as
\begin{equation}\label{ue_hat}
\hat u^\e(t,x):= \E[u_0(X_t^{x,\epsilon})\exp(\hat Y_t^\epsilon)]
\end{equation}
as $\e\to 0$.
\begin{proof}[Proof of Themrem \ref{HomoPDE}]
We only need to show $\hat u^\epsilon(t,x)\to u(t,x), \e\to 0$ for any $t\ge 0,x\in\R^d$. For the convenience of notation, we shall write $\Lambda_1^\e(c,g)_t$, $\Lambda_2^\e(\hat e, \tilde N^\alpha)_t$ as $\Lambda_1^\e(t)$, $\Lambda_2^\e(t)$, respectively. We fix a $t\in\R_+$.
Firstly, we prove the uniform integrability of the set $\{e^{\Lambda_2^\e(t)}|0<\e\le1\}$ for each $t\in\R_+$. This follows by proving that it is uniformly bounded in $L^2(\Omega,\P)$. Denoting the integrand in $\Lambda_2^\e(\hat e, \tilde N^\alpha)$ by
\begin{equation*}
\Gamma^\e(s,y):=\epsilon\left[\hat e_\epsilon\left(X_{s-}^{x,\epsilon}+ \sigma_\epsilon \left(X_{s-}^{x,\epsilon},y\right) \right)-\hat e_\epsilon\left(X_{s-}^{x,\epsilon}\right) \right].
\end{equation*}
Then by It\^o's formula,
\begin{equation*}
\begin{split}
e^{2\Lambda_2^\e(t)} =&\ 1-\int_0^t\int_{B^c} 2e^{2\Lambda_2^\e(s-)}\Gamma^\e(s,y) \nu^\alpha(dy)ds \\
&\ +\int_0^t\int_{\Bo} e^{2\Lambda_2^\e(s-)}\left(e^{2\Gamma^\e(s,y)}-1\right)\tilde N^\alpha(dy,ds) \\
&\ +\int_0^t\int_{B^c} e^{2\Lambda_2^\e(s-)} \left(e^{2\Gamma^\e(s,y)}-1\right) N^\alpha(dy,ds) \\
&\ +\int_0^t\int_{\Bo} e^{2\Lambda_2^\e(s-)} \left[e^{2\Gamma^\e(s,y)}-1- 2\Gamma^\e(s,y)\right] \nu^\alpha(dy)ds.
\end{split}
\end{equation*}
Since $\hat e$ is bounded, $\Gamma^\e$ has a uniform bound for all $\e>0$. Then there exists a large constant $C>0$ such that for each $\e>0$ and $t\in\R_+$,
\begin{equation*}
\E\int_0^t\int_{B^c}\left|e^{2\Lambda_2^\e(s-)} \left(e^{2\Gamma^\e(s,y)}-1-2\Gamma^\e(s,y)\right)\right|^2 \nu^\alpha(dy)ds \le C\nu^\alpha(B^c)\E\int_0^t e^{2\Lambda_2^\e(s-)} ds<\infty.
\end{equation*}
Hence combining these, there exists $\theta\in(0,1)$ such that
\begin{equation*}
\begin{split}
\E e^{2\Lambda_2^\e(t)} & =1+\E\int_0^t\int_{\Bo} e^{2\Lambda_2^\e(s-)} \left[e^{2\Gamma^\e(s,y)}-1- 2\Gamma^\e(s,y)\right] \nu^\alpha(dy)ds \\
&\le 1+\E\int_0^t e^{2\Lambda_2^\e(s-)} \int_{\Bo}2e^{2\theta\Gamma^\e(s,y)} |\Gamma^\e(s,y)|^2 \nu^\alpha(dy)ds \\
&\le 1+C(\|\hat e\|_0)\E\int_0^t e^{2\Lambda_2^\e(s-)} \int_{\Bo} |\Gamma^\e(s,y)|^2 \nu^\alpha(dy)ds.
\end{split}
\end{equation*}
As shown in the proof of part (iii) and (iv) in Lemma \ref{FCLT},
\begin{equation*}
\int_{\Bo} |\Gamma^\e(s,y)|^2 \nu^\alpha(dy) \le \e^2\|\hat e\|_1^2 \int_{\Bo} |\sigma_\e(X_{s-},y)|^2 \nu^\alpha(dy) \le \frac{\lambda(\S^{d-1})}{2-\alpha} \e^2\|\hat e\|_1^2 |\psi_\e(X_{s-})|^2.
\end{equation*}
Thus,
\begin{equation*}
\E e^{2\Lambda_2^\e(t)} \le 1+\e^2 C(\alpha,\lambda(\S^{d-1}),\|\hat e\|_1,\|\psi\|_{L^\infty})\int_0^t\E e^{2\Lambda_2^\e(s-)}ds.
\end{equation*}
By Gr\"onwall's inequality, the uniform boundness of $\{e^{\Lambda_2^\e(t)}|0<\e\le1\}$ in $L^2(\Omega,\P)$ follows.
Secondly,
the set $\{\Lambda_1^\e(t)| 0<\e\le1\}$ is bounded by virtue of the boundness of $c,g$ and $\hat e$. Also since $u_0$ is periodic and continuous, $\{u_0(X_t^\e)|0<\e\le1\}$ is bounded. Thus, the set $\{u_0(X_t^{x,\epsilon})\exp(\hat Y_t^\epsilon)|0<\e\le1\}$ is uniformly integrable.
Finally, we pass to the limit. It is easy to see that $e^{\hat Y_t^\e}\to e^{y(t)}$ in probability as $\e\to0$.
Then for any subsequence $\{\e_n\}\to0$, there exists a subsubsequence $\{\e_{n_k}\}\to0$ such that $e^{\hat Y_t^{\e_{n_k}}}\to e^{y(t)}$ almost uniformly (cf. \cite[Lemma 4.2]{Kal06}). That is, for any $\rho>0$, there exists a set $N\in\F$ with $\P(N)\le\rho$, such that
\begin{equation}\label{a.un.}
\left\|e^{\hat Y_t^{\e_{n_k}}}-e^{y(t)}\right\|_{L^\infty(N^c,\P)}\to 0, \quad k\to\infty.
\end{equation}
By the boundness of $u_0$, we know the set $\{u_0(X_t^{x,\epsilon})[\exp(\hat Y_t^\epsilon)-\exp(y(t))]|0<\e\le1\}$ is also uniformly integrable. Then for any $\delta>0$, there exist $\rho_0>0$ and $N_0\in\F$ with $\P(N_0)\le\rho_0$, such that
\begin{equation}\label{uni_abs_cont}
\E\left|u_0(X_t^\e)\left(e^{\hat Y_t^\e}- e^{y(t)}\right)\ind_{N_0}\right|<\delta.
\end{equation}
Now along the sequence $\{\e_{n_k}\}$, we combining \eqref{a.un.} with \eqref{uni_abs_cont} to get
\begin{equation*}
\begin{split}
\E\left|u_0(X_t^{\e_{n_k}})\left(e^{\hat Y_t^{\e_{n_k}}}- e^{y(t)}\right)\right| &\le \E\left|\cdots\ind_{N_0}\right| + \E\left|\cdots\ind_{N_0^c}\right| \\
&\le \delta+\|u_0\|_{L^\infty}\P(N_0^c)\left\|e^{\hat Y_t^{\e_{n_k}}}-e^{y(t)}\right\|_{L^\infty(N^c,\P)} \\
&\le 2\delta.
\end{split}
\end{equation*}
To summarize these together, for any subsequence $\{\e_n\}\to0$, there exists a subsubsequence $\{\e_{n_k}\}\to0$ such that
\begin{equation*}
\E\left|u_0(X_t^{\e_{n_k}})\left(e^{\hat Y_t^{\e_{n_k}}}- e^{y(t)}\right)\right|\to 0, \quad k\to \infty,
\end{equation*}
which implies that the convergence holds on the whole line $0<\e\le1$. On the other hand, by Proposition \ref{HomoSDE}, we know that $\E|u_0(X_t^\e)-u_0(X_t)|\to0$ as $\e\to0$. The result \eqref{u_conv} follows immediately.
\end{proof}
\begin{remark}
We close this section by some comments for the proof of Theorem \ref{HomoPDE}. In \cite{Par99}, the author applied Girsanov's transform to get rid of the stochastic integral term involved in $\hat Y^\e$, since this term may not possess the uniformly integrability. While in our case, since the stochastic integral term in $Y_t^\e$ has an infinitesimal integrand $\Gamma^\e(s,y)$, the uniform integrability of $\{\exp(\hat Y_t^\e)|0<\e\le1\}$ is easier to treat. \qed
\end{remark}
\bigskip
\textbf{Acknowledgements}. The research of J. Duan was partly supported by the NSF grant 1620449. The research of Q. Huang was partly supported by China Scholarship Council (CSC), and NSFC grants 11531006 and 11771449. The research of R. Song is supported in part by a grant from the Simons Foundation ($\#$ 429343, Renming Song).
{\footnotesize
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 245 |
Feuerman had always seen himself as a New Yorker, but "everything changed that day," he said.
"There really is no safe place," said Feuerman, whose children survived but lost friends in the massacre that killed 17 people at Marjory Stoneman Douglas High School.
About 30 weeks a year, Scott Dacey drives from his home near New Bern, N.C., to Washington for a few days. The 350-mile trips are a price the federal lobbyist pays for peace of mind after Sept. 11.
"It really made us have a wake-up call: 'How do we want to live our lives?'" Scott said. "Do we want to be up here in this rat race of Washington, D.C.?" Or raising kids somewhere that didn't feel so on-guard, somewhere closer to family in times of crisis?
The choice wasn't simple, particularly for a lobbyist. The couple's 2002 move to the New Bern suburb of Trent Woods meant extra costs, including a Washington apartment and a then-advanced phone system to make sure Scott wouldn't miss clients' calls to his office there.
Jennifer, already a lawyer, had to take a second bar exam in North Carolina.
New York and church had brought the couple together in the 1980s: she a Haitian-American from Brooklyn, he a white art student from Massachusetts. They had a small house and a full life. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,613 |
\section{Introduction}
Although
studies of spiral galaxies
commonly assume an axisymmetric
equilibrium state, possibly perturbed
by spiral arms,
there is growing evidence that
many galaxies lack this
symmetry.
Based on optical
appearance, approximately $30\%$ of disk
galaxies exhibit significant
``lopsidedness'' (Rix \& Zaritsky 1995;
Kornreich, Haynes, \& Lovelace 1998) which
supports the early findings of Baldwin,
Lynden-Bell, \& Sancisi (1980).
As many as $\sim50\%$
of spiral galaxies show
departures from the expected symmetric
two--horned global HI line profile
(Richter \& Sancisi 1994; Haynes
{\it et al.} 1998).
Furthermore, HI maps of several galaxies,
for example, NGC~3631 (Knapen 1997),
NGC~5474 (Rownd, Dickey, \& Helou 1994), and
NGC~7217 (Buta {\it et al.} 1995),
have revealed offsets between the
optical centers of light and
their kinematic centers.
Two examples of kinematically
lopsided galaxies
have recently
be discussed by Swaters {\it et al.} (1998).
Other recent
observations such as the
{\it Hubble Space Telescope\/}
observations
of the nucleus of M31
(Lauer {\it et al.} 1993)
and the Kitt Peak 0.9
meter observations of NGC~1073
(Kornreich {\it et al.} 1998)
indicate that even the optical
center of light may be displaced from
the center of the optical
isophotes of the major part of the galaxy in
a significant fraction of cases.
The origin of the lopsideness
in disk galaxies remains uncertain.
Baldwin
{\it et al.} (1980) proposed a
simple kinematic model in which
different rings of the galaxy,
assumed non-interacting, are initially
shifted from their centered
equilibrium positions.
The shifted rings precess in the overall
gravitational potential in a direction
opposite to that of the mass motion.
Because the precession rate
decreases in general with radial
distance, an initial disturbance tends
to ``wind up'' into a leading
spiral arm in a time appreciably
less than the Hubble time.
Miller and Smith (1988, 1992) have
made extensive computer simulation
studies of the unstable
eccentric motion of matter in the
nuclei of galaxies which they
suggest is pertinent to the off-center
nuclei observed in a number of
galaxies, for example, M31, M33, and M101.
Understanding of the origin
of the observed disk asymmetries
is important because it
provides clues to ongoing
accretion of gas and the distribution
of dark, unseen mass in the
halo.
Schoenmakers, Franx, and de Zeeuw (1997),
interpret optical
asymmetry as an indicator of
asymmetry in the overall galactic
potential, and therefore an
indicator of the spatial
distribution of the dark
matter in a galaxy which may have
a triaxial distribution.
By analyzing the spiral
components present in the
surface brightness or \ion{H}{1}
distribution, the velocity
gradient, and therefore, the shape of the
gravitational potential, may be uncovered.
Jog (1997)
has studied of the
orbits of stars and gas in a lopsided
potential, and shows that
lopsided potentials arising from disks
alone are not self-consistent;
rather, a stationary lopsided disk may
be responding to asymmetries in the halo.
Zaritsky and Rix (1997)
proposed that optical
lopsidedness arises from tidal
interactions and/or minor mergers.
Such mergers are often suggested as
the most likely contributors to galaxy
asymmetry, even when no
interacting companions are evident.
While galaxies such as
NGC~5474 are well-known to
be under the tidal
influence of neighbors,
the apparently long-lived kinematic
offsets of other relatively
isolated objects, and the common
asymmetries in flocculent
(as opposed to tidally induced ``grand
design'') spiral galaxies,
are not explained by simple tidal
interaction models, which
produce only transient asymmetric
features.
A further possibility is that an optical disk
may be in a quasi-stationary lopsided
state in a symmetric potential,
as discussed by Syer and Tremaine (1996).
In this model,
gaseous and stellar matter swirl about the
minimum of the halo potential
in a state not fully relaxed.
The result is a lopsided flow within a
symmetric mass distribution.
Numerical simulations of this situation
have been done by Levine and Sparke (1998) using
a gravitational $N-$body
tree-code method (see Barnes \&
Hut 1989) for disk galaxies
shifted from the center of the main halo
potential. The results are
suggestive of lopsidedness with large
lifetimes.
An $N-$body simulation study
of a rotating spheroidal
stellar system including the dynamics
of a massive central object by
Taga and Iye (1998a) indicates that
the central object goes into a
long lasting oscillation similar
to those found earlier by Miller and
Smith (1988, 1992) and
which may explain asymmetric
structures observed
in M31 and NGC 4486B.
A linear stability analysis of
a self-gravitating fluid disk
including a massive
central object also by Taga and Iye (1998b)
indicates a linear instability
(Taga \& Iye 1998b).
We comment on the relation of this
work to the present study in the
conclusions section of this work.
Here, we develop a theory of
the dynamics of eccentric perturbations
of a disk galaxy residing in a spherical
dark matter halo.
We represent
the disk as a large number $N$
of rings as suggested by Baldwin {\it et al.} (1980)
(and Lovelace (1998) for the treatment
of disk warping).
In contrast with Baldwin {\it et al.}, the
gravitational interactions between the
rings is fully accounted for.
We show that for general
eccentric perturbations,
the centers of the rings are shifted
{\it and} the azimuthal distribution
of matter in the rings is perturbed.
The ring representation is
analogous to the approach of Contopoulos and
Gr{\o}sbal (1986, 1988), where self-consistent
galaxy models are constructed from a finite
set of stellar orbits.
Section 2 develops a theory
for treating eccentric perturbations
of a disk galaxy.
The assumed equilibrium
is first discussed (\S 2.1), and a
description of the disk perturbations
is developed (\S 2.2).
The representation
of the disk in terms of a finite
number $N$ of rings
is presented (\S 2.3), and the ring
equations of motion are derived (\S 2.4).
We renormalize the ring
equations so as to reduce the nearest
ring interactions (\S 2.5).
The dynamics and influence
of the displacement of
the center of the galaxy, which may
include a massive black hole, is
discussed separately (\S 2.6).
We obtain an energy constant of the
motion for the dynamical equations (\S 2.7),
the Lagrangian,
and the
conserved total canonical angular
momentum (\S 2.8).
We discuss the nature
of the precession of
a single ring in \S 3.
In \S 4 we study the eccentric motion
of a disk consisting
of two rings and show that this
motion is unstable for sufficiently
large ring masses.
In \S 5 we study the eccentric motion
of one ring including the radial shift
of the central mass and show that
this situation is unstable for sufficiently
large mass of the center and/or of
the ring.
Section 6 presents numerical results
for the eccentric dynamics of disk
of many rings including the radial shift
of the central mass.
Section 7 summarizes the conclusions
of this work.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig1.eps}
\caption{
Sample disk rotation curve
$v_\phi(r)$
for the values
$M_d = 6\times 10^{10}M_\odot$ and
$r_d=4$ kpc for the disk,
$M_b=5 \times 10^{9}M_\odot$ and $r_b=1$ kpc for
the bulge, and $v_h=250$ km/s and
$r_h = 5$ kpc for the halo, using
expressions given
in \S 2.1.
}
\end{figure*}
\section{Theory}
\subsection{Equilibrium}
The equilibrium galaxy is assumed to be
axisymmetric and to consist of a thin disk of
stars and gas and a spheroidal
distributions consisting of
a bulge component and a halo of
dark matter.
We use an inertial cylindrical $(r,\phi,z)$
and Cartesian $(x,y,z)$ coordinate systems with the
disk and halo equatorial planes in the $z=0$ plane.
The
total gravitational potential is written as
\begin{equation}
\Phi(r,z)=\Phi_{d} +\Phi_b+\Phi_h~,
\end{equation}
where $\Phi_{d}$ is the potential due to the
disk, $\Phi_b$ is due to the bulge,
and $\Phi_h$ is that for the
halo.
The galaxy may have a central massive
black hole of mass $M_{bh}$ in which
case a term $\Phi_{bh}=-GM_{bh}/\sqrt{r^2+z^2}$ is
added to the right-hand side of (1).
The particle orbits in the equilibrium disk
are approximately circular with angular
rotation rate $\Omega(r)$, where
\begin{equation}
\Omega^2(r)={1\over r}
{{\partial \Phi}\over{\partial r}}\bigg|_{z=0} =
\Omega_{d}^2+\Omega_b^2+\Omega_h^2~.
\end{equation}
The equilibrium disk velocity is
${\bf v} = r \Omega(r) \hat{{\hbox{\tenbg\char'036}}~}.$
A central black hole is accounted for
by adding the term $\Omega_{bh}^2=GM_{bh}/r^3$
to the right-hand side of (2).
The surface mass density of the (optical) disk
is taken to be $\Sigma_d =\Sigma_{d0} {\rm exp}
(-r/r_d)$ with $\Sigma_{d0}$ and $r_d$
constants and $M_d=2\pi r_d^2\Sigma_{d0}$ the
total disk mass. The potential
due
to this disk matter is
$$\Phi_d(r,0)=-~{G M_d \over r_d}
R[I_0(R)K_1(R)
-I_1(R)K_0(R)] ~,
$$
and the corresponding angular velocity is
\begin{equation}
\Omega_d^2={1\over 2}{G M_d \over r_d^3}
\left[I_0(R)K_0(R)
-I_1(R) K_1(R) \right]~,
\end{equation}
where $R\equiv r/(2r_d)$
and the $I's$ and $K's$ are
the usual modified Bessel
functions (Freeman 1970; Binney \& Tremaine 1987, p.77).
Typical values are
$M_d = 6 \times 10^{10}{\rm M}_\odot$
and $r_d = 4$ kpc.
For these values, $v_d \equiv \sqrt{GM_d/r_d}
\approx 255$ km/s.
The potential due to the bulge component
is taken as a Plummer model
$$
\Phi_b =-~{G M_b \over (r_b^2+r^2+z^2)^{1/2}}~,
$$
where $M_b$ is the mass of the bulge and $r_b$
is its characteristic radius (Binney \& Tremaine
1987, p.42).
We have
\begin{equation}
\Omega_b^2 = { G M_b \over
(r_b^2 + r^2)^{3/2}}~.
\end{equation}
Typical values
are $M_b = 10^{10} {\rm M}_{\odot}$ and $r_b =
1$ kpc, and for these values $v_b \equiv \sqrt{GM_b/r_b}
\approx 208~ {\rm km/s}$.
The potential of the halo is taken to
be
$$\Phi_h = {1\over 2}v_{h}^2 ~
\ln(r_{h}^2+{ r}^2+z^2)~,
$$
where
$v_{h}=$ const is the circular
velocity at large distances and
$r_{h}=$ const is the core radius
of the halo.
We have
\begin{equation}
\Omega_h^2 = {v_{h}^2 \over r_{h}^2+{ r}^2}~.
\end{equation}
Typical values are $v_{h} \sim 200 -300$ km/s
and $r_{h}\sim 2-20$ kpc. Figure 1 shows
an illustrative rotation curve.
\placefigure{fig1}
\subsection{Perturbations}
We treat
the disk as fluid and use a
Lagrangian representation
for the perturbation
as developed by Frieman and Rotenberg (1960).
The position vector ${\bf r}$ of a fluid
element which at $t=0$ was at ${\bf r}_0$ is given
by
\begin{equation}
{\bf r} = {\bf r}_0 +{\hbox{\tenbg\char'030}}({\bf r}_0,t)~.
\end{equation}
That is, ${\bf r}_0(t)$ is the unperturbed and
${\bf r}(t)$ the perturbed orbit of a fluid element.
This description is applicable to {\it both}
the disk gas and the disk stars which are in
approximately laminar motion with circular
orbits. The perturbations of the
halo and bulge are assumed negligible compared
with that of the disk.
For these approximately spheroidal components,
the particle motion is highly non-laminar with
criss-crossing orbits with the result that
their response ``averages out'' the disk
perturbation.
Further, the perturbations
are assumed to consist
of small in-plane displacements or shifts
of the disk matter,
\begin{equation}
{\hbox{\tenbg\char'030}} =\xi_r \hat{{\bf r}}+
\xi_\phi\hat{{\hbox{\tenbg\char'036}}~}~,
\end{equation}
with azimuthal mode number $m=1$.
That is, $\xi_r$ and $\xi_\phi$ have
$\phi$-dependences proportional to
$\exp(i\phi)$.
From equation (6), we have
${\bf v}({\bf r},t)={\bf v}_0({\bf r}_0,t)+
\partial {\hbox{\tenbg\char'030}}/\partial t +
({\bf v} \cdot {\bf \nabla})
{\hbox{\tenbg\char'030}}$.
The Eulerian velocity perturbation is
$\delta {\bf v}({\bf r},t) \equiv {\bf v}({\bf r},t)
-{\bf v}_0({\bf r},t)$.
Therefore,
\begin{equation}
\delta{\bf v}({\bf r},t) ={\partial {\hbox{\tenbg\char'030}} \over \partial t}
+\left({\bf v} \cdot {\bf \nabla}\right) {\hbox{\tenbg\char'030}}
-\left({\hbox{\tenbg\char'030}} \cdot {\bf \nabla} \right) {\bf v}~.
\end{equation}
The components of this equation are
\begin{eqnarray}
\delta v_r &=& {\cal D} ~\xi_r~, \nonumber \\
\delta v_\phi &=& {\cal D} ~\xi_\phi -r \Omega^\prime~ \xi_r~,
\end{eqnarray}
where
$$
{\cal D} \equiv {\partial \over \partial t}
+\Omega(r){\partial \over \partial \phi}~,
$$
and
$\Omega^\prime \equiv \partial \Omega/\partial r$~.
The main equation of motion is
\begin{equation}
{d~ \delta {\bf v} \over dt} = \delta {\bf F}
= -{\bf \nabla } \delta \Phi~,
\end{equation}
where $\delta {\bf F}$ is the perturbation
in the gravitational force
(per unit mass), and $\delta \Phi$ is
the perturbation of the gravitational
potential.
The pressure force contribution
is small compared to $\delta {\bf F}$ by a factor
$(v_{th}/v_\phi)^2 \ll 1$ and is neglected, where
$v_{th}$ is the `thermal' spread of the velocities
of the disk matter.
Also, note that
\begin{equation}
{d~ \delta{\bf v} \over dt}=
{\partial ~\delta {\bf v} \over \partial t}
+\left({\bf v}\cdot {\bf \nabla} \right) \delta {\bf v}
+\left(\delta {\bf v} \cdot {\bf \nabla} \right) {\bf v}~.
\end{equation}
The components of (11) give
\begin{eqnarray}
\left({d~ \delta {\bf v} \over dt }\right)_r &=&
{\cal D} \delta v_r -2 \Omega \delta v_\phi~, \nonumber \\
&=& ({\cal D}^2 +2\Omega r \Omega^\prime)~ \xi_r
-2\Omega{\cal D}~\xi_\phi~, \nonumber\\
\left({d~ \delta {\bf v} \over dt} \right)_\phi
&=& {\cal D} \delta v_\phi
+(\kappa_r^2/2\Omega) \delta v_r~, \nonumber \\
&=& {\cal D}^2 ~\xi_\phi +2\Omega{\cal D} ~\xi_r~,
\end{eqnarray}
where $\kappa_r^2 \equiv (1/r^3)d(r^4\Omega^2)/dr$
is the radial
epicyclic frequency (squared).
The perturbation of the surface mass
density of the disk obeys
\begin{eqnarray*}
{\partial~\delta \Sigma \over \partial t}
= -{\bf \nabla}\cdot \left(\Sigma ~\delta {\bf v}
+\delta \Sigma ~{\bf v} \right)~,
\end{eqnarray*}
where $\Sigma(r)$ is the surface density of the
equilibrium disk.
Because ${\bf \nabla} \cdot
(\Sigma~{\bf v}_0) = 0$, this equation implies
\begin{equation}
\delta \Sigma = - {\nabla}\cdot
\left(\Sigma~{\hbox{\tenbg\char'030}}\right)~.
\end{equation}
The perturbation of the gravitational potential
is given by
\begin{equation}
\delta \Phi({\bf r},t) = - G \int d^2 r^\prime~
{\delta \Sigma({\bf r}^\prime,t)
\over \left| {\bf r} - {\bf r^\prime} \right|}~,
\end{equation}
where the integration is over the
surface area of the disk.
\subsection{Ring Representation}
We represent the disk
by a finite number $N$
of radially shifted plane circular rings.
The matter distribution around each
ring is also perturbed. An elliptical
distortion of a ring
corresponds to an $m=\pm2$ which is
not considered here.
This description is
{\it general} for small shifts
$( dr/r)^2 \ll1$, where
the linearized equations are applicable
(Lovelace 1998).
Of course, the orbit of a single
perturbed particle is not in general
closed in the inertial frame used. However,
the orbit {\it is} closed in an
appropriately rotating frame, and this
rotation rate is simply the angular
precession frequency of the ring $\omega$
discussed below (Baldwin {\it et al.} 1980).
The disk is assumed geometrically thin.
For the equilibrium disk we take
\begin{equation}
\Sigma (r) = \sum_{j=1}^N {M_j
\over 2\pi r \sqrt{2\pi} \Delta r_j}~
\exp \left[-~{(r-r_j)^2 \over 2\Delta r_j^2} \right]~,
\end{equation}
where $M_j$ is the mass of the $j^{th}$
ring, $r_j$ is its radius with
$0<r_1 < r_2 ~...~ r_N$, and $\Delta r_j \ll r_j$
is its width.
The motion of the central part of the
disk ($r<r_1$) is treated separately in \S 2.6
A physical choice for the rings distribution
will have the ring spacing of the order of
the disk thickness,
$r_{j+1}-r_j = {\cal O}( \Delta z)$.
Further, we assume
$(r_{j+1}-r_j)^2 \ll r_{j+1}r_j$ in order
to simplify the calculation of the ring
interaction as discussed in Appendix A.
For example, a possible choice is
$r_j=r_1+(j-1)\delta r$ with
$r_1=1$ kpc, $\delta r=0.5$ kpc,
and $M_j=2\pi r_j\delta r
\Sigma_d(r_j)$.
For $\Delta r_j =
\delta r/\sqrt{2\ln2}\approx \delta r/1.177$,
the profile of a ring falls to half its
maximum value at $\delta r$ so that
equation (15)
gives a fairly smooth
representation of $\Sigma_d(r)$.
The perturbation in the disk's
surface density is
\begin{equation}
\delta \Sigma(r,\phi,t) = \sum \delta \Sigma_j~,
\quad
\delta \Sigma_j=
-{\bf \nabla}\cdot (\Sigma_j {\hbox{\tenbg\char'030}})~,
\end{equation}
where $\Sigma_j \!\equiv \!
(M_j/2\pi r \sqrt{2\pi} \Delta r_j)
\exp[- (r-r_j)^2/(2\Delta r_j^2)]$.
We can express different moments of the
perturbed disk in terms of the rings.
For example, the center of mass of the
disk is
\begin{equation}
<{\bf r}>
={\sum M_j < {\bf r}_j >
\over \sum M_j}~,
\end{equation}
where
\begin{eqnarray}
M_j<{\bf r}_j>& =& \int d^2x~ {\bf r} ~\delta \Sigma_j~,
\nonumber \\
&=& M_j \oint {d\phi \over 2\pi}~ {\hbox{\tenbg\char'030}}_j~,
\end{eqnarray}
where an integration by parts has been made.
We can write in general
\begin{eqnarray}
\xi_{jr} &=& \epsilon_{jx}\cos\phi +
\epsilon_{jy}\sin\phi~,\nonumber \\
\xi_{j\phi}&=& -\delta_{jx}\sin\phi +
\delta_{jy}\cos\phi~.
\end{eqnarray}
Here, $\epsilon_{j x,y}$ and $\delta_{jx,y}$ are
the {\it ring displacement amplitudes}:
$\epsilon_{jx,y}$
represents the shift of the ring's center, and
$\delta_{jx,y}$ represents in general both
the shift of the ring's center {\it and} the
azimuthal displacement of the ring matter.
Firstly, notice that for $\epsilon_{jx,y}=0$ and
$\delta_{jx,y} \neq 0$, there is no shift of
the ring's center but rather an azimuthal
displacement of the ring matter. In this
case $\delta \Sigma_j =
-(\Sigma_j/r_j)(\partial \xi_{j\phi}/\partial \phi)$.
Secondly, notice that $\delta_{jx,y} = \epsilon_{jx,y}$
corresponds to a {\it rigid} shift of the
ring without azimuthal displacement of
the ring matter. For example, a rigid shift
in the $x-$direction has $\epsilon_{jx}=\delta_{jx}$
and $\epsilon_{jy}=0=\delta_{jy}$ so that
$\xi_{jr}=\epsilon_{jx}\cos\phi$ and $\xi_{j\phi}=
-\epsilon_{jx}\sin\phi$.
In this case, ${\bf \nabla \cdot}
{\hbox{\tenbg\char'030}}_j=0$ so that $\delta \Sigma_j =
-{\bf \nabla \cdot}
(\Sigma_j {\hbox{\tenbg\char'030}}_j) =
- \xi_{rj}(\partial \Sigma_j /\partial r)$.
Figure 2 shows the nature of
perturbations with a shift of the ring's center
and with an azimuthal displacement of
the ring matter.
\placefigure{fig2}
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig2.eps}
\caption{
Drawing of two perturbed
rings with
equilibrium radii $r_1=1$ and $r_2=2$ in
arbitrary units.
The
center of ring $1$ is rigidly shifted
in the $x-$direction from the origin by
$\epsilon_{1x}=0.4$ and $\epsilon_{1y}=0$.
The center of ring $2$ is at the origin while
the distribution of matter
around the ring is perturbed as
indicated by the small circles. For
this ring, $\delta_{2x}=0.4$ and $\delta_{2y}=0$.
}
\end{figure*}
Equation (18) now gives
\begin{eqnarray}
<r_{jx}>& =& {1\over 2}(\epsilon_{jx} + \delta_{jx})~,
\nonumber \\
<r_{jy}> &=& {1\over 2}(\epsilon_{jy} + \delta_{jy})~.
\end{eqnarray}
For the case of a rigid shift of a ring,
$\epsilon_{jx,y}=\delta_{jx,y}$, the center of
mass position is simply $<r_{jx,y}>=\epsilon_{jx,y}$,
as expected.
Similarly, the velocity perturbation of
the disk can be written as
\begin{equation}
<\delta {\bf v}>
={\sum M_j < \delta {\bf v}_j >
\over \sum M_j}~,
\end{equation}
where
\begin{equation}
<\delta {\bf v}_j > = {1\over M_j} \int d^2x~
\left(\delta \Sigma_j {\bf v} +
\Sigma_j \delta {\bf v}_j \right)~.
\end{equation}
Evaluation of (22) gives
\begin{eqnarray}
<\delta v_{jx}>\!&=&\!{1\over2}\left(
\dot{\epsilon}_{jx}
+\dot{\delta}_{jx} \right),\nonumber\\
<\delta v_{jy}>\!&=&\!{1\over2}\left(
\dot{\epsilon}_{jy}
+\dot{\delta}_{jy} \right)~.
\end{eqnarray}
The influence of the eccentric motion
on the rotation curves is discussed at
the end of \S 6.3.
\subsection{Ring Equations of Motion}
Equations (12) and (19) give the ring
equations of motion,
\begin{eqnarray}
\ddot{\epsilon}_x+2\Omega\dot{\epsilon}_y
-\tilde{\Omega}^2\epsilon_x
-2\Omega (\dot{\delta}_y -\Omega \delta_x)& =&
<\delta F_r^C>~,
\nonumber \\
\ddot{\epsilon}_y-2\Omega\dot{\epsilon}_x
-\tilde{\Omega}^2\epsilon_y
+2\Omega (\dot{\delta}_x +\Omega \delta_y)& =&
<\delta F_r^S>~,
\nonumber \\
\ddot{\delta}_x+2\Omega\dot{\delta}_y -\Omega^2\delta_x
-2\Omega(\dot{\epsilon}_y -\Omega {\epsilon}_x)& =&
\!\!\!-<\delta F_\phi^S>~,
\nonumber \\
\ddot{\delta}_y-2\Omega\dot{\delta}_x -\Omega^2\delta_y
+2\Omega(\dot{\epsilon}_x +\Omega {\epsilon}_y) &=&
<\delta F_\phi^C>~,
\nonumber\\
\end{eqnarray}
where the $j$ subscripts are implicit,
where the angular
brackets indicate the
average over the ring
$<(..)>\equiv 2\pi \int r dr (..) \Sigma_j(r)/M_j$,
where
$\tilde{\Omega}^2 \equiv
\Omega^2 -2\Omega r\Omega^\prime$, and
$$
\delta F_\alpha^{C,S}
\equiv \oint {d\phi \over \pi}~
\left[\cos \phi,~\sin \phi\right]\delta F_\alpha
~,
$$
with $\alpha = r,~\phi$.
We now evaluate the different force terms on
the right-hand side of (24). For this
it is useful to write $\delta \Sigma =
\delta \Sigma_a + \delta \Sigma_b$, where
\begin{equation}
\delta \Sigma_a =-{1\over r}{\partial (r\Sigma \xi_r)
\over \partial r}~,\quad \delta \Sigma_b =
-{1\over r}{\partial (\Sigma \xi_\phi) \over
\partial \phi}~,
\end{equation}
from equation (13).
The corresponding contributions to the potential,
$\delta \Phi = \delta \Phi_a + \delta \Phi_b$
evaluated at $(r,\phi)$
are from (13),
$$
\delta \Phi_a(r,\phi) =
G\sum_k M_k(\epsilon_{kx} \cos \phi
+\epsilon_{ky} \sin \phi)~ \times \quad \quad \quad
$$
\begin{equation}
\int_0^\infty r^\prime dr^\prime~
\bigg\{{{1 \over r^\prime}
{\partial \over \partial r^\prime}
\bigg[r^\prime S(r^\prime|r_k)\bigg]}\bigg\}
\oint{d \Psi^\prime \over 2\pi}~
{\cos \Psi \over R(r,r^\prime)}
~,
\end{equation}
and
$$
\delta \Phi_b(r,\phi) =
- G\sum_k M_k(\delta_{kx}\cos\phi
+\delta_{ky}\sin\phi)~\times \quad \quad \quad
$$
\begin{equation}
\int_0^\infty r^\prime dr^\prime~
{S(r^\prime|r_k) \over r^\prime}
\oint {d\Psi \over 2\pi}~
{\cos \Psi \over R(r,r^\prime)}
\end{equation}
where $R^2(r,r^\prime) \equiv
r^2+(r^\prime)^2-2rr^\prime \cos\Psi$,
$$
S(r|r_k) \equiv { 2\pi \Sigma_k(r) \over M_k}=
{\exp[-(r-r_k)^2/2\Delta r_k^2]
\over r \sqrt{2\pi} \Delta r_k}~,
$$
and
$\Psi \equiv \phi^\prime -\phi$.
Evaluation of the force on
the $j^{\rm th}$ ring due to the other rings gives
\begin{eqnarray}
M_j <\delta F_{rj}^C> &=&
\sum_{k}
\left(C_{jk} \epsilon_{kx}
+D_{jk} \delta_{kx}\right)~, \nonumber \\
M_j <\delta F_{rj}^S> &=&
\sum_{k}
\left( C_{jk} \epsilon_{ky}
+D_{jk}\delta_{ky}\right)~, \nonumber \\
-M_j<\delta F_{\phi j}^S> &=&
\sum_{k}
\left(E_{jk} \delta_{kx}
+D^\prime_{jk} \epsilon_{kx}\right)~,\nonumber \\
M_j<\delta F_{\phi j}^C> &=&
\sum_{k}
\left(E_{jk} \delta_{ky}
+D^\prime_{jk} \epsilon_{ky}\right),
\end{eqnarray}
where the `tidal coefficients' are
$$
C_{jk}=-GM_jM_k
\int \!\!\int r dr ~r^\prime dr^\prime ~\times
\quad\quad\quad\quad\quad\quad~~
$$
\begin{equation}
S(r|r_j)
{\partial \big[r^\prime S(r^\prime|r_k)\big]
\over r^\prime ~\partial r^\prime}
{\partial {\cal K}(r,r^\prime) \over \partial r}
~,
\end{equation}
$$
D_{jk}=GM_jM_k\int\!\!\int
r dr~r^\prime dr^\prime~ \times
\quad\quad\quad\quad\quad\quad\quad\quad
$$
\begin{equation}
{S(r|r_j)}
{S(r^\prime|r_k)}
{\partial {\cal K}(r,r^\prime) \over r^\prime~\partial r}
\end{equation}
$$
D_{jk}^\prime=-GM_jM_k
\int\!\!\int rdr~ r^\prime dr^\prime ~\times
\quad\quad\quad\quad\quad\quad\quad
$$
\begin{equation}
{S(r|r_j)\over r}
{\partial \big[r^\prime S(r^\prime|r_k)\big]
\over r^\prime \partial r^\prime}~
{\cal K}(r,r^\prime)
~,
\end{equation}
$$
E_{jk}=GM_jM_k\int\!\!\int rdr~
r^\prime dr^\prime ~\times
\quad\quad\quad\quad\quad\quad\quad\quad
$$
\begin{equation}
{S(r|r_j) }
{S(r^\prime|r_k) }~
{{\cal K}(r,r^\prime) \over r~r^\prime }~,
\end{equation}
where
\begin{equation}
{\cal K}(r,r^\prime) \equiv \oint
{d\Psi \over 2 \pi}
{\cos\Psi \over R(r,r^\prime)}~,
\end{equation}
and where the $r,r^\prime$ integrals
are all from $0$ to $\infty$.
Formal integration by parts of (29) gives
$$
C_{jk}=GM_jM_k
\int \!\!\int r dr ~r^\prime dr^\prime ~\times
\quad\quad\quad\quad\quad\quad~~
$$
\begin{equation}
S(r|r_j)
S(r^\prime|r_k)
{\partial^2 {\cal K}(r,r^\prime)
\over \partial r^\prime ~\partial r}
~,
\end{equation}
which shows that $C_{jk}=C_{kj}$.
Also, integration by parts of
(31) gives
$$
D_{jk}^\prime=GM_jM_k
\int\!\!\int rdr~ r^\prime dr^\prime ~\times
\quad\quad\quad\quad\quad\quad\quad
$$
\begin{equation}
{S(r|r_j)}
{ S(r^\prime|r_k)}~
{\partial {\cal K}(r,r^\prime)
\over r~ \partial r^\prime}
~,
\end{equation}
and this shows that
$D_{jk} = D^\prime_{kj}$.
Expressions for the tidal coefficients
in terms of elliptic
integrals are given in the Appendix.
Following the approach of Lovelace (1998),
we introduce the complex displacement
amplitudes
\begin{equation}
{\cal E}_j \equiv \epsilon_{jx} - i \epsilon_{jy}
=\epsilon_j(t) \exp[- i\varphi_j(t)]~,
\end{equation}
\begin{equation}
~~~~~\Delta_j \equiv \delta_{jx}- i \delta_{jy}
= \delta_j(t) \exp[-i \psi_j(t)]~.
\end{equation}
Here, $\epsilon_j \geq 0$ is the amplitude
of the shift of the ring's center, and
$\varphi_j$ is angle of the shift with
respect to the $x-$axis; $\delta_j \geq 0$
is the amplitude of the azimuthal displacement
of the ring matter, and $\psi_j$ is the angle
of the maximum of the ring density also
with respect to the $x-$axis. If $\varphi(t)$
and $\psi(t)$ increase with time, the ring
precesses in the same sense as the particle
motion and we refer to this as {\it forward
precession}. The opposite case, with $\varphi(t)$
and $\psi(t)$ decreasing with time, is termed
{\it backward precession}.
We now combine equations (24) and (28) to
obtain the ring equations of motion,
\begin{eqnarray}
M_j \left( ~\ddot{\cal E}_j +
2i\Omega_j \dot{\cal E}_j -
\tilde{\Omega}^2_j {\cal E}_j -
2i\Omega_j \dot{\Delta}_j +2\Omega_j^2 \Delta_j
\right) = \nonumber \\
\sum_{k}
\left( C_{jk} {\cal E}_k
+D_{jk} \Delta_k \right)~, \\
M_j \left( \ddot{\Delta}_j +
2i\Omega_j \dot{\Delta}_j -
\Omega_j^2 \Delta_j -
2i\Omega_j \dot{\cal E}_j
+2\Omega_j^2 {\cal E}_j\right)= \nonumber \\
\sum_{k }
\left(E_{jk} \Delta_k+
D^\prime_{jk} {\cal E}_k \right)~,
\end{eqnarray}
where $j=1..N$, $\Omega_j \equiv \Omega(r_j)$, and
$\tilde{\Omega}_j \equiv \tilde{\Omega}(r_j)$.
\subsection{Renormalization of Ring Equations}
Here, we redo the ring equations of
motion so as to diminish the strong
tidal interactions of nearest neighbor
rings due to the terms $\propto C_{j,j+k}$.
First,
we rewrite the right-hand
side of (38) as
\begin{eqnarray}
\sum_{k}
\left[C_{jk} \left({\cal E}_k-{\cal E}_j\right)
+D_{jk} \left({\Delta}_k-{\cal E}_j \right)\right]+\nonumber\\
{\cal E}_j\sum_{k}(C_{jk} + D_{jk})~,
\end{eqnarray}
Similarly, we rewrite the right-hand side of (39)
as
\begin{eqnarray}
\sum_{k}
\left[E_{jk} \left({\Delta}_k-{\Delta}_j\right)
+D_{jk}^\prime \left({\cal E}_k-
{\Delta}_j \right)\right]+\nonumber\\
{\Delta}_j\sum_{k}\left( E_{jk} + D_{jk}^\prime \right)~.
\end{eqnarray}
We define
\begin{equation}
\Omega_{{\cal E}j}^2 \equiv \tilde{\Omega}_j^2+
{1\over M_j}\sum_{k}
\left(C_{jk}+D_{jk} \right)~.
\end{equation}
Using the relations $\tilde{\Omega}^2=\Omega^2
-r(d\Omega^2/dr)$,
$\Omega^2=\Omega_d^2 +\Omega_b^2+\Omega_h^2$,
and equation (A14)
of the Appendix gives
\begin{equation}
\Omega_{{\cal E}j}^2 =
\Omega_j^2
-\bigg(r{\partial (\Omega_{b}^2+\Omega_h^2)
\over \partial r}\bigg)_j
+{1 \over M_j }
\sum_{k }(D^\prime_{jk} +E_{jk})~.
\end{equation}
(If there is a central black hole $M_{bh}$ then
the right-hand side of (43) also has the
term $-[r(\partial \Omega_{bh}^2/\partial r)]_j
=3GM_{bh}/r_j^3$.)
Similarly, we define
\begin{equation}
\Omega_{\Delta j}^2 \equiv \Omega_j^2 +
{1 \over M_j}\sum_{k }(D^\prime_{jk}+E_{jk})~.
\end{equation}
The ring equations of motion now become
$$
M_j \left( ~\ddot{\cal E}_j +
2i\Omega_j \dot{\cal E}_j -
\Omega_{{\cal E}j}^2 {\cal E}_j -
2i\Omega_j \dot{\Delta}_j +2\Omega_j^2 \Delta_j
\right) =
$$
\vspace{-0.12in}
\begin{equation}
\sum_{k}
\left[ C_{jk} \left({\cal E}_k-{\cal E}_j \right)
+D_{jk} \left(\Delta_k-{\cal E}_j\right) \right]~,
\end{equation}
$$
M_j \left( \ddot{\Delta}_j +
2i\Omega_j \dot{\Delta}_j -
\Omega_{\Delta j}^2 \Delta_j -
2i\Omega_j \dot{\cal E}_j
+2\Omega_j^2 {\cal E}_j\right)=
$$
\begin{equation}
\sum_{k }
\left[E_{jk} \left(\Delta_k-\Delta_j\right)+
D^\prime_{jk} \left({\cal E}_k-\Delta_j\right) \right]~.
\end{equation}
In the large $N$ limit,
the sums in equations (45) and (46)
go over to bounded integrals.
The
diagonal elements, $C_{jj}$
(and $E_{jj}$), are absent from (45) and (46).
Note also that the self-interaction of
a ring vanishes as it should for the case of a rigid
shift where ${\cal E}_j = \Delta_j$.
\subsection{Dynamics and Influence of ``Center''}
The dynamical equations (45) and (46) do
not account for the part of the disk
inside the innermost ring $r_1$. Also,
there may be a massive black
hole $M_{bh}$ near
the galaxy center.
We treat this central region
separately as a point
mass $M_0$,
\begin{equation}
M_0=M_{bh}+2\pi \int_0^{r_1^\prime}
rdr~\Sigma_d(r)~,
\end{equation}
where $r_1^\prime \equiv r_1-\delta r_1/2$.
The
horizontal displacement of the
``center'' is
\begin{equation}
\rvecepsilon_0(t)=\epsilon_{0x}\hat{\bf x}
+\epsilon_{0y}\hat{\bf y}~.
\end{equation}
The equation of motion for $\rvecepsilon_0$
taking into account the eccentric displacemnts
of the rings
is
\begin{equation}
M_0\bigg( { d^2 {\cal E}_0 \over dt^2} +
\Omega_0^2 {\cal E}_0\bigg)
=-\sum_{k=1}^N F_{0k}\left( {\cal E}_k -
{1\over 2} \Delta_k \right)~,
\end{equation}
where
\begin{equation}
{\cal E}_0 \equiv \epsilon_{0x} - i\epsilon_{0y}
=\epsilon_0\exp(-i\varphi_0)
\end{equation}
is the complex displacement amplitude of the ``center'';
$F_{0k} \equiv GM_0M_k/r_k^3$, $k=1..N$, are
the tidal coefficients between the ``center'' and the
rings; and
\begin{equation}
\Omega_0^2 =
\bigg({1\over r}{\partial \Phi \over \partial r}\bigg)_0
=\bigg({1\over r}{\partial (\Phi_b+\Phi_h)
\over \partial r}\bigg)_0
-{1\over 2 M_0}\sum_j F_{0j}
\end{equation}
is the angular oscillation frequency of a particle
at the galaxy center.
From \S 2.1, we
have $[(1/r)(\partial \Phi_b/\partial r)]_0
= GM_b/r_b^3$ and
$[(1/r)(\partial \Phi_h/\partial r)]_0
= v_{h}^2/r_{h}^2$.
For all of the considered conditions,
we find $\Omega_0^2>0$.
The influence of the displaced ``center'' on
the rings is included by adding the force
terms, due to the ``center's''
displacement, to the right-hand sides of equations
(45) and (46),
\begin{equation}
M_j \left( ~\ddot{\cal E}_j ~+~..\right)
=-~2F_{0j}{\cal E}_0~+~..~,
\end{equation}
\begin{equation}
M_j\left(~\ddot{\Delta}_j~+~.. \right)
=+~F_{0j}{\cal E}_0~+~..~,
\end{equation}
where the ellipses denote terms
in equations (45) and (46).
\subsection{Energy Conservation}
An energy constant of the motion of the ring
system can be obtained by multiplying
equation (52) by $\dot{\cal E}_j^*$
and (53) by $\dot{\Delta}_j^*$
(with $[...]^*$ denoting
the complex conjugate), adding
the two equations, summing over
$j$, and dividing by 2.
(This factor of $2$ makes the
kinetic energy of a rigidly
shifted ring with $\rvecepsilon_j=
\rvecdelta_j$ equal to
$(1/2)M_j \dot{\rvecepsilon}_j^2$.)
Further, we multiply equation (49)
by ${\cal E}_0^*$ and add the result to
the previous sum.
In this way we find $d E/dt =0$, where
\vspace{-0.05in}
$$
E\!=\!{1\over 4}\sum_j M_j
\left(\dot{\rvecepsilon}_j^2 -
{\Omega}^2_{{\cal E} j}\rvecepsilon^2_j +
\dot{\rvecdelta}_j^2 -
\Omega_{\Delta j}^2 \rvecdelta_j^2 +
4\Omega_j^2 \rvecepsilon_j \cdot \rvecdelta_j
\right)
$$
\vspace{-0.1in}
$$
+{1\over 8}\sum_{k\neq j}\sum_{j}
\bigg[
C_{jk} (\rvecepsilon_j - \rvecepsilon_k)^2
+2D_{jk}(\rvecepsilon_j- \rvecdelta_k)^2+
$$
\vspace{-0.1in}
$$
+E_{jk}(\rvecdelta_j - \rvecdelta_k)^2 \bigg]
$$
\begin{equation}
+{1\over 2}M_0 (\dot{\rvecepsilon}_0^2
+\Omega_0^2 {\rvecepsilon}_0^2)
+\sum_jF_{0j}\left(\rvecepsilon_0\cdot \rvecepsilon_j
-{1\over 2}\rvecepsilon_0\cdot \rvecdelta_j \right)~,
\end{equation}
and where the real vectors
$\rvecepsilon=\epsilon_x\hat{\bf x}
+\epsilon_y \hat{\bf y}$ and
$\rvecdelta=\delta_x\hat{\bf x}
+\delta_y \hat{\bf y}$ are useful here.
\subsection{Lagrangian}
By inspection, we find the Lagrangian
for the ring system ${\cal L}
(\epsilon_{jx},\dot{\epsilon}_{jx}, ..)$,
\vspace{-0.05in}
$$
{\cal L}={1\over 4}\sum_j M_j
\bigg\{ \dot{\rvecepsilon}_j^2 +
{\Omega}^2_{{\cal E}j}\rvecepsilon^2_j +
\dot{\rvecdelta}_j^2 +
\Omega_{\Delta j}^2 \rvecdelta_j^2 -
4\Omega_j^2 \rvecepsilon_j\cdot \rvecdelta_j
$$
\vspace{-0.15in}
$$
-2\Omega_j [(\rvecepsilon_j-\rvecdelta_j)
\times (\dot{\rvecepsilon}_j -\dot{\rvecdelta}_j)
]\cdot \hat{\bf z} \bigg \}+
{1\over 2}M_0
(\dot{\rvecepsilon}_0^2-\Omega_0^2\rvecepsilon_0^2)
$$
$$
- {1\over 8}\sum_{k\neq j}\sum_{j}
\bigg[
C_{jk} (\rvecepsilon_j - \rvecepsilon_k)^2
+2D_{jk}(\rvecepsilon_j- \rvecdelta_k)^2+
$$
\begin{equation}
\vspace{-0.1in}
+E_{jk}(\rvecdelta_j - \rvecdelta_k)^2 \bigg]
-\sum_j F_{0j}\left(\rvecepsilon_0\cdot \rvecepsilon_j
-{1\over 2}\rvecepsilon_0\cdot \rvecdelta_j \right)~,
\end{equation}
\noindent Because $\partial {\cal L}/\partial t = 0$,
the Hamiltonian,
\begin{equation}
{\cal H} \equiv \sum_j\left(\dot{\epsilon}_{jx}
{\partial {\cal L} \over
\partial \dot{\epsilon}_{jx}} + ... \right)
-{\cal L}~,
\end{equation}
is a constant of the motion.
It is readily verified that ${\cal H}=E$.
We can make a canonical transformation
$(\epsilon_{jx},\epsilon_{jy})$ $ \rightarrow
(\epsilon_j,\varphi_j)$,
$(\delta_{jx},\delta_{jy}) \rightarrow
(\delta_j,\psi_j)$ to obtain
the Lagrangian as ${\cal L}=
{\cal L}(\dot{\epsilon}_j,\dot{\delta}_j,
\epsilon_j, \delta_j, \varphi_j,\psi_j)$.
Note for example that $\dot{\rvecepsilon}^2_j
\rightarrow \dot{\epsilon}_j^2 + \epsilon_j^2
\dot{\varphi}_j^2$.
It is then clear from the azimuthal
symmetry of the equilibrium that ${\cal L}$ is
invariant under the simultaneous changes
$\varphi_j \rightarrow \varphi_j+\theta$,
$\psi_j \rightarrow \psi_j+\theta$ for
$j=1,..,N$, where $\theta$ is an arbitrary
angle. Thus
$$
\sum_j\left({\partial {\cal L}
\over \partial \varphi_j} +
{\partial {\cal L} \over \partial \psi_j} \right)
=0~,
$$
and consequently the total canonical angular
momentum of the ring system,
\begin{equation}
{\cal P}_\phi \equiv \sum_j
\left({\partial {\cal L} \over
\partial \dot{\varphi}_j} +
{\partial {\cal L} \over
\partial \dot{\psi}_j } \right)~,
\end{equation}
is another constant of the motion.
Evaluating (57) gives
\vspace{-0.05in}
$$
{\cal P}_\phi = {1\over 2}\sum_j M_j
\bigg[\epsilon_j^2(\dot{\varphi}_j-\Omega_j)
+\delta_j^2(\dot{\psi}_j-\Omega_j)
$$
\vspace{-0.1in}
\begin{equation}
+2\Omega_j\epsilon_j\delta_j\cos(\varphi_j-\psi_j)
\bigg]+M_0\epsilon_0^2\dot{\varphi}_0~,
\end{equation}
where the last term represents the angular momentum
of the galaxy center.
The last term within the square
brackets can also be written
as $2\Omega_j (\rvecepsilon_j
\cdot \rvecdelta_j)$.
The constants of the motion ${\cal H}$ and
${\cal P}_\phi$ are valuable for checking
numerical integrations of the equations
of motion (52) and (53).
\begin{figure*}[t]
\epsfscale=400
\plotone{Efig3.eps}
\caption{
The top panel ({\bf a}) of
the figure shows the radial
dependence of the four frequencies
$\omega_\alpha$
($\alpha=1,..,4$) of the modes of
oscillation of an eccentric
ring normalized by the disk's angular
rotation rate $\Omega(r)$.
The bottom panel ({\bf b}) shows the
corresponding radial dependence
of the mode amplitude
ratio ${\cal E}/\Delta$
as discussed in
the text.
The case shown is
for the galaxy parameters of
Figure 1 with innermost
ring of radius $r_1=1$ kpc.
The values
$\Omega_j$, $\Omega_{{\cal E}j}$,
and $\Omega_{\Delta j}$ were
obtained using equations (A11),
(43), and (44).
}
\end{figure*}
\section{Eccentric Motion of a Single Ring}
Consider the
eccentric motion
of a particular ring with the other rings
not excited.
This is {\it not}
a self-consistent limit because
gravitational interactions will
in general excite all of the rings.
However this limit
is informative.
With ${\cal E}_j \propto \Delta_j
\propto \exp(-i\omega t)$, $j=1,..,N$, where $\omega$
the ring precession frequency,
we get
$$
\left[(\omega-\Omega)^2 +
E\right]{\cal E}
+\left[2\Omega (\omega -\Omega )+d~\right]\Delta=0~,\quad~~
$$
\begin{equation}
\left[2\Omega(\omega-\Omega)+d~ \right]{\cal E}
+\left[(\omega-\Omega)^2+
D\right]\Delta
=0~,
\end{equation}
where the $j$ subscripts
are implicit,
$E \equiv \Omega_{{\cal E}j}^2 -\Omega_j^2-D_{jj}/M_j$,
$D \equiv \Omega_{\Delta j}^2-\Omega_j^2-D_{jj}/M_j$,
and $d \equiv D_{jj}/M_j$.
With $w \equiv \omega-\Omega$, we get
\begin{equation}
w^4+(D+E-4\Omega^2)w^2-4d\Omega w +ED-d^2=0~.
\end{equation}
For the limit of many rings, the terms
involving $d$ become negligible, and (60)
can be readily solved to give four
real roots if $0<ED<(4\Omega^2-D-E)^2/4$.
Figure 3a shows the radial dependence of
the four ring
precession frequencies $\omega_\alpha$
($\alpha =1,..,4$) corresponding to the
four modes of oscillation. These modes
are the analogues of the normal modes
of vibration of a non-rotating
system (see Lovelace
1998).
Three of the modes in Figure 3a
have positive frequencies ($\omega_\alpha >0$
for $\alpha =2,3,4$)
so that they have forward
precession with
\begin{equation}
\varphi_\alpha
=\omega_\alpha t+ {\rm const}~,\quad
{\rm and}\quad \psi_\alpha =
\omega_\alpha t+ {\rm const}'~.
\end{equation}
These relations follow from (36) and (37)
because $|{\cal E}|=\epsilon$
and $|\Delta|=\delta$ are constants.
The other mode ($\alpha=1$) has $\omega_1 <0$
and therefore backward precession.
Figure 3b shows that the two modes
$\alpha =2,3$ have
${\cal E}/\Delta $ small compared with
unity near
the inner radius of the disk, $r_1$.
Note that ${\cal E}/\Delta=0$ corresponds
to a pure azimuthal shift of the ring matter
without a shift of the ring center.
The mode $\alpha=1$ with
backward precession has ${\cal E}/\Delta \sim 1$.
As mentioned, ${\cal E}/\Delta =1$
corresponds to a rigid shift of the ring center.
Evaluation of the ring energy for the
four modes using equation (54)
shows that the modes $\alpha=1,2$ have
{\it negative energy} whereas
the modes $\alpha=3,4$ have positive energy.
The negative energy modes are unstable
in the presence of dissipation, for example,
the force due to dynamical friction
(see for example Lovelace 1998).
Note that for vanishing ring mass
($D \propto d \propto M_j\rightarrow 0$),
the middle two
roots approach $\omega
=\Omega \pm [D E (4\Omega^2-E)]^{1/2}$.
Thus for $M_j\rightarrow 0$,
there are only three different roots
of equation (60), $\omega=\Omega$
and $\omega=\Omega \pm (4\Omega^2-E)^{1/2}$.
Outside of the central region of
a galaxy we have
$\Omega \sim 1/r$.
Consequently,
the radial dependences of the mode
frequencies
$\omega_\alpha(r) \propto \pm 1/r$ will tend to
``wrap up'' an initially coherent asymmetry
into a tightly wrapped spiral (in
the absence of ring interactions).
The forward precessing modes ($\alpha=2-4$)
will give a {\it trailing} spiral wave,
$\varphi_\alpha \propto 1/r$,
(with respect to the azimuthal motion $v_\phi>0$),
whereas the backward precessing mode ($\alpha =1$)
will give a {\it leading} spiral wave,
$\varphi_1 \propto -1/r$.
The case of mode $\alpha=1$ for
non-interacting rings
was discussed earlier by Baldwin {\it et al.} (1980).
\section{Eccentric Motion of a ``Disk'' of Two Rings}
Here, we consider the eccentric motion
of a ``disk'' consisting
of two interacting rings,
one of mass
$M_1$ and radius $r_1$,
and the other of mass $M_2$,
radius $r_2$.
The values of $\Omega_j^2$,
$\Omega_{{\cal E}j}^2$,
$\Omega_{\Delta j}^2$ ($j=1,2$)
are given by equations
(A11), (43), and (44) with the bulge and halo
potentials as given in \S 2.1.
Thus the present treatment
is {\it self-consistent} (in contrast with the
previous subsection).
With ${\cal E}_j$ and
$\Delta_j$ proportional to
$\exp(-i\omega t)$, equations (45) and (46) give
$$
\big[-(\omega-\Omega_1)^2+\Omega_1^2-
\Omega_{{\cal E}1}^2 +D_{11}\big]{\cal E}_1
\quad \quad \quad \quad \quad \quad
$$
\vspace{-0.2in}
$$
\quad \quad \quad \quad
+~\big[2\Omega_1(\Omega_1-\omega)-D_{11}\big]\Delta_1
$$
\begin{equation}
=
\big[C_{12}({\cal E}_{2}-{\cal E}_1)
+ D_{12} (\Delta_2 -{\cal E}_1)\big]/M_1~,
\end{equation}
\vspace{0.01in}
$$
\big[-(\omega-\Omega_1)^2+
\Omega_1^2-\Omega_{\Delta 1}^2 +D^\prime_{11}\big]\Delta_1
\quad \quad \quad \quad \quad \quad
$$
$$ \quad \quad \quad \quad
+~[2\Omega_1(\Omega_1-\omega)-D^\prime_{11}\big]{\cal E}_1
$$
\begin{equation}
=\big[E_{12}(\Delta_2-\Delta_1)
+ D_{12}^\prime ({\cal E}_2-\Delta_1)\big]/M_1~,
\end{equation}
\vspace{0.01in}
$$
\big[-(\omega-\Omega_2)^2+\Omega_2^2 -
\Omega_{{\cal E}2}^2+D_{22}\big]
{\cal E}_2 \quad \quad \quad \quad \quad \quad
$$
$$
\quad \quad \quad \quad
+~\big [2\Omega_2(\Omega_2-\omega)-D_{22}\big]\Delta_2
$$
\begin{equation}
=\big[C_{12}({\cal E}_{1}-{\cal E}_2)
+ D_{12}^\prime( \Delta_1 - \Delta_2)\big]/M_2~,
\end{equation}
\vspace{0.01in}
$$
\big[-(\omega-\Omega_2)^2+
\Omega_2^2-\Omega_{\Delta 2}^2+D^\prime_{22}\big]\Delta_2
\quad \quad \quad \quad \quad \quad
$$
$$
\quad \quad \quad \quad
+~\big[2\Omega_2(\Omega_2-\omega){\cal E}_2-D^\prime_{22}\big]
$$
\begin{equation}
=\big[E_{12}(\Delta_1-\Delta_2)
+ D_{12}( {\cal E}_1-\Delta_2)\big]/M_2~.
\end{equation}
For a non-zero solution, the determinant
of the $4\times4$ matrix multiplying
$({\cal E}_1,~\Delta_1,~{\cal E}_2,~\Delta_2)$
must be zero.
This leads to an eighth
order polynomial in $\omega$ which
can be readily solved (with Maple R. 5) for the
frequencies of the $8$ modes ($\alpha=1..8$).
\begin{figure*}[t]
\epsfscale=400
\plotone{Efig4.eps}
\caption{
The figure shows
the eccentric instability of a
``disk'' of mass $M_d$ consisting of
two rings of
equal mass $M_d/2$
at equilibrium radii $r_1=3$
and $r_2=5$ kpc.
The top panel (${\bf a}$)
shows the dependence of the real
parts of the
frequencies $\omega_{r\alpha}$
($\alpha=1,..,8$)
on $M_d$.
The labels $u1,u2$ indicate
unstable branches where the
frequency is complex; $u1$ is
referred to in the text
as the ``first instability''
and $u2$ the ``second instability.''
Here,
$t_o\equiv 10^6$ yr.
The bulge has
$M_b=0.5\times 10^{10}M_\odot$ and $r_b=1$ kpc,
and the halo $v_h=250$ km/s and
$r_h = 5$ kpc in the
expressions given
in \S 2.1.
The bottom panel (${\bf b}$) shows
the dependence of the growth rate
$\omega_i$ on the ring mass. The
onset of instability corresponds to
the merging of the two real frequencies
in panel (${\bf a}$).
The tidal
coefficients, $\{C_{jk}\},$, etc., are
obtained using the equations of
the Appendix and
$\Delta r_j=2$ kpc.
}
\end{figure*}
Figure 4 shows the behavior, including
instability, of a system of two rings of
equal mass $M_1=M_2$ so that
the ``disk'' mass is $M_d = 2 M_1$.
For $M_d \rightarrow 0$, the modes
$\alpha=3,4$ approach $\Omega_2$ and
the modes $\alpha=5,6$ approach $\Omega_1$
which agrees with the behavior found in \S 3.
Note that for small $M_d$ modes $\alpha=1-4$ are
associated with ring $2$ while modes $\alpha=5-8$
are associated with ring $1$.
In the absence of interactions, the
rings are stable independent of their mass.
As the disk mass increases,
the modes $\alpha=4,5$ approach each other
and merge at $M_d\approx 0.276 \times 10^{10}
M\odot$ and give instability with
${\cal I}m(\omega)\equiv \omega_i >0$
as shown in Figure 4b.
We refer to this as the ``first instability.''
Notice that the onset of instability
corresponds to a merging of the positive energy
mode $\alpha=4$ of ring $2$ with the negative
energy mode $\alpha=5$ of ring $1$ (see \S 3).
The interaction of positive and negative
energy modes is a well-known instability
mechanism (see for example Lovelace, Jore,
\& Haynes 1997). At the instability
threshold, a dimensionless measure of the
ring self-gravity is
$GM_d/(\bar{r}^3 \bar{\Omega}^2) \sim 0.1$,
where $\bar{r}=4$ kpc and $\bar{\Omega}
\approx 0.0458/t_o$.
The dependence
of the growth rate is well fitted
by $\omega_i t_o
\approx 0.00842(M_d-M_{c1})^{0.67}$,
with the masses in units of $10^{10}M_\odot$
and $t_o \equiv 10^6$ yr.
For $M_d >3.12\times 10^{10}\equiv M_{c2}$,
there is a ``second instability'' with growth
rate $\omega_it_o \approx
0.0196(M_d-M_{c2})^{0.55}$.
The ratios of the complex perturbation
amplitudes can readily be obtained from
(62) - (65) once the $8$, possibly
complex frequencies are known.
For the ``first instability,'' for example,
for $M_d= 10^{10}M_\odot$ and the same
conditions as for Figure 4,
we find ${\cal E}_1 \approx 0.254-0.117i$,
$\Delta_1=1$ (by choice), ${\cal E}_2 \approx
0.0108-0.365i$, and $\Delta_2 \approx
0.673+1.30i$.
These values correspond to
$\varphi_1 \approx 24.7^\circ$,
$\varphi_2 \approx 91.7^\circ$,
$\psi_1=0$ (by choice), and
$\psi_2\approx 297^\circ$.
Thus the azimuthal density enhancement
in the outer ring {\it trails} the
density enhancement of the inner ring.
On the other hand,
the radial shift of the outer ring
{\it leads} the shift of the inner ring.
As a second example, for
$M_d =4 \times 10^{10}M_\odot$
both the ``first'' and ``second''
instability occur.
For the
``first'' instability, we again find that
the azimuthal density enhancement
of the outer ring {\it trails} that
of the inner ring, whereas
the radial shift of the
outer ring {\it leads} that
of the inner ring.
For the ``second'' instability
the situation is different in
that {\it both} the azimuthal density
enhancement and the radial shift
of the outer ring {\it trail} those
in the inner ring.
Specifically,
we find $\varphi_1\approx 178^\circ$,
$\varphi_2 \approx 89.3^\circ$,
$\psi_1=0$ (by choice),
and $\psi_2 \approx 283^\circ$.
Note that for both rings, the angle of
the radial shift is roughly $180^\circ$
displaced from the
azimuthal density enhancement.
The displacement of the center of mass
of the ring (equation 20) is
dominated by the azimuthal
displacement ($|{\cal E}_1|
\approx 0.70|\Delta_1|$
and $|{\cal E}_2|\approx 0.67|\Delta_2|
\approx 0.22|\Delta_1|$).
Thus having $\varphi_j$
and $\psi_j$ about $180^\circ$ out of phase
allows the center of mass of
each ring to be closer to the origin,
and this is a lower energy configuration.
Consider now the angular momentum of
the perturbed two ring system
which from (58) is
${\cal P}_\phi = {\cal P}_{1\phi}
+{\cal P}_{2\phi}={\rm const}$.
For the evaluation of
${\cal P}_\phi$, note that for each
of the eight modes,
$\dot{\Delta}_j = -i\omega_\alpha \Delta_j$,
and $\dot{\cal E}_j=-i\omega_\alpha {\cal E}_j$,
where $\alpha=1,..,8$ labels the mode.
This implies that
\begin{equation}
\omega_\alpha
=\dot{\psi}_j +i(\dot{\delta}_j/\delta_j)
~=~\dot{\varphi}_j +i(\dot{\epsilon}_j/\epsilon_j)~.
\end{equation}
Thus, for the ``first instability,''
for $M_d= 10^{10}M_\odot$ where $\omega t_o
\approx 0.0500+0.00719i$,
we have $\dot{\psi}_j=\dot{\varphi}_j=0.05/t_o$,
(which corresponds to forward precession), and
$\dot{\delta}_j/\delta_j =\dot{\epsilon}_j
/\epsilon_j = 0.00719/t_o$.
For an unstable mode, the coefficients
of the six terms in ${\cal P}_\phi$ are all
grow exponentially.
The only possible
way in which ${\cal P}_\phi =$const can
be maintained is to have ${\cal P}_\phi =0=
{\cal P}_{\phi1}+{\cal P}_{\phi2}$.
This relation provides a useful check
on the correctness of the calculations.
For the ``first instability'' ($M_d>M_{c1}$),
we find by evaluating
(58) that
${\cal P}_{\phi1}=
-{\cal P}_{\phi2} >0$.
This means that
the angular momentum of the inner ring increases
while that of the outer ring decreases.
Thus the instability
acts to transfer angular momentum {\it inward}.
In contrast, for the ``second instability''
($M_d>M_{c2}$), we find that
${\cal P}_{\phi1}=-{\cal P}_{\phi2} < 0$.
Thus, the ``second instability''
acts to transfer angular momentum
{\it outward}.
If $P_{\phi1}$ decreases and $P_{\phi2}$
increases, then the average radius of ring
1 must decrease and
that of ring 2 must increase.
Therefore, the ``second instability'' may be
important for the accretion of matter in
a gravitating disk.
\section{Eccentric Motion of
One Ring and ``Center''}
Here, we consider the eccentric motion of
a ``disk'' of one ring including the influence
of the eccentric motion of the ``center'' $M_0$
which is located at ${\bf r}=0$ in equilibrium.
The mass $M_0$ includes the mass of a central
black hole $M_{bh}$ if it is present.
The ring perturbation is described by ${\cal E}(t)$
and $\Delta(t)$ as given by (52) and (53), while
the ``center'' is described by ${\cal E}_0(t)$ which
is given by (49).
With the perturbations
$\propto \exp(-i\omega t)$, we find
$$
\left[(\omega-\Omega)^2 +
E\right]{\cal E}
+\left[2\Omega (\omega -\Omega )+d~\right]\Delta
=2\Omega_a^2{\cal E}_0~,
$$
$$
\left[2\Omega(\omega-\Omega)+d~ \right]{\cal E}
+\left[(\omega-\Omega)^2+
D\right]\Delta
=~\Omega_a^2 {\cal E}_0~,
$$
\begin{equation}
(\omega^2-\Omega_0^2){\cal E}_0=
\Omega_b^2 \bigg({\cal E}-{\Delta \over 2}\bigg)~,
\end{equation}
where $\Omega_a^2 \equiv G M_0/r_1^3$ and
$\Omega_b^2 \equiv G M_1/r_1^3$, with
$M_1$ and $r_1$ the mass and radius of the
ring.
For a non-zero solution, the determinant
of the $3\times 3$ matrix multiplying
$({\cal E}, \Delta, {\cal E}_0)$ must
be zero.
This leads to a sixth order
polynomial in $\omega$ or $w \equiv \omega-\Omega$
which can readily be solved (with Maple, R5)
for the frequencies of the $6$ modes
$\omega_\alpha$ ($\alpha=1..6$).
We obtain
$$
\left [\left (w+\Omega\right )^{2}
-{{\Omega_0}}^{2}\right]\left [\left ({w}^{2}+
E\right )\left ({w}^{2}+D\right )-\left (2
\Omega w+d\right )^{2}\right]
$$
\begin{equation}
-\Omega_{ab}^4\left ({\frac {
5{w}^{2}}{2}}+2D+{\frac {E}{2}}+2\Omega w
+d\right )=0~,
\end{equation}
where the strength of the interaction
between the ring and the ``center'' is
measured by
\begin{equation}
\Omega_{ab}^2 \equiv
\Omega_a \Omega_b ={ G \sqrt{M_0 M_1}\over r_1^3}~.
\end{equation}
Here, $D,~E,~d$ and $\Omega=\Omega_1$ are defined in
(59), and $\Omega_0$ is defined in (51).
Figure 5 shows the dependence of
the growth rate
$\omega_i={\cal I}m(\omega_\alpha)$
on the mass of the ``center'' $M_0$ with the
mass of the ring held fixed.
The associated real part of the
frequency is positive.
The onset of instability corresponds
to the point where two of the six modes
with frequencies given by (68) merge.
The merging is again of positive and
negative energy modes.
The growth rate has to
a good approximation
the dependence
$\omega_i t_o \approx 0.00764 (M_0-2.61)^{1/2}$,
with $M_0$ in units of $10^9 M_\odot$
and $t_o = 10^6$ yr.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig5.eps}
\caption{
The figure shows
the eccentric instability of a
ring of mass $M_1$
interacting gravitationally
with a displaced point
mass $M_0$ shifted from its
equilibrium position ${\bf r}=0$.
The halo and bulge potentials
are the same as in Figure 1.
The mass $M_0$ includes
the mass of a central black
hole $M_{bh}$ if present.
The mass of the ring is
$M_1= 6 \times 10^{10} M_\odot$
and its radius is $r_1 =3$ kpc.
Here,
$t_o\equiv 10^6$ yr. The tidal
coefficients are evaluated
using the expressions of the
Appendix with $\Delta r_1=2$ kpc.
}
\end{figure*}
The ratios of the complex amplitudes
follow from (67), and for the unstable
mode for $M_0 =2.75 \times 10^9M_\odot$
we find $\omega t_o \approx 0.155 +0.00289i$,
$\Omega t_o \approx 0.131$, $\Omega_0 t_o
\approx 0.142$, $\Omega_{ab} \approx 0.0465$,
${\cal E} \approx -0.817+0.0333i$,
$\Delta =1$ (by choice), and ${\cal E}_0
\approx -3.38+0.916i$ which give
$|{\cal E}|\approx 0.817$, $|{\cal E}_1|
\approx 3.51$, $\varphi \approx 182^\circ$,
and $\varphi_0 \approx 195^\circ$, where
$t_o=10^6$ yr.
Thus, the radial shift of the ring is roughly
$180^\circ$ away from the maximum of the
density enhancement which
has $\psi=0$ (by choice).
The shift of the center of mass of the
ring (equation 20) is dominated by the
azimuthal density enhancement.
Thus the radial shift and azimuthal
displacements are such that the center
of mass of the ring moves closer to
the origin which is a lower energy
configuration.
Note that the radial shift of
the ``center'' {\it trails}
the ring center of mass
by an angle $360^\circ-\varphi_0 \approx 165^\circ$.
Hence, the torque of the ``center'' on the ring
acts to {\it reduce} the angular momentum of the
ring as verified below. At the same time
the ring acts to increase the angular momentum
of the ``center.''
Consider now the angular momentum of the
perturbed ring plus ``center''
system which is given
by (58).
Following the
arguments of the prior section,
the sum of the angular momentum of
the ``center'' and that of the
ring must be zero for a growing
mode.
The angular momentum of the
``center'' is simply
$M_0\epsilon_0^2 \dot{\varphi}_0$.
For a pure mode, we have
${\cal R}e(\omega_\alpha)=\dot{\varphi}_0$.
As mentioned, the real part of the
frequency is positive for the
unstable mode and therefore the angular
momentum of the ``center'' increases
while the angular
momentum of the ring decreases.
Thus, there is a transfer of angular
momentum from the ring to the ``center.''
Due to the loss of angular momentum
the average radius of the ring will
decrease. Thus the instability may
be important for accretion of matter
to the galaxy center.
\section{Eccentric Motion of N Rings}
For the results presented here,
the rings are taken to
be uniformly spaced in $r$
with radii $r_j = 1 +(j-1)\delta r$ kpc
with $\delta r =0.5$ kpc
and with $j=1,..,N=31$.
The value $N=31$ gives good
spatial resolution over all but
the inner part of the disk.
The outer
radius $r_N$ (in the range
say $10 -20$ kpc) has
little influence on the eccentric
motion described here, as
verified by comparing
results with $r_N=16$ kpc
those obtained with
significantly larger $r_N$.
Also, the eccentric motion of
the outer disk, say, $r \gtrsim 3$ kpc, is
essentially independent of $\delta r$.
We first consider in \S 6.1 the case
where the ring masses correspond to
the exponential distribution
discussed in \S 2.1.
The inner part of the disk is found
to be strongly unstable to eccentric
motions and
therefore in \S 6.2 we consider a disk
with the mass of the innermost three rings
reduced. In \S 6.3 we consider disks
with a smooth reduction in $\Sigma_d(r)$
in the inner part of the disk, $r \lesssim r_d$.
We solve
(52) and (53) numerically
as eight first order
equations for
$\epsilon_{xj},~\dot{\epsilon}_{xj},~\epsilon_{yj},$
and $\dot{\epsilon}_{yj}$, and for
$\delta_{xj},~\dot{\delta}_{xj},~\delta_{yj},$
and $\dot{\delta}_{yj}$, $j=1,..,N$.
At the same time, we solve the two additional
equations,
\begin{equation}
{d\varphi_j \over dt~}= { \epsilon_{xj}\dot{\epsilon}_{yj} -
\epsilon_{yj}\dot{\epsilon}_{xj}
\over \epsilon_{xj}^2+\epsilon_{yj}^2}~,
\end{equation}
\begin{equation}
{d\psi_j \over dt~}= { \delta_{xj}\dot{\delta}_{yj} -
\delta_{yj}\dot{\delta}_{xj}
\over \delta_{xj}^2+\delta_{yj}^2}~,
\end{equation}
to give $\varphi_j(t)$, which is the
angle to the maximum of the radial shift, and
$\psi_j(t)$, which is the angle to the maximum
of the azimuthal density enhancement.
These angles are analogous
to the line-of-nodes angles for
the tilting of the rings of a disk
galaxy (Lovelace 1998).
Thus, we solve $10N$ first order equations.
In all cases, the
total energy (54)
and total canonical
angular momentum (58)
are accurately conserved.
The different frequencies $\Omega_j$,
$\Omega_{{\cal E}j}$, and $\Omega_{\Delta j}$,
and the tidal coefficients
$\{C_{jk}\}$, etc., are
evaluated using the equations
of the Appendix.
\subsection{Exponential Disk}
Here, we consider the eccentric
motion of the rings for the case
where
$M_j=2\pi r_j \delta r
\Sigma_d(r_j)$ with
$\Sigma_d(r)=\Sigma_{d0}\exp(-r/r_d)$.
The mass of the center is
assumed given by (47), which
gives $M_0\approx 1.06 \times 10^9$
for the parameters of Figure 1.
Alternatively, this value of $M_0$
could be due in part to a central black
hole.
Figure 6 shows the dependences
of the radial shifts $\epsilon_j(\varphi_j)$
and azimuthal
displacements $\delta_j(\psi_j)$ of
the rings ($j=1-31$) and the radial
shift of the
``center'' $\epsilon_0(\varphi_0)$ at
a short time, $100$ Myr after
an initial perturbation.
This type of plot is related to the
plots emphasized by Briggs (1990)
for characterizing the warps of
galactic disks (see also Lovelace 1998).
The angles
$\varphi$ and $\psi$ are
analogs of the line-of-nodes
angle the warp.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig6.eps}
\caption{
Polar plot of the radial
shift $\epsilon_j$ and
azimuthal displacement $\delta_j$
of ring matter as a function of
the angles $\varphi_j$ and $\psi_j$
($j=1-31$)
at time $t=100$ Myr.
The radial shift of the ``center''
$\epsilon_0$ is indicated by the
solid dot.
The conditions correspond
to the galaxy parameters of Figure 1.
The rings have radii
$r_j=1+\delta r(j-1)$ kpc and $\delta r=0.5$ kpc
for $j=1,..,31$,
masses $M_j=2\pi r_j \delta r \Sigma_d(r_j)$
with $\Sigma_d$ given in \S 2.1,
and $M_0$ given by (47),
which gives $M_0 \approx 1.06\times 10^9 M_\odot$.
The initial values of the shifts
and displacements are
$\epsilon_j =0.1(r_j/r_{max})=\delta_j$,
$\varphi_j=0=\psi_j$, and
$\epsilon_{0x} =
10^{-6}$ and $\epsilon_{0y}=0$.
The units of $\epsilon_j$
and $\delta_j$ are arbitrary in that
the equations are linear.
}
\end{figure*}
\begin{figure*}[b]
\epsfscale=500
\plotone{Efig7.eps}
\caption{
The plot shows
the exponential growth of center
shift $\epsilon_0$ and
radial shift $\epsilon_1$ and
azimuthal displacement $\delta_1$ of
the first ring at $r_1=1$ kpc for
the same conditions as for
Figure 6.
}
\end{figure*}
From Figure 6 note
that the azimuthal displacements
$\delta_j$ are larger than
the radial displacements
$\epsilon_j$ so that the
displacement of the center
of mass of a ring is
dominated by $\delta_j$.
The eccentric motion of the
inner rings, say, $j=1-4$ or $r_1=1$ to
$r_4 =2.5$ kpc,
show the most rapid, exponential
growth.
For these rings
the angles $\psi_j$ and $\varphi_j$
are approximately $180^\circ$
degrees out of phase, and
this agrees with the behavior
found for the ``second'' instability
of two rings discussed in \S 4.
As mentioned, this allows the
center of mass of each ring
to move closer to the origin,
which is a lower energy
configuration.
Note that with increasing
$j$, both $\varphi_j$ and
$\psi_j$ decrease for the
inner rings which corresponds
to a {\it trailing} pattern
the same as found for the
second instability of two rings.
Note also that the shift
of the ``center'' $\epsilon_0$
trails the shift of the center
of mass of the $j=1$ ring
in agreement with \S 5.
Thus the torque of the first
ring on the center acts to
increase the angular momentum
of the center while the
torque of the center on the
first ring decreases the
rings angular momentum.
Figure 7 shows the
exponential growth of the
azimuthal displacement $\delta_1$ and
radial shift $\epsilon_1$ of the first
ring and the simultaneous
growth of the radial shift
of the center $\epsilon_0$.
The $e-$folding time is about $29$ Myr.
For comparison, the period of
oscillation of the center is
$T_0 =2\pi/\Omega_0 \approx 46$
Myr for the conditions shown,
where $\Omega_0$ is given
by (51).
The growth of the eccentric
motion of the inner rings is
reduced somewhat if the mass
of the center is reduced
to $M_0 = 10^8 M_\odot$;
the $e-$folding time for in this
case is about $38$ Myr for ring 1.
Figure 8 shows the
perturbations of
the angular momentum of the center
$P_0$
and the rings $P_j$
at $t=100$ Myr
obtained from (58).
In agreement with \S 5
and the abovementioned direction
of the torque,
the angular momentum of the
center increases while that
of the first and second ring
decrease.
The decrease in angular
momenta of these rings will
result in their radii shrinking.
Note that the center rotates
in the same direction as the
disk matter.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig8.eps}
\caption{
Plot of
the perturbations of the
angular momentum of the center
$P_0$
and the rings $P_j$
at $t=100$ Myr
for the same conditions as Figure 6.
}
\end{figure*}
The present
linear theory does not address
the issue of saturation of
growth of the eccentric motion.
One possibility is that
the strong instability
of the inner rings of the disk
leads to the destruction of this
part of the disk.
\subsection{Exponential Disk with
Rings 1-3 Reduced}
Here, we consider the eccentric
motion of the rings for the case
where the ring masses
are the same
as in \S 6.1 {\it except} that
$M_1 \rightarrow 10^{-2} M_1$,
$M_2 \rightarrow 0.1 M_2$, and
$M_3 \rightarrow 0.3 M_3$,
which are the rings with
radii $r_1=1$, $r_2=1.5$, and
$r_3=2$ kpc. The mass
of the center is the same as
in \S 6.1,
$M_0\approx 1.06 \times 10^9 M_\odot$.
The disk mass is reduced by a factor
$\approx 0.84$ compared with an
exponential disk.
The aim of reducing $M_1-M_3$
is to reduce
the growth rate of the eccentric motion
of this part of the disk.
\begin{figure*}[b]
\epsfscale=500
\plotone{Efig9.eps}
\caption{
Polar plot of the radial
shift $\epsilon_j$ and
azimuthal displacement $\delta_j$
of ring matter as a function of
the angles $\varphi_j$ and $\psi_j$
of the maxima of the shift and
displacement at times $t=300$ and $305$ Myr.
The solid dot labeled by the
arrow indicates the shift of the
center at $t=300$ Myr; the other
dot is the shift at $305$ Myr.
The rings, the center, and
the initial values of the shifts
and displacements are the
same as in Figure 6 except that
$M_1 \rightarrow 10^{-2}M_1$,
$M_2\rightarrow 0.03M_2$, and
$M_3 \rightarrow 0.1M_3$.
Thus the conditions correspond
to the galaxy parameters of Figure 1
except that the disk mass is reduced
to $\approx 5.06 \times 10^{10}M_\odot$.
The shifts and displacements of the
rings $j=1,2$ are dynamically unimportant
and are not shown.
}
\end{figure*}
Figure 9 shows the essential
behavior in a polar plot
of the displacements and shifts
at two times.
The radial shift of the
center is negligible on
the scale of this figure.
The $e-$folding time for
ring 3 is about $49$ Myr.
Notice that the curves
traced out by $\delta_j$
and by $\epsilon_j$ are
approximated straight
lines from the origin
which rotate
rigidly in
the direction of motion
of the matter for
$j=4$ to about $j=11$,
which corresponds to
$r_3=2.5$ to $r_{11}=6$ kpc.
The instantaneous period
of rotation of
this pattern is
$\approx 60$ Myr, which is
longer that the
oscillation period at the
center, $T_0 \approx 46$ Myr.
This case is an example of the
{\it phase-locking} of the
eccentric motion of these
rings due to the self-gravity
between the rings.
This phase-locking is analogous
to that which occurs in
the tilting motion of the
rings representing a disk galaxy due
to self-gravity (Lovelace 1998).
In the case of tilting of
rings the phase-locking results
in the line-of-node angles of
the rings in the inner part of
the disk becoming the same.
Also in this case the growth
of the eccentric instability
of the inner rings is
sufficiently fast that it probably
further disrupts the inner
part of the disk.
\subsection{Reduced Inner Disk}
Here, we study the eccentric motion
of a disk the inner part of which
is attenuated relative to an
exponential disk.
Specifically, the rings masses are
$M_j=2\pi r_j\delta r$ $\hat{\Sigma}_d(r_j)$,
where
\begin{equation}
\hat{\Sigma}_d(r)=
\Sigma_{d0}\exp\left(-\beta{r_d \over r}\right)
\exp\left(-{r \over r_d}\right)~,
\end{equation}
with $\beta=$const.
We consider $\beta=1$.
The mass
of the center is the same as
in \S 6.1,
$M_0\approx 1.06 \times 10^9 M_\odot$.
However, the motion of
the center is negligible and
value $M_0$ has
little influence on the
eccentric motion of the disk described below.
The disk mass is smaller than
for an exponential disk
by a factor $\approx 0.47$.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig10.eps}
\caption{
Polar plot of the radial
shift $\epsilon_j$ and
azimuthal displacement $\delta_j$
of ring matter as a function of
the angles $\varphi_j$ and $\psi_j$
of the maxima of the shift and
displacement at times $t=300$ and $400$ Myr.
The radial shift of the center is
negligible on the scale of this plot.
The rings, the center, and
the initial values of the shifts
and displacements are the
same as in Figure 6
except that the ring masses
are obtained from (72).
The mass of the center is
the same as in Figure 6, $M_0\approx
1.06\times 10^9M_\odot$.
Thus the conditions correspond
to the galaxy parameters of Figure 1
except that the disk mass is reduced
to $\approx 2.82 \times 10^{10}M_\odot$.
The shifts and displacements of the
rings $j=1,2$ are dynamically unimportant
and are not shown.
}
\end{figure*}
\begin{figure*}[b]
\epsfscale=500
\plotone{Efig11.eps}
\caption{
Polar plot of the radius
to the maximum of the azimuthal
displacement $r_j(\psi_j)$ at
two times
for the same case as Figure 10.
}
\end{figure*}
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig12.eps}
\caption{
Radial dependence of
the perturbations in the ring
angular momenta $P_j$
at $t=300$ Myr from
equation (58). The angular
momentum of the center $P_0$
is negligible.
}
\end{figure*}
\begin{figure*}[b]
\epsfscale=500
\plotone{Efig13.eps}
\caption{
Profiles of fractional
density variations $10^2\delta \Sigma(x,0)
/\Sigma(x,0)$ (in percent) and variation of
azimuthal velocity
$10^3\delta v_y(x,0)/v_\phi(N)$
along the $x-$axis through the
middle of the galaxy at $t=400$
Myr for the same
case as Figure 10.
Here, $v_\phi(N)\approx 265$ km/s
is the disk rotation velocity
at $r=16$ kpc.
The vertical scale is
arbitrary in that
the equations solved are linear,
but the ratio
$\delta \Sigma/\delta v_\phi$
is fixed.
}
\end{figure*}
Figure 10 shows the essential
behavior in a polar plot of the shifts
and displacements at two times.
The shift of the center is
negligible on the scale of the
plot.
The magnitude of azimuthal
displacement of the outer ring
$\delta_N$ exhibits
an approximately {\it linear} growth
with time,
$\delta_N \approx {\rm const}+t/660{\rm Myr}$,
for the considered initial conditions
and $t\lesssim 1$ Gyr.
In contrast, the magnitude of
the radial shifts $\epsilon_j$
remains bounded by its initial
maximum value $\epsilon_N(t=0)$.
\begin{figure*}[t]
\epsfscale=500
\plotone{Efig14B.eps}
\caption{
Two-dimensional
appearance of the fractional
density variations $10^2 \delta \Sigma(x,y)
/\Sigma(x,y)$ at $t=400$ Myr for the
same case as Figure 13.
}
\end{figure*}
The patterns formed
by both $\delta_j$ and $\epsilon_j$
in Figure 10
are {\it trailing} spirals.
This is different from the proposal
of Baldwin {\it et al.} (1980) that
leading spirals should form.
The rotation of the outer point
on the spiral $\Delta_N$($r_N=16$ kpc)
in Figure 10 is in the direction
of rotation of the disk matter.
Its pattern speed $\Omega_p$ corresponds
to a period
$2\pi/\Omega_p\approx 460$ Myr, which is
a factor $\approx 1.24$
longer than the rotation period of matter at
this radius ($\approx 370$ Myr).
The pattern period of
say $\Delta_{15}(r=8{\rm kpc})$ is
$\approx 188$ Myr, which is a
less than the rotation period of the matter
at this radius ($\approx 206$ Myr).
Thus, it is evident that the
spiral is ``wrapping up'' as
time increases.
At the same time, the spiral
pattern propagates radially
outward. The outward speed is
about $10$ km/s at $r\sim 8$ kpc for
$t\sim 300$ Myr.
Figure 11 shows a polar plot
of the radius
to the maximum of the azimuthal
displacement $r_j(\psi_j)$ at
two times.
The curve is a trailing
spiral with
an approximate fit given by
\begin{equation}
\psi =
A(t)
\exp\left(- {r\over a(t)}
\right)~,
\end{equation}
where $A \approx 0.065$
$(t/{\rm kpc})$ rad, and
$a \approx 7.0+0.0044 $
$ (t/{\rm Myr})$ kpc
for $t\lesssim 1$ Gyr.
Thus the radial spacing between
spiral arms is
$\lambda_r \approx
(2\pi a/A)\exp(r/a)$ for $\lambda_r \ll a$.
For validity of the ring representation
we must have $\lambda_r \geq 2 \delta r(=1$ kpc),
where $\delta r$ the separation between
rings.
Figure 12 shows the radial variation
of the perturbations in
ring angular momenta. The perturbation
of the angular momentum of
the central mass is negligible.
This figure should be compared with
Figure 8 which gives the same plot
for an exponential disk.
Some simplification of
equations is possible in the
present case at long times due to the fact that
$\delta_j\gg \epsilon_j$.
Firstly, we have
$\delta{\bf v} \approx \delta v_\phi
\hat{{\hbox{\tenbg\char'036}}~}$ with
\begin{equation}
\delta v_\phi(r,\phi)
\approx -(\dot{\delta}_x +\Omega \delta_y)
\sin\phi +(\dot{\delta}_y-\Omega\delta_x)\cos\phi~.
\end{equation}
Secondly,
\begin{equation}
{\delta \Sigma(r,\phi) \over \Sigma(r)}
\approx{1\over r}
(\delta_x\cos\phi + \delta_y \sin\phi)~,
\end{equation}
for $\delta_j \gg r\epsilon_j/r_d$.
Figure 13 shows the profiles along
the $x-$axis through the
galaxy center of the fractional change
in the surface density
$\delta \Sigma/\Sigma$ and
the change in the azimuthal
velocity $\delta v_\phi$ obtained
from (74) and (75).
The opposite signs of $\delta v_\phi$
on the two sides of the galaxy would
of course make the rotation curves on
the two sides different as observed
in some cases (Swaters {\it et al.} 1998).
Note that in some regions the changes
$\delta \Sigma$ and $\delta v_\phi$
are correlated and in other regions
they are anticorrelated.
Figure 14 shows two-dimensional
appearance of the fractional surface
density variations from (75).
For long times $t>1$ Gyr, the
the azimuthal
displacements and shifts of the
inner rings ($2$ and $3$) start
to become large compared with
the values in the outer disk ($r>4$ kpc)
even though these rings have very
small masses.
At the same time,
the displacement of the center,
which has mass $M_0=1.06 \times 10^9M_\odot$,
grows, and at $t=1$ Gyr it is
$\epsilon_0 \approx 0.057$ for the
conditions of Figures 9-13.
If the mass of the center
is $M_0=10^6M_\odot$, then the displacements
and shifts of rings $2$ and $3$ at $t=1$ Gyr
are significantly reduced as is the shift
of the center which is
$\epsilon_0 \approx 0.014$.
\section{Conclusions}
The paper develops a theory of
eccentric ($m=\pm1$) linearized
perturbations of an
axisymmetric disk galaxy
residing in a spherical
dark matter halo and with a spherical
bulge component.
The disk is represented by
a large but finite
number $N$ of rings with shifted
centers {\it and} with perturbed
azimuthal matter distributions.
This description is appropriate
for a disk with small `thermal'
velocity spread $v_{th}$ where the
matter is in approximately laminar
circular motion.
The spread for a
thin disk has $(v_{th}/v_\phi)^2 \ll 1$,
but it is sufficent to give
a Toomre $Q(r) \gtrsim 1$.
Earlier, Baldwin {\it et al.} (1980)
discussed asymmetries in disk galaxies in terms
of shifted rings but without interactions
between the rings and without azimuthal
displacements of the ring matter.
Account is taken of the shift
of the matter at the
galaxy's center, which may include
a massive black hole.
The gravitational interactions
between the rings and between the
rings and the center is fully
accounted for, but the halo and
bulge components are treated as
passive gravitational field sources.
Equations of motion are derived for
the ring and the center, and from
these we obtain the Lagrangian for
the rings$+$center system.
For this system
we derive an energy
constant of the motion,
and a total canonical angular
momentum constant of the motion.
We first discuss the nature
of the precession of
a single ring with the other
rings fixed; this case although
not self-consistent is informative.
There are four modes, analogs to
the normal modes of a non-rotating
system, and two have negative energy
and two positive energy.
Negative energy modes are unstable
in the presence of dissipation such
as that due to dynamical friction.
We go on to study the eccentric motion
of a disk consisting
of two rings of different radii
but equal mass $M_d/2$.
Above a threshold value of $M_d$ the
two rings are unstable with
instability
due merging of positive
and negative energy modes.
This result is obtained by solving
the eighth order polynomial for the
frequencies of the eight modes.
Above a second, somewhat larger threshold
value of $M_d$, a second instability
appears, and in this case the ring
motion is such that the angular
momentum of the inner ring decreases
while that of the outer ring increases.
For the unstable motion, the maximum
of the azimuthal density enhancement
of a ring occurs at an angle about $180^\circ$
from the direction of the radial shift.
This allows the center of mass of the ring
to move closer to the center of mass of
the other ring and to the origin.
We also analyze the eccentric motion of
a disk of one ring interacting
with a radially shifted central mass.
This system has six modes, the frequencies
of which are obtained by solving a sixth
order polynomial.
In this case, instability sets
in above a threshold value of the central
mass (for a fixed ring mass), and it
acts to increase the angular momentum of
the central mass (which therefore rotates
in the direction of the disk matter), while
decreasing the angular momentum of the ring.
The instability is again due to the merging
of positive and negative energy modes.
We study the eccentric dynamics
of a disk with an exponential surface density
distribution represented by a large number
$N=31$ of rings and a central
mass $M_0 \sim 10^9M_\odot$
which may include the mass of a black hole.
The outer radius of the
disk is $r_N =16$ kpc;
we have checked that this value has
negligible affect
on the reported results.
In this case, we numerically integrate
the equations of motion.
A check on the validity of the integrations
is provided by monitoring the mentioned
total energy and total canonical angular
momentum, which are found to be accurately
constant in all presented results.
The inner part of the disk
$r\lesssim 2.5$ kpc is
found to be strongly unstable
with $e-$folding time $\sim 30$ Myr for the
conditions considered. The $e-$folding time
is somewhat longer if $M_0=0$.
Angular momentum of the rings is
transferred {\it outward}, {\it and} to
the central mass if it is present.
A {\it trailing} one-armed spiral
wave is formed in the disk.
This differs from the prediction of
Baldwin {\it et al.} (1980) of a
leading one-armed spiral.
The outer part of the disk $r \gtrsim r_d$
is stable and in this
region the angular momentum is
transported by the wave.
Thus our results appear compatible
with the theorem of Goldreich and
Nicholson (1989) regarding
angular momentum in {\it stable} rotating
fluids.
The instability found here
appears qualitatively
similar to that found
by Taga and Iye (1998b) for a fluid
Kuzmin disk with surface density
$\Sigma \propto 1/(1+r^2)^{3/2}$
with a point mass at the center where
unstable trailing one-armed spiral
waves are found.
The present
linear theory does not address
the issue of saturation of
growth of the eccentric motion.
One possibility is that
the strong instability
of the inner rings of the disk
leads to the destruction of this
part of the disk.
For this reason
we have studied a disk
with a modified exponential
density distribution where
the surface density of
the inner part of the disk is reduced.
However, the mass of the center of
the galaxy was kept the same as in
the case of an exponential disk,
$M_0 \sim 10^9 M_\odot$.
In this case we find much slower, linear -
as opposed to exponential -
growth of the eccentric motion of
the disk for times $t \lesssim 1$ Gyr.
A trailing one-armed spiral wave
forms in the disk and becomes more
tightly wrapped as time increases.
Angular momentum is transferred outward.
The motion of the central mass if present
is small compared with that of the disk
for $t \lesssim 1$ Gyr.
For long times $t>1$ Gyr, the
the azimuthal
displacements and shifts of the
inner rings start
to become large compared with
the values in the outer disk.
At the same time,
the radial shift of the center
grows.
This shift is significantly
reduced if the mass of the center
is changed from $\sim 10^9$
to $10^6 M_\odot$.
\acknowledgments{We thank M.S. Roberts
for valuable discussions which
stimulated this work.
We thank R.H. Miller
for bringing the work of Taga and Iye to
our attention and for
valuable comments on our work.
We thank M.M. Romanova for
valuable discussions and
an anonymous referee for valuable
comments.
This research has
been partially supported by NSF
grant AST 95-28860 to M.P.H. and
NASA grant NAG5 6311 to R.V.E.L.}
| {
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} | 9,596 |
Sparti, personaggi della mitologia greca
Sparti, nome greco dell'odierno comune di Sparta
Davide Sparti, filosofo e sociologo italiano | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 742 |
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ACCEPTED
#### According to
Index Fungorum
#### Published in
Trans. Sapporo nat. Hist. Soc. 15: 125 (1938)
#### Original name
Puccinia caricis-maximowiczii Homma
### Remarks
null | {
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UN Has Invaded 190 American Cities with Covert and Unscreened Islamic Jihadists
Home » UN Has Invaded 190 American Cities with Covert and Unscreened Islamic Jihadists
The United States is the equivalent of a four-letter word in the Middle East. Many people in that region of the world detest and loathe our very existence. Why? It's because our government, on behalf of the bankers and the oil companies, have plundered their resources, manipulated their governments, killed millions of Muslims without justification as the U.S. has invaded the region three times in recent history. The CIA may be responsible for funding most terrorism in the Middle East, but there is no shortage of willing participants to join such organizations as al-Qaeda, the Muslim Brotherhood, Hamas and now the latest threat, ISIS.
In short, the people in the Middle East hate America's guts. Yet, the Obama administration is placing hundreds of communities at risk by transplanting the victims of United States tyranny right into American backyards without a thought or care to the welfare of the citizens of this country. This article addresses the insanity and/or the treasonous nature of this issue.
Obama's Policy of Suicide by Immigration
Presently, there is a tsunami of unscreened Muslims into 190+ American communities from war-torn areas filled with the victims of American imperialism who believe that the only good American is a dead American. The most striking aspect of this silent invasion is that the Federal government has admitted under oath that they have no idea who these people are and what their intentions consist of.
When we juxtapose Obama's position to the multiple scenarios of Christian persecution and genocide in some areas of the Middle East, Obama has not lifted a finger to help Christians who are being slaughtered at the hands of Muslim extremist groups such as ISIS. Obama plays golf…
Meanwhile, our self-created enemies are engaged in getting even with innocent Christians.
Paul Sperry's "Infiltration" How Muslim Spies and Subversives have Penetrated Washington
Researcher, Paul Sperry, the author of Infiltration has detailed the type of immigrants that we are importing from the Middle East. Sperry details the threat to our communities while appearing on C-SPAN2.
Sperry cites how we know from the testimony offered from the FBI officials who are in charge of that type of vetting immigrants are not being allowed to perform their duties of protecting American communities from would-be unscreened terrorists as expressed in a recent radio interview in which FBI agents have admitted, under oath, that they have no idea who and what they are letting into the country.
The United Nations and the State Department Are Behind This Invasion of America
The following is one of the most stunning videos I have seen in some time with regard to the radical unscreened, extremist Muslim invasion of America. The program has its roots with the United Nations and the State Department is its willing accomplice.
The man's name is Antonio Guterres and the UN High Commissioner for Refugees. His job is the deculturalization of European nations and the United States. Leo Hohmann, from World Net Daily, has another in what is turning into a series on Obama's plan to change America by changing the people. This invasion has impacted 190 American communities and it is growing by the day. There are over 10,000 anti-American Somali refugees in ten years who have settled in Minnesota, alone. In the first 4 months of this year, we have admitted 4,425 Somalis to America.
ABC News reported the following:
"A mysterious ISIS recruiter known online as "Miski" was in close and repeated social media contact with Elton Simpson for months before the Sunday attack in Garland, Texas, an ABC News investigation has found". "Miski is well known to FBI officials who say his real name is Muhammed Hassan, a fugitive since 2009 when he fled Minneapolis as a teenager to join terror groups in Africa." This begs the question: Who are we keeping out?
Antonio Guterres is the head of the UNHCR and he is responsible for sending 9,000 Muslims from anti-America Syria to Boise and Twin Falls, Idaho. This man and his organization are your enemies!
Secretary of State, John Kerry and Anne C. Richards are the UN's accomplices in these devastating immigration policies.
Is this why we are seeing with ever-greater frequency these kinds of scenes inside of the United States?
….and this?
Fifth Column Watch
We have already had an ISIS attack on American soil in Garland, Texas. The FBI Director has admitted to the fact that the FBI has open ISIS in all 50 states. Last summer, I documented the flow of MS-13 gang members into the United States. For 30 months, I have detailed a Russian troop presence in our country, complete with pictures, eye witnesses, government documents and videos. The UN's Refugee/Resettlement is merely the latest in the implantation of 5th column forces inside of the United States. However, this invasion has reached 190 American cities and towns. In a future article, these groups will be tied together with regard to their common purpose as well as detailing how Jade Helm will factor into this scenario.
dave hodges islam jihad muslims refugees united nations
Gold & Silver – Exploding Demand/Diminishing Supplies
Obama Wants to Delay Immigration Reform by Executive Fiat till after Mid-Term Elections | {
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Entertainment, Gaming, Lifestyle, South Africa, Technology
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SA's Cybercrimes Bill: You can now face jail time for sending these malicious messages
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'use strict';
angular.module('tweets').controller('UserTimelineController', [
'$scope',
'$http',
'$stateParams',
function($scope, $http, $stateParams) {
$http.get('/friendships/show/' + $stateParams.username)
.success(function(response) {
$scope.profile = {
name: '',
screenName: $stateParams.username,
tweetCount: 4,
followerCount: 342,
followingCount: 13,
isFollowing: response.is_following,
isFollowed: response.is_followed
};
})
.error(function(response) {
$scope.error = response.message;
});
$http.get('/statuses/user_timeline/' + $stateParams.username)
.success(function(response) {
$scope.tweets = response;
})
.error(function(response) {
$scope.error = response.message;
});
$scope.follow = function(followUsername) {
$http.post('/friendships/follow', {
follow_username: followUsername
})
.success(function(response) {
$scope.profile.isFollowing = response.is_following;
})
.error(function(response) {
$scope.error = response.message;
});
};
$scope.unfollow = function(unfollowUsername) {
$http.post('/friendships/unfollow', {
unfollow_username: unfollowUsername
})
.success(function(response) {
$scope.profile.isFollowing = response.is_following;
})
.error(function(response) {
$scope.error = response.message;
});
};
}
]); | {
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\section{Introduction}
A recent development in motivic homotopy theory is the theory of framed transfers, introduced in \cite{voevodsky2001notes} and studied further in \cite{garkusha2014framed}, \cite{deloop1} and other works. The fundamental results of this theory are as follows. First, the \emph{reconstruction theorem} \cite[Theorem~3.5.12]{deloop1} states that every generalized motivic cohomology theory has a unique structure of (coherent) framed transfers, i.e., has covariance along finite syntomic maps of schemes equipped with a trivialization of the cotangent complex. Second, the \emph{motivic recognition principle} \cite[Theorem~3.5.14]{deloop1} shows that every $\P^1$-connective (also called ``very effective") motivic spectrum is a suspension spectrum on its infinite $\P^1$-loop space, when framed transfers on the space are taken into account. With these tools in hand, several important motivic spectra can be described geometrically as framed suspension spectra of certain moduli stacks of schemes. For example, the algebraic cobordism spectrum is the framed suspension spectrum of the stack of finite syntomic schemes \cite[Theorem~3.4.1]{deloop3}, and the effective algebraic K-theory spectrum is the framed suspension spectrum of the stack of finite flat schemes \cite[Theorem~5.4]{robbery}.
From the point of view of moduli of schemes, there are three natural conditions of interest: Cohen--Macaulay, Gorenstein, and local complete intersection,
see~\cite[Chapters~8--9]{HarDeform}. Being Cohen--Macaulay is automatic for finite
schemes, so by the discussion above, the first and third conditions relate to algebraic K-theory and
algebraic cobordism, respectively.
In this article, we connect the second condition, being Gorenstein, to another important cohomology theory, namely, hermitian K-theory. We show that, over a field of characteristic not $2$, hermitian K-theory is the framed suspension spectrum of the stack of finite Gorenstein algebras equipped with an \emph{orientation}, meaning a trivialization of the dualizing line bundle (Theorem~\ref{thm:KQ-kq-FGor}):
\[\kq \simeq \Sigma^\infty_\fr \FGor^\o.\]
This identification allows us to complete the description of all well-known generalized motivic cohomology theories as motivic spectra with framed transfers, which we summarize in \sectsign\ref{ssec:big} below (see also Theorem~\ref{thm:all mot spectra}).
The infinite $\P^1$-loop space of $\kq$ is motivically equivalent to the group completion of the stack $\Vect^\sym$ of vector bundles with a non-degenerate symmetric bilinear form.
Using the motivic recognition principle, the equivalence $\kq \simeq \Sigma^\infty_\fr \FGor^\o$ follows from the fact that the forgetful map of commutative monoids $\FGor^\o \to \Vect^\sym$ induces an $\A^1$-equivalence after group completion (Theorem~\ref{thm:main-gp}); the $\A^1$-inverse map is given by sending a symmetric vector bundle $V$ to $[V \oplus \Oo[x]/x^2] - [\Oo[x]/x^2] $. To prove this result, we employ several explicit $\A^1$-homotopies. A key ingredient is the intermediate notion of an \emph{isotropically augmented} oriented Gorenstein algebra, which is a condition that allows to split off a copy of the dual numbers in a controlled way.
We would like to emphasize that, although Theorem~\ref{thm:main-gp} is analogous to~\cite[Theorem~3.1]{robbery}, the proof in \emph{loc.\ cit.}\ cannot be applied in the hermitian context. Indeed, while the square-zero extension functor $\Vect_d \to \FFlat_{d+1}$ is an $\A^1$-equivalence, the analogous functor $\Vect^\sym_d \to \FGor^\o_{d+2}$, which adds a copy of the dual numbers, is not an $\A^1$-equivalence; we show this explicitly for the base field $\bR$ in Proposition~\ref{prop: square zero ext over R}. Instead, $\Vect^\sym_d$ is $\A^1$-equivalent to the substack of \emph{isotropic} oriented Gorenstein algebras $\FGor^{\o,0}_{d+2}$, defined by the condition that the orientation vanishes at the unit.
As a corollary of our results, we get a Hilbert scheme model for the Grothendieck–Witt space $\GWspace$ (Corollary~\ref{cor:Hilb-FGor}): we show that it is motivically equivalent to $\Z \times \Hilb_\infty^{\Gor,\o}(\A^\infty)$, where $\Hilb_d^{\Gor,\o}(\A^n)$ is the Hilbert scheme of degree $d$ finite Gorenstein subschemes of $\A^n$ equipped with an orientation.
We also give a new proof of the motivic equivalence $\GWspace \simeq \Z\times \GrO_\infty$ due to Schlichting and Tripathi \cite{SchlichtingTripathi}, where $\GrO_d$ is the orthogonal Grassmannian (see Corollary~\ref{cor: Vect^bil and GrO}).
A notable difference between these two models is that, when 2 is not invertible, the oriented Hilbert scheme gives the ``genuine symmetric'' variant of Grothendieck--Witt theory defined in \cite{HermitianI,HermitianII}, whereas the orthogonal Grassmannian corresponds to the ``genuine even'' variant.
\begin{rem}
None of the techniques in this paper require $2$ to be invertible. This assumption only appears in statements that involve the motivic spectra $\KQ$ or $\kq$, which so far have only been defined over $\Z[\tfrac 12]$-schemes (see however Remark~\ref{rem:2}).
\end{rem}
Our arguments have an unexpected direct application to complexity theory. One of the central problems there is bounding the asymptotic complexity of matrix multiplication~\cite{burgisser_clausen_shokrollahi, Landsberg__complexity_book}. The current algorithm giving the upper bound (called the \emph{laser method}) rests on the existence of certain tensors of minimal border rank, most notably the big Coppersmith--Winograd tensor $\mrm{CW}_q$. In Corollary~\ref{cor:degTensors} we show that \emph{every} $1$-generic tensor of minimal border rank degenerates to $\mrm{CW}_q$, thus the latter is the most degenerate one; it has the smallest value, which is very surprising given that the whole algorithm depends critically on the value of $\mrm{CW}_q$. Equally surprising is the fact that this result is seemingly not known to researchers in tensor world, even though it is much easier to obtain than our main results, since the degeneration need not be canonical.
Oriented Gorenstein algebras are also called commutative Frobenius algebras and appear in a different context: they classify 2-dimensional topological field theories~\cite[Section~1.1]{LurieTFT}.
\subsection{Framed models for motivic spectra}\label{ssec:big}
We give here an overview of the known framed models for motivic spectra. We work over a base field $k$ for simplicity and refer to Section~\ref{sec:framed-models} for more detailed statements and references.
Consider the following presheaves of $\infty$-groupoids on the category of smooth $k$-schemes:
\begin{itemize}
\item $\uZ$ is the constant sheaf with fiber $\Z$;
\item $\uGW$ is the sheaf of unramified Grothendieck--Witt groups;
\item $\Vect$ is the groupoid of vector bundles;
\item $\Vect^\sym$ is the groupoid of non-degenerate symmetric bilinear forms;
\item $\FFlat$ is the groupoid of finite flat schemes;
\item $\FGor^\o$ is the groupoid of oriented finite Gorenstein schemes;
\item $\FSyn$ is the groupoid of finite syntomic schemes;
\item $\FSyn^\o$ is the groupoid of oriented finite syntomic schemes;
\item $\FSyn^\fr$ is the $\infty$-groupoid of framed finite syntomic schemes \cite[3.5.17]{deloop1}.
\end{itemize}
Then the forgetful maps
\[\begin{tikzcd}
& \FSyn \ar{r} & \FFlat \ar{r} & \Vect \ar{r} & \uZ \\
\FSyn^\fr \ar{r} & \FSyn^\o \ar{u} \ar{r} & \FGor^\o \ar{r} \ar{u}& \Vect^\sym \ar{u}\ar{r} & \uGW\ar{u}
\end{tikzcd}\]
induce, upon taking framed suspension spectra, the canonical morphisms of motivic $\Einfty$-ring spectra over $k$ (assuming $\operatorname{char} k\neq 2$ for $\kq$):
\[\begin{tikzcd}
& \MGL \ar{r} & \kgl \ar["\simeq"]{r} & \kgl \ar{r} & \hz \\
\MonUnit \ar{r} & \MSL \ar{u} \ar{r} & \kq \ar["\simeq"]{r} \ar{u}& \kq \ar{u}\ar{r} & \hzmw \ar{u} \rlap .
\end{tikzcd}\]
Here, $\MonUnit$ is the motivic sphere spectrum, $\MGL$ (resp.\ $\MSL$) is the algebraic (resp.\ special linear) cobordism spectrum, $\kgl$ (resp.\ $\kq$) is the very effective algebraic (resp.\ hermitian) K-theory spectrum, and $\hz$ (resp.\ $\hzmw$) is the motivic cohomology (resp.\ Milnor–Witt motivic cohomology) spectrum.
One application of these geometric models is that they allow us to describe modules over these motivic ring spectra in terms of the corresponding transfers. For example, we prove in Theorem~\ref{thm:kq-modules} that modules over hermitian K-theory are equivalent (at least in characteristic 0) to generalized motivic cohomology theories with coherent transfers along oriented finite Gorenstein maps. The fact that hermitian K-theory has such transfers is well-known (see for example \cite[\sectsign 0]{GilleTransfers}), and this result characterizes hermitian K-theory as the universal such cohomology theory.
\subsection*{Acknowledgments} We are thankful to Tom Bachmann, Joseph M. Landsberg, Rahul Pandharipande and Burt Totaro for helpful discussions. We would like to thank SFB 1085 ``Higher invariants'' and Regensburg University for its hospitality. Yakerson was supported by a Hermann-Weyl-Instructorship and is grateful to the Institute of Mathematical Research (FIM) and to ETH Z\"urich for providing perfect working conditions.
\section{Oriented Gorenstein algebras}
In this section we introduce several types of algebras and their properties, which will later be used in the proof of the main theorem. All rings and algebras are assumed commutative.
Let $R$ be a base ring. An $R$-algebra $A$ is \emph{finite locally free} if it is finite, flat, and of finite presentation, or equivalently if $A$ is a locally free $R$-module of finite rank.
A finite locally free $R$-algebra $A$ is called a \emph{Gorenstein} $R$-algebra if its dualizing module $\omega_{A/R} = \Hom_R(A, R)$ is an invertible $A$-module~\cite[Proposition~21.5]{Eisenbud}.
We denote by $\FGor(R)$ the groupoid of Gorenstein $R$-algebras.
We emphasize that for us a Gorenstein algebra is by definition finite locally free; in this article we do not consider more general Gorenstein algebras, so this usage will not be ambiguous.
\begin{defn}\label{dfn:oriented-Gorenstein-algebra}
Let $A$ be a Gorenstein $R$-algebra. An \emph{orientation} of $A$ is an element $\varphi \in \omega_{A/R}$ that trivializes the dualizing module of $A$. Equivalently, $\varphi \colon A\to R$ is an $R$-linear homomorphism such that the bilinear form $B_\varphi(x,y)=\varphi(xy)$ on $A$ is non-degenerate (i.e., $B_\varphi$ induces an isomorphism $A \simeq A^\vee$). An \emph{oriented} Gorenstein $R$-algebra is a pair $(A, \varphi)$ where $\varphi$ is an orientation of the Gorenstein $R$-algebra $A$.
We denote by $\FGor^\o(R)$ the groupoid of oriented Gorenstein $R$-algebras.
\end{defn}
\begin{rem}
An oriented Gorenstein $R$-algebra is the same thing as a commutative Frobenius algebra in the category of $R$-modules, in the sense of \cite[Definition 4.6.5.1]{HA}.
\end{rem}
\begin{rem}
Let $R$ be a field or, more generally, a semilocal ring. Then every Gorenstein $R$-algebra can be oriented, but the choice of an orientation is usually far from unique. For a Gorenstein algebra $A$ over a more general ring $R$ an orientation may not exist, since $\omega_{A/R}$ may be a non-trivial line bundle.
\end{rem}
\begin{ex} \label{ex:kx/x2}
The algebra $R[x]/x^2$ can be equipped with the orientation $\varphi_0(r+sx)=s$, turning it into an oriented Gorenstein $R$-algebra.
The corresponding bilinear form $B_{\varphi_0}$ has matrix $\left(\begin{smallmatrix}0 & 1\\ 1& 0\end{smallmatrix}\right)$ in the basis $(1,x)$.
This algebra is therefore a refinement of the hyperbolic plane. As such, it will play a crucial role in Section~\ref{sec:gp}.
\end{ex}
\begin{ex}
More generally, consider the Gorenstein $R$-algebra $A = R[x]/x^{n+1}$. One can check that a functional $\varphi\colon A\to R$ is an orientation if and only if $\varphi(x^n) \in R^\times$.
\end{ex}
\begin{rem}\label{rem:FGoror}
The presheaves of groupoids $\FGor$ and $\FGor^\o$ are algebraic stacks.
To see this, consider the universal finite flat family $p\colon U \to \FFlat$.
By \cite[Tag 05P8]{stacks}, the locus $U_1\subset U$ where the sheaf $\omega_{U/\FFlat}$ is locally free of rank $1$ is open. By definition, $\FGor$ is the Weil restriction of $U_1$ along $p$, hence it is an open substack of $\FFlat$ \cite[\sectsign 7.6 Proposition 2(i)]{NeronModels}.
The forgetful map $\FGor^\o\to \FGor$ is similarly the Weil restriction of a $\bG_m$-torsor over $U\times_\FFlat\FGor$, hence it is affine. Alternatively, we can consider the vector bundle $E=\Spec\Sym(p_*\sO_U)$ over $\FFlat$. As a stack, $E$ classifies pairs $(A,\phi)$ where $A$ is a finite locally free $R$-algebra and $\phi\in\omega_{A/R}$. Thus, $\FGor^\o$ is the open substack of $E$ where the determinant of $B_\phi$ does not vanish. This shows moreover that the forgetful map $\FGor^\o\to \FFlat$ is affine, since the complement of $\FGor^\o$ in $E$ is cut out by a single equation (locally on $\FFlat$).
\end{rem}
\begin{lem}\label{lem:perpendiculars_to_ideals_are_intrinsic}
Let $R$ be a ring, $A$ a Gorenstein $R$-algebra and $I\subset A$ an
ideal. For every orientation $\varphi$ of $A$, we have $I^{\perp} = \Ann(I)$, where the orthogonal is taken with respect to $B_{\varphi}$.
\end{lem}
\begin{proof}
Take $a\in A$. If $aI = 0$ we have $B_{\varphi}(a, i) = \varphi(ai) =
0$ for every $i\in I$, hence $a\in I^{\perp}$. Conversely, if $aI\neq 0$ then, since $B_{\varphi}$ is
non-degenerate, there exist $i\in I$ and $a'\in A$ such that $B_{\varphi}(a', ai)\neq 0$. This means that
$B_{\varphi}(a,a'i)=\varphi(aa'i)=B_{\varphi}(a',ai)\neq 0$ so $a\not\in I^\perp$.
\end{proof}
Recall that for an $R$-algebra $A$ an \emph{augmentation} is an $R$-algebra map $e \colon A \to R$. When $(A,\varphi)$ is an oriented Gorenstein $R$-algebra, we denote by $e^*\colon R\to A$ the $R$-linear adjoint map, i.e., the map such that $B_\varphi(e^*\lambda,y)=\lambda e(y)$.
\begin{defn}\label{def:isotropic-aug}
Let $(A,\phi)$ be an oriented Gorenstein $R$-algebra. An augmentation $e\colon A\to R$ is called \emph{isotropic} if $e^*(1)$ is isotropic, i.e., if $B_\phi(e^*(1),e^*(1))=0$.
An \emph{isotropically augmented} oriented Gorenstein $R$-algebra is a triple $(A, \varphi, e)$ where $e$ is an isotropic augmentation of the oriented Gorenstein $R$-algebra $(A, \varphi)$. We denote by $\FGor^{\o,+}(R)$ the groupoid of isotropically augmented oriented Gorenstein $R$-algebras.
\end{defn}
\begin{prop}\label{prop:loosingorientation}
Let $(A,\phi)$ be an oriented Gorenstein $R$-algebra and $e\colon A\to R$ an augmentation. Then
$e$ is isotropic if and only if $\Ann(\ker e)\subset \ker e$.
\end{prop}
\begin{proof}
By definition, $e$ is isotropic if and only if the image of $e^*\colon R\to A$ is contained in $\ker e$.
By dualizing the short exact sequence of $R$-modules \[0\to\ker e\to A\stackrel e\to R\to 0,\] we see that $\Ima e^*=(\ker e)^\perp$.
By Lemma~\ref{lem:perpendiculars_to_ideals_are_intrinsic}, we get $\Ima e^*=\Ann(\ker e)$, whence the result.
\end{proof}
\begin{ex}\label{ex:isotropic}
Let $A$ be a finite algebra over a field $k$, so that $A$ is a finite product of local algebras
$A_\mathfrak{m}$. A choice of augmentation is then tantamount to a choice of $\mathfrak m$ such that $k\to \kappa(\mathfrak{m})$ is an isomorphism. This augmentation is isotropic (for any choice of orientation) if and only if $A_{\mathfrak{m}} \not\simeq k$.
\end{ex}
\begin{rem}
Suppose $R$ is a \emph{reduced} ring.
If $(A,\phi)$ is an oriented Gorenstein $R$-algebra with augmentation $e$, then $e$ is isotropic if and only it is isotropic after base change to the residue fields of $R$ (where isotropicity is a simple geometric condition, see Example~\ref{ex:isotropic}). Indeed, if $R$ is reduced then the map $R\to \prod_{\mathfrak p} \kappa(\mathfrak p)$ is injective, where $\mathfrak p$ ranges over the minimal primes of $R$.
\end{rem}
\begin{rem}
The condition $\Ann(\ker e)\subset \ker e$ in Proposition~\ref{prop:loosingorientation} is independent of the orientation $\phi$. Thus one could define isotropically augmented finite locally free algebras, but we do not know of an application for such a notion.
\end{rem}
\begin{defn}
Let $(A,\varphi,e) \in \FGor^{\o,+}(R)$. The element $e^\ast(1) \in A$ is called the \emph{local socle generator} of $(A,\varphi,e)$.
\end{defn}
By definition of $e^*$, the local socle generator $x=e^*(1)$ has the property that $B_\varphi(x,y)=e(y)$ for each $y\in A$. In particular, by the isotropy condition, we have $e(x) =B_\varphi(x,x) = 0$.
The name reflects the fact that $x$ generates the \emph{socle} (the annihilator of the maximal ideal) of the localization of $A$ at $\ker e$, see Lemma~\ref{lem:local_socle}(1) below.
\begin{ex}\label{ex:dualnumbers}
The oriented Gorenstein $R$-algebra $(R[x]/x^2, \varphi_0)$ from Example~\ref{ex:kx/x2} has a unique isotropic augmentation $e_0$ given by $e_0(r+sx) = r$. Its local socle generator is $x$.
\end{ex}
\begin{ex}\label{ex:Gorenstein-algebras-from-bilinear-forms}
Let $(V,B)$ be a non-degenerate symmetric bilinear form over a ring $R$. Then we can construct an algebra $A:=R[x]/x^2\oplus V$ with multiplication
\[(r+sx,v)\cdot(r'+s'x,v'):=(rr'+(rs'+r's+B(v,v'))x, r'v+rv')\,.\]
If we let $\varphi(r+sx,v)=s$ and $e(r+sx,v)=r$, then the triple $(A,\varphi,e)$ is an isotropically augmented oriented Gorenstein $R$-algebra, with local socle generator $x$. Note that the bilinear form $B_\phi$ is the direct sum of the hyperbolic form on $R[x]/x^2$ (with respect to the basis $(1,x)$) and the original form $B$ on $V$.
\end{ex}
\begin{lem}\label{lem:local_socle}
Let $x$ be the local socle generator of $(A,\varphi,e)\in \FGor^{\o,+}(R)$. Then
\begin{enumerate}
\item $x$ spans the $R$-module $\Ann(\ker e)$ (in particular, $R x \subset A$ is an
ideal);
\item $x^2=0$;
\item $\varphi(x)=1$.
\end{enumerate}
In particular, there is a canonical copy of $R[x]/x^2$ inside $A$, with an induced orientation given by $\varphi(r+sx)=\varphi(1)r+s$.
\end{lem}
\begin{proof}
By definition, $x$ spans $\Ima e^*=(\ker e)^\perp$ (see the proof of Proposition~\ref{prop:loosingorientation}), which is $\Ann(\ker e)$ by Lemma~\ref{lem:perpendiculars_to_ideals_are_intrinsic}.
By Proposition~\ref{prop:loosingorientation}, $\Ann(\ker e)$ is a square-zero ideal, hence $x^2=0$. Finally, $\phi(x)=B_\phi(x,1)=e(1)=1$, since $e$ is a ring homomorphism.
\end{proof}
\begin{defn}
A \emph{non-unital oriented Gorenstein} $R$-algebra is a pair $(V, B)$ where $V$ is a finite locally free non-unital (i.e., not necessariy unital) $R$-algebra and $B$ is a non-degenerate symmetric bilinear form on $V$ such that $B(xy,z)=B(x,yz)$ for all $x,y,z \in V$. We denote by $\FGor^{\nonu}(R)$ the groupoid of non-unital oriented Gorenstein $R$-algebras.
\end{defn}
\begin{rem}
If $(V,B)$ is a non-unital oriented Gorenstein $R$-algebra such that $V$ is unital, we can define an $R$-linear map $\varphi\colon V\to R$ by $\varphi(y) = B(y, 1)$. By construction, we then have $B_\varphi = B$, so $(V, \varphi)$ is an ordinary oriented Gorenstein $R$-algebra.
\end{rem}
If we view the moduli stack $\FGor^\o$ of oriented Gorenstein algebras as the hermitian counterpart of the moduli stack $\FFlat$ of finite locally free algebras, then $\FGor^{\o,+}$ is the hermitian counterpart of the moduli stack $\FFlat^\mrk$ of augmented finite locally free algebras, and $\FGor^\nonu$ that of the moduli stack $\FFlat^\nonu$ of non-unital finite locally free algebras. In the finite locally free case, the augmentation ideal defines an equivalence of presheaves of groupoids $\Aug\colon\FFlat^\mrk\to \FFlat^\nonu$, whose inverse is given by unitalization. We now investigate the hermitian analogue of this equivalence, which is slightly more complicated.
\begin{constr}\label{constr:non-unitalization}
If $(A, \varphi, e) \in \FGor^{\o,+}(R)$ has local socle generator $x$, there is a direct sum decomposition
\[
\ker e=Rx\oplus (R[x]/x^2)^\perp.
\]
Indeed, since $B_\phi(x,y)=e(y)$ we have $\ker e=(Rx)^\perp$, which contains the right-hand side as $x$ is isotropic. Conversely, if $y\in \ker e$, then $y-B_\phi(y,1)x$ is orthogonal to $1$ by Lemma~\ref{lem:local_socle}(3) and to $x$, hence it is orthogonal to $R[x]/x^2$.
In particular, $(R[x]/x^2)^\perp$ can be identified with the quotient $\ker(e)/Rx$.
Since $Rx$ is an ideal in $A$ by Lemma~\ref{lem:local_socle}(1), $((R[x]/x^2)^\perp, B_\varphi)$ is a non-unital oriented Gorenstein $R$-algebra with multiplication given by the multiplication in $A/Rx$ (or equivalently, by the multiplication in $A$ followed by the orthogonal projection).
\end{constr}
\begin{prop}\label{prop:nonunital=augmented}
The map
\[\Aug\colon\FGor^{\o,+}\to \A^1\times\FGor^{\nonu}\]
sending $(A,\varphi,e)$ to the pair $(\varphi(1),((R[x]/x^2)^\perp, B_\varphi))$ is an equivalence of presheaves of groupoids.
\end{prop}
\begin{proof}
We construct the inverse map $\Uni\colon \A^1\times\FGor^{\nonu} \to \FGor^{\o,+}$ by associating to $\lambda\in R$ and $(V, B) \in \FGor^{\nonu}(R)$ the $R$-module $R[x]/x^2\oplus V$ with the multiplication
\begin{equation}\label{eq:multiplication-in-augmented-algebras}
(r+sx,v)(r'+s'x,v')=(rr'+(sr'+s'r+B(v,v'))x, r'v+rv'+vv')\,,
\end{equation}
and augmentation and orientation given by
\[e(r+sx,v)=r,\qquad \varphi(r+sx,v)=r\lambda+s\,.\]
It is straightforward to check that this multiplication is associative. Clearly, this assignment is functorial, and $\Aug \circ \Uni \simeq \id$, so it remains to show that $\Uni \circ \Aug\simeq\id$.
Let $(A,\varphi,e) \in \FGor^{\o,+}(R)$ with the local socle generator $x$, and let $(V, B)$ be the orthogonal complement of $R[x]/x^2$ in $A$ equipped with the non-unital Gorenstein algebra structure of Construction~\ref{constr:non-unitalization}. Then there is a canonical orthogonal decomposition $A\simeq R[x]/x^2\oplus V.$
Since the kernel of the augmentation is exactly $(R\cdot x)^\perp$ (by definition of $x$), the augmentation is given by projecting onto the first summand and setting $x=0$. Moreover, $V$ is contained in the orthogonal complement of $R\cdot 1$ and so
\[\varphi(r+sx,v)=\varphi(r+sx,0)=r\lambda+s\,.\]
Finally, we need to check that the multiplication is given by \eqref{eq:multiplication-in-augmented-algebras}. Since $x$ annihilates $V$, the only question is what is the product of two elements of the form $(0,v)$ and $(0,v')$. The second component is, by definition, the product in $V$. Since the product still needs to be in $\ker e$, it follows that
\[(0,v)(0,v')=(bx,vv')\]
for some $b\in R$. But then
\[b=\varphi(bx,vv')=\varphi\left((0,v)(0,v')\right)=B_\varphi\left((0,v),(0,v')\right)=B(v,v')\]
as required.
\end{proof}
The following table summarizes the various presheaves of groupoids of finite flat algebras considered in this paper:
\begin{center}
\begin{tabular}{l l}
name & description\\
\toprule
$\FFlat$ & finite locally free algebras\\
$\FFlat^\mrk$ & augmented finite locally free algebras\\
$\FFlat^\nonu$ & non-unital finite locally free algebras\\
$\FGor$ & (finite) Gorenstein algebras\\
$\FGor^\o$ & oriented Gorenstein algebras\\
$\FGor^{\o,+}$ & isotropically augmented oriented Gorenstein algebras\\
$\FGor^\nonu$ & non-unital oriented Gorenstein algebras\\
$\FGor^{\o,0}$ & isotropic oriented Gorenstein algebras (see Section~\ref{sec:isotropic})
\end{tabular}
\end{center}
All of these groupoids of $R$-algebras extend by descent to groupoids of quasi-coherent $\Oo_S$-algebras for an arbitrary scheme $S$.
For example, $\FGor^\o(S)$ is the groupoid of finite locally free quasi-coherent $\sO_S$-algebras $\mathcal{A}$ equipped with an $\Oo_S$-linear map $\varphi\colon \mathcal{A}\to \Oo_S$ such that the induced bilinear form $B_\varphi\colon \sA\times\sA\to \sO_S$ is non-degenerate.
We will also identify quasi-coherent $\Oo_S$-algebras with $S$-schemes that are affine over $S$ (in particular in the discussion of Hilbert schemes). Thus, $\FGor^\o(S)$ can be identified with the groupoid of finite locally free $S$-schemes $p\colon Z\to S$ together with a suitable map $\varphi\colon p_*\Oo_Z\to \Oo_S$. In this situation, we will often abuse notation and regard $\sO_Z$ as a quasi-coherent $\sO_S$-algebra.
In order to construct some $\A^1$-homotopies in the proof of our main result, we will need a way to glue together objects of $\FGor^{\o,+}(S)$ along the basepoint.
Given $(Z_1, \varphi_1, e_1), (Z_2, \varphi_2, e_2) \in \FGor^{\o,+}(S)$,
the disjoint union $Z_1\sqcup Z_2$ inherits an orientation
\[
\phi\colon \sO_{Z_1\sqcup Z_2}=\sO_{Z_1}\times \sO_{Z_2}\to \sO_S,\quad \phi(a_1,a_2)=\phi_1(a_1)+\phi_2(a_2),
\]
which makes $\sO_{Z_1\sqcup Z_2}$ into the orthogonal sum of $\sO_{Z_1}$ and $\sO_{Z_2}$.
Gluing $Z_1$ and $Z_2$ along the basepoint means passing to the subalgebra $\sO_{Z_1\sqcup_SZ_2}=\sO_{Z_1}\times_{\sO_S} \sO_{Z_2}\subset \sO_{Z_1}\times \sO_{Z_2}$. However, the restriction of $\phi$ to $\sO_{Z_1\sqcup_SZ_2}$ is no longer an orientation:
if $x_i$ is the local socle generator of $(Z_i,\varphi_i,e_i)$, then $(x_1,-x_2)$ belongs to the radical of $B_\phi$ on $\sO_{Z_1\sqcup_SZ_2}$. It turns out that this is the only obstruction and that we obtain a well-defined orientation on the vanishing locus of $(x_1,-x_2)$:
\begin{prop}\label{prop:connectedSum}
Let $(Z_1, \varphi_1, e_1), (Z_2, \varphi_2, e_2) \in \FGor^{\o,+}(S)$ and let $x_i\in\sO(Z_i)$ be the corresponding local socle generators.
Let $Z_{12} \subset Z_1\sqcup_{S} Z_2$ be the closed subscheme given by the equation $(x_1,-x_2)$, let $e=e_1=e_2\colon S\to Z_{12}$, and let $\phi=\phi_1+\phi_2\colon \sO_{Z_{12}}\subset \sO_{Z_1}\oplus\sO_{Z_2}\to \sO_S$.
Then $Z_{12}$ is a finite Gorenstein $S$-scheme of degree $\deg(Z_1) + \deg(Z_2) - 2$, with orientation $\phi$ and isotropic augmentation $e$.
\end{prop}
\begin{proof}
First we need to prove that $Z_{12}$ is finite locally free over $S$. Since this property is local on the base we can assume $S=\Spec(R)$ is affine. Then we can write $Z_i=\Spec(A_i)$ and $Z_{12}=\Spec (A_1\times_R A_2)/(x_1,-x_2)$. We know that $A_1\times_R A_2$ is finite locally free over $R$ by \cite[Lemma~3.6]{robbery}. The inclusion of the ideal spanned by $x_i$ is a split inclusion $R\to A_i$, with the splitting given by $\varphi_i$, so the inclusion of the ideal spanned by $(x_1,-x_2)$ is also a split inclusion $R\to A_1\times_R A_2$ (split, say, by $\varphi_1\circ\mathrm{pr}_1$). In particular, the quotient $(A_1\times_R A_2)/(x_1,-x_2)$ is also finite locally free, and of the desired degree.
To show that $\phi$ is an orientation, we can again work in the affine case. We have to show that the radical of the form $B_\phi$ on $A_1\times_RA_2$ is precisely the ideal $R(x_1,-x_2)$.
On the one hand,
\[
B_\phi((x_1,-x_2),(y_1,y_2)) = e_1(y_1)-e_2(y_2) = 0
\]
for all $(y_1,y_2)\in A_1\times_RA_2$, so $(x_1,-x_2)$ is in the radical. By Lemma~\ref{lem:local_socle}, we can write $A_i=R[x_i]/x_i^2\oplus V_i$ with $V_i=(R[x_i]/x_i^2)^\perp$, whence
\[
A_1\times_RA_2 \simeq R[x_1,x_2]/(x_1,x_2)^2 \oplus V_1\oplus V_2.
\]
The restriction of $B_\phi$ to $R[x_1,x_2]/(x_1,x_2)^2$ is given by a matrix $\left(\begin{smallmatrix}b & 1 \\ 1 & 0\end{smallmatrix}\right)\oplus 0$ in the basis $(1,x_1,x_1-x_2)$, hence has radical $R(x_1-x_2)$. The latter contains the radical of $B_\phi$ since the restriction of $B_\phi$ to $V_1\oplus V_2$ is non-degenerate.
Finally, the augmentation $e$ is isotropic since $e^*(1)=(x_1,0)$ and hence $e(e^*(1))=e_2(0)=0$.
\end{proof}
\begin{defn}\label{def:connected sum}
In the setting of Proposition~\ref{prop:connectedSum}, $(Z_{12},\phi,e)\in\FGor^{\o,+}(S)$ is called the \emph{connected sum} of $(Z_1,\phi_1,e_1)$ and $(Z_2,\phi_2,e_2)$.
\end{defn}
\begin{rem}
In the affine case, Definition~\ref{def:connected sum} is a special case of the more general notion of connected sum of rings studied in \cite[Section 2]{Ananthnarayan_Connected_sums}. In particular, the fact that $Z_{12}$ is Gorenstein is also a consequence of \cite[Theorem~2.8]{Ananthnarayan_Connected_sums}.
\end{rem}
\begin{rem}
It is easy to show that the connected sum gives a commutative monoid structure on the stack $\FGor^{\o,+}$. Moreover, if $\FFlat^\mrk$ denotes the moduli stack of pointed finite locally free schemes with the commutative monoid structure given by the wedge sum, the morphism $\Hyp\colon \FFlat^\mrk\to \FGor^{\o,+}$ from Remark~\ref{rem:functor_hyp} is a morphism of commutative monoids. We shall not need these facts in the sequel.
\end{rem}
\section{$\A^1$-equivalence between the group completions of the stacks $\FGor^\ori$ and $\Vect^\sym$}
\label{sec:gp}
We denote by $\Vect^\sym$ the stack of finite locally free modules equipped with a non-degenerate symmetric bilinear form.
The direct sum and tensor product define an $\Einfty$-semiring structure on $\Vect^\sym$. Similarly, the disjoint union and cartesian product of schemes define an $\Einfty$-semiring structure on $ \FGor^\o$, and the forgetful map $\eta\colon\FGor^\o \to \Vect^\sym$, sending $(A,\varphi)$ to $(A,B_\varphi)$, is a morphism of $\Einfty$-semirings. We will describe these constructions more explicitly in Section~\ref{sec:kq}. The main result of this section is the following theorem.
\begin{thm}\label{thm:main-gp}
The map $\eta^\gp\colon \FGor^{\o,\gp} \to \Vect^{\sym,\gp}$ is an $\A^1$-equivalence of presheaves on the category of schemes, where $\gp$ stands for objectwise group completion.
\end{thm}
\begin{rem}\label{rem:functor_hyp}
Theorem~\ref{thm:main-gp} is analogous to~\cite[Theorem~2.1]{robbery}, which
states that the forgetful map from the stack of finite locally free schemes
$\FFlat$ to the stack of vector bundles $\Vect$ becomes an $\A^1$-equivalence
after group completion. The connection can be expressed precisely in the following way.
There is a commutative diagram of hyperbolic and forgetful functors
\[\begin{tikzcd}
\FFlat\ar[d,swap,"\Hyp"] \ar[r, "\eta"] & \Vect\ar[d,"\Hyp"]\\
\FGor^{\ori}\ar[r, "\eta"] \ar[d,swap,"\mathrm{forget}"] & \Vect^{\sym} \ar[d,"\mathrm{forget}"] \\
\FFlat \ar[r, "\eta"] & \Vect\rlap.
\end{tikzcd}\]
Here, the functor $\Hyp\colon \Vect\to\Vect^\sym$ sends $V$ to $\left(V\oplus V^\vee, \left(\begin{smallmatrix}
0 & I\\ I & 0\end{smallmatrix}\right)\right)$, and the functor $\Hyp\colon \FFlat\to \FGor^\o$ sends a finite locally free $R$-algebra $A$ to the square-zero extension $A\oplus\omega_{A/R}$ with the orientation $\varphi(a,f)=f(a)$.
\end{rem}
To proceed, we define the following stabilization maps:
\begin{align*}
\tau \colon \Vect^\sym \to \Vect^\sym, &\quad (V,B) \mapsto (V\oplus R^2, B \oplus B_{\Hyp}), \\
\sigma \colon \FGor^{\o} \to \FGor^{\o}, &\quad (A,\varphi) \mapsto (A\oplus R[x]/x^2, \varphi \oplus \varphi_0), \\
\sigma^+ \colon \FGor^{\o,+} \to \FGor^{\o,+}, &\quad (A,\varphi,e) \mapsto (A\oplus R[x]/x^2, \varphi \oplus \varphi_0, e\circ \pr_1),
\end{align*}
where $B_{\Hyp} = \left(\begin{smallmatrix} 0 & 1\\ 1& 0\end{smallmatrix}\right)$ is the hyperbolic form and $\phi_0$ is the orientation from Example~\ref{ex:kx/x2}.
We denote by $\Vect^{\sym,\st}$ the colimit of the sequence
\[
\Vect^\sym\xrightarrow{\tau} \Vect^\sym \xrightarrow{\tau }\Vect^\sym\to\dotsb,
\]
and we define $\FGor^{\o,\st}$ and $\FGor^{\o,+,\st}$ similarly.
By Example~\ref{ex:kx/x2}, the square
\[
\begin{tikzcd}
\FGor^{\o} \ar{r}{\eta} \ar{d}[swap]{\sigma} & \Vect^{\sym} \ar{d}{\tau} \\
\FGor^{\o} \ar{r}{\eta} & \Vect^{\sym}
\end{tikzcd}
\]
commutes, inducing a map $\eta^\st\colon \FGor^{\o,\st} \to \Vect^{\sym,\st}$ in the colimit. Similarly, the map $\theta\colon \FGor^{\ori,+}\to \FGor^{\ori}$ that forgets the augmentation stabilizes to a map $\theta^\st\colon \FGor^{\ori,+,\st}\to \FGor^{\ori,\st}$.
Note that there are canonical maps $\Vect^{\sym,\st}\to\Vect^{\sym,\gp}$ and $\FGor^{\o,\st}\to\FGor^{\o,\gp}$ from the telescopes to the group completions, induced by mapping the $n$th copy of $\Vect^\sym(R)$ resp.\ of $\FGor^{\o}(R)$ to the group completion via $(V,b)\mapsto (V,b)-n\cdot (R^2,B_{\Hyp})$ resp.\ $(A,\phi)\mapsto (A,\phi)-n\cdot (R[x]/x^2,\phi_0)$.
We shall deduce Theorem~\ref{thm:main-gp} from the following variant, which does not involve group completion:
\begin{thm}\label{thm:main-st}
The map $\eta^\st\colon \FGor^{\o,\st}\to \Vect^{\sym,\st}$ is an $\A^1$-equivalence.
\end{thm}
The strategy of the proof of Theorem~\ref{thm:main-st} is to use the stack $\FGor^{\o,+}$ and show that both $\theta^\st$ and $\eta^\st\circ \theta^\st$ are $\A^1$-equivalences.
For $\theta^\st$, the idea is to construct an inverse by stabilizing the map
\[
\gamma\colon\FGor^{\ori}\to \FGor^{\ori,+},\quad (A,\varphi)\mapsto (A\oplus R[x]/x^2, \varphi \oplus \varphi_0, e_0\circ \pr_2).
\]
However, this does not quite work: we have $\theta \circ \gamma \simeq \sigma$, but $\gamma \circ \theta \not\simeq \sigma^+$, since the maps $\gamma \circ \theta$ and $\sigma^+$ equip Gorenstein algebras with different augmentations. Construction~\ref{constr:bankrobbery} allows us to get around this obstruction; it is the main technical ingredient in the proof of Theorem~\ref{thm:main-st}.
For the convenience of the reader, the following table summarizes the various maps we will use in the proof of Theorem~\ref{thm:main-st}:
\begin{center}
\begin{tabular}{l l}
name & description\\
\toprule
$\eta\colon \FGor^\o \to \Vect^\sym$ & forgets the algebra structure\\
$\theta\colon \FGor^{\o,+} \to \FGor^\o$ & forgets the isotropic augmentation\\
$\tau\colon\Vect^\sym \to \Vect^\sym$ & adds a copy of the hyperbolic form\\
$\sigma\colon\FGor^\o \to \FGor^\o$ & adds a copy of the double point\\
$\sigma^+\colon \FGor^{\o,+}\to \FGor^{\o,+}$ & adds a copy of the double point without changing the augmentation\\
$\gamma\colon\FGor^\o \to \FGor^{\o,+}$ & adds a copy of the double point with its augmentation\\
$\varepsilon\colon\FGor^{\o,+} \to \FGor^{\o,+}$ & takes the connected sum with $R[x]/x^4$ \\
$\pi\colon\FGor^\nonu\to \Vect^\sym$ & forgets the algebra structure
\end{tabular}
\end{center}
\vskip\parskip
\begin{constr}
\label{constr:bankrobbery}
There is a zigzag of $\A^1$-homotopies
\[
\sigma^+ \stackrel{H^\const}\leftsquigarrow \varepsilon \stackrel{H^\mv}\rightsquigarrow \gamma \circ \theta
\]
and an isomorphism $\psi\colon \theta\circ H^\const\simeq\theta\circ H^\mv$ such that $\psi_0=\id_{\theta\circ\varepsilon}$ and $\psi_1$ is the canonical isomorphism $\theta\circ \sigma^+\simeq \sigma\circ\theta \simeq \theta\circ \gamma\circ \theta$.
\end{constr}
\begin{proof}
Consider the oriented Gorenstein $\Z[t]$-algebra
\[\robber = \Z[x,t]/((x-t)^2x^2),\quad \varphi_\robber(r_0 + r_1x + r_2x^2 + r_3x^3) = r_3,\]
where $r_i \in \Z[t]$.
Its fiber over $t=1$ is isomorphic to $(\Z[x]/x^2,\phi_0)\times(\Z[x]/x^2,\phi_0)$, where $\phi_0$ is the orientation of Example~\ref{ex:kx/x2}. Its fiber over $t=0$ is $\robber_0 =\Z[x]/(x^4)$, with orientation $\phi_{\robber_0}$ given by the same formula as $\phi_\robber$.
One can view $\robber$ as two copies of the dual numbers colliding at $t = 0$.
Each copy has a natural augmentation as in
Example~\ref{ex:dualnumbers}, which extends to an augmentation of $\robber$. Explicitly, we have two augmentations $e^\const, e^\mv\colon \robber \to \Z[t]$ given by sending $x$ to $0$ and to $t$. One computes that
\begin{align*}
(e^\const)^*(1) &= t^2x -2tx^2 + x^3,\\
(e^\mv)^*(1) &= -tx^2 + x^3,
\end{align*}
which shows that both $e^\const$ and $e^\mv$ are isotropic. We thus obtain two elements $(\robber,\varphi_\robber, e^\const)$ and $(\robber,\varphi_\robber, e^\mv)$ in
$\FGor^{\ori,+}(\Z[t])$, with local socle generators as above.
The augmentations $e^\const$ and $e^\mv$ agree at $t=0$ and define an element $(\robber_0, \varphi_{\robber_0},
e_{\robber_0})\in \FGor^{\ori,+}(\Z)$.
We let $\varepsilon \colon \FGor^{\ori,+} \to \FGor^{\ori,+}$ be the map that
sends $(Z,\phi,e)\in \FGor^{\ori,+}(S)$ to its connected sum with
$(\robber_0, \varphi_{\robber_0},
e_{\robber_0})_{S}$ (see Definition~\ref{def:connected sum}).
Given $(Z,\phi,e)\in \FGor^{\ori,+}(S)$, we denote by $(\tilde Z,\tilde \phi,\tilde e^\const)$ the connected sum of $\A^1_Z$ with $(\robber,\phi_\robber,e^\const)_S$ in $\FGor^{\ori,+}(\A^1_S)$. Explicitly, if $s\in\sO(Z)$ is the local socle generator, then $\tilde Z$ is the vanishing locus of the function $(-s, t^2x-2tx^2+x^3)$ on the pushout $\A^1_Z\sqcup_{\A^1_S}(\Spec\robber)_S$, which is defined using $e\colon S\to Z$ and $e^\const\colon \A^1\to\Spec\robber$.
Note that replacing $x$ by $t$ in $t^2x-2tx^2+x^3$ gives $0$, so the composite
\[
\A^1_S \xrightarrow{e^\mv} (\Spec\robber)_S \xrightarrow{\mathrm{can}} \A^1_Z\sqcup_{\A^1_S} (\Spec\robber)_S
\]
lands in $\tilde Z$. This defines another section $\tilde e^\mv\colon \A^1_S\to\tilde Z$, which is moreover isotropic.
Indeed, we have $(\tilde e^\mv)^*(1)=(0,(e^\mv)^*(1))$, hence $\tilde e^\mv((\tilde e^\mv)^*(1))=e^\mv((e^\mv)^*(1))=0$.
Let $H^{\const}\colon \FGor^{\ori,+}\to \FGor^{\ori,+}(\A^1\times -)$ be the map that
sends $(Z,\phi,e)$ to $(\tilde Z,\tilde \phi,\tilde e^\const)$, and let $H^{\mv}\colon \FGor^{\ori,+}\to \FGor^{\ori,+}(\A^1\times -)$ be the map that
sends $(Z,\phi,e)$ to $(\tilde Z,\tilde \phi,\tilde e^\mv)$.
Then it is clear that $H^\const_0=H^\mv_0\simeq\varepsilon$.
Moreover, we have $H^\const_1\simeq\sigma^+$ and $H^\mv_1\simeq\gamma\circ\theta$.
Indeed, the fiber of $(\tilde Z,\tilde\phi)$ over $t=1$ is the disjoint union of $(Z,\phi)$ and $(\Z[x]/x^2,\phi_0)_S$, with $\tilde e^\const_1$ being the given section $e$ to the first summand and $\tilde e^\mv_1$ the canonical section to the second summand.
Thus, $H^\const$ and $H^\mv$ are the desired $\A^1$-homotopies. By construction, the underlying oriented Gorenstein schemes of $H^\const(Z,\phi,e)$ and $H^\mv(Z,\phi,e)$ are the same, and we can take the isomorphism $\psi$ to be the identity.
\end{proof}
\begin{prop}\label{prop:bank-robbery}
The map $\theta^\st\colon \FGor^{\ori,+,\st}\to \FGor^{\ori,\st}$ is an $\A^1$-equivalence.
\end{prop}
\begin{proof}
Consider the diagram
\[
\begin{tikzcd}
\FGor^{\ori,+} \ar["\theta"]{r} \ar[swap,"\sigma^+"]{d} & \FGor^{\ori} \ar["\sigma"]{d} \ar[swap,"\gamma"]{dl} \\
\FGor^{\ori,+}\ar["\theta"]{r} & \FGor^{\ori} \rlap.
\end{tikzcd}
\]
By Construction~\ref{constr:bankrobbery} there is a homotopy $H\colon \Lhtp(\sigma^+)\simeq \Lhtp(\gamma\circ\theta)$ such that $\theta\circ H$ is the canonical isomorphism $\theta\circ \sigma^+\simeq \sigma\circ\theta\simeq \theta\circ\gamma\circ\theta$. It follows that the map $\Lhtp\gamma$ induces in the colimit a map $\Lhtp\FGor^{\o,\st} \to \Lhtp\FGor^{\o,+,\st}$, which is inverse to $\Lhtp\theta^\st$.
\end{proof}
\begin{prop}\label{prop:vect-nu}
The forgetful map $\pi\colon \FGor^{\nonu}\to \Vect^\sym$ is an $\A^1$-equivalence.
\end{prop}
\begin{proof}
Let $\nu\colon \Vect^\sym \to\FGor^{\nonu}$ be the map sending a symmetric space $(V,B)$ to the non-unital oriented Gorenstein algebra $(V, B)$ with zero multiplication.
Then $\pi\circ\nu$ is the identity, and the map
\[
\FGor^{\nonu} \to \FGor^{\nonu}(\A^1\times-), \quad (V, B) \mapsto (tV[t], (tp,tq)\mapsto B_{R[t]}(p,q)),
\]
where $B_{R[t]}$ is the $R[t]$-bilinear extension of $B$, is an $\A^1$-homotopy from $\nu\circ\pi$ to the identity of $\FGor^{\nonu}$.
\end{proof}
Recall that a symmetric space $(V,b)$ is said to be \emph{metabolic} if it has a Lagrangian, i.e., a direct summand $L\subset V$ such that $L=L^\perp$. For example, for every $V\in\Vect(R)$, the hyperbolic space $\Hyp V=\left(V\oplus V^\vee,\left(\begin{smallmatrix} 0 & I\\ I &0\end{smallmatrix}\right)\right)$ is metabolic, with Lagrangian $V\oplus 0 \subset V\oplus V^\vee$. When 2 is invertible, all metabolic spaces are in fact of this form. The following lemma shows that this is also the case up to $\A^1$-homotopy, even when 2 is not invertible.
\begin{lem}\label{lem:metabolic=hyperbolic}
Let $(V,b)$ be a symmetric space over a ring $R$.
If $(V,b)$ is metabolic with Lagrangian $L$, then the class of $(V,b)$ in $\pi_0(L_{\A^1}\Vect^\sym)(R)$ is equal to the class of $\Hyp L$.
\end{lem}
\begin{proof}
Let $W$ be a complement of $L$ in $V$. Then the map $W\to V\simeq V^\vee\to L^\vee$ is an isomorphism and we can identify $V$ as $L\oplus L^\vee$. Under this decomposition $b$ is given by the matrix
\[\begin{pmatrix} 0 & I\\ I & A\end{pmatrix},\]
where $A$ is some symmetric bilinear form on $L^\vee$. Then
\[\left(L[t]\oplus L[t]^\vee,\ \begin{pmatrix} 0 & I\\ I & tA\end{pmatrix}\right)\in \Vect^\sym(R[t])\]
is an $\A^1$-homotopy between $(V,b)$ and $\Hyp L$.
\end{proof}
\begin{lem}\label{lem:invert-hyp}
Let $\Hyp=\Hyp \Z\in \Vect^\sym(\Z)$, and let $\Vect^\sym[-\Hyp]$ be the commutative monoid obtained from $\Vect^\sym$ by additively inverting $\Hyp$.
\begin{enumerate}
\item The cyclic permutation of $\Hyp^{\oplus 3}$ is $\A^1$-homotopic to the identity.
\item The canonical map $\Vect^\sym[-\Hyp]\to \Vect^{\sym,\gp}$ is an $\A^1$-equivalence on affine schemes.
\item The canonical map $\Vect^{\sym,\st}\to\Vect^{\sym,\gp}$ is an $\A^1$-equivalence on affine schemes.
\end{enumerate}
\end{lem}
\begin{proof}
(1) The cyclic permutation of $\Hyp^{\oplus 3}$ is given by applying the functor $\Hyp$ to the cyclic permutation of $\Z^3$ in $\Vect$, which is $\A^1$-homotopic to the identity (being a product of elementary matrices).
(2) It suffices to show that every element of $\pi_0(L_{\A^1}\Vect^\sym[-\Hyp])(R)$ is additively invertible. Every object $(V,b)\in\Vect^\sym(R)$ is a summand of $(V,b)\oplus (V,-b)$, which is metabolic with Lagrangian given by the diagonal copy of $V$, so it suffices to show that every metabolic object is invertible. By Lemma~\ref{lem:metabolic=hyperbolic}, it then suffices to show that every object of the form $\Hyp V$ is invertible. But if $W$ is such that $V\oplus W=R^n$, we have $\Hyp V\oplus \Hyp W=\Hyp(R^n)=(\Hyp R)^{\oplus n}$, and so $\Hyp V$ is invertible.
(3) This follows from (1) and (2) by \cite[Proposition 5.1]{deloop4}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main-st}]
By Lemma~\ref{lem:local_socle}, there is a commutative square
\[
\begin{tikzcd}
\FGor^{\o,+} \ar{r}{\theta} \ar{d}[swap]{(\id\times\pi)\circ\Aug} & \FGor^\o \ar{d}{\eta} \\
\A^1\times \Vect^\sym \ar{r}{\tilde\tau} & \Vect^\sym\rlap,
\end{tikzcd}
\]
where $\Aug\colon \FGor^{\o,+}\to\A^1\times \FGor^{\nonu}$ was defined in Proposition~\ref{prop:nonunital=augmented} and $\tilde\tau(\lambda,-)$ adds a copy of the bilinear form $\left(\begin{smallmatrix}\lambda & 1 \\ 1 & 0 \end{smallmatrix}\right)$.
Moreover, this square fits in a commutative cube with the respective ``stabilization maps'' $\sigma^+$, $\sigma$, $\id_{\A^1}\times\tau$, and $\tau$, and in the colimit we obtain a commutative square
\[
\begin{tikzcd}
\FGor^{\o,+,\st} \ar{r}{\theta^\st} \ar{d}[swap]{((\id\times\pi)\circ\Aug)^\st} & \FGor^{\o,\st} \ar{d}{\eta^\st} \\
\A^1\times \Vect^{\sym,\st} \ar{r}{\tilde\tau^\st} & \Vect^{\sym,\st}\rlap.
\end{tikzcd}
\]
By Propositions \ref{prop:nonunital=augmented} and~\ref{prop:vect-nu}, the left vertical map is an $\A^1$-equivalence (already in the unstable square).
By Proposition~\ref{prop:bank-robbery}, $\theta^\st$ is an $\A^1$-equivalence. We have $\tilde\tau^\st(0,-)=\tau^\st$, where $\tau^\st\colon \Vect^{\sym,\st}\to\Vect^{\sym,\st}$ is the action of $\Hyp\in\Vect^\sym(\Z)$ on the $\Vect^\sym$-module $\Vect^{\sym,\st}$. Note that $\tau^\st$ is \emph{not} an equivalence, but it is an $\A^1$-equivalence by \cite[Proposition 5.1]{deloop4} since the cyclic permutation of $\Hyp^{\oplus 3}$ is $\A^1$-homotopic to the identity (Lemma~\ref{lem:invert-hyp}(1)). Thus, $\tilde\tau^\st$ is also an $\A^1$-equivalence. We conclude that $\eta^\st$ is an $\A^1$-equivalence, as desired.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main-gp}]
As above, let $\Vect^\sym[-\Hyp]$ be obtained from $\Vect^\sym$ by additively inverting $\Hyp$, and let $\FGor^\o[-\Z[x]/x^2]$ be obtained from $\FGor^\o$ by additively inverting the oriented Gorenstein algebra $(\Z[x]/x^2, \varphi_0)$ from Example~\ref{ex:kx/x2}. We have a commutative square
\[
\begin{tikzcd}
\FGor^{\o,\st} \ar{r} \ar[swap,"\eta^\st"]{d} & \FGor^\o[-\Z[x]/x^2] \ar{d} \\
\Vect^{\sym,\st} \ar{r} & \Vect^\sym[-\Hyp] \rlap.
\end{tikzcd}
\]
By Lemma~\ref{lem:invert-hyp}(1), the cyclic permutation of $\Hyp^{\oplus 3}$ becomes the identity in $\Lhtp\Vect^\sym$, which by \cite[Proposition 5.1]{deloop4} implies that the lower horizontal map is an $\A^1$-equivalence.
By Theorem~\ref{thm:main-st}, the left vertical map is an $\A^1$-equivalence. In particular, the cylic permutation of $(\Z[x]/x^2)^{\times 3}$ also becomes the identity in $\Lhtp\FGor^{\o,\st}$. It then follows again from \cite[Proposition 5.1]{deloop4} that the upper horizontal map is an $\A^1$-equivalence. Hence, the right vertical map is an $\A^1$-equivalence.
Since the functor $\Lhtp$ commutes with group completion \cite[Lemma 5.5]{HoyoisCdh}, we deduce that $ \FGor^{\o,\gp}\to\Vect^{\sym,\gp}$ is an $\A^1$-equivalence.
\end{proof}
\begin{rem}\label{rem:FGor-gp}
Combining Theorems~\ref{thm:main-gp} and \ref{thm:main-st} with Lemma~\ref{lem:invert-hyp}(3), we deduce that the canonical map $\FGor^{\o,\st}\to \FGor^{\o,\gp}$ is an $\A^1$-equivalence on affine schemes.
\end{rem}
\section{Consequences in complexity theory}
In this short section we derive consequences of Proposition~\ref{prop:vect-nu}
for structure tensors of finite algebras. This section is an interesting side
application of our methods: it has no relation to the subsequent sections, however it is of
interest for complexity theory. Because of this target audience,
we strive to be more explicit than elsewhere. In this section we work over an
algebraically closed field $k$.
For $q\geq 1$ the \emph{G-fat} point~\cite{Casnati_Notari_6points} is the spectrum of a
locally free $k$-algebra $A_q$ of degree $q+2$ presented as
\[
A_q := \frac{k[y_1, \ldots ,y_q]}{(y_iy_j\ |\ i\neq j) + (y_i^2 - y_j^2\
|\ i\neq j)+(y_1^3)}.
\]
This is an oriented Gorenstein algebra with orientation $\varphi_0 :=
(y_1^2)^*\in \Hom_k(A_q, k)$, and its unique augmentation is isotropic by
Example~\ref{ex:isotropic}. In the equivalence from
Proposition~\ref{prop:nonunital=augmented} the triple $(A_q, \varphi, e)$
corresponds to $(0, V_0)$, where $V_0$ is spanned by self-dual elements $y_1,
\ldots ,y_q$ and equipped with a trivial multiplication.
Let $(C, \varphi, e)$ be another isotropically augmented Gorenstein algebra of
degree $q+2$.
As in Definition~\ref{dfn:oriented-Gorenstein-algebra}, we have a
non-degenerate form $B_{\varphi}$ on $C$ and a local socle generator $x\in C$.
(If $C$ is local, then $x$ is exactly its socle generator, see
Lemma~\ref{lem:local_socle}). Let $V = 1_C^{\perp}\cap x^{\perp} \subset C$.
Then $B_{\varphi}$
restricts to a non-degenerate form on $V$, so as a $k$-vector space we have $C = k\cdot 1_C\oplus kx \oplus V$.
A $\bG_m$-equivariant \emph{degeneration} of a finite $k$-algebra $C$ to a finite $k$-algebra $A$
is a finite flat family $f\colon \mathcal{X}\to \A^1$ together with a
$\bG_m$-action on $\mathcal{X}$ making $f$ equivariant with respect to the usual
$\bG_m$-action on $\A^1$ and such that $\mathcal{X}_1 \simeq \Spec(C)$ and
$\mathcal{X}_0 \simeq \Spec(A)$. The $\bG_m$-equivariance implies that the fiber
of $\mathcal{X}$ over every nonzero $k$-point of $\A^1$ is isomorphic to $\Spec(C)$.
Proposition~\ref{prop:vect-nu} and Proposition~\ref{prop:nonunital=augmented}
contain a construction of a degeneration of $C$ to $A_q$. Explicitly, it is given by $f\colon\Spec(\mathcal{C})\to
\Spec(k[t])$, where $\mathcal{C}$
is a free $k[t]$-module $(k[t]\cdot 1 \oplus k[t]x)\oplus V[t]$ with
multiplication given by
\[
(r+sx,v)\cdot (r'+s'x,v')=(rr'+(sr'+s'r+B_{\varphi}(v,v'))x, r'v+rv'+tvv'),
\]
where $v,v'\in V[t]$ and $B_{\varphi}$ extends $k[t]$-linearly.
The algebra $\mathcal{C}$ admits a $\bG_m$-action given by $t\cdot (r+sx, v) =
(r+st^{-2}x, t^{-1}v)$ which proves that $f$ is a degeneration.
\begin{prop}\label{prop:abstractDegenerations}
Let $q\geq 1$ and let $C$ be a Gorenstein $k$-algebra of degree $q+2$. Then $C$ admits a $\bG_m$-equivariant
degeneration to $A_q$.
\end{prop}
\begin{proof}
Choose an orientation of $C$. If $C$ has nonzero nilpotent radical, then by
Example~\ref{ex:isotropic} it admits an isotropic augmentation and the
associated degeneration $\mathcal{C}$ above proves our claim.
It remains to consider reduced $C$. Since $k$ is algebraically closed, we
have $C = \prod_{q+2} k$.
In this case, choose a set $\Gamma$ of $q+2$ general lines through the origin of
$\A^{q+1}$ and a hyperplane $H = (y = 1)$ intersecting them transversely.
Then $\Gamma\cap (y = t)$ is a degeneration over $\A^1$ with
parameter $t$. Since a general tuple of $q+2$ points on $\P^{q}$ is
arithmetically Gorenstein, the special fiber of the degeneration is
Gorenstein. This fiber has $q$-dimensional tangent space, which implies
that it is $\Spec(A_q)$.
\end{proof}
\begin{rem}
Proposition~\ref{prop:abstractDegenerations} in the special case when $C$
is local was obtained
in~\cite{Casnati_Notari_6points}, by completely different means.
This proposition also shows that the Gorenstein locus of the Hilbert scheme of
$d$ points on $\A^n$ is
connected by rational curves whenever $n\geq d-2$. In general, even
topological connectedness of the Gorenstein locus is open.
\end{rem}
For the terminology on tensors below, we refer
to~\cite[5.6.1]{Landsberg__complexity_book}.
\begin{cor}\label{cor:degTensors}
Let $m\geq 3$ and let $T\in k^m\otimes k^m\otimes k^m$ be a $1$-generic tensor of minimal
border rank (that is, of border rank $m$). Then $T$ degenerates to the big
Coppersmith--Winograd tensor $\mrm{CW}_{m-2}$.
\end{cor}
\begin{proof}
As explained in~\cite[5.6.2.1]{Landsberg__complexity_book}, the tensor $T$
is isomorphic to a structure tensor of a Gorenstein algebra $C$. The tensor
$\mrm{CW}_{m-2}$ is isomorphic to the structure tensor of $A_{m-2}$. The claim
follows directly from Proposition~\ref{prop:abstractDegenerations}.
\end{proof}
\begin{rem}
The assumptions of Corollary~\ref{cor:degTensors} can be much weakened. We
only need to assume that $T$ is $1$-generic and satisfies Strassen's
equations, see~\cite[\sectsign 2.1]{Landsberg_Michalek__Abelian_Tensors} for
their definition. Indeed, to such a $T$ we associate a commuting tuple of
matrices~\cite[Definition 2.7]{Landsberg_Michalek__Abelian_Tensors}, hence a
module over a polynomial ring~\cite[Introduction]{Jelisiejew_Sivic}. Since $T$ is $1$-generic, it
is $1_B$-generic, hence this module is cyclic and can be viewed as a $k$-algebra.
By~\cite[5.6.2.1]{Landsberg__complexity_book} this algebra is
Gorenstein and we argue as in the proof of~Corollary~\ref{cor:degTensors}.
\end{rem}
\section{Oriented Hilbert scheme and orthogonal Grassmannian}
\begin{defn}
Let $X\to S$ be a morphism of schemes. The \emph{oriented Hilbert scheme}
\[\Hilb^{\Gor,\o}(X/S)\colon \Sch_S^\op\to\Set\]
is the pullback
\[\Hilb^{\Gor,\o}(X/S):=\Hilb(X/S)\times_{\FFlat} \FGor^\o\,.\]
That is, $\Hilb^{\Gor,\o}(X/S)$ classifies finite locally free subschemes of $X$ that are Gorenstein and equipped with an orientation.
\end{defn}
\begin{rem}
If $X\to S$ is such that $\Hilb(X/S)$ is a scheme (for example, $X$ is quasi-projective over $S$) or an algebraic space (for example, $X$ is separated over $S$), then so is $\Hilb^{\Gor,\o}(X/S)$. Indeed, the forgetful map $\Hilb^{\Gor,\o}(X/S)\to \Hilb(X/S)$ is a base change of $\FGor^\o\to \FFlat$, which is representable by schemes by Remark~\ref{rem:FGoror}.
\end{rem}
From now on we will be interested in the ind-scheme $\Hilb^{\Gor,\o}(\A^\infty) := \colim_n \Hilb^{\Gor,\o}(\A^n)$ over $\Spec\Z$.
Note that we have a coproduct decomposition
\[\Hilb^{\Gor,\o}(\A^\infty)\simeq\coprod_{d\ge0} \Hilb^{\Gor,\o}_d(\A^\infty)\]
where $\Hilb^{\Gor,\o}_d(\A^\infty)$ classifies the oriented Gorenstein subschemes of $\A^\infty$ of degree $d$.
\begin{prop}\label{prop:Hilb-FGor}
The forgetful map $\Hilb^{\Gor,\o}(\A^\infty)\to \FGor^{\o}$ is a universal $\A^1$-equivalence on affine schemes.
\end{prop}
\begin{proof}
By definition, this map is a base change of the forgetful map $\Hilb(\A^\infty)\to \FFlat$, which is a universal $\A^1$-equivalence on affine schemes by~\cite[Proposition~4.2]{robbery}.
\end{proof}
Consider the map $\sigma\colon\Hilb^{\Gor,\o}(\A^\infty)\to \Hilb^{\Gor,\o}(\A^\infty)$ sending an oriented Gorenstein subscheme $Z\subset \A^\infty$ to
\[(\Spec \Oo[x]/x^2\times\{0\})\sqcup (\{0\}\times Z)\subset \A^1\times \A^\infty,\]
where $\Spec \Oo[x]/x^2$ is equipped with the orientation of Example~\ref{ex:kx/x2} and is embedded in $\A^1$ at $1$. Then we define
\[\Hilb^{\Gor,\o}_\infty(\A^\infty) := \colim(\Hilb^{\Gor,\o}_2(\A^\infty)\xrightarrow{\sigma}\Hilb^{\Gor,\o}_4(\A^\infty)\xrightarrow{\sigma}\cdots).\]
\begin{cor}\label{cor:Hilb-FGor}
The forgetful maps
\[
\uZ\times \Hilb^{\Gor,\o}_\infty(\A^\infty) \to \FGor^{\o,\gp} \to \Vect^{\sym,\gp}
\]
are $\A^1$-equivalences on affine schemes.
\end{cor}
\begin{proof}
The second map is an $\A^1$-equivalence on all schemes by Theorem~\ref{thm:main-gp}.
The first map factors as
\[
\uZ\times \Hilb^{\Gor,\o}_\infty(\A^\infty) \to \FGor^{\o,\st} \to \FGor^{\o,\gp},
\]
where the second map is an $\A^1$-equivalence on affine schemes by Remark~\ref{rem:FGor-gp}.
On quasi-compact schemes, the presheaf $\uZ\times \Hilb^{\Gor,\o}_\infty(\A^\infty)$ is the colimit of
\[\Hilb^{\Gor,\o}(\A^\infty)\xrightarrow{\sigma}\Hilb^{\Gor,\o}(\A^\infty)\xrightarrow{\sigma}\cdots,\]
while $\FGor^{\o,\st}$ is the colimit of
\[\FGor^\o\xrightarrow{\sigma}\FGor^\o\xrightarrow{\sigma}\cdots.\]
Hence, the result follows from Proposition~\ref{prop:Hilb-FGor}.
\end{proof}
Let $(V,B)\in \Vect^\sym(S)$ be a vector bundle over $S$ with a non-degenerate symmetric bilinear form. Recall from \cite{SchlichtingTripathi} that the \emph{orthogonal Grassmannian} $\GrO_d(V,B)$ is the open subscheme of the Grassmannian $\Gr_d(V)$ given by the rank $d$ subbundles $W\subset V$ such that the restriction of $B$ to $W$ is non-degenerate. We consider the smooth ind-scheme
\[\GrO_d=\GrO_d(\Hyp^\infty):=\colim_n \GrO_d(\Hyp^n)\]
over $\Spec\Z$. Forgetting the embedding into $\Hyp^\infty$ defines a canonical map $\GrO_d\to \Vect^\sym_d$.
A bilinear form $B$ on an $R$-module $V$ is called \emph{even} if it is symmetric and $B(x,x)\in 2R$ for all $x\in V$. Clearly, if $2\in R^\times$, every symmetric form is even. If $V$ is projective, a symmetric bilinear form $B$ on $V$ is even if and only if there exists a bilinear form $B'$ on $V$ such that $B(x,y)=B'(x,y)+B'(y,x)$.
\begin{lem}\label{lem:Hyp-embedding}
Let $R$ be a ring, $V$ a finite locally free $R$-module, and $B$ an even symmetric bilinear form on $V$.
Then there exists an isometric embedding $V\hookrightarrow \Hyp(R^n)$ for some $n\geq 0$.
\end{lem}
\begin{proof}
Choose an isomorphism $V\oplus W\simeq R^n$ and let $A$ be the matrix representing $B'\oplus 0$. Then we can take the composition
\[
V\hookrightarrow R^n \to \Hyp(R^n),
\]
where the second map has components $(A, \id_{R^n})$.
\end{proof}
Note that every hyperbolic space $\Hyp V$ is even.
It follows that the forgetful map $\GrO_d\to\Vect^\sym_d$ lands in the subpresheaf $\Vect^\ev_d\subset \Vect^\sym_d$ of even forms.
\begin{prop}\label{prop:Vect^bil and Gr^o}
For any $d\ge 0$, the forgetful map
\[\GrO_d\to \Vect^\ev_d\]
is a universal $\A^1$-equivalence on affine schemes.
\end{prop}
\begin{proof}
We will apply \cite[Lemma~4.1(2)]{robbery}, or rather its proof.
Unlike $\Vect^\sym_d$, the presheaf $\Vect^\ev_d$ for $d\geq 2$ does not satisfy closed gluing (in particular, it is not an algebraic stack).
Indeed, consider a closed pushout of affine schemes $X\sqcup_ZY$ on which there is a noneven function $f$ whose restriction to both $X$ and $Y$ is even (for example: $X=Y=\Spec\Z[x,y]$, $Z=\Spec\Z[x,y]/(2x-2y)$, and $f=(2x,2y)$); then the non-degenerate symmetric bilinear form $\left(\begin{smallmatrix} f & 1\\ 1 & 0\end{smallmatrix}\right)$ is not even, even though its restriction to both $X$ and $Y$ is.
However, $\Vect^\ev_d$ does nevertheless transform the colimit $\partial \A^n_R=\colim_{\Delta^k\hook \partial\Delta^n} \A^k_R$ into a limit, since the presheaf $\Spec R\mapsto 2R$ does.
It therefore suffices to check that for every commutative square
\[\begin{tikzcd}
\Spec R/I \ar[r] \ar[d] & \GrO_d\ar[d]\\
\Spec R\ar[r]\ar[ur,dashed] & \Vect^\ev_d
\end{tikzcd}\]
with $R=R_0[t_0,\dotsc,t_n]/(\sum_i t_i-1)$ and $I=(t_0\dotsm t_n)$, there exists a diagonal arrow making both triangles commute. Unwrapping the definitions of $\Vect^\ev_d$ and $\GrO_d$, this means that for every even bilinear space $(V,B)$ over $R$ and every isometric embedding $F\colon V/IV\hook \Hyp(R/I)^N$, we must find an isometric embedding $\tilde F\colon V\hook \Hyp(R)^N$ lifting $F$, after possibly increasing $N$.
First, let $\tilde F\colon V\to \Hyp(R)^N$ be any lift of $F$ (possibly not isometric), which exists since $V$ is projective over $R$. If we write
\[\tilde B(x,y)\coloneqq B(x,y)-B_{\Hyp}\left(\tilde Fx, \tilde Fy\right),\]
then $\tilde B$ is an $I$-valued even form on $V$. Our first claim is that we can change $\tilde F$ so that $\tilde B$ has values in $I^2$.
To do so, note that since $I\cap 2R=2I$, every $I$-valued even form on $V$ is an $I$-linear combination of even forms. Hence, we can find an $I$-valued bilinear form $\tilde B'$ such that $\tilde B(x,y)=\tilde B'(x,y)+\tilde B'(y,x)$.
Since $B$ is non-degenerate, there exists $r\colon V\to IV$ such that
\[\tilde B'(x,y)=B(x,r(y))\,.\]
Then it is easy to check that replacing $\tilde F$ with $\tilde F + \tilde Fr$ has the desired effect.
Without loss of generality, we can therefore assume that $\tilde B$ takes values in $I^2$. Since $I^2\cap 2R=2I^2$, we can then write
\[\tilde B=\sum_i a_i b_i C_i\]
for some $a_i, b_i\in I$ and some even bilinear forms $C_i\colon V\times V\to R$.
Using Lemma~\ref{lem:Hyp-embedding}, we can find for each $i$ an $R$-linear map $G_i\colon V\to \Hyp(R)^{N_i}$ such that
\[C_i (x,y)=B_{\Hyp}(G_ix, G_iy)\,.\]
Then an isometric lift of $F$ is given by
\[\left(\tilde F,\left((a_i,b_i)\circ G_i\right)_i\right)\colon V\to \Hyp(R)^{N+\sum_i N_i}\,,\]
where $(a_i,b_i)\colon\Hyp(R)^{N_i}\to \Hyp(R)^{N_i}$ is the map given on each component by the matrix $\left(\begin{smallmatrix} a_i & 0\\ 0 & b_i\end{smallmatrix}\right)$.
\end{proof}
Let $\tau\colon\GrO_d\to \GrO_{d+2}$ be the map sending $V\subset \Hyp^\infty$ to $\Hyp \oplus V\subset \Hyp^{1+\infty}$, and define the infinite orthogonal Grassmannian as
\[\GrO_\infty\coloneqq \colim(\GrO_2\xrightarrow\tau \GrO_4\xrightarrow\tau \cdots)\,.\]
Using Proposition~\ref{prop:Vect^bil and Gr^o} we obtain the following result, which generalizes a theorem of Schlichting and Tripathi \cite[Theorem~5.2]{SchlichtingTripathi} to rings that are not necessarily regular nor $\Z[\tfrac 12]$-algebras.
\begin{cor}\label{cor: Vect^bil and GrO}
The forgetful map
\[
\uZ\times \GrO_\infty \to \Vect^{\ev,\gp}
\]
is an $\A^1$-equivalence on affine schemes.
\end{cor}
\begin{proof}
This map factors as
\[
\uZ\times \GrO_\infty \to \Vect^{\ev,\st} \to \Vect^{\ev,\gp},
\]
where the second map is an $\A^1$-equivalence on affine schemes by Lemma~\ref{lem:invert-hyp} (in which we may replace $\Vect^\sym$ by $\Vect^\ev$).
On quasi-compact schemes, the presheaf $\uZ\times \GrO_\infty$ is the colimit of
\[\coprod_{d\ge0}\GrO_d\xrightarrow{\tau}\coprod_{d\ge0}\GrO_d\xrightarrow{\tau}\cdots,\]
where the coproducts are taken in $\Pre_\Sigma(\Sch)$.
On the other hand, $\Vect^{\ev,\st}$ is the colimit of
\[\Vect^\ev\xrightarrow{\tau}\Vect^\ev\xrightarrow{\tau}\cdots.\]
Hence, the result follows from Proposition~\ref{prop:Vect^bil and Gr^o}.
\end{proof}
\begin{rem}
The spaces $\Vect^\sym(R)^\gp$ and $\Vect^\ev(R)^\gp$ appearing in Corollaries \ref{cor:Hilb-FGor} and~\ref{cor: Vect^bil and GrO} are equivalent to the ``genuine symmetric'' and ``genuine even'' Grothendieck--Witt spaces $\GWspace^\mathrm{gs}(R)$ and $\GWspace^\mathrm{ge}(R)$ defined in \cite[Section 4]{HermitianII}; this is proved in \cite{HebestreitSteimle}.
We note that the forgetful map $\GWspace^\mathrm{ge}\to\GWspace^\mathrm{gs}$ does not induce an equivalence in $\H(S)$ if $2$ is not invertible on $S$. Both motivic spaces are stable under base change (by Corollary~\ref{cor: Vect^bil and GrO} and \cite[Example A.0.6(5)]{deloop3}, respectively), so it suffices to consider a field $k$ of characteristic $2$. Then every even form over $k$ is alternating (by definition), and non-degenerate alternating forms have even rank \cite[I, Corollary 3.5]{HM}, so the composition
\[
\pi_0\GWspace^\mathrm{ge}(k)\twoheadrightarrow\pi_0(L_\mot\GWspace^\mathrm{ge})(k)\to \pi_0(L_\mot\GWspace^\mathrm{gs})(k)\stackrel{\rk}\twoheadlongrightarrow \Z
\]
has image $2\Z\subset\Z$.
\end{rem}
\section{Isotropic Gorenstein algebras}
\label{sec:isotropic}
Inspired by~\cite[Theorem~2.1]{robbery}, one is tempted to ask whether the map
\[\alpha\colon\Vect^\sym_{d-2}\to \FGor^\o_d\]
sending $(V,B)$ to the oriented Gorenstein algebra $(R[x]/x^2\oplus V,\phi)$ of Example~\ref{ex:Gorenstein-algebras-from-bilinear-forms} is an $\A^1$-equivalence. Unfortunately this turns out not to be the case.
\begin{prop}\label{prop: square zero ext over R}
For any formally real field $k$ and any $d\geq 2$, the map
\[\alpha\colon\Vect^\sym_{d-2}\to \FGor^\o_d\]
is not a motivic equivalence on smooth $k$-schemes.
\end{prop}
\begin{proof}
Let us consider the composition
\[\FGor^\o_d\to \Vect^\sym_d\to \uW\,,\]
where $\uW$ is the Zariski sheafification of the presheaf of Witt groups on smooth $k$-schemes.
It is well-known that $\uW$ is an $\A^1$-invariant Nisnevich sheaf, for example by \cite[Theorem 3.4.11]{deloop1}, since Witt groups have framed transfers and are $\A^1$-invariant on regular noetherian $\Z[\tfrac 12]$-schemes (see the discussion in the introduction of \cite{GilleWittGroups}). Hence, this composition factors through a map
\[\pi_0\left(L_\mot\FGor^\o_d\right)\to \uW\,.\]
Let us now consider the composition
\[\pi_0\left(L_\mot\Vect^\sym_{d-2}\right)(k)\xrightarrow\alpha \pi_0\left(L_\mot\FGor^\o_d\right)(k)\to \uW(k)\to \Z\,,\]
where the last map is the signature associated with some real closure of $k$. We claim that the image of this composition is contained in the subset $\{n\in\Z\mid -d+2\le n\le d-2\}$. Indeed, $\pi_0\left(L_\mot\Vect^\sym_{d-2}\right)(k)$ is a quotient of $\pi_0\Vect^\sym_{d-2}(k)$ \cite[\sectsign 2, Corollary 3.22]{MV}, so it suffices to consider the image of the latter. By definition of $\alpha$, the composition
\[
\Vect^\sym_{d-2}\xrightarrow{\alpha}\FGor^\o_d\to \Vect^\sym_d
\]
adds a hyperbolic form, and in particular does not change the signature, which is therefore bounded by the rank $d-2$.
Now if $\alpha$ were a motivic equivalence, this would imply that the image of
\[\pi_0\left(L_\mot\FGor^\o_d\right)(k)\to \uW(k)\to \Z\]
is also contained in the subset of integers of absolute value $\leq d-2$.
But the oriented Gorenstein algebra $k^d$ with the orientation $\varphi(x_1,\dots,x_d)=x_1+\cdots+x_d$ has signature $d$, thus providing a contradiction.
\end{proof}
\begin{rem}
The algebra $k^d$ seems to be the only example leading to a contradiction in Proposition~\ref{prop: square zero ext over R}.
More precisely, consider the complement $\mathcal{Z}$ of the open substack
of smooth schemes in
$\FGor^\o_d$. The map $\alpha$ factors through $\mathcal{Z}$, and it seems
possible that $\alpha\colon\Vect^\sym_{d-2}\to \mathcal{Z}$ is an
$\A^1$-equivalence.
\end{rem}
We now modify Proposition~\ref{prop: square zero ext over R} so that it becomes a positive statement.
\begin{defn}
An oriented Gorenstein $R$-algebra $(A,\varphi)$ is called \emph{isotropic} if $\varphi(1)=0$. We write $\FGor^{\o,0}_d\subset \FGor^\o_d$ for the substack of isotropic oriented Gorenstein algebras of rank $d$.
\end{defn}
\begin{rem}
We warn the reader that the notion of isotropic oriented Gorenstein algebra is not directly related to that of isotropically augmented oriented Gorenstein algebra from Definition~\ref{def:isotropic-aug}. In particular, the forgetful functor $\FGor^{\o,+}\to\FGor^\o$ does not land in $\FGor^{\o,0}$. Nevertheless, there is a zigzag of $\A^1$-equivalences
\[
\FGor^{\o,0}_{\geq 2} \leftarrow \FGor^{\o,0}\times_{\FGor^\o} \FGor^{\o,+} \to \FGor^{\o,+},
\]
where the middle term is equivalent to $\FGor^\nonu$. Indeed, the right-hand map can be indetified with the zero section $\FGor^\nonu\hook \A^1\times\FGor^\nonu$ via Proposition~\ref{prop:nonunital=augmented}, and the left-hand map fits in a commuting triangle
\[
\begin{tikzcd}
\FGor^{\o,0}_{\geq 2} \ar{dr} &[-35pt] &[-35pt] \FGor^\nonu \ar{ll} \ar{dl}{\pi} \\
& \Vect^\sym &
\end{tikzcd}
\]
where the diagonal maps are $\A^1$-equivalences (Propositions \ref{prop:vect-nu} and \ref{prop: Vect^bil and FGor}).
\end{rem}
Note that the image of the map $\alpha$ lies in the substack $\FGor^{\o,0}_d\subset \FGor^{\o}_d$.
Moreover, for $(A,\varphi) \in \FGor^{\o,0}_d(R)$, the underlying symmetric bilinear form $(A,B_\varphi)$ is equipped with a canonical isotropic subspace $R\subset A$. Therefore we can apply algebraic surgery to it and obtain the non-degenerate symmetric bilinear form $\left(R^\perp/R,\bar B_\varphi\right)$.
The following proposition was independently obtained by Burt Totaro.\footnote{Private communication}
\begin{prop}\label{prop: Vect^bil and FGor}
For any $d\geq 2$, the maps
\[\alpha\colon\Vect^\sym_{d-2}\to \FGor^{\o,0}_d\]
sending $(V,B)$ to the oriented Gorenstein algebra $(R[x]/x^2\oplus V,\phi)$ of Example~\ref{ex:Gorenstein-algebras-from-bilinear-forms} and
\[\FGor^{\o,0}_d\to \Vect^\sym_{d-2}\]
sending $(A,\phi)$ to $\left(R^\perp/R,\bar B_\varphi\right)$ are inverse up to $\A^1$-homotopy.
\end{prop}
\begin{proof}
It is clear that the composition
\[\Vect^\sym_{d-2}\to \FGor^{\o,0}_d\to \Vect^\sym_{d-2}\]
is isomorphic to the identity, as the orientation $\phi$ of the algebra $R[x]/x^2\oplus V$ satisfies $B_\phi((0,v),(0,v'))=\phi(B(v,v')x,0)=B(v,v')$ by definition. Following the proof of~\cite[Theorem~2.1]{robbery}, we will use the Rees algebra to construct an $\A^1$-homotopy of the other composition to identity. The difference with the proof in \textit{loc.\ cit.}\ is that we will use the orientation $\varphi$ to refine the filtration. Indeed, let us consider the natural filtration
\[R\subset R^\perp\subset A\]
of $A$. The corresponding Rees algebra is the $R[t]$-algebra given by
\[\Rees(A,\varphi)=\{a\in A[t]\mid a_0\in R,\ \varphi(a_1)=0\}.\]
By \cite[Lemma~2.2]{robbery}, it is a finite locally free $R[t]$-algebra such that
\[\Rees(A,\varphi)/(t-1)\simeq A\quad\text{and}\quad \Rees(A,\varphi)/(t)\simeq R\oplus R^\perp/R\oplus A/R^\perp\,.\]
We can equip $\Rees(A,\varphi)$ with an orientation $\tilde\varphi \colon \Rees(A,\varphi)\to R[t]$ given by $a\mapsto \frac{1}{t^2}\varphi(a)$ (note that $\varphi(a_0)=\varphi(a_1)=0$ by definition of the filtration, so $\varphi(a)$ is indeed divisible by $t^2$). We claim this gives $\Rees(A,\varphi)$ the structure of an oriented Gorenstein $R[t]$-algebra. Since this is a property local on the base, we can assume that $A$ is free as an $R$-module. Then let us choose a basis of $A$ of the form
\[(1,e_1,\dots,e_{d-2},x)\,,\]
where $1,e_1,\dots,e_{d-2}$ is a basis of $\ker \varphi$ and $x$ is such that $\varphi(x)=1$ and $\varphi(xe_i)=0$. Then the matrix of the bilinear form $B_\varphi$ is
\[\begin{pmatrix} 0 & 0 & 1\\
0 & D & 0\\
1 & 0 & \varphi(x^2)\end{pmatrix},\]
where $D$ is an invertible matrix (since $B_\varphi$ is non-degenerate). Then the collection
\[(1,e_1t,\dots,e_{d-2}t,xt^2)\]
is a basis for $\Rees(A,\varphi)$, and the matrix of the symmetric bilinear form $B_{\tilde \varphi}$ in this basis is
\[\begin{pmatrix} 0 & 0 & 1\\
0 & D & 0\\
1 & 0 & \varphi(x^2)t^2\end{pmatrix},\]
which is invertible (since $D$ is). Hence, sending $(A,\varphi)$ to $(\Rees(A,\varphi),\tilde\varphi)$ defines a natural transformation
\[\FGor^{\o,0}_d\to \FGor^{\o,0}_d(\A^1\times-)\,,\]
which provides the desired $\A^1$-homotopy. Indeed, at $t=0$ we have an isomorphism of $R$-algebras
\[
R\oplus R^\perp/R\oplus A/R^\perp \simeq R[x]/x^2\oplus R/R^\perp
\]
identifying $A/R^\perp$ with $Rx$ via $a\mapsto \phi(a)x$, under which the orientation $\tilde\phi_0=\phi\circ \pr_3$ of the left-hand side corresponds to the orientation of Example~\ref{ex:Gorenstein-algebras-from-bilinear-forms}, which extracts the coefficient of $x$.
\end{proof}
\begin{cor}
For any $d \geqslant 2$, there is a canonical $\A^1$-equivalence
\[\FGor^{\o,0}_d \simeq \GrO_{d-2}\]
on affine $\Z[\tfrac 12]$-schemes.
\end{cor}
\begin{proof}
Combine Propositions~\ref{prop:Vect^bil and Gr^o} and~\ref{prop: Vect^bil and FGor}.
\end{proof}
\section{The motivic hermitian K-theory spectrum}\label{sec:kq}
Fix a base scheme $S$.
By a \emph{presheaf with framed transfers} on $\Sm_S$ we mean a $\Sigma$-presheaf on the $\infty$-category $\Span^\fr(\Sm_S)$ constructed in \cite{deloop1} (i.e., a presheaf that takes finite coproducts in $\Span^\fr(\Sm_S)$ to products of spaces). We refer to \cite[\sectsign 3]{deloop1} for the definition of the $\infty$-categories $\H^\fr(S)$ and $\SH^\fr(S)$ of framed motivic spaces and framed motivic spectra over $S$.
The general reconstruction theorem \cite[Theorem 18]{framed-loc} states that the ``forget transfers'' functor implements an equivalence
\[
\SH^\fr(S)\simeq \SH(S)
\]
between the symmetric monoidal $\infty$-categories of framed motivic spectra and of motivic spectra. In particular, for any $E\in\SH(S)$, the presheaf of spaces $\Omega^\infty_\T E$ has a canonical structure of framed transfers.
We will abusively denote by $\Sigma^\infty_{\T,\fr}$ the composite functor
\[
\Pre_\Sigma(\Span^\fr(\Sm_S)) \to \H^\fr(S) \xrightarrow{\Sigma^\infty_{\T,\fr}} \SH^\fr(S)\simeq \SH(S).
\]
Let $\Span^{\fgor,\o}(\Sch_S)$ be the symmetric monoidal $(2,1)$-category of oriented finite Gorenstein correspondences: its objects are $S$-schemes and its morphisms are spans
\[
\begin{tikzcd}
& Z \ar[swap]{ld}{f}\ar{rd}{g} & \\
X & & Y
\end{tikzcd}
\]
where $Z\in\FGor^\o(X)$, that is, $f$ is finite locally free and equipped with a trivialization $\omega_f\simeq \sO_Z$ (see Definition~\ref{dfn:oriented-Gorenstein-algebra}). The symmetric monoidal structure is given by the product of $S$-schemes (for two finite locally free morphisms $f\colon Z\to X$ and $f'\colon Z'\to X'$ over $S$, the dualizing sheaf $\omega_{f\times_S f'}$ is the external tensor product $\omega_f\boxtimes_S \omega_{f'}$, so trivializations of $\omega_f$ and $\omega_{f'}$ induce a trivialization of $\omega_{f\times_Sf'}$). The wide subcategory where $f$ is finite syntomic is the $(2,1)$-category $\Span^{\fsyn,\o}(\Sch_S)$ of oriented finite syntomic correspondences considered in \cite[\sectsign 4.2]{deloop3}. We write $\Span^{\fgor,\o}(\Sm_S)$ for the full subcategory spanned by the smooth $S$-schemes.
The presheaf on $\Span^{\fgor,\o}(\Sch_S)$ represented by $S$ is the presheaf of groupoids $\FGor^\o$, which is therefore a commutative monoid in $\Pre_\Sigma(\Span^{\fgor,\o}(\Sch_S))$ (with respect to the Day convolution).
We observe that the presheaf of groupoids $\Vect^\sym$ can also be promoted to a commutative monoid object in $\Pre_\Sigma(\Span^{\fgor,\o}(\Sch_S))$. Given a finite Gorenstein morphism $f\colon Z\to X$ with orientation $\phi\colon f_*\sO_Z\to\sO_X$, we have a pushforward functor
\[
f_*\colon \Vect^\sym(Z) \to \Vect^\sym(X), \quad (\sE,B)\mapsto (f_*\sE, \phi\circ f_*B).
\]
This pushforward is compatible with composition, base change, and the tensor product (it satisfies a projection formula) up to canonical isomorphisms. Since $\Vect^\sym$ is an ordinary groupoid, it is straightforward to check the coherence of these canonical isomorphisms, which gives $\Vect^\sym$ the structure of a commutative monoid in $\Pre_\Sigma(\Span^{\fgor,\o}(\Sch_S))$.
Finally, the forgetful map $\eta\colon \FGor^\o\to \Vect^\sym$ can be uniquely promoted to a morphism of commutative monoids in $\Pre_\Sigma(\Span^{\fgor,\o}(\Sch_S))$, since $\FGor^\o$ is the initial such object. We will write $\FGor^\o_S$ and $\Vect^\sym_S$ for the restrictions of these presheaves to smooth $S$-schemes.
By Lemma~\ref{lem:det(L)} below (see also \cite[Theorem A.1]{BachmannWickelgren}), the determinant $\det\colon K_{\rk=0}\to\Pic$ defines a symmetric monoidal forgetful functor
\[
\Span^\fr(\Sch_S) \to \Span^{\fsyn,\o}(\Sch_S)\subset \Span^{\fgor,\o}(\Sch_S).
\]
We can therefore regard $\FGor^\o$ and $\Vect^\sym$ as commutative monoids in $\Pre_\Sigma(\Span^\fr(\Sch_S))$.
\begin{lem}\label{lem:det(L)}
Let $f$ be a finite syntomic map, $L_f$ its cotangent complex, and $\omega_f$ its relative dualizing sheaf. Then there is a canonical isomorphism
\[\det(L_f) \simeq \omega_f .\]
Moreover, this isomorphism is compatible with base change and composition in the obvious way.
\end{lem}
\begin{proof}
By \cite[Tag 0FKD]{stacks}, there is such a family of isomorphisms for $f$ a finite syntomic morphism between noetherian schemes. Since they are compatible with base change, they uniquely determine the desired isomorphisms for $f$ between qcqs schemes (by noetherian approximation), whence between arbitrary schemes by descent.
\end{proof}
\begin{rem}
Lemma~\ref{lem:det(L)} holds more generally if $f$ is any quasi-smooth morphism of derived schemes, with $\omega_f=f^!(\sO)$, but the proof is more complicated (in fact, the functor $f^!$ seems not to have been defined in this generality, although one can also define $\omega_f$ more directly).
\end{rem}
In \cite{Schlichting3}, Schlichting defined the Grothendieck--Witt spectrum of a qcqs $\Z[\tfrac 12]$-scheme $S$; we shall denote by $\GWspace(S)$ its underlying space.\footnote{In \emph{loc.\ cit.}, Schlichting works with schemes admitting an ample family of line bundles, but remarks that this assumption can be removed if one uses perfect complexes instead of strictly perfect complexes. In any case, Schlichting's setting suffices for our purposes, since it suffices to define the motivic localization of $\GWspace$.}
The presheaf $\GWspace$ is then a Nisnevich sheaf \cite[Theorem 9.6]{Schlichting3}, and it is $\A^1$-invariant on regular schemes \cite[Theorem 9.8]{Schlichting3}.
By the example in~\cite[p.~1781]{Schlichting3} and by~\cite[Remark~4.13]{Schlichting1}, for $R$ a $\Z[\tfrac 12]$-algebra there is a natural equivalence
\begin{equation*}\label{eqn:GW-Vect-sym}
\GWspace(R) \simeq \Vect^\sym(R)^\gp .
\end{equation*}
In particular, we have an equivalence $\GWspace \simeq L_\Zar\Vect^{\sym,\gp}$ making $\GWspace$ into a presheaf of $\Einfty$-ring spaces.
By \cite[Remark~5.9 and Theorem~9.10]{Schlichting3}, we have a chain of equivalences of $\GWspace$-modules:
\[\GWspace\simeq \GWspace^{[-4]}\simeq \Omega^4_{\P^1}\GWspace.\]
The element $1$ in the left-hand side gives a canonical element
$\betah\colon (\P^1)^{\wedge 4} \to \GWspace$, called the \emph{hermitian Bott element}.
The above equivalence means that $\GWspace$ is $\betah$-periodic in the sense of \cite[Section 3]{HoyoisCdh}.
Let $S$ be a scheme with $2\in \sO(S)^\times$.
We define the motivic hermitian K-theory spectrum $\KQ_S\in\SH(S)$ as the motivic spectrum associated with the $\T^{\wedge 4}$-prespectrum
\[
(L_\mot\GWspace_S, L_\mot\GWspace_S,\dotsc),
\]
where $\GWspace_S$ is the restriction of $\GWspace$ to $\Sm_S$, with structure maps $L_\mot\GWspace_S \to \Omega_\T^4 L_\mot\GWspace_S$ induced by the hermitian Bott element. Since the structure maps are $\GWspace_S$-module maps, we can regard $\KQ_S$ as an object of $\Mod_{\GWspace}(\SH(S))$.
We write $\kq_S$ for the very effective cover of $\KQ_S$.
Note that when $S$ is regular, then $\GWspace_S\simeq L_\mot\GWspace_S$ and the above prespectrum is already a spectrum; in that case, $\KQ_S$ was originally defined by Hornbostel \cite{Hornbostel:2005}. We refer to Proposition~\ref{prop:KQ=GW} below for a justification of our definition in general.
\begin{rem}\label{rem:2}
There is work in progress by B.~Calmès, Y.~Harpaz and the third author on constructing the motivic hermitian K-theory spectrum over base schemes where $2$ need not be invertible \cite{CalmesNardinHarpaz}. All of the results below remain valid without that assumption (in Proposition~\ref{prop:KSp}(2), one must however use skew-symmetric instead of alternating forms).
\end{rem}
\begin{lem}\label{lem:KQ-is-E-infinity}
Let $S$ be a regular $\Z[\tfrac 12]$-scheme. Then the object $\KQ_S\in \Mod_{\GWspace}(\SH(S))$ has a unique structure of $\Einfty$-algebra lifting the $\Einfty$-ring structure of $\GWspace_S$.
\end{lem}
\begin{proof}
Since $S$ is regular we have $\KQ_S=(\GWspace_S,\GWspace_S,\dotsc)$.
The action of $\betah$ on $\KQ_S$ is given levelwise by its action on $\GWspace_S$, and hence it is invertible. The motivic spectrum $\KQ_S$ is thus $\betah$-periodic in the sense of \cite[Section 3]{HoyoisCdh}. By \cite[Proposition 3.2]{HoyoisCdh}, there is an equivalence of symmetric monoidal $\infty$-categories
\[
\Sigma^\infty_\T(-)[\betah^{-1}]: P_\betah\Mod_{\GWspace}(\H_*(S)) \rightleftarrows P_\betah \Mod_{\GWspace}(\SH(S)): \Omega^\infty_\T,
\]
where $P_\betah\Mod_{\GWspace}$ on either side denotes the subcategory of $\betah$-periodic $\GWspace$-modules.
Since $\Omega^\infty_\T \KQ_S\simeq \GWspace_S$, this proves the claim.
\end{proof}
By construction, there is a $\T^{\wedge 4}$-prespectrum of presheaves on \emph{all} $\Z[\tfrac 12]$-schemes whose restriction to $\Sm_S$ gives $\KQ_S$ for all $S$. This implies that $S\mapsto \KQ_S$ is a section of the cocartesian fibration over $\Sch_{\Z[\frac 12]}^\op$ classified by $\SH(-)$. In particular, for any morphism of $\Z[\tfrac 12]$-schemes $f\colon T\to S$, we have a canonical comparison map $f^*(\KQ_S)\to \KQ_T$ in $\SH(T)$.
\begin{lem}\label{lem:KQ-bc}
For any morphism of $\Z[\tfrac 12]$-schemes $f\colon T\to S$, the canonical map $f^*(\KQ_S)\to \KQ_T$ is an equivalence.
\end{lem}
\begin{proof}
The functor $f^*$ is compatible with spectrification, motivic localization, and group completion. Hence, it suffices to show that the canonical map $f^*(\Vect^\sym_S)\to \Vect^\sym_T$ is a motivic equivalence of presheaves on $\Sm_T$. In fact, it is a Zariski-local equivalence by \cite[Proposition A.0.4]{deloop3}, since $\Vect^\sym$ is a smooth algebraic stack with affine diagonal.
\end{proof}
Combining Lemmas~\ref{lem:KQ-is-E-infinity} and \ref{lem:KQ-bc}, we obtain a canonical $\Einfty$-ring structure on $\KQ_S$ for any $\Z[\tfrac 12]$-scheme $S$, which is compatible with base change (an $\Einfty$-ring structure on $\KQ_S$ was also constructed in this generality by Lopez-Avila in \cite{AvilaThesis}; we suspect but do now know that it is equivalent to ours).
Next, we show that the motivic spectrum $\KQ_S$ represents Karoubi–Grothendieck–Witt theory made $\A^1$-homotopy invariant:
\begin{prop}\label{prop:KQ=GW}
Let $S$ be a qcqs scheme with $2\in\sO(S)^\times$ and let $n\in\Z$. Then there is a natural equivalence of spectra (and of $\Einfty$-ring spectra if $n=0$)
\[
\Gamma(S,\Sigma^n_\T\KQ) \simeq (\Lhtp\KGW^{[n]})(S),
\]
where $\KGW^{[n]}$ is the $n$th shifted Karoubi–Grothendieck–Witt spectrum \cite[Definition 8.6]{Schlichting3}.
\end{prop}
\begin{proof}
We start with the observation that the restriction of the Bott element $\beta\in \GWspace^{[1]}(\P^1)$ defined in \cite[\sectsign 9.4]{Schlichting3} to either $\A^1$ or $\P^1-0$ is canonically null-homotopic. This gives rise to the following four variations on the domain of the Bott element, which are all motivically equivalent:
\[
\begin{tikzcd}
\T=\A^1/\bG_m \ar{r} \ar{d} & \P^1/(\P^1-0) \ar{dr}{\beta} & \P^1/\infty \ar{l} \ar{d}{\beta} \\
\Sigma\bG_m \ar{rr}{\beta} & & \GWspace^{[1]}\rlap.
\end{tikzcd}
\]
Let us write $\GWspectrum$ for the Grothendieck–Witt spectrum and $\GWspectrum_{\geq 0}$ for its connective cover. We consider the following four ``periodic'' prespectra with structure maps given by $\betah=\beta^4$:
\begin{itemize}
\item[(A)] $(\GWspace,\GWspace,\dotsc)$,
\item[(B)] $(\GWspectrum_{\geq 0},\GWspectrum_{\geq 0},\dotsc)$,
\item[(C)] $(\GWspectrum,\GWspectrum,\dotsc)$,
\item[(D)] $(\KGW,\KGW,\dotsc)$.
\end{itemize}
Here, (A) is a prespectrum in presheaves of pointed spaces, which defines $\KQ$, whereas (B)–(D) are prespectra in presheaves of spectra. Note also that (A), (C), and (D) are $(\P^1)^{\wedge 4}$-spectra, by the projective bundle formula \cite[Theorem~9.10]{Schlichting3}.
Since $\KGW^{[n]}$ is a Nisnevich sheaf of spectra on qcqs schemes \cite[Theorem 9.6]{Schlichting3}, its motivic localization is given by $\Lhtp\KGW^{[n]}$. Moreover, since $\Lhtp$ commutes with $\Omega_{\P^1}$ and $\Omega_{\P^1}\KGW^{[n]}\simeq \KGW^{[n-1]}$, we see that the motivic spectrum defined by (D) satisfies the desired conclusion. It will thus suffice to show that (A)–(D) all define the same motivic spectrum. The fact that we get an equivalence of $\Einfty$-rings when $n=0$ follows by repeating the proof of Lemma~\ref{lem:KQ-is-E-infinity} with $\Lhtp\KGW$ instead of $\GWspace$.
The maps $(B)\to (C)\to(D)$ become equivalences after $(\Sigma\bG_m)^{\wedge 4}$-spectrification, since the maps \[\Omega_{\Sigma\bG_m}^{n}\GWspectrum_{\geq 0}\to \Omega_{\Sigma\bG_m}^n\GWspectrum\to \Omega_{\Sigma\bG_m}^n\KGW\] induce isomorphisms on $\pi_i$ for $i\geq -n$ \cite[Proposition 9.3(1)]{Schlichting3}. Hence, the motivic spectra associated with (B), (C), and (D) are the same. Furthermore, since $\Lhtp$ commutes with spectrification, we see that the motivic spectrum associated with (B) can be obtained in two steps: first apply $\Lhtp$ and then $(\Sigma\bG_m)^{\wedge 4}$-spectrify. As $\Omega^\infty$ preserves sifted colimits of \emph{connective} spectra, this explicit formula for the localization of (B) implies that the motivic spectra associated with (A) and (B) are identified under the equivalence between $\T^{\wedge 4}$-spectra in motivic spaces and in motivic $S^1$-spectra (which is implemented by the functor $\Omega^\infty$ levelwise). This completes the proof.
\end{proof}
\begin{samepage}
\begin{prop}\label{prop:KQ-kq-Vect^bil}
Assume $2\in\sO(S)^\times$.
\begin{enumerate}
\item There is an equivalence of $\Einfty$-ring spectra
\[
\KQ_S\simeq (\Sigma^\infty_{\T,\fr}\Vect^\sym_S)[\betah^{-1}].
\]
\item If $S$ is regular over a field, there is an equivalence of $\Einfty$-ring spectra
\[
\kq_S\simeq \Sigma^\infty_{\T,\fr}\Vect^\sym_S.
\]
\end{enumerate}
\end{prop}
\end{samepage}
\begin{proof}
Since $\GWspace$ is a Nisnevich sheaf, which is given by $\Vect^{\sym,\gp}$ on affine schemes, we have $\GWspace=L_\Nis\Vect^{\sym,\gp}$. Since Nisnevich sheafification preserves presheaves with oriented finite Gorenstein transfers (cf.\ \cite[Proposition 3.2.4]{deloop1}), the $\T^{\wedge 4}$-prespectrum defining $\KQ_S$ is a $\GWspace_S$-module with oriented finite Gorenstein transfers. In particular, it defines a $\GWspace_S$-module in framed motivic spectra whose underlying motivic spectrum is $\KQ_S$. Repeating the proof of Lemma~\ref{lem:KQ-is-E-infinity} in the framed setting, we see that $\KQ_S \simeq (\Sigma^\infty_{\T,\fr}\Vect^\sym_S)[\betah^{-1}]$ as $\Einfty$-rings when $S$ is regular. By Lemma~\ref{lem:KQ-bc}, the left-hand side is stable under base change. The right-hand side is stable under base change as well by \cite[Lemma~16]{framed-loc}, which proves (1) in general.
The proof of (2) uses the motivic recognition principle and is identical to the proof of \cite[Corollary 5.2]{robbery}.
\end{proof}
We have analogous results with the shifted Grothendieck–Witt space $\GWspace^{[2]}$.
For a $\Z[\tfrac 12]$-scheme $S$, we define $\KSp_S\in\Mod_\GWspace(\SH(S))$ to be the motivic spectrum associated with the $\T^{\wedge 4}$-prespectrum
\[
(L_\mot\GWspace^{[2]}, L_\mot\GWspace^{[2]},\dotsc),
\]
with structure maps induced by $\betah$. We write $\ksp_S$ for the very effective cover of $\KSp_S$.
For a $\Z[\tfrac 12]$-algebra $R$, we have
\[
\GWspace^{[2]}(R)\simeq \Vect^\alt(R)^\gp
\]
where $\Vect^\alt$ is the stack of non-degenerate alternating bilinear forms. Since the latter is a smooth algebraic stack with affine diagonal, we deduce as in Lemma~\ref{lem:KQ-bc} that $\KSp_S$ is stable under arbitrary base change. From the equivalence $\Omega_{\P^1}^2\GWspace^{[2]}\simeq \GWspace$, it follows immediately that
\[\KSp_S\simeq \Sigma^2_\T\KQ_S\]
when $S$ is regular, whence in general by base change.
Arguing exactly as in Proposition~\ref{prop:KQ-kq-Vect^bil}, we obtain the following result:
\begin{prop}\label{prop:KSp}
Assume $2\in\sO(S)^\times$.
\begin{enumerate}
\item There is an equivalence of $\KQ_S$-module spectra
\[
\KSp_S\simeq (\Sigma^\infty_{\T,\fr}\Vect^\alt_S)[\betah^{-1}].
\]
\item If $S$ is regular over a field, there is an equivalence of $\kq_S$-module spectra
\[
\ksp_S\simeq \Sigma^\infty_{\T,\fr}\Vect^\alt_S.
\]
\end{enumerate}
\end{prop}
\begin{rem}
Combining Propositions~\ref{prop:KQ-kq-Vect^bil} and \ref{prop:KSp}, we find
\[
\KQ_S\oplus \KSp_S \simeq \Sigma^\infty_{\T,\fr}\big(\Vect^\sym_S\times \Vect^\alt_S\big)[\betah^{-1}].
\]
One can easily check that the groupoid $\Vect^\sym\times \Vect^\alt$ has a symmetric monoidal structure given by tensoring bilinear forms, which is compatible with the oriented finite Gorenstein transfers. We deduce that $\KQ_S\oplus \KSp_S$ has an $\Einfty$-ring structure such that the summand inclusion $\KQ_S\to \KQ_S\oplus \KSp_S$ is an $\Einfty$-map.
\end{rem}
\begin{rem}\label{rem:SL-orientation}
Recall that $\MSL_S\simeq \Sigma^\infty_{\T,\fr}\FSyn^\o_S$ \cite[Theorem 3.4.3]{deloop3}. Hence, by Proposition~\ref{prop:KQ-kq-Vect^bil}, the forgetful map $\FSyn^\o\to \Vect^\sym$ induces a morphism of $\Einfty$-ring spectra
\[
\MSL_S \to \KQ_S
\]
for any $\Z[\tfrac 12]$-scheme $S$, which factors through $\kq_S$ since $\MSL_S$ is very effective.
\end{rem}
\begin{rem}
The oriented finite Gorenstein transfers in hermitian K-theory are known to not fully depend on the trivialization of the dualizing line $\omega_f$, but only on a choice of square root $\omega_f\simeq \mathcal L^{\otimes 2}$. Correspondingly, the coherent $\SL$-orientation of $\KQ_S$ from Remark~\ref{rem:SL-orientation} can be improved to an $\SL^c$-orientation.
Define the presheaf $K^{\SL^c}\colon \Sch^\op\to \Spc$ by the pullback square
\[
\begin{tikzcd}
K^{\SL^c} \ar{r} \ar{d} & K \ar{d}{\det} \\
\Pic \ar{r}{2} & \Pic\rlap,
\end{tikzcd}
\]
where $\Pic(X)$ is the groupoid of invertible sheaves on $X$.
The motivic $\Einfty$-ring spectrum $\MSL^c$ is the Thom spectrum associated with the composition \[K^{\SL^c}\times_{\uZ} \{0\}\to K\to \Pic(\SH),\] in the sense of \cite[Section 16]{norms}. By \cite[Theorem 3.3.10]{deloop3}, there is an equivalence of $\Einfty$-rings
\[
\MSL^c\simeq \Sigma^\infty_{\T,\fr}\FSyn^{c}
\]
where $\FSyn^c(X)$ is the groupoid of triples $(Z,\sL,\lambda)$ with $Z$ a finite syntomic $X$-scheme, $\sL$ an invertible sheaf on $Z$, and $\lambda\colon \omega_{Z/X}\simeq \sL^{\otimes 2}$. There is a map $\FSyn^c\to\Vect^\sym$ sending $(Z,\sL,\lambda)$ to $f_*(\sL)$, where $f\colon Z\to X$ is the structure map, with the symmetric bilinear form
\[
f_*(\sL) \otimes f_*(\sL) \to f_*(\sL^{\otimes 2}) \stackrel\lambda\simeq f_*(\omega_{Z/X}) \to\sO_X.
\]
This induces an $\Einfty$-$\SL^c$-orientation $\MSL^c\to\KQ$ refining the $\SL$-orientation of Remark~\ref{rem:SL-orientation}.
\end{rem}
Combining Proposition~\ref{prop:KQ-kq-Vect^bil} and Theorem~\ref{thm:main-gp}, we obtain a description of the motivic hermitian K-theory spectrum in terms of oriented Gorenstein algebras:
\begin{thm}\label{thm:KQ-kq-FGor}
Assume $2\in\sO(S)^\times$.
\begin{enumerate}
\item There is an equivalence of $\Einfty$-ring spectra
\[\KQ_S\simeq (\Sigma^\infty_{\T,\fr}\FGor^\o_S)[\betah^{-1}],\]
where $\betah\in\pi_{8,4}\Sigma^\infty_{\T,\fr}\FGor^\o_S$ is transported through the equivalence of Theorem~\ref{thm:main-gp}.
\item If $S$ is regular over a field, there is an equivalence of $\Einfty$-ring spectra
\[
\kq_S \simeq \Sigma^\infty_{\T,\fr} \FGor^\o_S.
\]
\end{enumerate}
\end{thm}
\section{The Milnor--Witt motivic cohomology spectrum}\label{sec:MW}
Let $S$ be a scheme that is pro-smooth over a field or over a Dedekind scheme (i.e., $S$ is locally a cofiltered limit of smooth schemes; for example, by Popescu's theorem~\cite[Tag 07GC]{stacks}, $S$ is regular over a field). The \emph{Milnor–Witt motivic cohomology spectrum} $\hzmw_S\in\SH(S)$ is defined by
\[\hzmw_S=\underline\pi^\eff_0(\MonUnit_S),\]
where $\underline\pi^\eff_*$ are the homotopy groups in the effective homotopy $t$-structure on $\SH^\eff(S)$ (whose connective part is the subcategory $\SH^\veff(S)$ of very effective motivic spectra).
This definition is due to Bachmann \cite{BachmannSlices,BachmannDedekind}, and it is known to be equivalent to the original definition of Calmès and Fasel over a perfect field of characteristic not $2$~\cite{BachmannFasel}.
Note that $\hzmw_S$ is an $\Einfty$-ring spectrum (even a normed spectrum, see \cite[Proposition 13.3]{norms}) that is stable under pro-smooth base change.
For $S$ as above, we define a presheaf of rings $\uGW_S$ on $\Sm_S$ in two cases:
\begin{enumerate}
\item if $S$ is over a perfect field $k$, $\uGW_S$ is the Zariski sheafification of the left Kan extension of the sheaf of unramified Grothendieck–Witt rings over $\Sm_k$ defined by Morel \cite[\sectsign 3.2]{Morel};
\item if $2\in\sO(S)^\times$, $\uGW_S$ is the Nisnevich sheafification of the presheaf of Grothendieck–Witt rings, i.e., $\uGW_S=L_\Nis\pi_0\Vect_S^{\sym,\gp}$.
\end{enumerate}
Note that these two definitions agree when they both apply (see the proof of \cite[Theorem 10.12]{norms}).
We can promote $\uGW_S$ to a commutative monoid in presheaves with framed transfers as follows.
In case (1), since $\uGW=(\underline{K}_3^{MW})_{-3}$, \cite[Theorem 5.19]{BachmannYakerson} implies that $\uGW_k$ admits a unique structure of presheaf with framed transfers compatible with its $\uGW_k$-module structure and extending Morel's transfers for monogenic field extensions \cite[\sectsign 4.2]{Morel}. In case (2), $\uGW_S$ inherits oriented finite Gorenstein transfers from $\Vect^\sym$. The uniqueness of the framed transfers in (1) implies that they agree with those in (2) when $2$ is invertible.
\begin{rem}
Over a field of characteristic $2$, there is a canonical epimorphism of Nisnevich sheaves $L_\Nis\pi_0\Vect^{\sym,\gp}\to\uGW$, but we do not know if it is an isomorphism (see \cite[Remark 4.7]{deloop4}).
\end{rem}
\begin{lem}\label{lem:hzmw-transfers}
Let $S$ be pro-smooth over a field or a Dedekind scheme. If $S$ has mixed characteristic, assume also that $2\in\sO(S)^\times$. Then there is an equivalence $\Omega^\infty_{\T,\fr}\hzmw_S\simeq \uGW_S$ of commutative monoids in $\Pre_\Sigma(\Span^\fr(\Sm_S))$.
\end{lem}
\begin{proof}
In the equicharacteristic case, it suffices to prove the result over a perfect field.
By \cite[Proposition~4(3)]{BachmannSlices}, $\hzmw$ is the effective cover of $\underline\pi_0(\1)_*$, and hence we have isomorphisms of rings $\Omega^\infty_\T\hzmw\simeq\underline\pi_0(\1)_0\simeq \uGW$. It remains to compare the framed transfers on either side. By \cite[Corollary 5.17]{BachmannYakerson}, it suffices to compare the transfers induced by a monogenic field extension $K\subset L$ with chosen generator $a\in L$. This was done in the proof of \cite[Proposition 4.3.17]{deloop2}.
Assume now that $2\in\sO(S)^\times$. By \cite[Definition 4.1 and Corollary 4.9]{BachmannDedekind}, we have an isomorphism $\Omega^\infty_{\T}\hzmw_S\simeq \uGW_S$ such that the following square commutes, where $\kq^\fr_S=\Sigma^\infty_{\T,\fr}\Vect^{\sym,\gp}_S$:
\[
\begin{tikzcd}
\Vect^{\sym,\gp}_S \ar{r}{\mathrm{unit}} \ar{d} & \Omega^\infty_{\T}\kq^\fr_S \ar{d} \\
\uGW_S \ar{r}{\simeq} & \Omega^\infty_{\T}\hzmw_S\rlap.
\end{tikzcd}
\]
The top and left arrows are morphisms of presheaves with framed transfers by definition, and so is the right vertical arrow since it is $\Omega^\infty_{\T}$ of the morphism $\kq_S^\fr\to \underline\pi^\eff_0\kq_S^\fr\simeq \hzmw_S$. Using the compatibility of the Nisnevich topology with framed transfers \cite[Proposition 3.2.14]{deloop1}, we deduce that the bottom isomorphism is compatible with framed transfers.
\end{proof}
The next theorem is the analogue of \cite[Theorem 21]{framed-loc} for Milnor–Witt motivic cohomology. We are grateful to Tom Bachmann for providing the rigidity argument in the mixed characteristic case.
\begin{thm}\label{thm:HtildeZ}
Let $S$ be pro-smooth over a field or a Dedekind scheme. If $S$ has mixed characteristic, assume also that $2\in\sO(S)^\times$. Then there is an equivalence of motivic $\Einfty$-ring spectra $\hzmw_S \simeq \Sigma^\infty_{\T,\fr} \uGW_S$ in $\SH(S)$.
\end{thm}
\begin{proof}
The equivalence $\Omega^\infty_{\T,\fr}\hzmw_S\simeq \uGW_S$ of Lemma~\ref{lem:hzmw-transfers} induces a canonical $\Einfty$-map
\[
\Sigma^\infty_{\T,\fr}\uGW_S\to \hzmw_S
\]
in $\SH(S)$. Since both sides are compatible with pro-smooth base change, it is enough to show that it is an equivalence when $S$ is a perfect field or a Dedekind domain. In the former case, the result follows directly from the motivic recognition principle \cite[Theorem~3.5.14]{deloop1}, since $\hzmw_S$ is very effective.
Let us therefore assume that $S$ is a Dedekind domain with $2\in\sO(S)^\times$. By \cite[Proposition B.3]{norms}, it suffices to show that both $\Sigma^\infty_{\T,\fr}\uGW_S$ and $\hzmw_S$ are stable under base change to the residue fields. For $\hzmw_S$, this holds by \cite[Theorem 4.4]{BachmannDedekind}. For $\Sigma^\infty_{\T,\fr}\uGW_S$, in view of \cite[Lemma 16]{framed-loc}, it suffices to show that the canonical map $s^*(\uGW_S)\to \uGW_{\kappa(s)}$ in $\Pre(\Sm_{\kappa(s)})$ is a motivic equivalence for all $s\in S$.
Since $\uGW\simeq\uW\times_{\uZ/2}\uZ$ and Witt groups satisfy rigidity for henselian local $\Z[\tfrac 12]$-algebras \cite[Lemma 4.1]{Jacobson}, this follows from Lemma~\ref{lem:rigidity} below.
\end{proof}
\begin{rem}
The proof of Theorem~\ref{thm:HtildeZ} shows that $\hzmw_S \simeq \Sigma^\infty_{\T,\fr} \uGW_S$ for any $\Z[\tfrac 12]$-scheme $S$, if one defines $\hzmw_S$ by base change from $\Spec\Z$ and $\uGW_S$ as $L_\Nis\pi_0\Vect^{\sym,\gp}_S$.
\end{rem}
We say that a presheaf $\sF\in \Pre(\Sch_S)$ is \emph{finitary} if it transforms limits of cofiltered diagrams of qcqs schemes with affine transition maps into colimits. We say that $\sF$ \emph{satisfies rigidity} for a henselian pair $(A,I)$ if the map $\sF(A)\to \sF(A/I)$ is an equivalence.
\begin{lem}[Bachmann]
\label{lem:rigidity}
Let $\sF\in\Pre(\Sch_S)$ be a finitary presheaf satisfying rigidity for all pairs $(A,I)$ where $A$ is an essentially smooth henselian local ring over $S$.
Suppose either that $S$ is locally of finite Krull dimension or that $\sF$ is truncated.
For $X\in\Sch_S$, denote by $\sF_X$ the restriction of $\sF$ to $\Sm_X$.
Then, for every morphism $f\colon Y\to X$ in $\Sch_S$, the canonical map
\[
f^*(\sF_X) \to \sF_Y
\]
in $\Pre(\Sm_Y)$ is a motivic equivalence.
\end{lem}
\begin{proof}
Note that the statement depends only on the Nisnevich sheafification of $\sF$, so we are free to assume that $\sF(\emptyset)=*$.
By 2-out-of-3, we can assume $X=S$. Since the question is local on $Y$ and $\sF$ is finitary, we can further assume that $Y$ is a closed subscheme of $\A^n_S$. Replacing $S$ by $\A^n_S$, we see that it suffices to prove the result for a closed immersion $i\colon Z\hook S$. Let $j\colon U\hook S$ be its open complement and consider the commutative square
\[
\begin{tikzcd}
j_\sharp\sF_U \ar{r} \ar{d} & \sF_S \ar{d} \\
U \ar{r} & i_*\sF_Z
\end{tikzcd}
\]
in $\Pre(\Sm_S)$. We claim that this square is a Nisnevich-local pushout square.
If $\sF$ is truncated, this is a square of truncated presheaves, so the claim can be checked on stalks; the same is true if $S$ is locally of finite dimension by \cite[Corollary 3.27]{ClausenMathew}. Using that $\sF$ is finitary, the stalks over an essentially smooth henselian local scheme $X$ are given by
\[
\begin{tikzcd}
\sF(X) \ar{r} \ar{d} & \sF(X) \ar{d} \\
* \ar{r} & \sF(\emptyset)
\end{tikzcd}
\quad\text{or}\quad
\begin{tikzcd}
\emptyset \ar{r} \ar{d} & \sF(X) \ar{d} \\
\emptyset \ar{r} & \sF(X_Z)\rlap,
\end{tikzcd}
\]
depending on whether $X_Z$ is empty or not. Since $\sF$ is rigid and satisfies $\sF(\emptyset)=*$, this proves the claim. On the other hand, by the Morel–Voevodsky localization theorem, the square
\[
\begin{tikzcd}
j_\sharp\sF_U \ar{r} \ar{d} & \sF_S \ar{d} \\
U \ar{r} & i_*i^*\sF_S
\end{tikzcd}
\]
is motivically cocartesian (see \cite[\S3 Theorem 2.21]{MV} or \cite[Corollary 5]{framed-loc}). It follows that the canonical map $i_*i^*\sF_S\to i_*\sF_Z$ is a motivic equivalence. Since $i_*$ commutes with $L_\mot$ for presheaves satisfying $\sF(\emptyset)=*$ and since $i_*\colon \H(Z)\to \H(S)$ is fully faithful, we deduce that $i^*\sF_S\to\sF_Z$ is a motivic equivalence, as desired.
\end{proof}
\section{Modules over hermitian K-theory}
We begin with a straightforward adaptation of Bachmann's cancellation theorem for finite flat correspondences \cite{BachmannFFlatCancellation} to Gorenstein and oriented Gorenstein correspondences.
Let $k$ be a perfect field and $\sC$ be a motivic $\infty$-category of correspondences over $k$, in the sense of \cite[Definition 2.1]{BachmannFFlatCancellation} (our primary interest is the example $\sC = \Span^{\fgor,\o}(\Sm_k)$).
We denote by $\h^\sC(X)\in \Pre_\Sigma(\sC)$ the presheaf represented by a smooth $k$-scheme $X$, by $\H^\sC(k)\subset \Pre_\Sigma(\sC)$ the full subcategory of $\A^1$-invariant Nisnevich sheaves, and by $\SH^\sC(k)$ the $\infty$-category of $\T$-spectra in $\H^\sC(k)_*$.
Recall that:
\begin{itemize}
\item $\sC$ \emph{satisfies cancellation} if the functor $\h^\sC(\bG_m,1)\otimes(-)\colon \H^\sC(k)^\gp\to \H^\sC(k)^\gp$ is fully faithful;
\item $\sC$ \emph{satisfies rational contractibility} if the presheaf $\h^\sC((\bG_m,1)^{\wedge n})^\gp$ on $\Sm_k$ is rationally contractible for all $n\geq 1$.
\end{itemize}
The following result is only new in the cases of Gorenstein and oriented Gorenstein correspondences, but the same proof works in all cases.
\begin{prop}\label{prop: cancellation + rat contract}
Let $k$ be a perfect field. Let $\sC$ be the $\infty$-category of smooth $k$-schemes and correspondences of one of the following types:
\begin{enumerate}
\item finite flat;
\item finite Gorenstein;
\item finite syntomic;
\item oriented finite Gorenstein;
\item oriented finite syntomic;
\item framed finite syntomic.
\end{enumerate}
Then $\sC$ satisfies cancellation and rational contractibility.
\end{prop}
\begin{proof}
The case of finite flat correspondences is \cite[Theorem 3.5 and Proposition 3.8]{BachmannFFlatCancellation}. Essentially the same argument applies to all the other cases.
For cancellation, we use the criterion \cite[Proposition 2.16]{BachmannFFlatCancellation} with $G=\h^\sC(\bG_m)$.
The object $\h^\sC(\bG_m,1)$ is symmetric in $\H^\sC(k)$, because $\H^\sC(k)$ is prestable \cite[Lemma 2.10]{BachmannFFlatCancellation} and $\Sigma(\bG_m,1)$ is already symmetric in $\H(k)_*$.
We then need to construct, for each $Y\in\Sm_k$, a map
\[
\rho\colon \Hom(\bG_m,L_\mot\h^\sC(\bG_m\times Y)^\gp) \to L_\mot\h^\sC(Y)^\gp.
\]
in $\H^\sC(k)^\gp$ satisfying some conditions.
For $m,n\geq 0$, we define
\begin{align*}
g_m^+\colon \bG_m\times\bG_m\to\A^1,& \quad g_m^+(t_1,t_2)=t_1^m+1,\\
g_m^-\colon \bG_m\times\bG_m\to\A^1,& \quad g_m^+(t_1,t_2)=t_1^m+t_2,\\
h_{m,n}^\pm\colon \A^1\times\bG_m\times\bG_m\to\A^1,&\quad h_{m,n}^\pm(s,t_1,t_2) = sg_n^\pm(t_1,t_2)+(1-s)g_m^\pm(t_1,t_2).
\end{align*}
(Thus, $h_{m,n}^\pm$ is the straight-line homotopy from $g_m^\pm$ to $g_n^\pm$.)
Given a span $\bG_m\times X\leftarrow Z\to \bG_m\times Y$ and $m,n\geq 0$, we let
$Z_m^{\pm}\subset Z$ and $Z_{m,n}^\pm\subset \A^1\times Z$ be the \emph{derived} vanishing loci of the functions
\begin{equation*}
Z\to\bG_m\times \bG_m \xrightarrow{g_m^\pm} \A^1\quad\text{and}\quad
\A^1\times Z\to \A^1\times\bG_m\times \bG_m \xrightarrow{h_{m,n}^\pm} \A^1.
\end{equation*}
The fibers of $Z_{m,n}^\pm$ over $0$ and $1$ in $\A^1$ are $Z_m^\pm$ and $Z_n^\pm$, respectively.
By \cite[Corollary 3.4]{BachmannFFlatCancellation}, if $Z$ is finite flat over $\bG_m\times X$, then $Z_{m,n}^\pm$ is finite flat over $\A^1\times X$ for $m,n$ large enough. For $i\geq 0$, let
\[
F_i^\sC(Y)\subset \Hom(\bG_m,\h^\sC(\bG_m\times Y))
\]
be the subpresheaf consisting of spans $\bG_m\times X\leftarrow Z\to \bG_m\times Y$ such that $Z_{m,n}^+$ and $Z_{m,n}^-$ are finite flat over $\A^1\times X$ for all $n,m\geq i$; this is an exhaustive filtration of $\Hom(\bG_m,\h^\sC(\bG_m\times X))$ in $\Pre_\Sigma(\sC)$.
Since $Z_{m,n}^\pm$ is cut out by a single equation in $\A^1\times Z$, the closed immersion $Z_{m,n}^\pm\hookrightarrow \A^1\times Z$ is quasi-smooth with trivialized conormal sheaf. On the other hand, the projection $\A^1\times\bG_m\times X\to \A^1\times X$ is smooth with trivialized cotangent sheaf. Hence, if $Z\to \bG_m\times X$ is
framed quasi-smooth, oriented quasi-smooth, quasi-smooth, or has trivialized or invertible dualizing sheaf, then the same holds for $Z_{m,n}^\pm\to \A^1\times X$ (this is essentially the only new observation needed compared to the finite flat case).
For $m,n\geq i$, we can therefore define
\[
\rho_{m}^\pm\colon F_i^\sC(Y)\to \h^\sC(Y)
\quad\text{and}\quad
\rho_{m,n}^\pm\colon F_i^\sC(Y)\to \h^\sC(Y)^{\A^1}
\]
by sending a $\sC$-correspondence $\bG_m\times X\leftarrow Z\to\bG_m\times Y$ to the $\sC$-correspondences
$X\leftarrow Z_{m}^\pm\to Y$ and
$\A^1\times X\leftarrow Z_{m,n}^\pm\to Y$, respectively. Note that $\rho_{m,n}^\pm$ is an $\A^1$-homotopy between $\rho_m^\pm$ and $\rho_n^\pm$. The morphisms
\[
\rho_i^\pm\colon F_i^\sC(Y)\to \h^\sC(Y)
\]
and the $\A^1$-homotopies $\rho_{i,i+1}^\pm$ induce in the colimit a pair of morphisms
\[
\rho^\pm\colon \Hom(\bG_m,\h^\sC(\bG_m\times Y))\to \Lhtp \h^\sC(Y).
\]
We let
\[
\rho =\rho^+-\rho^-\colon \Hom(\bG_m,\h^\sC(\bG_m\times Y))\to \Lhtp \h^\sC(Y)^\gp.
\]
By \cite[Proposition 2.8]{BachmannFFlatCancellation}, the canonical map
\[
\Hom(\bG_m,\h^\sC(\bG_m\times Y))^\gp \to \Hom(\bG_m,L_\mot\h^\sC(\bG_m\times Y)^\gp)
\]
is a motivic equivalence, so we obtain an induced morphism
\[
\rho\colon \Hom(\bG_m,L_\mot\h^\sC(\bG_m\times Y)^\gp)\to L_\mot \h^\sC(Y)^\gp.
\]
We now check that $\rho$ satisfies conditions (1)–(3) of \cite[Proposition 2.16]{BachmannFFlatCancellation}.
Conditions (1) and (2) follow from the corresponding facts about $\rho_{m,n}^\pm$, namely, the commutativity of the triangle
\[
\begin{tikzcd}
\h^\sC(U)\otimes F_i(Y) \otimes \h^\sC(\A^1) \ar{dr}{\id\otimes\rho_{m,n}^\pm} \ar{d} & \\
F_i(U\times Y)\otimes\h^\sC(\A^1) \ar{r}[swap]{\rho_{m,n}^\pm} & \h^\sC(U\times Y)
\end{tikzcd}
\]
and the naturality of $\rho_{m,n}^\pm$ in $Y\in\Sm_k$.
As in the proof of \cite[Theorem 3.5]{BachmannFFlatCancellation}, condition (3) reduces to the existence of an $\A^1$-homotopy
\[
\rho^+_2(\id_{\bG_m}) \simeq_{\A^1} \id + \rho^-_2(\id_{\bG_m})
\]
between endomorphisms of $\Spec k$ in $\sC$. Here, $\rho_2^+(\id_{\bG_m})$ and $\rho_2^-(\id_{\bG_m})$ are the framed finite syntomic $k$-schemes $\Spec k[t]/(t^2+1)$ and $\Spec k[t]/(t^2+t)$, respectively.
Let $H\subset \A^2=\Spec k[s,t]$ be the vanishing locus of $t^2+st+1-s$. Then $H$ is framed finite syntomic over $\A^1=\Spec k[s]$ and defines an $\A^1$-homotopy from $\rho_2^+(\id_{\bG_m})$ to $\id + \rho^-_2(\id_{\bG_m})$, as desired.
For rational contractibility, the proof of \cite[Proposition 3.8]{BachmannFFlatCancellation} applies with no significant changes. Indeed, one can replace $\h^\fflat$ by $\h^\sC$ in the statement and proof of \cite[Proposition 3.7]{BachmannFFlatCancellation}: it suffices to note that the morphism $s\colon \h^\fflat(X)\to \hat C_1\h^\fflat(X)$ constructed in \emph{loc.\ cit.}\ extends in an obvious way to a morphism $s\colon \h^\sC(X)\to \hat C_1\h^\sC(X)$.
\end{proof}
We can now prove the universality of hermitian K-theory as a generalized motivic cohomology theory with oriented finite Gorenstein transfers, in the following strong sense.
\begin{thm}\label{thm:kq-modules}
Let $k$ be a field of exponential characteristic $e \ne 2$. Then there is an equivalence of symmetric monoidal $\infty$-categories
\[\Mod_\kq \SH(k)[\tfrac{1}{e}] \simeq \SH^{\fgor,\o}(k)[\tfrac{1}{e}],\]
which is compatible with the forgetful functors to $\SH(k)$.
\end{thm}
\begin{proof}
The symmetric monoidal forgetful functor $\gamma\colon \Span^\fr(\Sm_k)\to\Span^{\fgor,\o}(\Sm_k)$ gives rise to an adjunction
\[
\gamma^*: \SH(k)\simeq\SH^\fr(k) \rightleftarrows \SH^{\fgor,\o}(k) : \gamma_*
\]
where the left adjoint $\gamma^*$ is symmetric monoidal. We claim that the right adjoint $\gamma_*$ sends the unit to the $\Einfty$-algebra $\mathrm{kq}$. To prove this, since $\gamma_*$ commutes with pro-smooth base change, we can assume that $k$ is perfect. Then the cancellation and rational contractibility properties of the category of oriented Gorenstein correspondences obtained in Proposition~\ref{prop: cancellation + rat contract}, together with \cite[Corollary~2.20]{BachmannFFlatCancellation}, imply that
\[ \gamma_* (\MonUnit) \simeq \Sigma^\infty_{\T, \fr} \FGor^\o_k. \]
As we showed in Theorem~\ref{thm:KQ-kq-FGor}, $\Sigma^\infty_{\T, \fr} \FGor^\o_k \simeq \kq$.
We therefore obtain an induced adjunction
\[
\Mod_{\mathrm{kq}}(\SH(k))\rightleftarrows \SH^{\fgor,\o}(k),
\]
which we claim is an equivalence after inverting $e$. Since the right adjoint is conservative, it suffices to show that the left adjoint is fully faithful. By construction, the unit of the adjunction is an equivalence on the unit object $\mathrm{kq}$, hence on any dualizable object. But $\SH(k)[\tfrac 1e]$ is generated under colimits by dualizable objects \cite[Theorem 3.2.1]{shperf}, so the claim follows.
\end{proof}
\section{Summary of framed models for motivic spectra}
\label{sec:framed-models}
In this final section, we offer a summary of the known geometric models for the most common motivic spectra. For simplicity, we first state the results in the regular equicharacteristic case; for the state of the art see Remark~\ref{rem:general base}.
\begin{thm}\label{thm:all mot spectra}
Let $S$ be a regular scheme over a field. Then the forgetful maps of $\Einfty$-semirings
\[\begin{tikzcd}
& \FSyn \ar{r} & \FFlat \ar{r} & \Vect \ar{r} & \uZ \\
\FSyn^\fr \ar{r} & \FSyn^\o \ar{u} \ar{r} & \FGor^\o \ar{r} \ar{u}& \Vect^\sym \ar{u}\ar{r} & \uGW\ar{u}
\end{tikzcd}\]
induce, upon taking framed suspension spectra, canonical maps of motivic $\Einfty$-ring spectra over $S$ (assuming $2\in\sO(S)^\times$ for $\kq$):
\[\begin{tikzcd}
& \MGL \ar{r} & \kgl \ar["\simeq"]{r} & \kgl \ar{r} & \hz \\
\MonUnit \ar{r} & \MSL \ar{u} \ar{r} & \kq \ar["\simeq"]{r} \ar{u}& \kq \ar{u}\ar{r} & \hzmw \ar{u} \rlap .
\end{tikzcd}\]
\end{thm}
\begin{proof}
The functor $\Sigma^\infty_{\T,\fr}$ is symmetric monoidal, so it takes the unit $\FSyn^\fr$ in presheaves with framed transfers to the unit $\MonUnit$ in motivic spectra.
The leftmost vertical arrow is a consequence of \cite[Theorems~3.4.1 and~3.4.3]{deloop3}. The top row is contained in \cite[Corollary~5.2, Theorem~5.4]{robbery} and \cite[Theorem~21]{framed-loc}, while the bottom row follows from Proposition~\ref{prop:KQ-kq-Vect^bil}, Theorem~\ref{thm:KQ-kq-FGor}, and Theorem~\ref{thm:HtildeZ}.
\end{proof}
\begin{rem}\label{rem:general base}
The statements about $\MonUnit$, $\MSL$, $\MGL$, and $\hz$ in Theorem~\ref{thm:all mot spectra} are in fact proved over general base schemes in the given references.
The statement about $\hzmw$ holds under the assumption of Theorem~\ref{thm:HtildeZ}, and the statements about $\kgl$ and $\kq$ were recently extended by Bachmann to the same generality \cite{BachmannKGL}.
\end{rem}
\begin{cor}
On the level of infinite $\P^1$-loop spaces, the diagram of motivic spectra in Theorem~\ref{thm:all mot spectra} is the motivic localization of the following diagram of forgetful maps (up to the Nisnevich-local plus construction for the left half in characteristic zero):
\[\begin{tikzcd}
&\Z\times \Hilb_\infty^\lci(\A^\infty) \ar{r} & \Z\times \Hilb_\infty(\A^\infty) \ar{r} & \uZ \\
\Z\times \Hilb_\infty^\fr(\A^\infty) \ar{r} & \Z\times \Hilb_\infty^{\lci,\o}(\A^\infty) \ar{u} \ar{r} & \Z\times \Hilb_\infty^{\Gor,\o}(\A^\infty) \ar{r} \ar{u} & \uGW\ar{u}\rlap.
\end{tikzcd}\]
\end{cor}
\begin{proof}
This follows from Theorem~\ref{thm:all mot spectra} using \cite[Theorems 1.2 and~1.5]{deloop4}, \cite[Corollary~4.5]{robbery}, and Corollary~\ref{cor:Hilb-FGor}.
\end{proof}
Let us conclude with some comments on the ``canonical maps'' in Theorem~\ref{thm:all mot spectra}.
First, we note that the $\Einfty$-maps to $\hzmw$ and $\hz$ in the diagram are all \emph{unique}. For $\hzmw$, this is because $\hzmw=\underline\pi^\eff_0(\1)$ and the unit maps of all the spectra in the lower row induce equivalences on $\underline\pi^\eff_0$ (cf.\ \cite[Corollary~3.6.7]{MuraMSL}). Similarly, we have $\hz=s_0(\1)$ \cite[Theorem 10.5.1]{Levine:2008} and the unit maps of all spectra except $\hzmw$ induce equivalences on $s_0$ (see \cite[Remark 10.2]{SpitzweckHZ}, \cite[Example 16.35]{norms}, and \cite[Theorem 3.2]{kq-slices}). The $\Einfty$-map $\hzmw\to \hz$ is also unique since $\hz$ is in the heart of the effective homotopy $t$-structure.
The $\Einfty$-map $\MGL\to\kgl$ in Theorem~\ref{thm:all mot spectra} was shown in \cite[Proposition~6.2]{robbery} to induce the usual Thom classes in algebraic K-theory, i.e., it is an $\Einfty$ refinement of the usual orientation map obtained from the universal property of $\MGL$ as a homotopy commutative ring spectrum \cite[Theorem 2.7]{Panin:2008}. A similar argument will show that the $\Einfty$-map $\MSL\to\kq$ induces the usual Thom classes of oriented vector bundles in Grothendieck–Witt theory, once one promotes the forgetful map $\FQSm^{\o,4*}\to \Perf^\sym$ to a morphism of framed $\Einfty$-semirings (which requires some work but should present no essential difficulty). By \cite[Theorem 5.9]{PaninWalter}, it is known that there exists at least one unital morphism $\MSL\to\kq$ inducing the usual Thom classes in Grothendieck–Witt theory, but its uniqueness and multiplicativity properties are unclear.
To the best of our knowledge, the existence of $\Einfty$-maps $\MGL\to \kgl$ and $\MSL\to\kq$ was not known before the main result of \cite{deloop3}, which describes $\MGL$ and $\MSL$ as framed suspension spectra.
\let\mathbb=\mathbf
{\small
\newcommand{\etalchar}[1]{$^{#1}$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,945 |
Kratos, Amaterasu, Asura... How do game gods compare to their real inspirations?
By Samuel James Riley 2015-08-14T13:41:23.225Z Feature
Gaming God
It's impossible to deny the epic scale of religious histories. That may help to explain why so many video games take such inspiration in their theologies (here's looking at you multiple Jesus-clone messiah protagonists). In essence, they're a ready-made source of rich and detailed adventures with profound resonance, their enduring popularity and name-recognition long-since assured. What publisher wouldn't want that?
Of course, the other side of the equation looks a little less rosy. The odds of offending large swathes of your potential player base only increase when tapping into existing religions, meaning most developers using them play around with their presentation to a greater or lesser degree. The following seven games all opted to take that risk, representing complex belief systems via the interactive medium. Some edged closer to the sacred scripture than others, but all shall be judged of their worthiness. Do any of these games offer a worthwhile introduction to the great gamut of the gods? Let's find out.
Amazon Prime Day deals start Monday: find out more at TechRadar.
Too Human (Norse)
Developer Silicon Knights envisaged Too Human as an attempt to rationalise complex Norse theology via futuristic technology. Here the gods of the pantheon are not born, but made, granted their astounding powers through advanced cybernetic augmentation. In place of the nine realms of existence, Too Human supposes a singular planet, broken up into numerous distinct sectors, such as the human city of Midgard. Their implacable opposition - the giant folk of Jtnar, are now sentient machines, while the likes of Beowulf's Grendel and the menacing Dark Elves become simple boss and enemy types.
As for the game's hero, Baldur - god of light, joy, purity and presumably also lollypops - Too Human opts to introduce us to a much gruffer deity. Expanding upon the most famous of all Baldur stories - his untimely death (as orchestrated by the mischievous half-god Loki), Too Human takes both this and other existing legends as the basis for an entirely new saga, one based upon rebirth and ultimately revenge. As an introduction to Norse theology, Too Human represents a strong and surprisingly accurate start, familiarising players with the names, relationships and dwelling places of these Germanic-Scandinavian gods. If you can ignore the sci-fi trimmings, shoddy gameplay and freshly expanded lore, then Too Human is a good a way as any to begin your ongoing course in Norse.
See Also - Age of Mythology, Jotun (2016), Viking: Battle for Asgard
God of War (Ancient Greek)
Strange as it may sound, the single most inaccurate element of the entire God of War franchise is the game's ongoing depiction of central character Kratos. Indeed, far from being the frenzied father-stabber that he appears in the games, the theological Kratos is instead counted among the most zealously loyal of Zeus' retinue. He's also considered the God of Strength, as opposed to the demi-god and later God of War he's portrayed as in the games. That being the case, it seems highly likely that Sony Santa Monica chose the name Kratos as an ironic nod to the original character, inverting not only his loyalties but also on occasion his actions. Case in point, the scene in which a virtual Kratos frees the heroic titan Prometheus, having done just the opposite according to the original Greek narrative.
Like many of the games on this list, the God of War franchise positively excels in taking existing theological legends and using them to create new and exciting adventures. One such example is the renewed Titan vs. God war, the big round two of an event that dominates much of the ancient theology. While Kratos' actions may not bear much relation to the stories upon which his world is based, God of War remains a fine foreword to the rest of ancient Greek mythology. After all, what better way to become accustomed to the characters, places and epic tales than by cutting a bloody swathe right through them?
See Also - Age of Mythology, Disney's Hercules, NyxQuest: Kindred Spirits, Spartan: Total Warrior, Titan Quest
Okami (Shinto)
Despite some rather obvious parallels Okami's exact relationship to the Shinto theology remains somewhat difficult to define. That's because the team at Clover Studios has always tended towards a vague and imprecise language when discussing that facet of the project. Game director Hideki Kamiya argues that Okami's characters, in particular Amaterasu, are not necessarily the same as those found in the Shinto faith, though they do share a great deal in common. It's possible that this tack was taken simply to avoid offending believers, though it could also indicate an intentional level of ambiguity.
Familiarities (and differences) include the aforementioned Amaterasu, goddess of the sun in both Shinto tradition and in Okami. Interestingly however, no mention of wolves or wolf-form is ever made in reference to the Shinto deity, while the virtual version is neither strictly male nor female, at least not in the Japanese version of the game (translation difficulties forced the team to pick an English pronoun i.e. he or she, thereby confusing the issue). As for the game's celestial brush gods, none appear to correspond directly with Shinto deities, but are instead based upon the Far Eastern Zodiac signs. Susano meanwhile appears somewhat similar to the Shinto hero Susano-o - both of whom are famous for slaying (or helping to slay) an 8-headed serpent named Orochi. As a guide to Shinto deities, Okami is hardly authoritative, not would the game's designers have you believe that it is. All told, those wishing to learn more about Shinto via the game should probably support their playthrough with the odd spot of research.
See Also - Okamiden
Asura's Wrath (Hinduism/ Buddhism)
Basing a game on the tenets of an active religion is always going to prove tricky. Basing a game on two even more so. Asura's Wrath manages to monkey around with details of both the Hindu and Buddhist faiths (the two groups share a number of common legends, including those drawn from the Indian epic The Rigveda). That's not to say that either side became too irate about it, most just balking at the perceived inaccuracies. So what exactly did the game get right, and more importantly, what did it get wrong? Well for starters, the game - much like Too Human - opts to take the sci-fi reinvention route, trading in powerful demi-gods for massively upgraded cyber soldiers.
Speaking of demi-gods, the term Asura actually applies to all beings of this type, not just to any one individual. The game also leaves out the more benevolent set of deities known as the Devas, although the game's final boss Chakravartin may be considered as such. While religious tradition holds that the two sets of gods waged war upon each other, Asura's Wrath finds these so-called 'Guardian Generals' (i.e. the Asura) battling it out against the Gohma, a race of hideous monsters led by Vlitra, a planet-devouring serpent similar to Vritra, an evil demigod from Hindu Vedic tradition. The Gohma, meanwhile, appear to have little to no basis in religious texts. All things considered the game does present a solid introduction to the Asura deities, not individually perhaps but rather as a group, nailing their look, temperament and abilities as recounted in various religious texts. The rest, however remains somewhat less authentic.
Sphinx & the Cursed Mummy (Ancient Egyptian)
You might expect that there'd be a litany of titles covering the ancient Egyptian pantheon, and you'd be right, though precious few of those exist outside of the god game and RTS genres. Consider Sphinx and the Cursed Mummy a rare exception to that rule, dealing as it does with the likes of Ra, Set, Osiris and co. through the format of an action-platformer. The game stars Sphinx, an upright, decidedly more human take on the beastly man-lion hybrids that guard many an Egyptian temple. Truth be told, this incarnation is little more than a teenager with a tail, though other ancient icons ring truer. Horus has his falcon head, Anubis is a jackal and Tutankhamen is covered from head to tow in bandages. Likewise, King Tut's mummification also results in the storage of four sacred organs, just as it would have in ancient days.
Interestingly however, the game also chooses to shift around several well-established relationships. Tutankhamen's mother and father become his lover and brother, respectively, head honcho Ra is created via the union of Osiris and Set - as opposed to being his own independent deity - and Osiris' son Horus is now working for Set, instead of actively trying to murder him. These elements aside, Sphinx and the Cursed Mummy does an entertaining job of familiarising fans with the unique iconography - think ankhs, amulets and sarcophagi - as well as many of the major players of the ancient Egyptian tradition.
See Also - Age of Mythology, Pharaoh, Ankh: Battle of the Gods
Dante's Inferno (Christian)
Alright, first things first - Dante Alighieri's Inferno isn't exactly considered canon by the Christian church. However the epic poem does make use of, and even helped to inform, certain elements of popular Christian theology. Suffice to say, this blood and guts actioner shouldn't be taken as a literal guide to Christian tradition, but rather as the retelling of an influential Christian myth. So, how does it fare? Well, much like the gameplay itself, largely hit and miss. Alighieri's epic poem stars none other than the author himself, indulging in a brisk walking tour of the nine levels of Hell. Digital Dante, meanwhile finds himself recast as a fighting knight of the third Crusade, wading through the underworld so as to free his beleaguered lover from Satan's icy grip.
The game does get a lot right, including the presence of Roman poet Virgil, who acts as guide to both iterations of Dante. Likewise, many of the levels of Hell, and the punishments performed therein, reflect those originally envisaged by the writer. Other translations prove less authentic, with the knife-wielding babies of the purgatory level proving particularly false. So too the use of Cleopatra as a willing servant of Satan, as opposed to the simple prisoner she appears in the poem. Strangest of all however, is the inclusion of an absolve/punish mechanic for beaten enemies. Somehow it seems difficult to believe that the Church would sign off on an admitted sinner forgiving other offenders their sins. All in all then, a decent visual trip through quasi-Christian tradition, chock full of crosses, demons and holy iconography. Just take it with a massive pinch of salt.
See Also - Super Noah's Ark 3D?
Folklore (Celtic)
A surprisingly popular tradition, at least where video game settings are concerned, the world of Celtic theology entirely informs the 2007 RPG Folklore. The title takes place in a small country village that just so happens to act as a functioning gateway to a bizarre realm of the dead. In keeping with Celtic tradition, this fissure only appears once every year - on the night of Samhain, to be precise, a major inspiration for modern Hallowe'en - leading the spirits of the dead to revisit our world, as adventurous peoples head off in the other direction. Folklore's depiction of this netherworld is largely in keeping with that of the ancient Celts, drawn an underground paradise - known alternatively as Mag Mell or Tr na ng - made up of strange and fantastical creatures.
These inhabitants, or 'folk' will occasionally attempt to kill the player (somewhat less authentic, though as the game explains, other adventurers have previously 'broken' the paradise) and are largely modelled after fabled Celtic creatures. Taken as a whole, the game provides a strong overview of ancient Celtic theology, though crucially it does gloss over the role of the actual gods, including Balor, Crom and Morrigan. As such, it might better be described as a game of mythology, or as the title itself states, folklore.
See Also - Bloodforge (much greater focus on the Gods - terrible, terrible game), Hellblade (2016), Sorcery
Too Human | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,604 |
An Ant Called Amy
Cruinniú na nÓg 2021
As part of Cruinniú na nÓg 2021, Roscommon Arts Centre presented an online performance of AN ANT CALLED AMY, created and performed by Julie Sharkey on June 12th & 13th.
This new play for children was first conceived by actor Julie during her time as "Theatre Artist in Residence" at Roscommon Arts Centre (2017/2018), during which a work in progress performance took place as part of the Bookworms Festival.
Since then, and with the ongoing support of Roscommon Arts Centre, The Arts Council and mentors Martin Drury and Mikel Murfi, Julie continued to develop the piece and under the direction of Raymond Keane and with the support of the Creative Ireland Roscommon programme, "An Ant Called Amy" made her debut as part of our 2021 Crinniu na nÓg programme.
AN ANT CALLED AMY is a very special story about an ant called Amy (an award winning ant), her brother Andy and a Brown Spider. A story about a busy little ant who learns to slow down. Suitable for Ages 5- 8
Cruinniú na nÓg is Ireland's national day of free creativity for children and young people under 18. It is a flagship initiative of the Creative Ireland Programme's Creative Youth plan to enable the creative potential of children and young people.
Theatre Early Years Family
Little Folk on the Road
Kyle Riley
Saturday 28th January | 11am | €10 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,973 |
\section{Introduction}
\label{sec:introduction}
Commercial buildings account for about one-third of global energy consumption, with HVAC (Heating Ventilating and Air-Conditioning) units being the major contributor. An HVAC typically comprises of few Air Handling Units (AHUs), which heat or cool the air to a specified setpoint temperature, and Variable Air Volume (VAV) units that control the volume of air flowing into each thermal zone. In most commercial buildings, HVAC maintains a desired \emph{set point} temperature during working hours (9 AM to 6 PM) and a \emph{set back} temperature during non-working hours. Unfortunately, given the stochastic nature of building occupancy, a static schedule either leads to energy wastage or occupant discomfort~\cite{dawson2013boss,erickson2013poem}.
HVAC energy optimization is an active area of research. In the literature, studies have proposed several control strategies which broadly fall into two categories: \emph{reactive}~\cite{jain2016non, gyalistras2010use} and \emph{predictive}~\cite{jain2017portable+, oldewurtel2010energy}. In a reactive controller, AHUs and VAVs respond to measured occupancy in a zone. Here, occupancy is measured using motion, $CO_2$ sensors, or by monitoring building's WiFi infrastructure~\cite{trivedi2017ischedule}. Since buildings typically take some time to respond to control inputs, better performance can be obtained using predictive control strategy where the controller selects the optimal trajectory of set points for a finite time horizon~\cite{jain2018energy}. Of the predictive control techniques, perhaps the best-known approach is Model Predictive Control (MPC)~\cite{garcia1989model}.
In a typical MPC, a known building thermal model estimates the future system state using forecasts of model inputs, such as building occupancy and outside air temperature. However, the effectiveness of this approach depends on the accuracy of the predictions. As prediction accuracy deteriorates, MPC performance - in terms of occupant comfort and building energy use - degrades and may get even worse than conventional techniques. In recent work, Oldewurtel et al.~\cite{oldewurtel2011stochastic} extensively studied the influence of errors in weather forecast on HVAC energy consumption and occupants' comfort and quantified the impact of mis-predictions. However, the work neither addressed errors in occupancy prediction nor studied the ways to mitigate the influence of prediction errors.
In this paper, we address this gap. We study the influence of occupancy errors on MPC performance using a custom-built building simulator. We also model and analyze the impact of personal environmental control system (PEC) in the presence of prediction errors. A PEC could be an off-the-shelf desktop fan or a heater to provide individual thermal comfort~\cite{brager2015evolving}. We find that PEC when used with model predictive control, can reduce both - the variability in energy consumption and the occupants' discomfort.
Our contributions are as follows:
\begin{enumerate}
\item We present the design and development of a building thermal simulator that models conventional schedule-based, reactive occupancy-based, and predictive MPC-based HVAC controllers.
\item We extend the MPC-based control strategy proposed by Kalaimani et al.~\cite{Rac17} and allow PEC to react between any two consecutive states of the system.
\item We quantify the impact of occupancy prediction errors on two MPC-based control strategies - with and without PEC. For analysis, we use occupancy data from forty-five volunteers over three months and simulations of a test building in both heating and cooling seasons.
\item It is important that occupancy forecast errors are realistic; thus, we propose a method to systematically introduce realistic occupancy errors into MPC predictions using real-world occupancy data.
\end{enumerate}
The rest of the paper is organized as follows. Section~\ref{sec:related} discusses the literature and studies conducted in the past. In Section~\ref{sec:background}, we outline the control strategies studied in the paper. In Section~\ref{sec:architecture}, we present the detailed architecture and design of the thermal simulator followed by detailed analysis in Section~\ref{sec:evaluation}. In Section~\ref{sec:discussions}, we discuss several limitations and possible future directions of the study and conclude the paper.
\section{Related Work}
\label{sec:related}
\subsection{Central HVAC Controllers}
In the past, researchers have extensively studied the optimization of HVAC controllers to minimize the aggregate energy consumption and maximize user comfort~\cite{jain2016data, jain2014pacman}. Agarwal et al.~\cite{agarwal2011duty} studied aggressive duty cycling of HVAC based on occupancy patterns within the building. Lu et al.~\cite{lu2010smart} proposed a smart thermostat to automate HVAC control by sensing occupancy and sleeping patterns in residential buildings. The occupancy-based control allows buildings to operate outside of comfort regimes when unoccupied, thus reducing energy usage~\cite{erickson2011observe}. Henceforth, several other studies also explored the use of occupancy information to optimize the HVAC energy operations~\cite{erickson2013poem,balaji2013sentinel,erickson2011observe,Nest,aswani2012reducing,iyengar2015iprogram,kleiminger2014smart,koehler2013therml,scott2011preheat,yang2012living}. However, centralized HVAC controllers divide a building into thermal zones comprising of private and shared spaces. Within each zone, these control strategies maintain ASHRAE standard while assuming each zone as either occupied or unoccupied; thus, ignoring individual comfort requirements.
\subsection{Personal Environmental Control}
For personalized comfort, studies proposed to use personal environmental control systems (PECs), especially in shared spaces~\cite{brager2015evolving,bauman1998field,zhang2010comfort,gao2013spot,gao2013optimal,rabbani2016spot}. Unlike conventional centrally-controlled HVAC system, where people share the same set point temperature~\cite{dear2013progress, jain2017decision}, PEC systems can meet the comfort requirements of all occupants, albeit at the cost of additional energy expenditure. Kalaimani et al.~\cite{Rac17} merged PEC with model predictive control to further minimize the HVAC energy consumption and maximize the user comfort.
Though advanced predictive control strategies (such as MPC) have the potential to optimize HVAC operations significantly, none of the studies mentioned above quantify the influence of the prediction errors on the energy consumption of HVAC and on the occupants' comfort.
\subsection{Error Analysis}
Oldewurtel et al.~\cite{oldewurtel2011stochastic} studied the influence of errors in weather forecast on MPC-controlled HVAC operations, and their results indicate that the quality of weather predictions highly correlates with the performance of the model predictive controller. However, the study only focused on prediction errors in the weather forecast and the evaluation was limited to ``pure" MPC-based HVAC controller. Given that occupancy prediction is also an input to MPC, it is essential to analyze the influence of occupancy prediction errors on HVAC operations. Besides, the study~\cite{oldewurtel2011stochastic} is limited to HVAC and does not incorporate the impact of PECs in satisfying the comfort requirements of occupants.
In this paper, we extend the work in~\cite{oldewurtel2011stochastic} and in \cite{Rac17} by first analyzing the effect of prediction errors in occupancy and later exploring the benefits of PECs in mitigating (or minimizing) the influence of prediction errors on HVAC operations. Our study indicates that predictive control strategies make HVAC operations highly unreliable. High variability has discouraged building managers to use advanced HVAC control strategies, and thus, they have continued using conventional HVAC controllers.
\section{HVAC Control Strategies}
\label{sec:background}
In a typical commercial building, spaces are either private (such as offices) or shared (such as cafeteria, corridor), and a set of private and shared spaces constitutes a zone. Within each zone, there exists a VAV unit that takes air from AHU at a particular temperature~($u(t)$) and supplies it across the rooms at a specific rate~($v^{ij}{(t)}$) to maintain the room temperature close to the set point temperature. Here, $j$ indicates the room number in the $i^{th}$ zone of the building. To ensure a consistent supply of fresh air, AHU recirculates only a limited amount of used air~($r{(t)}$) and ejects the remaining air in the open environment.
Defined in Table~\ref{table:hvac_cvars}, $u(t)$, $v^{ij}{(t)}$, and $r{(t)}$ are key HVAC control parameters and their values are typically decided by the control strategy. In this section, we discuss the four control strategies, implemented to analyze the influence of occupancy prediction errors on HVAC operations. The first two methods are non-predictive, and building managers widely use these strategies in commercial buildings today; when employed, the HVAC operations are independent of prediction errors. The last two are MPC-based control strategies. In the paper, we use non-predictive control strategies as the baseline strategies for predictive control strategies when occupancy prediction is not perfect.
\subsection{Schedule-based control}
In a schedule-based control of HVAC, the building manager starts the HVAC at a fixed time in the morning and shuts it down in the evening (typically $9~AM$ to $6~PM$). On any day, AHU supplies air at a static temperature which is chosen based on the season (summer/winter), and the set point temperature does not vary within a day. Based on ASHRAE standards\footnote{American Society of Heating, Refrigeration and Air-Conditioning - a global organisation that publishes standards and guidelines related to HVAC.}, we set the supply air temperature ($u^{(t)}$) to $15^{\circ}$C for summers and $20^{\circ}$C for winters. For both seasons, the ratio of reuse air ($r$) and rate of flow of supply air ($v^{(t)}$) is constant at $0.8$ and $0.236~m^3/s$, respectively. The approach is naive but widely used by building managers in commercial buildings.
\subsection{Reactive control}
In reactive control strategies, VAV cools or heats the space only if people are present in the corresponding VAV zone. In the past, studies have suggested several direct and indirect HVAC control strategies to estimate occupancy; we use the occupancy data and implement the strategy proposed by Ardakanian et al.~\cite{ardakanian2016non} for benchmarking.
\begin{table}[h!]
\centering
\caption{List of HVAC control variables}
\ra{1.5}
\begin{tabular}{@{}lm{6.5cm}rl@{}}\toprule[0.3ex]
\textbf{Symbol} & \textbf{Description} & \textbf{Unit} \\
\hline
$u{(t)}$ & Supply air temperature at time $t$ & $^\circ$C \\
$v^{ij}{(t)}$ & Rate of flow of supply air in room $j$ of a VAV zone $i$ at time $t$ & $m^3/s$ \\
\bottomrule[0.3ex]
\end{tabular}%
\label{table:hvac_cvars}
\end{table}
\subsection{Model Predictive Control}
Model predictive control (MPC) is a recent approach for HVAC where controller can compute the room temperature over a finite time horizon~\cite{mpcsurvey}. Typically, a thermal model using occupancy estimates and weather forecast determines the future system state over a time horizon. In our implementation of MPC, we used Equation~\ref{eq:tm_hvac} as the thermal model that considers the influence of HVAC, atmospheric temperature, heating load by the occupants, and other heating or cooling loads present in the room~\cite{riederer2002room}. Table~\ref{table:param_desc_mpc} lists all symbols of the thermal model and their default values.
\begin{equation}
\begin{split}
\frac{T^{ij}{(t+1)} - T^{ij}{(t)}}{\tau} \times C^{ij} &= \frac{\rho\sigma}{n_r^i} \times v^{ij}{(t)} \times (u{(t)} - T^{ij}{(t)}) \\ & + \alpha_{ex}^{ij} \times (T_{ex}{(t)} - T^{ij}{(t)}) \\ &+ (Q_{oc}^{ij} + Q_{ap}^{ij}) \times O^{ij}{(t)}
\end{split}
\label{eq:tm_hvac}
\end{equation}
For the time horizon, the controller computes $u{(t)}$, $v^{ij}{(t)}$, and $r{(t)}$, by solving an optimization problem using the current state of the system, with an objective to minimize the total energy consumption (Equation~\ref{eq:min_power_hvac}).
\begin{equation}
\begin{split}
Po{(t)} &= V{(t)} \times \eta_{h} \times (u{(t)} - T_{cu}{(t)}) + V{(t)} \times V{(t)} \times \eta_{\mathit{f}}\\ &+ V{(t)} \times \eta_{c} \times (T_{mx}{(t)} - T_{cu}{(t)})
\end{split}
\label{eq:min_power_hvac}
\end{equation}
where,
\begin{equation}
\begin{split}
V{(t)} = \sum_{i=1}^{n_z}\sum_{j=1}^{n_r^i}{v^{ij}{(t)}}
\end{split}
\label{eq:vt}
\end{equation}
$V{(t)}$ depicts the total air supplied across all the rooms within a building, $\eta_h$ and $\eta_c$ indicate the efficiency of the heating and cooling unit, respectively. $T_{cu}(t)$ and $T_{mx}(t)$ denote the temperature of air coming from the cooling and mixing unit, respectively. $\eta_f$ is the efficiency of the VAV fan which is supplying air to the room. Details about the power consumed by the supply fan can be found in Rabbani et al.~\cite{rabbani2016spot}.
The optimization problem constraints the comfort index to remain within the specified bounds to ensure user comfort. In this study, we use widely used metric PMV - Predicted Mean Vote, to measure user comfort~\cite{fanger1973assessment}. Other constraints include time-scale limitations, thermal dynamics (Equations \ref{eq:tm_hvac}), and constraints dictated by the system setup (such as thermal comfort and HVAC operation should remain within a desired range). In this paper, we implemented MPC with two time-scales where the controller updates the supply air temperature every hour and the supply air volume every 10 minutes. The above time-scales are typically determined by the physical limitation of an HVAC unit. For more details about this
specific formulation of the optimization problem,
please refer to Kalaimani et al.~\cite{kalkesros16}.
\begin{table}[ht!]
\caption{List of symbols used in the thermal model}
\ra{1.5}
\begin{tabular}{@{}lm{6cm}rl@{}}\toprule[0.3ex]
\textbf{Symbol} & \textbf{Description} & \textbf{Default} & \textbf{Unit} \\
\hline
$\rho$ & Density of air & $1.204$ & $kg/m^3$ \\
$\sigma$ & Specific heat of air & $1.003$ & $kJ/(kg.K)$ \\
$\tau$ & Sampling interval & $-$ & $s$ \\
$n_z$ & Total number of VAV zones in a building & $-$ & $-$ \\
$n_r^i$ & Total number of rooms in VAV zone $i$ & $-$ & $-$ \\
$O^{ij}{(t)}$ & Occupancy in room $j$ of zone $i$ at time $t$ & $-$ & $-$ \\
$T^{ij}{(t)}$ & Temperature in room $j$ of zone $i$ at time $t$ due to HVAC & $-$ & $^\circ C$ \\
$T_{ex}{(t)}$ & External temperature at time $t$
& $-$ & $^\circ C$ \\
$C^{ij}$ & Thermal capacity of room $j$ in zone $i$
& $2000$ & $kJ/K$ \\
$\alpha_{ex}^{ij}$ & Heat transfer coefficient between outside and room $j$ in zone $i$ & $0.048$ & $kJ/(K.s)$ \\
$Q_{ap}^{ij}$ & Heat load due to heating/cooling equipments in room $j$ of zone $i$ & $0.1$ & $kW$ \\
$Q_{oc}^{ij}$ & Heat load due to occupant in room $j$ of zone $i$
& $0.1$ & $kW$ \\
\bottomrule[0.3ex]
\end{tabular}%
\label{table:param_desc_mpc}
\end{table}
\subsection{MPC with Personal Environment Controller}
Recently, Kalaimani et al.~\cite{Rac17} proposed a hybrid HVAC controller and combined MPC with a personal environmental control system. In the study, \cite{Rac17} used SPOT - an off-the-shelf desktop fan/heater with local temperature sensing and a computer-controlled actuator to provide individual thermal comfort. Assuming perfect prediction of occupancy and outside temperature, the study shows that combining MPC with SPOT is effective in reducing the total energy consumption by choosing appropriate thermal setbacks during the intervals of sparse occupancy.
At the time of partial occupancy, HVAC runs at a base temperature which is slightly higher (in summers) or lower (in winters) than the desired temperature. Equation~\ref{eq:tm_hvac} depicts the base temperature (due to HVAC) which depends on HVAC, external weather conditions, and occupants within the space.
\begin{equation}
\begin{split}
\frac{T_{hv}^{ij}{(t+1)} - T_{hv}^{ij}{(t)}}{\tau} \times C^{ij} &= \frac{\rho\sigma}{n_r^i} \times v^{ij}{(t)} \times (u^{ij}{(t)} - T_{hv}^{ij}{(t)}) \\ & + \alpha_{ex}^{ij} \times (T_{ex}{(t)} - T_{hv}^{ij}{(t)}) + Q_{ap}^{ij} \times O^{ij}{(t)}
\end{split}
\label{eq:tm_thvac}
\end{equation}
In the proposed approach, we assume that room is divided into two regions:
\textit{occupied }- the part of the room where the occupant is present; and
\textit{unoccupied} - the other part of the room.
In the occupied region, SPOT provides the offset comfort to attain the comfort requirements of the occupant. In Equation~{\ref{eq:tm_thvac}}, we show how the room temperature changes when taking into account the impact
of HVAC, heat exchange with the outside, external weather conditions, and other heating/cooling loads present in the room. In Equation~\ref{eq:tm_delta}, we then calculate the change in temperature due to SPOT, occupant, and other heat exchanging load present in the room, followed by temperature in the occupied part of the room in Equation~\ref{eq:tm_toc}. On the other hand, in the unoccupied portion (Equation~\ref{eq:tm_tun}), both SPOT and the occupants \textit{indirectly} influence the room temperature due to thermal coupling between the two zones, modeled by the heat transfer coefficient $\alpha_{in}$.Table~\ref{table:param_desc_spot} lists the new notations used in the extended model.
\begin{equation}
\begin{split}
\frac{\Delta_{oc}^{ij}{(t+1)} - \Delta_{oc}^{ij}{(t)}}{\tau} \times C_{oc}^{ij} &= Q_{oc}^{ij} \times O^{ij}{(t)} + Q_{he}^{ij} \times S_{he}^{ij}{(t)}\\& - \alpha_{in}^{ij} \times \Delta_{oc}^{ij}{(t)}
\end{split}
\label{eq:tm_delta}
\end{equation}
\begin{equation}
T_{oc}^{ij}{(t+1)} = T_{hv}^{ij}{(t+1)} + \Delta_{oc}^{ij}{(t+1)}
\label{eq:tm_toc}
\end{equation}
\begin{equation}
T_{un}^{ij}{(t+1)} = T_{hv}^{ij}{(t+1)} + \frac{\tau \times \alpha_{in}}{C^{ij} - C_{oc}^{ij}} \times \Delta_{oc}^{ij}{(t)}
\label{eq:tm_tun}
\end{equation}
The revised objective function has additional parameters $S_{f}$ for fan and $S_{he}$ for heater (Equation~\ref{eq:min_power_spot}). A fan consumes negligible power, thus
the objective function only considers the power consumption of SPOT's heater.
\begin{equation}
\begin{split}
Po{(t)} &= V{(t)} \times \eta_{h} \times (u{(t)} - T_{cu}{(t)}) + V{(t)} \times V{(t)} \times \eta_{\mathit{f}}\\ &+ V{(t)} \times \eta_{c} \times (T_{mx}{(t)} - T_{cu}{(t)}) + \sum_{i=1}^{n_z}\sum_{j=1}^{n_r^i}{S_{he}^{ij}{(t)}}
\end{split}
\label{eq:min_power_spot}
\end{equation}
The controller determines HVAC control parameters (Table~\ref{table:hvac_cvars}) on a 10-minute timescale and in between, fan/heater (of SPOT) reacts to occupancy every 30 seconds. By doing so, SPOT assists the controller in regulating the discomfort that might arise due to mis-predictions; thus ensuring both - personalized comfort and minimal influence of prediction errors on HVAC operations. Next, we discuss the simulator.
\begin{table}[h!]
\caption{Notations used in the revised thermal model}
\ra{1.5}
\begin{tabular}{@{}lm{5cm}rl@{}}\toprule[0.3ex]
\textbf{Symbol} & \textbf{Description} & \textbf{Default} & \textbf{Unit} \\
\hline
$T_{hv}^{ij}{(t)}$ & Temperature in room $j$ of zone $i$ at time $t$ due to HVAC & $-$ & $^\circ C$ \\
$\Delta_{oc}^{ij}{(t)}$ & Change in temperature of occupied region of room $j$ in zone $i$ at time $t$ & $-$ & $^\circ C$ \\
$T_{oc}^{ij}{(t)}$ & Temperature in occupied region of room $j$ in zone $i$ at time $t$ & $-$ & $^\circ C$ \\
$T_{un}^{ij}{(t)}$ & Temperature in unoccupied region of room $j$ in zone $i$ at time $t$ & $-$ & $^\circ C$ \\
$C_{oc}^{ij}$ & Thermal capacity of occupied region of room $j$ in zone $i$ & $200$ & $kJ/K$ \\
$\alpha_{in}$ & Heat transfer coefficient between occupied and unoccupied regions of room $j$ in zone $i$ & $0.1425$ & $kJ/(K.s)$ \\
$Q_{he}^{ij}$ & Heat load due to SPOT in room $j$ of zone $i$
& $0.7$ & $kW$
\\
\bottomrule[0.3ex]
\end{tabular}%
\label{table:param_desc_spot}
\end{table}
\section{Simulator Software Architecture}
\label{sec:architecture}
To evaluate the impact of forecast errors on the different HVAC controllers, we built a custom open source
thermal simulator called ThermalSim~\cite{jain2017thermalsim}. ThermalSim is a lightweight C/C++ based simulation platform, whose focus is to study the influence of prediction errors on HVAC operations.
Figure~\ref{fig:architecture} outlines the architecture of ThermalSim. It consists of four major modules:
\begin{enumerate}
\item Master - to handle data I/O and preprocessing,
\item Error Management - to inject \textit{unbiased} errors in the occupancy streams,
\item Simulator - to simulate room temperature for a given thermal model and control logic, and
\item Analyser - to compute energy consumption, occupant comfort, and analyze simulated data streams.
\end{enumerate}
In the current version, Simulator module incorporates AMPL~\cite{AMPL} -- an algebraic modeling language for the mathematical programming -- to compute the control parameters.
\subsection{Master Module}
The Master module takes as input historical weather and occupancy data in CSV (Comma Separated Values) format, a user-generated description of the building, and simulation control parameters (Figure~\ref{fig:input}) including start and stop time of the simulation, parameters of the thermal model, control strategy, among others. Before executing the simulations, the Master module pre-processes the data, and after completion saves the output of simulation in the CSV format.
\begin{figure}[h!]
[width=0.8\linewidth]{"fig1_architecture"}
\caption{\csentence{Architecture.}
ThermalSim is a lightweight C/C++ based building simulation platform that focuses on analysing the influence of prediction errors on HVAC operations.}
\label{fig:architecture}
\end{figure}
\begin{figure}[h!]
[width=\linewidth]{"fig2_input"}
\caption{\csentence{Input format for ThermalSim.}}
\label{fig:input}
\end{figure}
\subsection{Modeling Occupancy Prediction Errors}
ThermalSim represents occupancy data for a day as a string of consecutive 0's (for unoccupied workspaces) and 1's (for occupied spaces). We consider only two states of occupancy because a majority of occupancy prediction algorithms use occupancy as a two-state variable. We call this string an \emph{occupancy string}. The length of a single occupancy string depends upon the sampling rate of the occupancy data. Data sampled every ten minutes will generate an occupancy string of length 144 characters, and if the sampling rate is thirty seconds, the string will be 2880 characters long.
\subsubsection{Error Matrix}
It is important that occupancy forecast errors be realistic.
For example, it does not make sense to randomly flip occupancy states,
since this may result in forecasting occupancy during the middle of the night, which is very unlikely.
Our key insight is that a likely outcome of an errored forecast is to forecast \textit{another valid occupancy string}, with the observation that the higher the error rate, the larger the distance, in an appropriate
metric space, between the true and the errored strings.
We use the following approach: For a dataset with $n$ occupancy strings, each cell of an \emph{error matrix} depicts the Hamming Distance between any two occupancy strings -- the number of mismatching characters~\cite{hamming1950error}. To normalize, we divide value in each cell by the length of occupancy string. The \emph{error matrix} is a
symmetric matrix of size $n^2$ which helps in systematically injecting unbiased errors in the occupancy data.
To illustrate, consider a scenario where we want to analyze different control strategies with 10\% prediction error in the occupancy data. The error management module will refer \emph{error matrix} for an occupancy string which is closest to the day of analysis. We term the selected occupancy string as the \emph{reference} string. The module will then look into the \emph{error matrix} to find all those strings that have 10\% error as compared to the \emph{reference string} and randomly select one. We call the selected one an {erroneous} string. If the day (\emph{reference} string) was 30\% occupied, then the occupancy in the \emph{erroneous} string may fall anywhere in between 20\%-40\%.
\begin{figure}[h!]
[width=0.8\linewidth]{"fig3_simulation_error"}
\caption{\csentence{Simulation Error.}
ThermalSim can simulate daily room temperature with an RMSE of $1.52^\circ C (\sigma = 0.18^\circ C)$.}
\label{fig:simulation_error}
\end{figure}
\begin{figure}[h!]
[width=0.6\linewidth]{"fig3_1_simulator_results"}
\caption{\csentence{Simulation Results.}
The hard line indicates the actual room temperature and dotted line indicates the predicted room temperature. }
\label{fig:simulation_results}
\end{figure}
\subsection{Simulator}
The simulator module takes input from the master and error management modules to simulate the room temperature. It comprises two major blocks -
\begin{enumerate*}
\item thermal model - depicts various thermal interactions occurring within a room, and
\item control module - to compute the control parameters.
\end{enumerate*}
In the current version, we have implemented two thermal models -
\begin{enumerate*}
\item single region - no partition exists within a room (Equation~\ref{eq:tm_hvac}), and
\item two regions - the occupied area is separated from the unoccupied portion by a thin layer of air (Equations~\ref{eq:tm_thvac}-\ref{eq:tm_tun}).
\end{enumerate*}
As discussed in Section~\ref{sec:background}, we have implemented four HVAC controllers in ThermalSim -
\begin{enumerate*}
\item schedule-based,
\item reactive,
\item model predictive control (no SPOT device present), and
\item SPOT-aware model predictive control.
\end{enumerate*}
In the rest of the paper, we will use NS as an acronym for No-SPOT model predictive control and SA for SPOT-Aware MPC.
\subsubsection*{Simulator Validation}
To quantify the accuracy of \emph{ThermalSim} in simulating room temperature, from a room in residential apartment, we collected temperature data for $17$ days and carried out leave-p-out cross validation with $p=5$. In such an approach, we validate the model on $p$ observations and use the remaining observations for training. We used a non-linear solver whose objective was to minimize the residual between predicted and actual room temperature. The simulator tunes following model parameters -
\begin{enumerate}
\item thermal capacity of the room ($C$),
\item heat transfer coefficient between outside and room ($\alpha_{ex}$),
\item coefficient of heating/cooling ($\rho \sigma$)
\item heat load due to occupants ($Q_{ac}$), and
\item heat load due to heating/cooling appliances ($Q_{ac}$).
\end{enumerate}
Our analysis (in Figure~\ref{fig:simulation_error}) indicates that \emph{ThermalSim} can simulate the daily room temperature with an RMSE (Root Mean Square Error) of $1.52^\circ C (\sigma = 0.18^\circ C)$. Figure~\ref{fig:simulation_results} depicts the average (solid line) and predicted (dashed line) room temperature. Note that though the predicted room temperature follows the pattern of actual room temperature, it fails to align perfectly. Though misalignment does increase the RMSE at some time instances,
we found that it has little overall impact on total energy consumption and occupants' comfort.
\subsection{Metrics}
\subsubsection{Energy Consumption} Equation~\ref{eq:energy} computes the total energy consumption of a building for a day. Here, $Po(t)$ denotes the power consumption of HVAC and other heating/cooling devices, $\tau$ is the sampling rate, and $n_t$ is the number of daily samples.
\begin{equation}
E = \sum_{t=0}^{n_t} Po{(t)} \times \frac{\tau}{3600}
\label{eq:energy}
\end{equation}
\subsubsection{Occupant Discomfort} ThermalSim leverages Predicted Mean Vote (PMV)~\cite{ASHRAE} to estimate the comfort level of the occupants~(Equation~\ref{eq:pmv}). At a given time instant $t$, if PMV ($P^{ij}(t)$) lies within the comfort requirements ($[P_{ll}, P_{ul}]$) of an individual then we mark the room as comfortable, else uncomfortable. $D_{\%}^{ij}$ denotes the percentage of time instances in a day when the user was uncomfortable in the room.
\begin{equation}
P^{ij}{(t)} = P1 \times T_{oc}^{ij}{(t)} - P2 \times v_{a}^{ij}{(t)}+ P3 \times v_{a}^{ij}{(t)} \times v_{a}^{ij}{(t)} - P4
\label{eq:pmv}
\end{equation}
\begin{equation}
D^{ij}{(t)} = max(0, P_{ll} - P^{ij}(t), P^{ij}(t) - P_{ul})
\label{eq:dc}
\end{equation}
\begin{equation}
D_{\%}^{ij} = \frac{\sum_{t=0}^{n_t}[{D^{ij}{(t)} \ne 0}]}{\sum_{t=0}^{n_t}[{O^{ij}{(t)} = 1}]}
\label{eq:dcp}
\end{equation}
\begin{figure}[h!]
[width=0.8\linewidth]{"fig4_illustration"}
\caption{\csentence{Illustration.}
As error increases, the energy consumption and occupants' discomfort vary depending on the \emph{nature} and the \emph{timing} of prediction errors. 5\% errors on the left and 20\% on the right. Large circles/triangles indicate a perfect prediction scenario and small circles/triangles correspond to those scenarios when occupancy prediction was erroneous.}
\label{fig:illustration}
\end{figure}
\subsubsection{Robustness}
Prediction errors are stochastic in nature and their impact on energy consumption and occupant comfort depends on two factors:
\textbf{Nature of the Error:} If the prediction algorithm mispredicts occupancy for short time intervals (say for a minute or so), we term the prediction errors as point errors, otherwise we call them burst errors. For a particular error percentage, an erroneous occupancy string can have point errors, burst errors, or a mix of both; resulting in different values of energy consumption and occupants' discomfort for the \textit{same} error percentage.
\textbf{Timing of the Error:} The occupancy prediction algorithm can make errors at any time of the day - such as during peak or non-peak time. Consider the situation where the occupancy prediction has 15\% error during the peak hours and the controller assumes one of the five rooms to be occupied though it was unoccupied.
In this situation there is a high chance that the HVAC might be already running during that time. Given the fact that the other four rooms are occupied, this particular prediction error will have an insignificant impact on the HVAC operations. However, during night time, the same error percentage might waste significant energy.
This illustrates that the \textit{timing} of the prediction errors has a significant impact on both
comfort and energy consumption.
For a specific error percentage, depending on the nature and timing of the errors, the energy consumption and user discomfort may either increase or decrease, potentially destabilizing HVAC operations. For a specific example, consider the big circle and triangle in Figure~\ref{fig:illustration}, which depict the energy consumption and user discomfort for NS and SA controllers respectively for perfect occupancy predictions in a particular simulation scenario.
For a specific error percentage, the small circles (NS) and triangles (SA) depict the energy consumption and user discomfort for fifteen different erroneous occupancy strings. We noticed that as prediction error increases from $5\%$ (left) to $20\%$ (right), the points indicating erroneous strings start moving away from the results obtained from perfect prediction.
Note that the circles (NS) are more scattered than the triangles (SA). In the case of NS, the system decides the control parameters such that the desired room temperature (which is the same for each room) is achieved across all the rooms. In case of a sudden change in the occupancy, NS updates the control parameters, but it takes significant time to re-attain the energy-discomfort tradeoff setpoint. In contrast, in SA, the controller knows the current state of SPOT; thus, the controller chooses a set point such that HVAC provides a certain level of comfort to the occupants and SPOT provides the necessary additional offset. SPOT, being responsive in nature, keeps the comfort level of individuals within the desired range with insignificant increase in aggregate energy consumption. Therefore, even if the error percentage increases, the energy and discomfort stays close to the perfect prediction for SA whereas NS becomes highly unstable.
To capture this phenomenon, Equation~\ref{eq:robust} defines a $robust$ ($cs \in \{NS, SA\}$) metric which quantifies the robustness of a particular control strategy $cs$ towards the prediction errors. It computes the number of instances that stay within the desired limits of the building manager.
\begin{equation}
robust_{cs}~(\%) = \frac{\text{\# of instances within limits}}{\text{total \# of instances}} \times 100
\label{eq:robust}
\end{equation}
For concreteness,
we use $\pm 20~kWh$ and $\pm 5\%$ as the acceptable limits for energy consumption and occupants' discomfort, respectively, as shown by the rectangles in the figure. For the given scenario (in Figure~\ref{fig:illustration}), when the error percentage is increasing from $5\%$ to $20\%$, NS is less $robust$ towards the prediction error ($60\% \rightarrow 0\%$), however, SA remains consistent ($100\% \rightarrow 93\%$). For a predictive control strategy, a PEC system (like SPOT) mitigates the effect of prediction errors to make the HVAC operations more reliable and robust. Whenever there is an unexpected occupancy in the room, SPOT can react quickly as compared to central HVAC system which has a slower time-scale.
\section{Evaluation}
\label{sec:evaluation}
\subsection{Test Building Description}
For our evaluation, we consider a single zone in a typical building comprising of five rooms where each room is surrounded by walls on three sides and has a window exposed to weather conditions on the fourth (Figure~\ref{fig:setup}). We assume an AHU and a VAV unit in the building. Though the structure is hypothetical, it is a typical architecture for faculty offices in Universities where thick brick walls separate the rooms. We believe that the key insights obtained for the study are well representative of more complicated building architectures. Note that ThermalSim can also deal with more complicated structures, should that be desired.
When we evaluate MPC with a reactive controller, we also consider effect of SPOT heater/fan on the room temperature. For the stated scenario, we next discuss the dataset.
\subsection{Dataset}
ThermalSim requires real-world occupancy data to generate an \emph{error matrix}. We leveraged an existing deployment from our university campus and gathered occupancy data (along with other information) from more than fifty volunteers -- including students, faculty, and the staff members every 30 seconds for a year.
\begin{figure}[h!]
[width=0.8\linewidth]{"fig5_rooms"}
\caption{\csentence{Setup.}
For evaluation, we considered a hypothetical building consisting of 5 rooms separated by walls.}
\label{fig:setup}
\end{figure}
\begin{figure}[h!]
[width=0.9\linewidth]{"fig6_pocc_counts"}
\caption{\csentence{Variation in Occupants' Schedule.}
We sample occupancy every 30 seconds; in every 10-minute interval, there exist 20 measurements of occupancy information. Here, the color indicates the number of 30 seconds instances in a 10-minute interval when the room was occupied. Notice that room would be marked occupied for all the three scenarios, however, the percentage of instances when the room was occupied for less than 2 minutes (in the range of (0, 5]) is relatively low.}
\label{fig:variation}
\end{figure}
\begin{figure}[h!]
\begin{minipage}[t]{0.44\linewidth}
[width=0.9\linewidth]{"fig7_day_com_scatt"}
\caption{\csentence{Summers.} Energy-discomfort plot when prediction is perfect. The arrow indicates the performance degradation, in terms of energy consumption and user comfort, when we move from predictive to non-predictive control strategies.}
\label{fig:day_com_scatt}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
[width=0.9\linewidth]{"fig8_day_com_ewise"}
\caption{\csentence{Summers.}
As error increases $5\% \rightarrow 20\%$, SA stays more robust than NS. Error bars indicate the variation in different simulated scenarios. For system to be more robust, the length of error bar should be smaller.}
\label{fig:err_wise_csummers}
\end{minipage}
\end{figure}
\begin{figure}[h!]
[width=0.9\linewidth]{"fig9_day_com_m2"}
\caption{\csentence{Summers.} For $20\%$ prediction error in occupancy, SA is more reliable and robust NS across all the 25 days of summer.}
\label{fig:day_wise_csummers}
\end{figure}
An MPC requires occupancy information in every 10-minute to compute the control parameters 24 hour time horizon; therefore, we upsample the occupancy data from 30-seconds to 10-minutes by applying the following rule - \textit{``Mark a 10-minute interval unoccupied if all the 30-second instances indicate the room to be unoccupied, else mark the space as occupied''.} However, with this strategy, even if a single instance in the 10-minute interval is occupied, the controller will mark the space as occupied for the whole duration. To understand whether such a bias is limiting or not, we analyzed the occupancy data and our analysis indicates that data has only 3\% 10-minute instances where the room is occupied for less than 2 minutes (Figure~\ref{fig:variation}). Therefore, we only mark a 10-minute interval unoccupied if the room was occupied at all the 30-second instances within that interval.
\subsection{Evaluation Setup}
Our research hypothesis is that
the benefits of using a PEC system like SPOT along with HVAC controller
mitigates the influence of prediction errors on MPC-based HVAC operation.
We validate this hypothesis assuming occupants in all the five rooms have similar comfort requirements: $[23^\circ$C$, 25^\circ$C$]$ in summers and $[21^\circ$C$, 23^\circ$C$]$ in winters.
For the given setup, we compare the performance of predictive
and non-predictive HVAC controllers for 25 days, both in summers and winters.
\begin{figure}[h!]
\begin{minipage}[t]{0.44\linewidth}
[width=0.9\linewidth]{"fig10_wday_com_scatt"}
\caption{\csentence{Winters.}
Energy-discomfort plot when prediction is perfect. The arrow indicates the performance degradation, in terms of energy consumption and user comfort, when we move from predictive to non-predictive control strategies.}
\label{fig:wday_com_scatt}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
[width=0.9\linewidth]{"fig11_wday_com_ewise"}
\caption{\csentence{Winters.}
Even with slow heater, SA is better or comparable than NS. Error bars indicate the variation in different simulated scenarios. For system to be more robust, the length of error bar should be smaller.}
\label{fig:err_wise_cwinters}
\end{minipage}
\end{figure}
\begin{figure}[h!]
[width=0.9\linewidth]{"fig12_wday_com_m2"}
\caption{\csentence{Winters.}
For error percentage as high as $20\%$, note that SA has less deviation in HVAC operations than NS.}
\label{fig:day_wise_cwinters}
\end{figure}
For each day, we select an occupancy string from the \emph{error matrix} that deviates (from the current day) by the error percentage specified in the system. For instance, if we wish to introduce 10\% error in the current day occupancy string, we search for another occupancy string in historical data where 288 out of 2880 instances (for a data sampled every 30 seconds) have a mismatch with the current day occupancy string. \emph{ThermalSim} utilizes both actual and erroneous occupancy string to simulate the building (depicted in Figure~\ref{fig:setup}) for all the four control strategies and compare their performance.
To mitigate any bias in the selection of erroneous occupancy strings, \emph{ThermalSim} evaluates fifteen different erroneous occupancy patterns for each day and error percentage. Furthermore, a separate analysis for each of the two seasons provides better understanding of the influence of seasonal variations.
\subsection{Insights \textcolor{white}{updated the title and text of this whole section}}
\label{sec:sim}
In the jurisdiction corresponding to
our temperature data set, i.e., Southwestern Ontario,
we find that for all control strategies,
the HVAC system consumes less energy in summers when compared to winters (see Figures~\ref{fig:day_com_scatt}, \ref{fig:wday_com_scatt}).
In our setting, the outside temperature in summers is only a few degrees higher than the desired room temperature and so the HVAC has to put in
less effort to achieve the desired comfort. On the other hand, in winters, the HVAC energy consumption is significantly higher because the outside
temperature is quite cold. In winters, all control strategies attempt
to maintain a room temperature in the range of $21^\circ C$ and $23^\circ C$ which is much higher than the outside temperature (approx. $-10^\circ C$). Consequently, HVAC has to expend more energy in winters than in summers
to attain the desirable comfort conditions in the occupied zones.
\subsubsection*{User Experience}
The schedule-based and reactive controllers can make occupants uncomfortable and yet consume significant energy, even with perfect prediction. When set to follow a fixed schedule, HVAC supplies air at a constant flow and temperature, and does not consider occupants' schedules or
daily temperature changes. For pictorial representation, we use energy-discomfort plot where x-axis denotes the daily energy consumption of the building and y-axis represents the total discomfort for the users.
Consequently, with a \textit{schedule-based} control strategy,
user experience lies in the top-right corner of the energy-discomfort plot with maximum energy consumption along with notable discomfort for both the seasons (see Figures~\ref{fig:day_com_scatt}, \ref{fig:wday_com_scatt}). On the other hand, a reactive controller with occupancy information is marginally better or equivalent to the schedule-based controller. Model predictive control (with no SPOT) shows significant improvement in minimizing both energy consumption and occupants' discomfort. Given the weather forecast and occupancy prediction, MPC keeps updating the temperature and volume of supply air at regular time intervals.
As central HVAC unit cannot cater to the dynamic schedule of the occupants, discomfort in NS is slightly higher than the hybrid control strategy that integrates SPOT with MPC to satisfy the comfort requirements of each individual in the building. In SA, the central HVAC system is aware of the SPOT system, therefore, the controller choses the set point temperature such that HVAC can provide minimal comfort, and SPOT can offset the individual comfort requirements. This results in additional savings in energy when there is partial occupancy is in line with the results from previous study by Rachel et al.~\cite{Rac17}. Next, we observed that the discomfort is negligible for summer as opposed to winter. The fan assists the occupant in quickly achieving her desired comfort level as opposed to a heater which takes comparatively more time to increase the temperature to provide the offset. In conclusion, irrespective of the season, both SA and NS strategies improve comfort and energy compared to schedule-based and reactive, with SA outperforming NS.
\subsubsection*{Error Analysis}
When occupancy predictions are erroneous, depending upon the \emph{nature} and \emph{timing} of errors, energy consumption and occupants' discomfort vary, hence HVAC operations become highly variable. When analysed over 25 days each for 15 different occupancy patterns, we find that the SA control strategy is more robust than NS even with a high error percentage. As the prediction error increase from $5\%$ to $20\%$, the performance of NS drops while SA performance remains quite consistent (Figure~\ref{fig:err_wise_csummers}). For $20\%$ prediction error, SA ($\sigma=5\%$) is $12\%$ more robust than NS ($\sigma=16\%$).
Next, Figure~\ref{fig:day_wise_csummers} shows that the SA is consistently robust across all the days as compared to NS for $20\%$ prediction error in the occupancy. Highly unreliable HVAC operations lead to significant variations in the energy consumption and the user comfort. Though the fan makes insignificant impact on the room temperature, it quickly achieves the desired comfort level by providing a cooling perception to the user. Thus, the fan is very helpful in dealing with the unexpected changes in the occupancy of the room while NS alone fails to do so.
We noticed that a fan is more effective and quicker than a heater in mitigating the effect of prediction errors on both energy consumption and discomfort. When a room gets occupied, a heater slowly increases the room temperature to achieve the comfort requirements of the occupants. This results in few intervals of discomfort for the user. This effect is visible in Figures~\ref{fig:err_wise_cwinters}, \ref{fig:day_wise_cwinters}. For $20\%$ prediction error, we noticed that the SA is now even less robust than NS on few days due to the slow reaction of the SPOT heater. However, the average performance of SA is still better or comparable than NS.
\section{Discussion and Conclusion}
\label{sec:discussions}
In this work, we analysed the influence of prediction errors in occupancy on the HVAC operations while leveraging a custom-built building simulator - \emph{ThermalSim}. In this section, we summarize our results, discuss various limitations of the study followed by research questions which are open for the community.
Our insights include the following: First, our dataset indicates that aggregate energy consumption is higher in winters than in summers. Second, integrating a PEC like SPOT with a predictive HVAC controller is definitely better or comparable than a pure MPC based approach. Third, for SA controller, fast reactive device (such as fan) is $20\%$ better than the heater, in terms of occupants discomfort. Finally, NS typically fails to satisfy the comfort requirements on any day.
Our work suffers from two main limitations.
First, while the thermal model (of \emph{ThermalSim}) considered the effect of numerous sources (such as weather, occupancy) affecting the room temperature, there still exist various other factors (such as humidity) which are critical for such analysis. We plan to explore such factors and enrich the data for a deeper analysis in future.
Second, we carried out the study through a dataset collected from a particular part of the world. Climate, users' attitude (towards energy savings), and many other factors differ significantly across the geographies. Though the results indicate that SA is more robust than NS, there can be considerable discrepancy across (and within) the countries. A real-world implementation of the technology is critical to understand its effectiveness in achieving the desired goals.
We find that mitigating the effect of prediction errors possess considerable potential in optimising the HVAC operations with predictive controllers. While model predictive control (MPC) is one of the most promising state of the art HVAC control strategies, its performance is limited by the accuracy of the weather and occupancy predictions. Therefore, we designed a custom-built building simulator -- \emph{ThermalSim} -- to analyse the influence of prediction errors on HVAC operations. We also proposed a method to introduce realistic errors in occupancy for the analysis. Our initial analysis indicates that prediction error (in occupancy) of $20\%$ can make the HVAC operations highly unstable in terms of both energy consumption and occupants' comfort. Recent literature shows that it is feasible to use a personal thermal comfort system -- SPOT -- along with predictive strategy to ensure personalised comfort in personal and shared spaces. We observed that while SPOT is effective in attaining better personalised comfort, it also strengthens the predictive strategies by mitigating the influence of predictions errors on energy consumption and occupants' comfort because it works at a finer time-scale than the MPC-based HVAC. Employing a personal thermal comfort system, such as SPOT, we stay in the acceptable region 95\% of the times as oppose to 83\% of the times even for the prediction errors as high as $20\%$, in the occupancy; thus, motivating a reliable control strategy across the commercial buildings.
\begin{backmatter}
\section*{Availability of data and materials}
On request, the authors will supply occupancy data to interested researchers. The simulator is open source and available at \url{https://github.com/milanjain81/SBS_MakefileProject}.
\section*{Competing interests}
The author(s) declare(s) that they have no competing interests.
\section*{Funding}
This work was supported by Cisco Systems Canada and the Canadian Natural Sciences and Engineering Research Council under a Cooperative Research and Development grant. The funding body played no role in the design of the study, collection, analysis, and interpretation of data, and in writing the manuscript.
\section*{Author's contributions}
Together, all the authors devised the project, the main conceptual ideas and outline for the analysis. Milan Jain carried out the implementation and analysed the data. Prof. Kalaimani, Prof. Keshav, and Prof. Rosenberg worked out the optimisation problem for Model Predictive Control and helped in writing the manuscript.
\section*{Acknowledgements}
We would like to acknowledge Alimohammad Rabbani (then a Masters student at University of Waterloo) and Costin Ograda-Bratu (currently a lab technician at University of Waterloo) for collecting and sharing occpancy data. We would also like to acknowledge the volunteers who participated (and are even now participating) in data collection.
\bibliographystyle{bmc-mathphys}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,912 |
Q: What do call() and apply() actually do to fool methods into working on array-like objects There's a lot of information out there of differences between call, apply and, bind but I'm struggling to find info on how exactly the call and apply methods fool existing functions—normally accepting arrays—into accepting array-like objects.
To my understanding, the full reference to such a method is an instruction on where to find the method.
The call/apply methods then change the 'this' param to point to the array-like object.
I've searched round for JS source code to no avail.
In Math.max.apply(Math,arguements), if part of the reason we can repurpose a function expecting an array to instead work on an array-like object is due to updating the 'this' param, how does it make sense to just give Math as it's context.
What is special about call and apply that tricks methods into working on array-like objects??
<script type="text/javascript">
function multiMax(multi){
return multi * Math.max.apply(Math,
Array.prototype.slice.call(arguments,1));
}
assert(multiMax(3, 1, 2, 3) == 9, "3*3=9 (First arg, by largest.)");
</script>
A: call and apply can't fool functions.
Instead, when you use them to call array methods on array-like objects, that works because array methods are intentionally generic.
The [whatever] function is intentionally generic; it does not require that
its this value be an Array object. Therefore it can be transferred
to other kinds of objects for use as a method.
In the case of Math functions, since they don't use the this value at all, you can use whatever value you want.
A: All arrays are objects. The only thing that would prevent a method from working on an array-like object would involve explicitly checking if the object is an array. This is simply not done by the methods that can work on array-like objects. In fact you don't even need to use call or bind to accomplish this; You could attach the method to an array-like object and call it like you normally would (however I probably wouldn't recommend it).
A: the apply and call do nothing and fool nothing to work on array-like elements. They simply do (work), because they expose indexes of their members.
Array methods access members by index; and if a member is of a right type for a given array method it will operate on it with the same ease it does with its own. But if the member is not of a type that can be handled by a given array method you will naturally get an error.
In earlier versions of JavaScript you could pass the object of interest as an argument to the property function of say Array object without the need of using the call or apply methods.
As in:
Array.split( collection );.
A: I think other answers described it quite well, but maybe this illustrative code could shed some light on this topic as well. It is a tale about generic push and its artificial evil twin:
// create plain object
var obj = {}
// give it ability to `push` by simple assignment from some poor one-time array instance
obj.PSH = [].push
// poor array will be lost in garbage: we should have used Array.prototype instead, but nevermind
console.log(obj)
// => Object {}
// (Chrome console does not show method, but we can see it in inspection or in for-in loop:)
for(var prop in obj) console.log(prop,':',obj[prop])
// => PSH : push() { [native code] }
// ah, 'native code', how enigmatic.
// let's use it
obj.PSH('zero')
obj.PSH('one')
console.log(obj)
// => Object {0: "zero", 1: "one", length: 2}
// now we see it created numbered properties and `length` property on our object
// what happens if we alter that `length` and call PSH after?
obj.length = 10
obj.PSH('TEN')
console.log(obj)
// => Object {0: "zero", 1: "one", 10: "TEN", length: 11}
// ah, predictable
// now we know what that native code most probably does, so we can create evil twin
function prankpush (what) {
var where = this.length || 0
this[where] = what // insert that to to the last index
this.length = where + 2 // but lets make it more interesting
}
// this time we will call it, so we will not taint out obj with another method
prankpush.call(obj,'prank2')
prankpush.call(obj,'prank3')
console.log(obj)
// => Object {0: "zero", 1: "one", 10: "TEN", 11: "prank1", 13: "prank2", length: 15}
// no 12 and length is bigger, what an evil success!
// but we could have as well do one time method assignment, call (like we did with native push in the beginning) …
obj.prankpush = prankpush
obj.prankpush('prank4')
// … and this time cover our tracks so prankpush will not be present in for-in-loop
delete obj.prankpush
console.log(obj)
// => Object {0: "zero", 1: "one", 10: "TEN", 11: "prank2", 13: "prank3", 15: "prank4", 17: "prank4", length: 19}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 360 |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{T}{he} Technical University of Munich has been participating in the Roborace competition since 2018. Many parts of our software stack are already available on an open source basis \cite{ChairofAutomotiveTechnology2020} including the code of the algorithm in this work \cite{ChairofAutomotiveTechnology2020b}. This paper explains an optimization-based \gls{NLMPP}, mathematically formulated as a \gls{mpSQP} \cite{Morari2009}, to calculate the velocity profiles during a race. The velocity planner inputs are the offered race paths (``performance'' and ``emergency''), stemming from our graph-based path-planning framework \cite{Stahl2019}, see Fig. \ref{fig:architecture}. The presented velocity optimization in combination with the path framework span our local trajectory planner that will be used within the competition. The trajectory planner's output is forwarded to the underlying vehicle controller \cite{Heilmeier2019, Betz2019b}, transforming the target trajectory into actuator commands for the race car, called ``DevBot 2.0'', see Fig. \ref{fig:tum_devbot}. A huge motivation behind setting up an optimization-based velocity planner was to be able to handle information about the locally and temporally varying friction potential on the race track \cite{Hermansdorfer2019}, and utilize the information provided by the race \gls{ES} \cite{Herrmann2020} as a vehicle's velocity profile has a significant influence on it's energy consumption and power losses \cite{Ozatay2017}. The friction potential estimation and the calculation of the \gls{ES} are handled by separate modules in our software stack. Their outputs, the friction potential and variable power limits, are then considered in the presented velocity optimization algorithm.
\begin{figure}[!tb]
\centering
\input{./ressources/overview/architecture.tex}
\caption{Software environment of presented velocity optimization module.}
\label{fig:architecture}
\end{figure}
To achieve real-time-capable calculation times, we build local approximations of the nonlinear velocity-planning problem, resulting in convex \gls{mpQP} \cite{Liniger2015} that can be solved iteratively using a \gls{SQP} method. We evaluated different open-source \gls{QP} solvers and compared their solution qualities and calculation times to a direct solution of the \gls{NLP}. We chose the \gls{OSQP} \cite{Stellato2020} solver as it outperformed its competitors on a standard x86-64 platform as well as on the DevBot's automotive-grade \gls{ECU}, the ARM-based NVIDIA Drive PX2.
\begin{figure}[!tb]
\centering
\includegraphics[width=\columnwidth]{ressources/db/tum_db_gray.eps}
\caption{TUM Roborace DevBot 2.0 on the race track.}
\label{fig:tum_devbot}
\end{figure}
\subsection{State of the art}
The field of trajectory planning of vehicles at the limits of handling is attracting growing attention in research. The scenarios where the car is required to be operated at the limits of its driving dynamics will become more and important as we see the spread of cars equipped with self-driving functionality, and even fully autonomous vehicles. Through this, complex scenarios with self-driving vehicles on the road will occur more frequently. Research is also being carried out on the race track where these challenging scenarios can deliberately be created in a safe environment \cite{Betz2019}.
In the field of global trajectory optimization for race tracks, different mathematical concepts are applied. In the work of Ebbesen et al. \cite{Ebbesen2018} a \gls{SOCP} formulation is used to calculate the optimal power distribution within the hybrid powertrain of a Formula One race car leading to globally time-optimal velocity profiles for a given path. For the same racing format, Limebeer and Perantoni \cite{Limebeer2015} took into account the 3D geometry of the race track within their formulation of an \gls{OCP} to solve a \gls{MLTP}. In a similar approach, Tremlett and Limebeer \cite{Tremlett2016} consider the thermodynamic effects of the tires. Christ et al. \cite{Christ2019} consider spatially variable but temporally fixed friction coefficients along the race track to calculate time-optimal global race trajectories for a sophisticated \gls{NDTM}. They show a significant influence of the variable friction coefficients on the achievable lap time when considered during the trajectory optimization process. A minimum-curvature \gls{QP} formulation, calculating the quasi-time-optimal trajectory for an autonomous race car on the basis of an occupancy grid map, is given by Heilmeier et al. \cite{Heilmeier2019}. Their advantage in comparison to \cite{Christ2019} is the computation time, but the resulting trajectories are suboptimal in terms of lap time. Also, Dal Bianco et al. \cite{DalBianco2018} formulate an \gls{OCP} to find the minimum lap time for a GP2 car and include a detailed multibody vehicle dynamics model with 14 degrees of freedom. However, none of these approaches are intended to work in real-time on a vehicle \gls{ECU}, but to deliver detailed and close-to-reality results for lap time or for the sensitivity analysis of vehicle setup parameters.
A further necessary and important part in a software stack for autonomous driving is the online re-planning of trajectories to avoid static and dynamic obstacles at high speeds. The literature can be structured into the three fields:
\begin{itemize}
\item ``separated/two-step trajectory planning'', where the velocity calculation is a subsequent process of the path planner \cite{Huang2020, Meng2019, Zhang2018}.
\item ``combined trajectory planning'', optimizing both path and velocity at the same time \cite{Subosits2019, Svensson2019, Mercy2018}.
\item ``\gls{MPC} approaches'' taking into account the current vehicle state \cite{Liniger2015, Carvalho2013, Alrifaee2018, Williams2016, Alcala2020}.
\end{itemize}
In the following, we evaluate the literature according to the implemented features regarding
\begin{itemize}
\item spatially and temporally varying friction coefficients,
\item powertrain behavior,
\item applicability at the limits of vehicle dynamics through fast computation times.
\end{itemize}
In the spline-based approach of Mercy et al. \cite{Mercy2018} trajectories for robots operating at low velocities are optimized. The calculation times of the general \gls{NLP}-solver \gls{IPOPT} \cite{Wachter2006} range up to several hundred milliseconds, which is too long for race car applications. Another general \gls{NLP}-formulation is done by Svensson et al. \cite{Svensson2018}. The latter describe a planning approach for safety trajectories of automated vehicles, which they validate experimentally in simulations for maximum velocities of \SI{30}{\kilo\meter\per\hour}, leveraging the general nonlinear optimal control toolkit ACADO \cite{Houska2011}.
Huang et al. \cite{Huang2020} describe a two-step approach: first determining the path across discretized available space and then calculating a sufficient velocity. Also, Meng et al. \cite{Meng2019} leverage a decoupled approach using a quadratic formulation for the speed profile optimization, reaching real-time capable calculation times below \SI{0.1}{\second} in this step. Nevertheless, both publications deal with low vehicle speeds in simulations of max. \SI{60}{\kilo\meter\per\hour}. Furthermore, Zhang et al. \cite{Zhang2018} implement a two-step algorithm where they use MTSOS \cite{Lipp2014} for the speed profile generation within several milliseconds for path lengths of up to \SI{100}{\meter}. The speed-profile optimization framework MTSOS developed by Lipp and Boyd \cite{Lipp2014} works for fixed paths leveraging a change of variables. As in the aforementioned publications, they consider a static friction coefficient and neglect the maximum available power of the car. The same is true for the \gls{MPC} algorithm by Carvalho et al. \cite{Carvalho2013}. They plan trajectories considering the driving dynamics of a bicycle model, neglecting physical constraints stemming from the powertrain, like maximum available torque or power. This is a major drawback for our application, as the DevBot 2.0 is often operating at the power limit of its electric machines.
Subosits and Gerdes \cite{Subosits2019} formulate a \gls{QCQP} replanning path and velocity of a race car at spatially fixed points on the track to avoid static obstacles. They consider a constant friction potential and the maximum available vehicle power. However, the obstacles need to be known in advance before the journey commences, and must be placed at a decent distance from the replanning points to allow the algorithm to find a feasible passing trajectory, given the physical constraints. In order to reach fast calculation times, Alrifaee et al. \cite{Alrifaee2018} use a sequential linearization technique for real-time-capable trajectory optimization. They consider the friction maxima, with included velocity dependency that they determine beforehand. This dependency is assumed to be globally constant, thus neglecting the true track conditions during driving. Their experimental results stem from simulations with peak computation times of several hundred milliseconds on a desktop PC.
Considering variable friction on the road is attracting more attention, as it is an emergency-relevant feature for passenger cars and a performance-critical topic for race cars. Therefore, Svensson et al. \cite{Svensson2019} describe an adaptive trajectory-planning and optimization approach. They pre-sample trajectory primitives to avoid local optima in subsequent \gls{SQP}s stemming mainly from avoidance maneuvers to the suboptimal side of an obstacle. The vehicle adaptively reacts to a varying friction potential on the road at speeds of up to \SI{100}{\kilo\meter\per\hour}. The resulting problem is solved using simulations in MATLAB, so no information about the calculation speed on embedded hardware is given.
Stahl et al. \cite{Stahl2019} describe a two-step, multi-layered graph-based path planner. This approach allows for functionalities such as following other vehicles and overtaking maneuvers, also in non-convex scenarios at a high update rate. We use this path planner to generate the inputs for the velocity-optimization algorithm.
\subsection{Contributions}
In this paper we contribute to the state of the art in the field of real-time-capable trajectory planning with the following content.
(1) We formulate a tailored \gls{mpSQP} algorithm capable of adaptive velocity planning for race cars operating at the limits of handling, and at velocities above \SI{200}{\kilo\meter\per\hour}. The planner computes velocity profiles for various paths using the path planner \cite{Stahl2019} in real time on the target hardware, an NVIDIA Drive PX2 \cite{Denton2020} being an \gls{ECU} already proven for autonomous driving. The adaptivity refers to the multi-parametric input to the planner, depending on the vehicle's environment. The quadratic subproblems within the \gls{mpSQP} are handled using the \gls{OSQP} \cite{Stellato2020} solver. Its primal and dual infeasibility detection for convex problems \cite{Banjac2019} was integrated to flag up (as fast as possible) offered paths which could not feasibly be driven \cite{Stahl2019}.
(2) With the formulation of an \gls{mpSQP} optimization algorithm, it is possible to integrate our race \gls{ES}, described in our previous works \cite{Herrmann2020, Herrmann2019}. The necessary variable power parameters are forwarded to the velocity planner and considered as a hard constraint, see Fig. \ref{fig:architecture}. In the case of electric race cars, such an \gls{ES} is vital in order not to overstress the powertrain thermodynamically.
(3) We further allow the friction coefficient on the race track to vary spatially as well as temporally \cite{Hermansdorfer2019}. Therefore, global limits of the allowed longitudinal and lateral acceleration of the vehicle are omitted. This improves the achievable lap time significantly as the tires are locally exploited to their maximum. Via the temporal variation of the friction limits, we take into account varying grip due to, e.g., warming tires or changing weather conditions.
(4) To boost the solver selection for similar projects dealing with trajectory optimization within the community, we compare the efficiency of different solver types regarding calculation speed and solution quality. Therefore, we solve the quadratic subproblems in our \gls{mpSQP} using a first-order \gls{ADMM} implemented in the \gls{OSQP}-framework. Its results are compared to the active-set solver \gls{qpOASES} \cite{Ferreau2014}. We contrast both \gls{SQP}s with a direct solution of the nonlinear velocity optimization problem with the open-source, second-order interior point solver \gls{IPOPT} \cite{Wachter2006}, interfaced by the symbolic framework CasADi \cite{Andersson2019}.
Section \ref{sec:preliminaries} introduces the mathematical background of an \gls{SQP} method to solve an \gls{NLP}. In the following Section \ref{sec:optV}, the nonlinear equations of our velocity planner are introduced. We explain their efficient incorporation within an \gls{mpSQP} and explain details about the recursive feasibility of our optimization problem. The Results section shows the realization of the \gls{ES} and the handling of variable friction by our velocity-optimization algorithm. Furthermore, we contrast different solvers in terms of their runtime and solution quality.
\section{Preliminaries}
\label{sec:preliminaries}
In this section, the mathematical background to an \gls{SQP} optimization method to solve local approximations of an \gls{NLP} with objective function $J(\boldsymbol{o})$, $h_b(\boldsymbol{o})$ and $g_c(\boldsymbol{o})$, denoting equality and inequality constraints of scalar quantity $b$ and $c$, and optimization variables $\boldsymbol{o}$ is introduced.
The standard form of a \gls{NOCP} is given by \cite{Andersson2019}, \cite{Luenberger2008}:
\begin{align}
\min~& J(\underbrace{x(s), u(s)}_{\boldsymbol{o}})\\
\mathrm{s.t.}~\frac{\mathrm{d}x(s)}{\mathrm{d}s} &= f(x(s), u(s))\\
h_b(\boldsymbol{o}) &= 0\\
g_c(\boldsymbol{o}) &\leq 0.
\label{eq:NLPP}
\end{align}
The independent space variable $s$ describes the distance along the vehicle's path in our problem. The function $f(x(s), u(s))$ specifies the derivatives of the state variable $x(s)$ as a function of the state $x(s)$ and the control input $u(s)$.
The standard form of a \gls{QP} is expressed as \cite{Boyd2004}
\begin{align}
&\min{\quad\frac{1}{2} \boldsymbol{z}^T_\mathrm{qp} \boldsymbol{P} \boldsymbol{z}_\mathrm{qp} + \boldsymbol{q}^T \boldsymbol{z}_\mathrm{qp}} \nonumber \\
&\mathrm{s.t.} \quad \boldsymbol{l} \leq \boldsymbol{A}\boldsymbol{z}_\mathrm{qp} \leq \boldsymbol{u},
\label{eq:QP}
\end{align}
where $\boldsymbol{z}_\mathrm{qp}$ is the optimization vector, matrix $\boldsymbol{P}$ is the Hessian matrix of the discretized objective $J(\boldsymbol{o}^k)$ and the vector $\boldsymbol{q}^T$ equals the Jacobian of the discretized objective $\nabla J(\boldsymbol{o}^k)$ with iterate $k$. Matrix $\boldsymbol{A}$ contains the linearized versions of the constraints $h_b$ and $g_c$ in the optimization problem. Their upper and lower bounds are summarized in both vectors, $\boldsymbol{l}$ and $\boldsymbol{u}$.
In an \gls{SQP} method, the linearization point $\boldsymbol{o}^k$ is updated after every \gls{QP} iteration $k$ using \cite{Boggs1995}
\begin{align}
\boldsymbol{o}^{k+1} &= \boldsymbol{o}^k + \alpha \boldsymbol{z}_{\mathrm{qp}
\label{eq:zSQPUpdate} \\
\boldsymbol{\lambda}^{k+1} &= \boldsymbol{\lambda}^k_\mathrm{qp} \\
\boldsymbol{z}_{\mathrm{qp}} &= \boldsymbol{o} - \boldsymbol{o}^k.
\end{align}
In the quadratic subproblem, a solution for $\boldsymbol{z}_{\mathrm{qp}}$ is computed. We chose to initialize the Lagrange multiplier vector $\boldsymbol{\lambda}^{k+1}$ using the previous \gls{QP} solution as stated in the local SQP algorithm in \cite{Nocedal2006}.
On the one hand, the steplength parameter $\alpha$ must be calculated in order to perform a large step in the direction of the optimum $\boldsymbol{o}^*$ for fast convergence. On the other, $\alpha$ must be small enough to not skip or oscillate around $\boldsymbol{o}^*$. It is therefore necessary to define a suitable merit function, taking into account the minimization of the objective function as well as the adherence of the constraints \cite{Luenberger2008}, \cite{Boggs1995}. As it is hard to find such a merit function, we use the \gls{SQP} \gls{RMSE} $\bar{\varepsilon}_\mathrm{SQP}$ as well as the \gls{SQP} infinity norm error $\hat{\varepsilon}_\mathrm{SQP}$ to determine whether a stepsize $\alpha$ is suitable or not,
\begin{align}
\bar{\varepsilon}_\mathrm{SQP} &= \frac{1}{K}
\begin{Vmatrix}
\boldsymbol{o}^{k+1} - \boldsymbol{o}^k
\end{Vmatrix}_2 \leq \bar{\varepsilon}_\mathrm{SQP,tol} \label{eq:eps_sqp_bar}\\
\hat{\varepsilon}_\mathrm{SQP} &=
\begin{Vmatrix}
\boldsymbol{o}^{k+1} - \boldsymbol{o}^k
\end{Vmatrix}_\infty \leq \hat{\varepsilon}_\mathrm{SQP,tol},
\label{eq:eps_sqp_hat}
\end{align}
where $K$ denotes the number of elements in $\boldsymbol{o}^k$. In case one of the two errors $\varepsilon_\mathrm{SQP}$ increases when applying (\ref{eq:zSQPUpdate}), the counting variable $\gamma$ is increased and, therefore, $\alpha$ is reduced until the tolerance criteria values $\bar{\varepsilon}_\mathrm{SQP,tol}$ and $\hat{\varepsilon}_\mathrm{SQP,tol}$ are met:
\begin{align}
\alpha = \beta^{\gamma}.
\end{align}
The parameter $\beta \in \left]0, 1\right[$ is to be tuned problem-dependent as the Armijo rule states \cite{Luenberger2008} with $\gamma \in \left[0; 1; 2; ... \right]$.
To bring the objective $J(\boldsymbol{o})$ and the necessary nonlinear constraints $h_b(\boldsymbol{o})$ and $g_c(\boldsymbol{o})$ into the mathematical form of a \gls{QP} (\ref{eq:QP}), they are discretized and approximated quadratically or linearly, respectively, using Taylor series expansions in the form
\begin{align}
J(\boldsymbol{o}) \approx &\frac{1}{2} (\boldsymbol{o} - \boldsymbol{o}^k)^T \boldsymbol{P}(\boldsymbol{o}^k) (\boldsymbol{o} - \boldsymbol{o}^k) +\nonumber \\
&\nabla J(\boldsymbol{o}^k) (\boldsymbol{o} - \boldsymbol{o}^k) + J(\boldsymbol{o}^k)
\label{eq:J_approx}
\end{align}
and
\begin{align}
g_c(\boldsymbol{o}) \approx \nabla g(\boldsymbol{o}^k) (\boldsymbol{o} - \boldsymbol{o}^k) + g(\boldsymbol{o}^k).
\label{eq:linConstraints}
\end{align}
\section{Optimization-Based Velocity Planner}
\label{sec:optV}
This section describes the implemented point mass model, the used objective function, and the constraints necessary to optimize the velocity on the available paths. The point mass model was chosen, as it is commonly used to describe the driving dynamics in the automotive context. Due to its simplicity, it delivers a small number of optimization variables and constraints. Therefore, quick solver runtimes can be achieved. It still delivers quite accurate results for the task of pure velocity optimization \cite{Ebbesen2018}.
The concept of the optimization-based algorithm is to plan velocities with inputs from other software modules, cf. Section \ref{sec:introduction}. We do not deal with sensor noise in the planner but in the vehicle dynamics controller \cite{Heilmeier2019}, which receives the trajectory input. The trajectory planning module, consisting of a path planner \cite{Stahl2019} and the presented velocity optimization algorithm, always keeps the first discretization points of a new trajectory constant with the solution from a previous planning step. After matching the current vehicle position to the closest coordinate in the previously planned trajectory, a new plan starting from this position is made within the remaining part of a new trajectory. Through this, two control loops can be omitted, and prevent from unnecessary inferences in the planning and the control module.
\subsection{Nonlinear problem}
\label{subsec:NLP}
This subsection presents the nonlinear velocity optimization problem, structured into its system dynamics, equality and inequality constraints as well as its objective function.
\subsubsection{System dynamics}
Let us first introduce the system dynamics of the point mass model for our physical vehicle state $v(s)$. Newton's second law for a point mass $m_\mathrm{v}$ states
\begin{equation}
m_\mathrm{v} v(s) \frac{\mathrm{d} v(s)}{\mathrm{d} s} = m_\mathrm{v} a_\mathrm{x}(s).
\end{equation}
With the derivative of the kinetic energy,
\begin{equation}
\frac{\mathrm{d}E_\mathrm{kin}(s)}{\mathrm{d} s} = \frac{1}{2} m_\mathrm{v} \frac{\mathrm{d}v^2(s)}{\mathrm{d}s} = m_\mathrm{v} v(s) \frac{\mathrm{d} v(s)}{\mathrm{d} s},
\end{equation}
the system dynamics are given by the longitudinal acceleration
\begin{equation}
a_\mathrm{x}(s) = \frac{1}{m_\mathrm{v}} \frac{\mathrm{d} E_\mathrm{kin}(s)}{\mathrm{d} s}.
\end{equation}
The force $F_\mathrm{x,p}(s)$ applied by the powertrain to move the point mass model can be calculated by
\begin{equation}
F_\mathrm{x,p}(s) = m_\mathrm{v} a_\mathrm{x}(s) + c_\mathrm{r} v^2(s)
\label{eq:FsAero}
\end{equation}
where $c_\mathrm{r}$ is the product of the air density $\rho_\mathrm{a}$, the air resistance coefficient $c_\mathrm{w}$ and the vehicle's frontal area $A_\mathrm{v}$,
\begin{equation}
c_\mathrm{r} = \frac{1}{2} \rho_\mathrm{a} c_\mathrm{w} A_\mathrm{v}.
\end{equation}
\subsubsection{Equality and inequality constraints}
To improve numerical stability and avoid backward movement, the velocity $v(s)$ is constrained,
\begin{equation}
0 \leq v(s) \leq v_\mathrm{max} (s)
\label{eq:vmaxConstraint}
\end{equation}
To ensure that the optimization remains feasible in combination with a moving horizon, the terminal constraint
\begin{equation}
v(s_\mathrm{f}) \leq v_\mathrm{end}
\label{eq:vendConstraint}
\end{equation}
on the last coordinate point $s_\mathrm{f}$ within the optimization horizon is leveraged. Here, $v_\mathrm{end}$ denotes the minimal velocity the vehicle can take in the case of maximum specified track curvature $\kappa_\mathrm{max}$ at the vehicle's technically maximum possible lateral acceleration $a_\mathrm{y,max}$. Therefore,
\begin{equation}
v_\mathrm{end} = \sqrt{ \frac{a_\mathrm{y,max}}{\kappa_\mathrm{max}} }.
\label{eq:v_recursiveInfeas}
\end{equation}
At the beginning of the optimization horizon, the velocity and acceleration must equal the vehicle's target states of the currently executed plan $v_\mathrm{ini}$ and $a_\mathrm{x,ini}$,
\begin{align}
v(s_\mathrm{s}) &= v_\mathrm{ini},\nonumber\\
a_\mathrm{x,ini} - \delta_\mathrm{a} \leq a_\mathrm{x}(s_\mathrm{s}) &\leq a_\mathrm{x,ini} + \delta_\mathrm{a},
\end{align}
where $s_\mathrm{s}$ denotes the first coordinate within the moving optimization horizon and $\delta_\mathrm{a}$ a small tolerance to account for numerical imprecision.
As the vehicle's maximum braking as well as driving forces are technically limited, the resulting constraints are
\begin{equation}
F_\mathrm{min} \leq F_\mathrm{x,p}(s) \leq F_\mathrm{max}.
\end{equation}
The negative force constraint $F_\mathrm{min}$ does not affect the optimization-problem feasibility, as the DevBot's braking actuators can produce more negative force than the tires can transform.
The electric machine's output power $P(s)$ is computed using
\begin{equation}
P(s) = F_\mathrm{x,p}(s) v(s),
\end{equation}
limited by the available maximum
\begin{equation}
P(s) \leq P_\mathrm{max}(s).
\label{eq:EnergyConstraint}
\end{equation}
We highlight that $P_\mathrm{max}(s)$ is a space-dependent parameter in contrast to the constant maximum force $F_\mathrm{max}$. By this, the given race \gls{ES} based on our previous works \cite{Herrmann2020, Herrmann2019} is realized.
To further integrate the tire physics, we interpret the friction potential as a combined, diamond-shaped acceleration limit for the vehicle \cite{Hermansdorfer2019} given by the inequality
\begin{equation}
\begin{Vmatrix}
\left( \hat{a}_{\mathrm{x}}(s), \hat{a}_{\mathrm{y}}(s) \right)
\end{Vmatrix}_1 \leq 1 + \epsilon(s)
\label{eq:frictionConstraint}
\end{equation}
where $\left|\left| \cdot \right|\right|_1$ denotes the $l^1$-norm. Furthermore, the normalized longitudinal $\hat{a}_\mathrm{x}(s)$ as well as the lateral tire utilizations $\hat{a}_\mathrm{y}(s)$ are given by
\begin{equation}
\hat{a}_{\mathrm{x}}(s) = \frac{F_\mathrm{x,p}(s)}{m_\mathrm{v}}\frac{1}{\bar{a}_{\mathrm{x}}(s)}, \quad \hat{a}_{\mathrm{y}}(s) = \frac{a_{\mathrm{y}}(s)}{\bar{a}_{\mathrm{y}}(s)}.
\label{eq:var_accel_pars}
\end{equation}
Here, we use $\bar{a}_{\mathrm{x/y}}(s)$ to indicate a variable, space-dependent acceleration potential in both longitudinal and lateral direction, which is to be leveraged \cite{Hermansdorfer2019}. The lateral acceleration $a_\mathrm{y}(s)$ reads \cite{Braghin2008}
\begin{equation}
a_\mathrm{y}(s) = \kappa(s) v^2(s)
\label{eq:AccLat}
\end{equation}
accounting for the target path geometry by the variable road curvature parameter $\kappa(s)$.
In (\ref{eq:frictionConstraint}) the slack variable $\epsilon(s)$ ensures the recursive feasibility of the optimization problem: details are given in Subsection \ref{subsec:slackVariable}. We constrain the slack variable $\epsilon(s)$ by
\begin{equation}
0 \leq \epsilon(s) \leq \epsilon_\mathrm{max}
\label{eq:EpsilonConstraint}
\end{equation}
to prohibit negative values and additionally keep the physical tire exploitation within a specified maximum.
Similarly to (\ref{eq:v_recursiveInfeas}), the longitudinal and lateral acceleration limits at the end of the optimization horizon $\bar{a}_\mathrm{x}(s_\mathrm{f})$ and $\bar{a}_\mathrm{y}(s_\mathrm{f})$ must be set to the lowest physically possible acceleration limits $\bar{a}_\mathrm{x/y,min}$ for the current track conditions,
\begin{equation}
\bar{a}_\mathrm{x/y}(s_\mathrm{f}) \leq \bar{a}_\mathrm{x/y,min}.
\label{eq:acc_recursive}
\end{equation}
\subsubsection{Continuous objective function}
With the help of the introduced symbols and equations we can now formulate the objective function $J(x(s))$ to minimize the traveling time along the given path:
\begin{align}
J(x(s)) =& \int_{0}^{s_\mathrm{f}}{\frac{1}{v(s)}\mathrm{d}s} +
\frac{\rho_{\mathrm{j}}}{s_\mathrm{f}}\int_{0}^{s_\mathrm{f}}{\left( \frac{\mathrm{d}^2 v(s)}{\mathrm{d}^2 t}\right)^2} \mathrm{d}s +\nonumber\\
& \frac{\rho_{\mathrm{\epsilon,l}}}{s_\mathrm{f}} \int_{0}^{s_\mathrm{f}}{\epsilon(s)}\mathrm{d}s + \frac{\rho_{\mathrm{\epsilon,q}}}{s_\mathrm{f}} \int_{0}^{s_\mathrm{f}}{\epsilon^2(s)}\mathrm{d}s.
\label{eq:nonlinearObjective}
\end{align}
We chose the optimization variables $\boldsymbol{o}$ to be the state velocity $v(s)$ as well as the slacks $\epsilon(s)$. The control input to the vehicle $u(s) = F_\mathrm{x,p}(s)$ doesn't occur explicitly in the objective function but can be recalculated from the state trajectory $v(s)$, cf. (\ref{eq:FsAero}).
Minimizing the term $\frac{1}{v(s)}$ is equivalent to the minimization of the lethargy $\frac{\mathrm{d}t}{\mathrm{d}s}$, which can be interpreted as the time necessary to drive a unit distance \cite{Ebbesen2018}. To weight the different terms, the penalty parameters $\rho$ are used. These include a jerk penalty $\rho_{\mathrm{j}}$, a slack weight $\rho_{\mathrm{\epsilon,l}}$ on their integral and a penalty $\rho_{\mathrm{\epsilon,q}}$ on the integral of the squared slack values. The linear penalty term on the slack variable $\epsilon(s)$ is necessary to achieve an exact penalty maintaining the original problem's optimum $\left[v^*(s)~\epsilon^*(s)\right]$ if feasible \cite{Kerrigan2000}. Similar to a regularization term, the integral of the squared slacks $\epsilon(s)$ is additionally added to improve numerical stability and the smoothness of the results.
\subsection{Multi-parametric Sequential Quadratic Problem}
\label{subsec:efficient_implementation}
This chapter gives details about the implementation of the \gls{NLP} given in Subsection \ref{subsec:NLP} as an \gls{mpSQP} in order to efficiently solve local approximations of the velocity planning problem. We describe how to approximate the nonlinear objective function $J(x(s))$ (\ref{eq:nonlinearObjective}) to achieve a constant and tuneable Hessian matrix within our tailored \gls{mpSQP} algorithm. Furthermore, we present a method to reduce the number of slack variables $\epsilon(s)$ and the slack constraints (\ref{eq:EpsilonConstraint}) therefore necessary.
Our optimization vector $\boldsymbol{z}=\boldsymbol{o}^k$ in a discrete formulation transforms into
\begin{align}
\boldsymbol{z} =
&\begin{bmatrix}
\underbrace{v_1(s_1) \ldots v_{M - 1}(s_{M-1})}_{\boldsymbol{v}} & \underbrace{\epsilon_0(s_0) \ldots \epsilon_{N - 1}(s_{N - 1})}_{\boldsymbol{\epsilon}}
\end{bmatrix}^T \\
&\in \mathbb{R}^{K \times 1} \nonumber
\end{align}
where $K = M - 1 + N$ where $M$ denotes the number of discrete velocity points $v_m$ and $N$ the number of discrete slack variables $\epsilon_n$ used in the tire inequality constraints within one optimization horizon. We drop the dependency of $\boldsymbol{z}$ on $s_m$ in the following for the sake of readability. The velocity variable $v_0$ is removed from the vector $\boldsymbol{z}$ as it is a fixed parameter equaling the velocity planned in a previous \gls{SQP} $l - 1$ for the current position.
To reduce the problem size, we apply one slack variable $\epsilon_n$ to multiple consecutive discrete velocity points $v_m$. This is done uniformly and leads to:
\begin{align}
\begin{bmatrix}
\underbrace{
v_1~\ldots~v_{\tilde{N}}}_{\epsilon_0
& \underbrace{
v_{\tilde{N} + 1}~\ldots~v_{2\tilde{N}}}_{\epsilon_1
& \underbrace{v_{2\tilde{N} + 1}~\ldots~v_{M - 1}}_{\ldots}
\label{eq:slackVisualization}
\end{bmatrix}
\end{align}
Here, $\tilde{N}$ is a problem-specific parameter setting a trade off between the number of optimization variables and therefore the calculation speed and accuracy in the solution.
From domain knowledge we know that the objective function can be approximated in the form
\begin{align}
J(\boldsymbol{z}) \approx
&\underbrace{
\begin{Vmatrix}
\boldsymbol{v} - \boldsymbol{v}_\mathrm{max}
\end{Vmatrix}^2_2}_{J_\mathrm{v}}
\underbrace{\rho_{\mathrm{j}} \begin{Vmatrix} \Delta \boldsymbol{v} \end{Vmatrix}^2_2}_{J_\mathrm{j}} + \nonumber\\
&
\underbrace{
\rho_{\mathrm{\epsilon,l}} \begin{Vmatrix} \zeta \boldsymbol{\epsilon} \end{Vmatrix}^2_1}_{J_\mathrm{\epsilon,l}} +
\underbrace{
\rho_{\mathrm{\epsilon,q}} \begin{Vmatrix} \zeta \boldsymbol{\epsilon} \end{Vmatrix}^2_2}_{J_\mathrm{\epsilon,q}}.
\label{eq:discreteObjective}
\end{align}
The slack variables are transformed via the constant factor $\zeta$; how this is selected is discussed at the end of this section. By using the $l^2$-norm of the vector difference of $\boldsymbol{v}$ and $\boldsymbol{v}_\mathrm{max}$, the solution tends to minimize the travel time along the path. Still, this formulation in combination with (\ref{eq:vmaxConstraint}) makes the car keep a specified maximum velocity $v_\mathrm{max}(s)$ dependent on the current position $s$ to react, e.g., to other cars. To control the vehicle's jerk behavior, we add the Tikhonov regularization term $\rho_{\mathrm{j}} \begin{Vmatrix} \Delta \boldsymbol{v} \end{Vmatrix}^2_2$ \cite{Boyd2004} that approximates the second derivative of $\boldsymbol{v}$. The tridiagonal Toeplitz matrix $\Delta\in \mathbb{R}^{M - 3 \times M - 1}$ contains the diagonal elements $\left(\begin{smallmatrix}1 & -2 & 1\end{smallmatrix}\right)$ \cite{Boyd2004}. By the $l^1$-norm within $J_\mathrm{\epsilon,l}$, the summation of the absolute values of the slack variable vector entries in $\boldsymbol{\epsilon}$ is achieved. To improve the numerical conditioning of the problem, their $l^2$-norm is added additionally by $J_\mathrm{\epsilon,q}$.
For the specific choice of cost function in (\ref{eq:J_approx}), the Hessian matrix $\boldsymbol{P} \in \mathbb{R}^{K \times K}$ does not depend on $\boldsymbol{z}$. The condition number $\sigma_\mathrm{H}$ of the Hessian $\boldsymbol{P}$ is tuned to be as close to 1 as possible via the penalties $\rho_{\mathrm{j}}$ and $\rho_{\mathrm{\epsilon,l}}$ as well as $\zeta$ denoting the unit conversion factor of the tire slack variable values in $\boldsymbol{\epsilon}$ to SI units:
\begin{equation}
\renewcommand{\arraystretch}{1.1}
\boldsymbol{P} =
\left[
\begin{array}{ccc|cc}
\ddots & \ddots & 0 & & \\
\ddots & c_j\rho_\mathrm{j} & \ddots & \multicolumn{2}{c}{0} \\
0 & \ddots & \ddots & & \\
\hline
\multicolumn{3}{c|}{\multirow{2}{*}{0}} & 2\rho_{\mathrm{\epsilon,q}} \zeta^2 & 0\\
& & & 0 & \ddots\\
\end{array}
\right]
\label{eq:Hess}
\end{equation}
The function $c_j \rho_\mathrm{j}$ represents different constant entries $j$ that are linearly dependent on $\rho_\mathrm{j}$. This upper left part of $\boldsymbol{P}$ is a bisymmetric matrix with constant entries on its main diagonal, as well as on its first and second ones.
By using the approach of multi-parametric programming, we can vary several problem parameters online in the \gls{SQP} (Section \ref{sec:preliminaries}) without changing the problem size. These parameters include the
\begin{itemize}
\item spatial discretization length $\Delta s_m$.
\item curvature of the local path $\kappa (s_m)$ \cite{Stahl2019}.
\item maximum allowed velocity $v_\mathrm{max} (s_m)$.
\item power limitations $P_\mathrm{max} (s_m)$ stemming from a global race strategy taking energy limitations into account \cite{Herrmann2020, Herrmann2019}.
\item longitudinal and lateral acceleration limits $\bar{a}_\mathrm{x}(s_m)$, $\bar{a}_\mathrm{y}(s_m)$ \cite{Hermansdorfer2019}.
\end{itemize}
\subsection{Variable acceleration limits}
\label{sec:variable_friction}
To fully utilize the maximum possible tire forces, a time- and location-dependent map of the race track, containing the maximum possible accelerations, is generated. The acceleration limits can be interpreted as vehicle-related friction coefficients, cf. Subsection \ref{subsec:NLP}. The 1D map along the global coordinate $s_\mathrm{glo}$ with variable discretization step length stores the individual acceleration limitations $\Sigma_{\bar{a}}(s_\mathrm{glo})$ in longitudinal and lateral directions. The acceleration limits are used in the selected local path as the parameters $\bar{a}_\mathrm{x/y}(s)$ (\ref{eq:var_accel_pars}). It is important to know that while the vehicle proceeds, the target path is updated constantly, i.e., the global coordinates $s_\mathrm{glo}$ selected as the local path $s$ vary permanently in subsequent velocity optimizations. The path planning guarantees to re-use the first few global coordinates $s_\mathrm{glo}$ from a previous timestep $t^0$ as the starting coordinates $s_m$ of the subsequently chosen path at $t^1$. Still, the path coordinates $s_m$ at the end of the planning horizon are not guaranteed to precisely match all of the previously used global coordinates $s_\mathrm{glo}$ since the path might change. Therefore, the requested coordinates in the acceleration map do also vary slightly, but are matched to the same local coordinate indices in $s_m$ within the local path in subsequent timesteps. This leads to differences in the acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ in subsequent planning iterations for identical local path indices $m$ and therefore probably to infeasible problems in terms of optimization, see Section \ref{subsec:slackVariable}.
The nature of the discretization problem is illustrated in Fig.~\ref{fig:map_discretization}. The given example provides stored acceleration limits $\Sigma_{\bar{a}}(s_\mathrm{glo})$ in \SI{10}{\meter} steps. The planning horizon ranges from $s_\mathrm{glo}$ = \SIrange{0}{300}{\meter}. Therefore, we show a snippet of the end of the planning horizon ($s_\mathrm{glo}$ = \SIrange{200}{270}{\meter}) as the discretization issues are clearly visible here. At timestep $t^0$, the planning algorithm requests the stored acceleration limits $\Sigma_{\bar{a}}(s_\mathrm{glo})$ every \SI{5.5}{\meter}, starting at $s_\mathrm{glo}=\SI{0}{\meter}$. Within the depicted path snippet, the subsequent iteration at $t^1$ starts at a shift of \SI{2.0}{\meter} with the same stepsize.
The simple approach of directly obtaining the local acceleration limits from the stored values $\Sigma_{\bar{a}}(s_\mathrm{glo})$ by applying zero-order hold comes with drawbacks. A slight shift in the global coordinate selection can lead to situations where the acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ differ between subsequent timesteps ($t^0$, $t^1$) in the local path. If the subsequent acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ at $t^1$ are smaller, they can lead to infeasibility. In Fig. \ref{fig:map_discretization}, the gray areas highlight the situations where the obtained acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ at $t^1$ are smaller compared to $t^0$ for identical local path indices $m$.
To mitigate the discretization effects, we propose an interpolation scheme leading to the values $\tilde{\Sigma}_{\bar{a}}(s_\mathrm{glo})$. It applies linear interpolation between the stored acceleration limits $\Sigma_{\bar{a}}(s_\mathrm{glo})$ but acts cautiously in the sense that it always underestimates the actual values $\Sigma_{\bar{a}}(s_\mathrm{glo})$. This can be seen from $s_\mathrm{glo}$ = \SIrange{200}{210}{\meter}, where the value is kept constant instead of interpolating between \SI{11}{} and \SI{12}{\meter\per\square\second}, and from $s_\mathrm{glo}$ = \SIrange{230}{240}{\meter} where the algorithm adapts to the decreasing values although the stored value is \SI{13}{\meter\per\square\second}. Then, zero-order hold is applied to this conservatively interpolated line. The limits obtained at $t^1$ often lie above $\bar{a}_\mathrm{x/y}(s_m)$ at $t^0$, which allow higher accelerations $a_\mathrm{x/y}(s_m)$ than expected at $t^0$ and thus compensating for overestimated areas~(gray areas).
The acceleration limits $\Sigma_{\bar{a}}(s_\mathrm{glo})$ are constantly updated by an estimation algorithm \cite{Hermansdorfer2019} and are therefore also considered time-variant. During the update process, it must be guaranteed that the update does not lead to an infeasible vehicle state for the velocity-planning algorithm, e.g., when the vehicle is approaching a turn already utilizing full tire forces under braking, and the acceleration limits are suddenly decreased in front of the vehicle. Therefore, the updates only take place outside the planning horizon of the algorithm.
The difference between subsequently obtained acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ at identical local path coordinates $s_m$ can be controlled via the maximum change between the stored values $\Sigma_{\bar{a}}(s_\mathrm{glo})$. The slope of the interpolated values $\tilde{\Sigma}_{\bar{a}}(s_\mathrm{glo})$ can be used to calculate the maximum error when applying a particular step size $\Delta s_m$ in path planning.
\begin{figure}[!tb]
\centering
\resizebox{1.025\columnwidth}{!}{
\input{./ressources/frictionmap/discretization_frictionmap.tex}}
\caption{Diagram of acceleration limits for two subsequently planned paths; with pure readout values $\Sigma_{\bar{a}}(s_\mathrm{glo})$ (top) and with the proposed interpolation scheme $\tilde{\Sigma}_{\bar{a}}(s_\mathrm{glo})$ (bottom). Gray areas show where the subsequently planned path receives decreased maximum acceleration limits $\bar{a}_\mathrm{x/y}(s_m)$ due to tolerances in the spatial discretization.}
\label{fig:map_discretization}
\end{figure}
\subsection{Recursive feasibility}
\label{subsec:slackVariable}
The minimum-time optimization problem tends to produce solutions with many active constraints, as it maximizes the tire utilization. It follows from this property, that ensuring recursive feasibility is a highly relevant aspect for the application of such an algorithm, and should be achieved via the terminal constraints (\ref{eq:v_recursiveInfeas}) and (\ref{eq:acc_recursive}) by making a worst-case assumption about the curvature $\kappa(s_\mathrm{f})$ and acceleration limits $\bar{a}_\mathrm{x/y}(s_\mathrm{f})$ at the end of the optimization horizon. This property holds as long as the optimization problem is shifted by an integer multiple of the discretization $\Delta s_m$ while the relations of the local path coordinates $s_m$ with the curvature $\kappa(s_m)$ and the acceleration limits $\bar{a}_\mathrm{x}(s_m)$ and $\bar{a}_\mathrm{y}(s_m)$ remain constant. This cannot be ensured since the path planner \cite{Stahl2019} might slightly vary the target path due to an obstacle entering the planning horizon, or due to discretization effects. This leads to a deviation in the curvature profile $\kappa(s_m)$ and deviations in the admissible accelerations $\bar{a}_{x}(s_m)$ and $\bar{a}_{y}(s_m)$ since the local coordinate $s_m$ might refer to a different point in global coordinates $s_\mathrm{glo}$ now, see Fig. \ref{fig:map_discretization}.
To mitigate this deficiency, we introduce slack variables $\epsilon$ based on the exact penalty function approach \cite{Kerrigan2000}. This strategy ensures that the hard-constrained solution is recovered if it is feasible, and therefore the solution is not altered by addition of the slacks unless it is mandatory. The nature of the combined acceleration constraint (\ref{eq:frictionConstraint}) allows for a straightforward interpretation of the slack variables as a violation of $\epsilon$ in \SI{}{\percent}. Together with the upper bound on the slack variables in (\ref{eq:EpsilonConstraint}), we can therefore state that the optimization problem is always feasible as long as the maximum required violation is limited to $\epsilon_{\mathrm{max}}$. In case no solution is found within the specified tolerance band, a dedicated failure-handling strategy is employed within the trajectory planning framework. We wish to point out that a suitable scaling of the slack variables is crucial to achieve sufficiently tight tolerances $\varepsilon_\mathrm{QP,tol}$ when using a numerical \gls{QP} solver. We therefore employ a variable transformation with $\epsilon = \zeta\epsilon_n$ and optimize over $\epsilon_n$ instead. Realistic values for the maximum slack variable $\epsilon_\mathrm{max}$ were found to be around \SI{3}{\percent} in extensive simulations on different race tracks (Berlin (Germany), Hong Kong (China), Indianapolis Motor Speedway (USA), Las Vegas Motor Speedway (USA), Millbrook (UK), Modena (Italy), Monteblanco (Spain), Paris (France), Upper Heyford (UK), Zalazone (Hungary)) and obstacle scenarios. We consider this to be an acceptable tolerance level and believe it will be difficult to achieve significantly tighter guarantees in the face of the scenario complexity we tackle in \cite{Stahl2019}.
\section{Results}
In this section, the results achieved with the presented velocity \gls{mpSQP} will be presented. We conducted the experiments on our Hardware-in-the-Loop simulator, which consists of a Speedgoat Performance real-time target machine, where validated physics models of the real race car in combination with realistic sensor noise are implemented. An additional NVIDIA Drive PX2 receives this sensor feedback and calculates the local trajectories. A Speedgoat Mobile real-time target machine transforms this trajectory input into low level vehicle commands to close the loop to the physics simulation. Therefore, we used the DevBot 2.0 data: $m_\mathrm{v} = \SI{1160}{\kilogram}$, $P_\mathrm{max} = \SI{270}{\kilo\watt}$, $F_\mathrm{max} = \SI{7.1}{\kilo\newton}$, $F_\mathrm{min} = \SI{-20}{\kilo\newton}$, $c_\mathrm{r} = \SI{0.85}{\kilogram\per\meter}$. The results in this section have been produced with the velocity planner parametrizations given in Table \ref{tab:keyfacts_SQP}.
\begin{table}[!tb]
\renewcommand{\arraystretch}{1.1}
\caption{Emergency- and Performance-SQP parametrization.}
\label{tab:keyfacts_SQP}
\centering
\begin{tabular}{|c c c c c c|}
\hline
& Parameter & Unit & \multicolumn{2}{c}{Value} & \\
& & & Performance & Emergency & \\
\hline
& $M$ & - & 115 & 50 & \\
& $N$ & - & 12 & 5 & \\
& $\delta_\mathrm{a}$ & \SI{}{\meter\per\second\squared} & 0.1 & inactive & \\
& $\epsilon_\mathrm{max}$ & \SI{}{\percent} & 3.0 & 3.0 & \\
& $\rho_{\mathrm{j}}$ & - & $3e^2$ & 0.0 & \\
& $\rho_{\mathrm{\epsilon,l}}$ & \SI{}{\meter\squared\per\second\squared} & $1e^5$ & $5e^4$ & \\
& $\rho_{\mathrm{\epsilon,q}}$ & \SI{}{\meter\squared\per\second\squared} & $1e^4$ & $1e^3$ & \\
& $n_{\mathrm{SQP,max}}$ & - & 20 & 20 & \\
& $\Delta t_{\mathrm{max}}$ & \SI{}{\milli\second} & $300$ & $100$ & \\
& $\beta$ & - & $0.5$ & $0.5$ & \\
& $\bar{\varepsilon}_\mathrm{SQP,tol}$ & - & $1e^0$ & $1.5e^0$ & \\
& $\hat{\varepsilon}_\mathrm{SQP,tol}$ & - & $1e^0$ & $1.5e^0$ & \\
& $\varepsilon_\mathrm{QP,tol}$ & - & $1e^{-2}$ & $1e^{-2}$ & \\
\hline
\end{tabular}
\end{table}
We show results for two types of offered paths: performance and emergency. The emergency path is identical to the performance one, except for a coarser spatial discretization, and the fact that the velocity planner tries to stop as soon as possible on the emergency line. The optimization of the emergency line requires therefore fewer variables $M + N$. This formulation reduces the necessary calculation time of the emergency line that must be updated more frequently for safety reasons.
\subsection{Objective function design}
To explain the chosen values of the penalty weights $\rho$, we show the values of the single objective function terms in $J(\boldsymbol{z})$ being minimized during the calculation of the performance velocity profile in Fig. \ref{fig:costPerf}. The symbol $l$ denotes the number of the optimized velocity profiles during the driven lap (including the race start) as well as the vehicle's stopping scenario. It can clearly be seen that the velocity term $J_\mathrm{v}$ has the highest relative impact on the optimum solution. Its value range is at least two orders of magnitude higher than the slack penalty terms $J_\mathrm{\epsilon,l}$, $J_\mathrm{\epsilon,q}$ (not displayed) and the jerk penalty $J_\mathrm{j}$. The penalty weight $\rho_\mathrm{\epsilon,l}$ on the linear slack term was chosen to increase the value of $J_\mathrm{\epsilon,l}$ to be one order of magnitude higher than $J_\mathrm{v}$ if $\epsilon_\mathrm{max}$ was fully exploited on all the slack variables $\epsilon_n$. Therefore, $J_\mathrm{\epsilon,l}$ prevents the solver from permanent usage of tire slack $\epsilon$ for further lap time gains. As $\rho_\mathrm{\epsilon,q}$ is applied to the squared values $\epsilon_n$, it is sufficient to keep the magnitude of $\rho_\mathrm{\epsilon,q}$ one order smaller than $\rho_\mathrm{\epsilon,l}$. To provide a smooth velocity profile, we set the jerk penalty $\rho_\mathrm{j}$ to increase the value of $J_\mathrm{j}$ to be higher than the slack penalty terms during normal operation. By this, effects on a possible lap time loss stay as small as possible, whilst a smoothing effect in the range of numerical oscillations on the velocity and acceleration profile is still visible.
\begin{figure}[!tb]
\centering
\input{./ressources/objective_function/SQP_OSQP_objective_perf.tex}
\caption{Cost terms of the objective function $J$ being minimized within the performance velocity profile. $J_\mathrm{v}$ shows the most significant influence on the solution due to its value range compared to the other objective terms $J_\mathrm{j}$, $J_\mathrm{\epsilon,l}$ and $J_\mathrm{\epsilon, q}$ (not displayed as it equals almost 0). Symbol $l$ denotes the number of the optimized velocity profiles.}
\label{fig:costPerf}
\end{figure}
We further integrated a calculation time limit $\Delta t_\mathrm{max}$ for the velocity optimization and a maximum \gls{SQP} iteration number $n_\mathrm{SQP,max}$. In case of reached limits, the algorithm would return the last suboptimal but driveable solution. The \gls{SQP} never reached these limits during our experiments, and they can be considered as safety limitations. Instead, the \gls{SQP}-algorithm always terminated due to the reached tolerance criteria $\bar{\varepsilon}_\mathrm{SQP,tol}$ and $\hat{\varepsilon}_\mathrm{SQP,tol}$, cf. (\ref{eq:eps_sqp_bar}) and (\ref{eq:eps_sqp_hat}).
\subsection{Energy Strategy}
As stated in Subsection \ref{subsec:efficient_implementation}, the presented \gls{mpSQP} is able to implement our global race \gls{ES} \cite{Herrmann2020, Herrmann2019}. The \gls{ES} is pre-computed offline and re-calculated online due to disturbances, unforeseen events during the race and model uncertainties. Through this, we account for the limited amount of stored battery energy and the thermodynamic limitations of the electric powertrain. Therefore, the \gls{ES} delivers the maximum permissible power $P_\mathrm{max} (s_\mathrm{glo})$ in order to reach the minimum race time, see Fig. \ref{fig:architecture}. It takes the following effects into account:
\begin{itemize}
\item the vehicle dynamics in the form of an \gls{NDTM} including a nonlinear tire model;
\item the electric behavior of battery, power inverters and electric machines, i.e., the power losses of these components during operation;
\item the thermodynamics within the powertrain transforming power losses into temperature contribution.
\end{itemize}
Fig. \ref{fig:optimalPowerUsage} depicts the output of the \gls{ES} computed offline (top). We varied the amount of energy available for one race lap by the three values \SI{100}{\percent} ($E_\mathrm{glo,100}(s_\mathrm{glo})$), \SI{80}{\percent} ($E_\mathrm{glo,80}(s_\mathrm{glo})$) and \SI{60}{\percent} ($E_\mathrm{glo,60}(s_\mathrm{glo})$). The optimal power usage $P_\mathrm{glo}(s_\mathrm{glo})$ belonging to these energy values $E_\mathrm{glo}(s_\mathrm{glo})$ is depicted in the first diagram. The positive values in $P_\mathrm{glo}(s_\mathrm{glo})$ become the parametric input of the power constraint within the velocity optimization (\ref{eq:EnergyConstraint}). By this formulation, we only restrict the vehicle's acceleration power but leave the braking force unaffected. This experiment consists of one race lap with constant maximum acceleration values $\bar{a}_\mathrm{x/y}$ on the Modena (Italy) race circuit.
The power usage $P_\mathrm{loc,80}(s_\mathrm{glo})$ locally planned by the \gls{mpSQP} is shown in the middle plot of Fig. \ref{fig:optimalPowerUsage}. The positive power values in $P_\mathrm{loc,80}(s_\mathrm{glo})$ remained below the maximum power request allowed by the global strategy $P_\mathrm{glo,80}(s_\mathrm{glo})$. Differences between the globally optimal power usage $P_\mathrm{glo}(s_\mathrm{glo})$ and the locally transformed power $P_\mathrm{loc}(s_\mathrm{glo})$ stem from different model equations in both - offline \gls{ES} and online \gls{mpSQP} - optimization algorithms. The velocity planner's point mass is more limited in its combined acceleration potential due to the diamond-shaped acceleration constraint (\ref{eq:frictionConstraint}) compared to the vehicle dynamics model (\gls{NDTM}) in the \gls{ES}. The \gls{NDTM} overshoots the dynamical capability of the car slightly in edge cases due to parameter-tuning difficulties. Furthermore, the effect of longitudinal wheel-load transfer is considered within the \gls{NDTM}. For these reasons, the point mass model accelerates less but meets the maximum admissible power $P_\mathrm{glo}(s_\mathrm{glo})$ on the straights.
\begin{figure}[!tb]
\centering
\input{./ressources/energy_strategy/EMS_power.tex}
\caption{Top: Optimal power usage $P_\mathrm{glo} (s_\mathrm{glo})$ from the \gls{ES} that should be requested whilst driving on the Modena (Italy) race track. Mid: Power planned locally $P_\mathrm{loc,80}(s_\mathrm{glo})$ compared to $P_\mathrm{glo,80} (s_\mathrm{glo})$ resulting in an absolute drift in the energy demand between $E_\mathrm{glo,80}(s_\mathrm{glo})$ and $E_\mathrm{loc,80}(s_\mathrm{glo})$ of approx. \SI{15.8}{\percent} within one race lap (bottom).}
\label{fig:optimalPowerUsage}
\end{figure}
The accumulated error in this experiment can be expressed by the energy demand $E_\mathrm{loc}$ resulting from
\begin{equation}
E_\mathrm{loc} = \int{F_\mathrm{x,p}(s_\mathrm{glo}) \mathrm{d}s_\mathrm{glo}}.
\end{equation}
In total, an energy amount of $E_\mathrm{glo} = \SI{1.27}{\kilo\watt\hour}$ was allowed whereas $E_\mathrm{loc} = \SI{1.07}{\kilo\watt\hour}$ was used, implying \SI{15.8}{\percent} drift. With the help of a re-calculation strategy adjusting the \gls{ES} during the race, this error can be significantly reduced. This feature was switched off during this experiment to isolate the working principle of the race \gls{ES}, i.e., the interaction between global and local planners.
The effect of the \gls{ES} on the vehicle speed $\boldsymbol{v}_\mathrm{P80}$ is depicted in Fig. \ref{fig:ES_local}. The left part (A) of both plots consists of a scenario where the race car is accelerating with the maximum available machine power of $P_\mathrm{max} = \SI{270}{\kilo\watt}$ and a subsequent straight where the \gls{ES} forces the vehicle to coast ($P_\mathrm{glo,80}(s_m) = P_\mathrm{loc,80}(s_m) = \SI{0}{\kilo\watt}$). Acceleration without any power restriction on this straight would have resulted in the velocity curve $\boldsymbol{v}_\mathrm{P100}$ meaning a slightly higher top speed by approx. $\SI{4}{\percent}$ or $\SI{8}{\kilo\meter\per\hour}$. The second part (B) of the shown planning horizon ensures recursive feasibility. Here, we force the velocity variable $v(s_N)$ to reach $v_\mathrm{end}$ (\ref{eq:vendConstraint}).
\begin{figure}[!tb]
\centering
\input{./ressources/energy_strategy/EMS_lift-cost_ModenaE80.tex}
\caption{The effect of the \gls{ES} on one solution of the velocity-planning problem on the vehicle speed $\boldsymbol{v}_\mathrm{P80}$. In part (A), the race car is accelerating with the maximum available machine power of $P_\mathrm{\Sigma} = \SI{270}{\kilo\watt}$ and is forced to coast ($P_\mathrm{glo}(s_m) = P_\mathrm{loc}(s_m) = \SI{0}{\kilo\watt}$) thereafter. Acceleration without any power restriction is denoted by $\boldsymbol{v}_\mathrm{P100}$. Part (B) of the planning horizon ensures recursive feasibility.}
\label{fig:ES_local}
\end{figure}
\subsection{Variable acceleration limits}
\label{subsec:VarFriction}
Fig. \ref{fig:tpamap} shows a locally variable acceleration map to determine the values of $\bar{a}_\mathrm{y} (s)$ along the driven path.
\begin{figure}[!tb]
\centering
\input{./ressources/variable_friction/tpa.tex}
\caption{Race track map displaying the spatially variable acceleration potential $\Sigma_{\bar{a}}(s_\mathrm{glo})$ including the driven path and markers starting from the global $s$-coordinate of \SI{1400}{\meter} at \SI{100}{\meter} gaps.}
\label{fig:tpamap}
\end{figure}
We conducted three experiments with a constant acceleration potential of $\bar{a}_\mathrm{x}$ = \SI{12.5}{\meter\per\second\squared} and varied $\bar{a}_\mathrm{y}$ in three different ways:
\begin{itemize}
\item $\bar{a}_\mathrm{y,cst}^+$ has a constant value of \SI{12.5}{\meter\per\second\squared} along the entire track, describing a high friction potential.
\item $\bar{a}_\mathrm{y,cst}^-$ has a constant value of \SI{6.5}{\meter\per\second\squared} along the entire track, describing a low friction potential.
\item $\bar{a}_\mathrm{y,var}(s)$ has a variable value in the range of \SIrange{6.5}{12.5}{\meter\per\second\squared} with the steepest gradients between $s_\mathrm{glo}$ = \SI{1520}{} - \SI{1550}{\meter} and $s_\mathrm{glo}$ = \SI{1670}{} - \SI{1700}{\meter}, as shown in Fig. \ref{fig:tpamap}.
\end{itemize}
The results of these experiments can be seen in Fig. \ref{fig:velocityComp_varFriction}. We depicted the acceleration potentials $\bar{a}_\mathrm{y}$ of the three scenarios in combination with the planned lateral acceleration $a_\mathrm{y}(s_\mathrm{glo})$ (top) and the vehicle velocity $v(s_\mathrm{glo})$ (bottom).
\begin{figure}[!tb]
\centering
\input{./ressources/variable_friction/var_friction_comparison.tex}
\caption{Comparison of velocity $v(s_\mathrm{glo})$ and lateral acceleration $a_\mathrm{y}(s_\mathrm{glo})$ profiles resulting from spatially and temporally varying acceleration coefficients $\bar{a}_\mathrm{y}(s_\mathrm{glo})$.}
\label{fig:velocityComp_varFriction}
\end{figure}
In both scenarios with constant $\bar{a}_\mathrm{y,cst}$ values, the planner leverages the entire admissible lateral potentials which can be seen between $s_\mathrm{glo}$ = \SI{1500}{} - \SI{1600}{\meter} and $s_\mathrm{glo}$ = \SI{1650}{} - \SI{1700}{\meter}. The results achieved using the variable $\bar{a}_\mathrm{y,var}(s)$ are interesting: The planned lateral acceleration $a_\mathrm{y}(s)$ stayed within the boundaries of the low- and high-friction experiments. The planner handles the drop of $\bar{a}_\mathrm{y,var}(s)$ of \SI{50}{\percent} at $s_\mathrm{glo}$ = \SI{1520}{\meter} by reducing $a_\mathrm{y}(s)$ in advance, while still fully leveraging $\bar{a}_\mathrm{y,var}(s)$. This results in a vehicle velocity of $v(s_\mathrm{glo})$ in the low-friction scenario. The same is true for $s_\mathrm{glo}$ = \SI{1670}{\meter}. Also during the rest of the shown experiment, the oscillating values of $\bar{a}_\mathrm{y,var}(s)$ can be handled by the \gls{mpSQP} algorithm. This behavior allows us to fully exploit the dynamical limits of the race car on a track with variable acceleration potential.
A further indicator that the velocity planner utilizes the full acceleration potential is shown in Fig. \ref{fig:friction_diamond}. Here, we depicted the vehicle's optimized operating points $\mu_\mathrm{o}(s)$ regarding combined acceleration $a_\mathrm{x}(s)$ and $a_\mathrm{y}(s)$ within the planning horizon of $s_\mathrm{glo}$ = \SI{1500}{} - \SI{1800}{\meter}. Both solid diamond shapes express the given constant acceleration limits of the high-friction scenario, $\bar{\mu}_\mathrm{h} = \bar{a}_\mathrm{x}$ \& $\bar{a}_\mathrm{y,cst}^+$ = \SI{12.5}{} \& \SI{12.5}{\meter\per\second\squared}, and the low-friction scenario $\bar{\mu}_\mathrm{l} = \bar{a}_\mathrm{x}$ \& $\bar{a}_\mathrm{y,cst}^-$ = \SI{12.5}{} \& \SI{6.5}{\meter\per\second\squared}. The horizontal red dashed line indicates the maximum available electric machine acceleration $\bar{a}_\mathrm{x,m} = \frac{F_\mathrm{max}}{m_\mathrm{v}}$. From this diagram we see that the vehicle operates at the limits of the given acceleration constraints in both scenarios. This means the \gls{mpSQP} leverages the maximum acceleration potential in combination with fully available cornering potential.
\begin{figure}[!tb]
\centering
\input{./ressources/variable_friction/diamond/friction_diamond.tex}
\caption{Acceleration limit model including operating points at low- and high-friction experiments ($\bar{\mu}_\mathrm{l}$ and $\bar{\mu}_\mathrm{h}$) for a planning horizon ranging from the global coordinate of $s_\mathrm{glo}$ = \SIrange{1500}{1800}{\meter}.}
\label{fig:friction_diamond}
\end{figure}
\subsection{Solver comparison}
The number of variables in the performance profile within one \gls{QP} for the performance trajectory is \SI{126}{}, including \SI{810}{} constraints. From the problem formulation (\ref{eq:discreteObjective}), a small number of non-zero entries in the matrices $\boldsymbol{A}$ and $\boldsymbol{P}$ of \SI{2295}{} in total arises with constant entries in the problem's Hessian $\boldsymbol{P}$.
Fig. \ref{fig:solution_SQPvsNLP} contains a comparison
of the \gls{ADMM} solver \gls{OSQP} \cite{Stellato2020} and the active set solver \gls{qpOASES} \cite{Ferreau2014} as \gls{QP} solvers for an \gls{mpSQP} method as proposed in this paper. Furthermore, we solve the original \gls{NLP} using the interior point solver \gls{IPOPT} \cite{Wachter2006} interfaced via CasADi \cite{Andersson2019} to obtain a measure of solution quality for the proposed \gls{mpSQP}. Note that \gls{IPOPT} is widely used to benchmark the solution quality even if it is not specifically designed for embedded optimization. We chose a scenario where the vehicle is heading towards a narrow right-hand turn at a high velocity of almost \SI{200}{\kilo\meter\per\hour}. The optimization horizon spans this turn including the consecutive straight where positive acceleration occurs. Therefore, the solvers have to deal with high gradients for the longitudinal force $F_\mathrm{x}(s_m)$ and lateral acceleration $a_\mathrm{y}(s_m)$ stemming from curve entry and exit. The velocity plot shows that the optimal solution $\boldsymbol{z}^*$ almost equals the initial guess $\boldsymbol{z}_\mathrm{ini}$ for both the velocity vector $\boldsymbol{v}$ and the slack values $\boldsymbol{\epsilon}$. This behavior is expected, as the previous \gls{SQP} solution $\boldsymbol{z}^{l-1}(s_m)$ is shifted by the traveled distance and used as initialization $\boldsymbol{z}_\mathrm{ini}$. The optimization outputs $\boldsymbol{z}_\mathrm{OSQP}^*$ and $\boldsymbol{z}_\mathrm{IPOPT}^*$ overlap except at the end of the planning horizon where \gls{IPOPT} initially allows more positive longitudinal force $F_\mathrm{x}(s_m)$, resulting in more aggressive braking to fulfill the hard constraint $v_\mathrm{end}$. \gls{qpOASES}'s solution oscillates in the force $F_\mathrm{x}(s_m)$ at the steep gradients. All the algorithms keep the initially given longitudinal force $F_\mathrm{x}(s_\mathrm{s})$ within the specified tolerance of $\pm \SI{0.1}{\kilo\newton}$, with \gls{OSQP} matching the exact value (see magnified section in the second plot). The slack values $\boldsymbol{\epsilon}$ are close to zero. However, the \gls{OSQP}-solution shows small numerical oscillations within a negligible range of approx. $\pm \SI{0.04}{\percent}$. Nevertheless this behavior is typical for an \gls{ADMM} algorithm, and is thus noteworthy.
\begin{figure}[!tb]
\centering
\input{./ressources/solver_benchmark/solution_SQPvsNLP.tex}
\caption{Comparison of the solution of the \gls{mpSQP} (internal \gls{QP}s solved by the \gls{ADMM} solver \gls{OSQP} \cite{Stellato2020}) with the general \gls{NLP} interior point solver \gls{IPOPT} \cite{Wachter2006} interfaced by CasADi \cite{Andersson2019} and the active set solver \gls{qpOASES} \cite{Ferreau2014}. The chosen scenario includes the vehicle heading towards a narrow right-hand bend at a high velocity of almost \SI{200}{\kilo\meter\per\hour}. The optimization horizon spans the curve, and includes the subsequent straight.}
\label{fig:solution_SQPvsNLP}
\end{figure}
Apart from the solution qualities, we further analyzed the velocity optimization runtimes. The scenario consisted of two race laps, including a race start and coming to a standstill after the second lap on the Monteblanco (Spain) race circuit with a variable acceleration potential along the circuit, cf. Subsection \ref{subsec:VarFriction}. The histograms in Fig. \ref{fig:runtime_solvers} display the calculation times $\Delta t_\mathrm{sol}$ for the number of calls $C$ to optimize a speed profile. $C_\mathrm{P}$ and $C_\mathrm{E}$ denote the calls for the performance and the emergency lines, respectively. Their mean values are given by $\Delta \tilde{t}_\mathrm{sol}$. We wish to point out that the algorithm runtimes shown refer to an entire \gls{SQP} optimization process for a speed profile consisting of the solution of several \gls{QP}s in the case of the used solvers \gls{OSQP} or \gls{qpOASES}, which have been warm-started. The optimization runtimes on the specific CPUs are also summarized in Table \ref{tab:runtimes} where the ARM A57 is the NVIDIA Drive PX2 CPU. As we selected \gls{OSQP} for our application on the target hardware, we do not show additional solver times of \gls{IPOPT} or \gls{qpOASES} for the ARM A57 CPU.
Our \gls{mpSQP} in combination with the \gls{QP} solver \gls{OSQP} reaches nearly equal mean runtimes of \SIrange{6}{7}{\milli\second} for both velocity profiles on an Intel i7-7820HQ CPU and \SIrange{32}{34}{\milli\second} on an A57 ARM CPU. An amount of \SIrange{2}{5}{} \gls{SQP} iterations for the performance line was sufficient to reach the defined tolerances $\varepsilon_\mathrm{SQP}$. To optimize the emergency speed profile, a higher amount of \gls{SQP} iterations in the range of 5 - 10 was necessary. Therefore, the computational effort for the iterative linearizations increased on this profile. In contrast, it was possible to solve the single \gls{QP}s in less calculation time, as less than half the number of optimization variables $M + N$ in comparison to the performance profile are present. With maximum computation times of \SI{16.9}{\milli\second} (\SI{73.3}{\milli\second} A57 ARM) on the performance profile and \SI{15.0}{\milli\second} (\SI{77.1}{\milli\second} A57 ARM) for the emergency line, we managed to stay far below our predefined process-timeouts, see Table \ref{tab:keyfacts_SQP}.
The same optimization problem was formulated with the CasADi-language \cite{Andersson2019} as a general \gls{NLP} and passed to the interior point solver \gls{IPOPT}. The \gls{IPOPT} mean solver runtimes are at least approximately five times higher, with maxima of around \SI{0.1}{\second} on the emergency line being too long for vehicle operations at velocities beyond \SI{200}{\kilo\meter\per\hour}. On the emergency profile, the active set solver \gls{qpOASES} beats \gls{IPOPT} slightly in terms of its mean runtime $\Delta \tilde{t}_\mathrm{sol}$ but consumes four times the \gls{IPOPT} computation time to generate the performance speed profile, see Table \ref{tab:runtimes}. This behavior is rational, as a higher number of optimization variables and active constraints increases the calculation speed of an active set solver significantly. Nevertheless, maximum computation times for the \gls{mpSQP} of around \SI{0.5}{\second} on the performance line exclude the \gls{QP} solver \gls{qpOASES} for this type of application.
\begin{figure}[!tb]
\centering
\input{./ressources/solver_benchmark/runtime_SQPvsNLP.tex}
\caption{Solver runtimes for the \gls{mpSQP} with the local \gls{QP}s solved by \gls{OSQP} \cite{Stellato2020} or by the active set solver \gls{qpOASES} \cite{Ferreau2014}, compared to the problem solved by the general \gls{NLP} formulation passed to the interior point solver \gls{IPOPT} interfaced by CasADi \cite{Andersson2019}.}
\label{fig:runtime_solvers}
\end{figure}
\begin{table}[!tb]
\renewcommand{\arraystretch}{1.1}
\caption{Solver mean runtimes.}
\label{tab:runtimes}
\centering
\begin{tabular}{|l c c c c|}
\hline
CPU & \multicolumn{3}{c}{Intel i7-7820HQ} & A57 ARM\\
Prob. formulation & mpSQP & NLP & mpSQP & mpSQP\\
Solver & OSQP & IPOPT & qpOASES & OSQP\\
Performance in \SI{}{\milli\second} & 6.20 & 40.4 & 164 & 32.4\\
Emergency in \SI{}{\milli\second} & 6.95 & 29.8 & 26.1 & 34.2\\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
In this paper we presented a tailored \gls{mpSQP} algorithm capable of adaptive velocity planning in real time for race cars operating at the limits of handling, and velocities above \SI{200}{\kilo\meter\per\hour}. The planner can deal with performance and emergency velocity profiles. Furthermore, the optimization handles multi-parametric input, e.g., from the friction estimation module or the race \gls{ES}. We also specified the boundaries of maximum variation within these parameters to keep the problem feasible. Additionally, we compared different solvers applied to our problem formulation to compare calculation times as well as the solution qualities. Here, our \gls{mpSQP} in combination with the \gls{ADMM} solver \gls{OSQP} outperformed the active set strategy \gls{qpOASES} and the general \gls{NLP} solver \gls{IPOPT} in terms of calculation time, but reached nearly the same solution quality as \gls{IPOPT}. This indicates that the first order \gls{ADMM} in \gls{OSQP} shows its strength for the
minimum-time optimization problem as it handles the large set of active constraints well.
In future work we will apply the presented algorithm in autonomous races and implement a tailored trajectory optimization module based on the presented results and techniques for comparison.
\section*{Contributions \& Acknowledgments}
T. H. initiated the idea of the paper and contributed
significantly to the concept, modeling, implementation and results. A. W. contributed to the design and feasibility analysis of the optimization problem. L. H. contributed essentially to the integration of variable acceleration limits. J. B. contributed to the whole concept of the paper.
M. L. provided a significant contribution to the concept
of the research project. He revised the paper critically for
important intellectual content. M. L. gave final approval for
the publication of this version and is in agreement with all
aspects of the work. As a guarantor, he accepts responsibility
for the overall integrity of this paper.
We would like to thank the Roborace team for giving us the opportunity to work with them and for the use of their vehicles for our research project. We would also like to thank the Bavarian Research Foundation (Bayerische Forschungsstiftung) for funding us in connection with the ``rAIcing'' research project. This work was also conducted with basic research fund of the Institute of Automotive Technology from the Technical University of Munich.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
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June 30, 2016 News Releases Events Diversity and Inclusion
Kandice Head Selected for #NABJNAHJ16 Student Multimedia Project
The National Convention Will Be Held in Washington on Aug. 1-6
Washington (June 30, 2016) — Kandice Head, BJ '16, is among the nationwide pool of students who will participate in the Student Multimedia Project during the National Association of Black Journalists (NABJ) and the National Association of Hispanic Journalists (NAHJ) Convention and Career Fair in Washington, D.C.
Kandice Head
Head, along with the other college students, will receive an all-expenses-paid fellowship to cover the 2016 convention and local stories in the host city. This year's program will take place Aug. 1-6 at the Marriott Wardman Park Hotel.
This will be the second year that Head has been selected for the public relations team. Her responsibilities include covering the conference via social media, writing press releases, directing media availabilities, doing video editing and production, and assisting in media escorts. In Washington, as at the 2015 Minneapolis convention, she says "These are amazing experiences that teach me so much."
A Chicago native, Head studied strategic communication. She is currently an account management intern at the Possible agency in New York. Other work experiences and honors include Fluent360 production intern, a Louis Carr Internship Foundation finalist, a NABJ Public Relations Fellow and a Solomon Public Relations Fellow in 2015, as well as a Teach for America campus campaign coordinator intern in 2014.
NABJ President Sarah Glover said the Student Multimedia Project offers its student members the opportunity to receive on-the-job training from their dedicated professional journalists.
"I'm a proud 'NABJ Baby' from the 1995 student projects and am happy to see this crop of students get a first-class training opportunity at #NABJNAHJ16," Glover said. "The program gives both NABJ and NAHJ students an opportunity to showcase their talents to the membership and help further their journalistic skills."
Mekahlo Medina, NAHJ President, said, "As a former student projects participant and mentor, I know firsthand the value of the great project, and I'm happy we are continuing to develop the next generation of Latino journalists."
The students produce both breaking news and long-form multimedia stories, which are featured in the convention's daily newspaper (print and online), and on a daily newscast. Additionally, the students promote special events, programming, and sponsor-related information on the organization's social media platforms.
Sep 24, 2018 Humans of Strat Comm: Kelsie Wilkins
Feb 06, 2018 Missouri School of Journalism NABJ-MU Student Chapter Visits Leading Washington D.C. Media Outlets
Feb 24, 2017 6 Missouri Students Selected for National Multicultural Advertising Program
Feb 22, 2017 National Association of Black Journalists Visit Chicago for Annual Media Tour
Nov 30, 2016 5th Annual Unity Conference Empowers Millennials in Media Industry
Jun 30, 2016 Kandice Head Selected for #NABJNAHJ16 Student Multimedia Project
May 27, 2016 Francisco Vara-Orta Is One of Five Selected for Inaugural ProPublica Scholarship
Mar 28, 2016 Missouri-Hurley Symposium Offers Insights into Diversity, First Amendment
Dec 21, 2015 Junior Aaron Reiss Selected for National Sports Journalism Institute
Jun 09, 2014 Missouri Chapter of the National Association of Black Journalists Named a Finalist for Student Chapter of the Year
Oct 25, 2013 Master's Student Selected for Alfred Fleishman Diversity Fellows Program at FleishmanHillard
Jan 09, 2013 2 Missouri Journalism Students among the 12 Selected for the National Sports Journalism Institute
Mar 10, 2010 Latina Journalist Cecilia Alvear to Share Her Experiences and Advice with Aspiring Journalists
Mar 03, 2010 NABJ Students Visit The Big Apple
Sep 05, 2007 Senior Wins $5,000 Newhouse Foundation Scholarship from NABJ
Oct 16, 2006 Journalism Student Named Winner in National Hispanic Essay Contest | {
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Learn how Ombudsman empowers students to succeed in school and life.
Providing alternative education programs for middle and high school students who have dropped out or who are at risk of dropping out of school.
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News > Medscape Medical News > Psychiatry News
Iron Deficiency Linked to Psychiatric Disorders in Kids
Megan Brooks
Children and adolescents with iron deficiency anemia (IDA) are at increased risk for psychiatric disorders, including depressive disorder, bipolar disorder, anxiety disorder, and autism, new research shows.
"When encountering those with IDA in clinical practice, prompt iron supplementation should be considered to prevent possible psychiatric sequelae, and vice versa, psychiatrists should check the iron level in those children and adolescents with psychiatric disorders," study author Ya-Mei Bai, MD, PhD, Department of Psychiatry, Taipei Veterans General Hospital and National Yang-Ming University, in Taiwan, told Medscape Medical News.
The study was published online June 4 in BMC Psychiatry.
Impact on Cognitive Development
IDA is prevalent in children, adolescents, and women in nonindustrialized countries, and iron deficiency is the most prevalent nutritional deficiency in industrialized countries, the investigators note.
Iron plays a key role in brain development, including myelination of white matter and the development and functioning of the different neurotransmitter systems, including the dopamine, norepinephrine, and serotonin systems.
"There is well-documented evidence in the literature that IDA has a significant influence on cognitive development, intelligence, and developmental delay. However, the association between IDA and childhood/adolescence-onset psychiatric disorders is rarely investigated," first author Mu-Hong Chen, MD, told Medscape Medical News.
Dr. Mu-Hong Chen
Using the Taiwan National Health Insurance Database and a case-control method, the investigators set out to clarify the association between IDA and various psychiatric disorders. Participants included 2957 children and adolescents in the database with a diagnostic code of iron deficiency anemia and 11,828 healthy control individuals matched for age and sex.
In multiple logistic regression analysis adjusting for demographic data and risk factors related to IDA, children and adolescents with IDA had a higher risk for several psychiatric disorders.
Table. Association Between IDA and Psychiatric Disorders
Disorder Odds Ratio (95% CI)
Unipolar depressive disorder 2.34 (1.59 - 3.46)
Bipolar disorder 5.80 (2.24 - 15.05)
Anxiety disorder 2.17 (1.49 - 3.16)
Autism spectrum disorder 3.08 (1.79 - 5.28)
Attention-deficit/hyperactivity disorder 1.67 (1.29 - 2.17)
Tic disorder 1.70 (1.03 - 2.78)
Delayed development 2.45 (2.00 - 3.00)
Mental retardation 2.70 (2.00 - 3.65)
CI, confidence interval
"This is an interesting and thought-provoking study," said Betsy Lozoff, MD, who studies the effects of iron deficiency in infants at the University of Michigan in Ann Arbor but who was not involved in the study.
"The researchers appropriately note that the observed associations cannot tell if iron deficiency contributed to the psychiatric problems or vice versa," she told Medscape Medical News. "For instance, eating patterns and problems often observed in children with psychiatric disturbances could contribute to nutritional deficiencies, such as iron deficiency. So it is important to avoid making causal inferences. Nonetheless, the findings of this important study warrant further investigation," Dr. Lozoff explained.
Dr. Ya-Mei Ba
Dr. Bai agreed and noted that "further well-designed cohort studies are needed to elucidate the causality or comorbid effect between IDA and psychiatric disorders."
Hopefully, added Dr. Chen, the results will "inspire further studies to investigate the role of iron or IDA in brain development of cognitive function, intelligence, social cognition, and behavioral and emotional regulation."
The authors and Dr. Lozoff report no relevant financial relationships.
BMC Psychiatry. Published online June 4, 2013. Full article
Cite this: Iron Deficiency Linked to Psychiatric Disorders in Kids - Medscape - Jun 24, 2013.
Megan Brooks is a freelance writer for Reuters Health.
Disclosure: Megan Brooks has no relevant financial relationships to disclose.
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{"url":"https:\/\/www.varsitytutors.com\/high_school_math-help\/how-to-find-the-length-of-the-diagonal-of-a-square","text":"High School Math : How to find the length of the diagonal of a square\n\nExample Questions\n\nExample Question #1 : Squares\n\nWhat is the length of a diagonal of a square with a side length ? Round to the nearest tenth.\n\nExplanation:\n\nA square is comprised of two 45-45-90 right triangles. The hypotenuse of a 45-45-90 right triangle follows the rule below, where\u00a0 is the length of the sides.\n\nIn this instance, is equal to 6.\n\nExample Question #1 : Squares\n\nA square has sides of . What is the length of the diagonal of this square?\n\nExplanation:\n\nTo find the diagonal of the square, we effectively cut the square into two triangles.\n\nThe pattern for the sides of a is .\n\nSince two sides are equal to , this triangle will have sides of .\n\nTherefore, the diagonal (the hypotenuse) will have a length of .\n\nExample Question #2 : How To Find The Length Of The Diagonal Of A Square\n\nA square has sides of . What is the length of the diagonal of this square?\n\nExplanation:\n\nTo find the diagonal of the square, we effectively cut the square into two triangles.\n\nThe pattern for the sides of a is .\n\nSince two sides are equal to , this triangle will have sides of .\n\nTherefore, the diagonal (the hypotenuse) will have a length of .\n\nExample Question #4 : Squares\n\nWhat is the length of the diagonal of a square with a side length of ?\n\nExplanation:\n\nTo find the diagonal of a square, we must use the side length to create a 90 degree triangle with side lengths of , and a hypotenuse which is equal to the diagonal.\n\nPythagorean\u2019s Theorem states , where a and b are the legs and c is the hypotenuse.\n\nTake \u00a0and \u00a0and plug them into the equation for \u00a0and\u00a0:\n\nAfter squaring the numbers, add them together:\n\nOnce you have the sum, take the square root of both sides:\n\nSimplify to find the answer: , or .\n\nExample Question #1 : How To Find The Length Of The Diagonal Of A Square\n\nWhat is the length of the diagonal of a 7-by-7 square? (Round to the nearest tenth.)\n\nExplanation:\n\nTo find the diagonal of a square we must use the side lengths to create a 90 degree triangle with side lengths of 7 and a hypotenuse which is equal to\u00a0the diagonal.\n\nWe can use the Pythagorean Theorem here to solve for the hypotenuse of a right triangle.\n\nThe Pythagorean Theorem states , where a and b are the sidelengths and c is the hypotenuse.\n\nPlug the side lengths into the equation as\u00a0 and\u00a0:\n\nSquare the numbers:\n\nAdd the terms on the left side of the equation together:\n\nTake the square root of both sides:\n\nTherefore the length of the diagonal is 9.9.\n\nExample Question #1 : How To Find The Length Of The Diagonal Of A Square\n\nThe perimeter of a square is 48. What is the length of its diagonal?\n\nExplanation:\n\nPerimeter = side * 4\n\n48 = side * 4\n\nSide = 12\n\nWe can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.\n\nTherefore, we can use the Pythagorean Theorem to solve for the diagonal:","date":"2018-06-19 03:09:51","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8861650824546814, \"perplexity\": 257.2459362064204}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267861752.19\/warc\/CC-MAIN-20180619021643-20180619041643-00136.warc.gz\"}"} | null | null |
Biografia
Figlio del critico cinematografico Lino, inizia come assistente e operatore di macchina di Arnaldo Catinari nei primi anni novanta.
Il suo primo cortometraggio come regista è Baci proibiti, con un giovanissimo Pierfrancesco Favino e Elda Alvigini, la storia di un uomo che segue coppie di amanti che si baciano per strada. Il corto è presentato alla 53ª Mostra del Cinema di Venezia (sezione Finestra sulle Immagini) e vince il Premio Vittorio Mezzogiorno. Esce in sala nel 1997, all'interno del film Corti stellari.
Dal 2000 al 2004, inizialmente con Simona Ercolani e poi da solo, Miccichè cura la regia di Sfide, il programma culto di Rai 3 che ha inventato un nuovo modo di raccontare lo sport. Sfide vince nel 2002 il Premio Flaiano e il Premio Regia Televisiva. Nello stesso anno cura la regia delle fiction comiche di Convenscion (Rai 2) con Max Tortora e Tullio Solenghi, ed è regista per Rai 3 di alcuni episodi di Un posto al sole e La squadra 7.
Nel 2007 firma il soggetto e la regia di Liberi di giocare, una miniserie per Rai 1 con Pierfrancesco Favino, Isabella Ferrari, Edoardo Leo e Sabrina Impacciatore. Nel 2008 dirige 5 serate della seconda stagione di Medicina generale con Nicole Grimaudo e Andrea Di Stefano, l'anno successivo alcuni episodi de La nuova squadra, con Rolando Ravello e Marco Giallini.
Per Canale 5 nel 2009 realizza la terza stagione de I liceali con Massimo Poggio e Christiane Filangieri, successivamente la seconda e terza stagione di R.I.S. Roma - Delitti imperfetti con Euridice Axen e Fabio Troiano. Dirige nel 2012 la prima stagione di Benvenuti a tavola - Nord vs Sud, con Fabrizio Bentivoglio e Giorgio Tirabassi, un successo di critica e pubblico per Mediaset. Torna a lavorare per Canale 5 nel 2022 curando la regia della serie tv Anima Gemella (in lavorazione), con Daniele Liotti e Chiara Mastalli.
Nel 2013 scrive, dirige, monta e produce il documentario sul padre: Lino Miccichè, mio padre - Una visione del mondo . Presentato alla 70a Mostra del cinema di Venezia, ottiene una Menzione speciale ai Nastri d'argento per i documentari. Del 2016 è Socialismo. L'ultima utopia, realizzato per Rai 3 e La grande storia. Del 2014 è La Tavola dell'Alleanza (insignito della Medaglia di Rappresentanza del Presidente della Repubblica) e del 2018 è Il Filo dell'Alleanza, documentari che raccontano la realizzazione di due progetti dell'artista Daniela Papadia. Sergio Marchionne per Rai Doc va in onda in prima serata su Rai 3 nel 2021, mentre nel 2022 è presentato alla Festa del Cinema di Roma Er gol de Turone era bono realizzato per Raicinema, che riceve una Menzione Speciale ai Nastri D'Argento 2023.
Miccichè è uno dei più prolifici registi di docufiction e docufilm in Italia. Nel 2016 divide con Giovanni Filippetto la regia di Io sono Libero, primo esperimento di docufiction in prima serata su Rai 1. Al racconto degli ultimi anni di vita di Libero Grassi, ne seguiranno altri, tutti in prima serata su Rai1: Paolo Borsellino - Adesso tocca a me, scritto con Sandrone Dazieri e Giovanni Filippetto e interpretato da Cesare Bocci.; Aldo Moro - Il professore con Sergio Castellitto, scritto con Franco Bernini e Giovanni Filippetto; Figli del destino, realizzato assieme a Marco Spagnoli che vince il Premio Speciale ai Nastri d'argento Doc 2020; La scelta di Maria , scritto con Marco Videtta, interpretato da Sonia Bergamasco, Cesare Bocci e Alessio Vassallo, nominato come miglior docufilm ai Nastri d'Argento 2022 . Nel 2022 scrive con Salvatore De Mola e dirige le docufiction Arnoldo Mondadori, i libri per cambiare il mondo con Michele Placido, nel 2023 andrà in onda Raul Gardini con Fabrizio Bentivoglio e Pilar Fogliati.
Il primo film di Miccichè per il cinema è Loro chi?, con Marco Giallini e Edoardo Leo, codiretto con Fabio Bonifacci e candidato al David di Donatello 2016 nella categoria Miglior regista esordiente. A settembre 2018 esce Ricchi di fantasia con Sergio Castellitto e Sabrina Ferilli , mentre del 2019 è Compromessi sposi con Vincenzo Salemme e Diego Abatantuono.
Filmografia
Cinema
Baci proibiti, cortometraggio che fa parte di Corti stellari (1997)
Loro chi? (2015), Picomedia per Warner Bros Italia
Ricchi di fantasia (2018), Italian International Film per Raicinema
Compromessi sposi (2019), Camaleo e Rinho per Vision Distribution
Fiction
Un posto al sole (dal 2003), soap opera, Rai 3
La squadra 7 (2006), serie tv, Rai 3
Liberi di giocare (2007), miniserie tv, Rai 1
Medicina generale 2 (2008), serie tv, Rai 1
La nuova squadra 2 (2009), serie tv, Rai 3
I liceali 3 (2010), serie tv, Canale 5
R.I.S. Roma 2 - Delitti imperfetti (2011), serie tv, Canale 5
L'olimpiade nascosta (2012), miniserie tv Rai 1
Benvenuti a tavola - Nord vs Sud (2012), serie tv, Canale 5
R.I.S. Roma 3 - Delitti imperfetti (2012), serie tv, Canale 5
Io sono Libero (2016), docufiction, Rai 1
Paolo Borsellino - Adesso tocca a me (2017), docufiction, Rai 1
Aldo Moro - Il professore (2018), docufiction, Rai 1
Figli del destino (2019), docufiction, Rai 1
Io ricordo, Piazza Fontana (2019), docufiction, Rai 1
Io, una giudice popolare al Maxiprocesso (2020), docufiction, Rai 1
La scelta di Maria (2021), docufilm, Rai 1
Riparare il tempo (2021), docufiction, Raicom (soggetto e sceneggiatura)
Arnoldo Mondadori - I libri per cambiare il mondo (2022) docu-drama, Rai 1
Documentari
Mezzano, il passato dal fondo (1995)
Australian Adventure (1998)
La via dei Fori Imperiali (2000), RaisatArte
Sfide (1999-2004), Rai 3
Ice Badile: la libertà costa poco (2011)
Lino Miccichè, mio padre - Una visione del mondo (2013), Rai Movie
La Tavola dell'Alleanza (2014), Istituto Luce
Socialismo. Ultima utopia (2016), Rai 3
Il Filo dell'Alleanza (2018), Istituto Luce
Sergio Marchionne (2021), Raidoc, Rai 3
Er gol de Turone era bono (2022), Raicinema
Note
Collegamenti esterni
Registi cinematografici italiani
Registi televisivi italiani | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,532 |
{"url":"http:\/\/www.sciencemadness.org\/talk\/viewthread.php?tid=15150","text":"Sciencemadness Discussion Board \u00bb Special topics \u00bb Energetic Materials \u00bb Detonation Safety Distance Select A Forum Fundamentals \u00a0 \u00bb Chemistry in General \u00a0 \u00bb Organic Chemistry \u00a0 \u00bb Reagents and Apparatus Acquisition \u00a0 \u00bb Beginnings \u00a0 \u00bb Responsible Practices \u00a0 \u00bb Miscellaneous \u00a0 \u00bb The Wiki Special topics \u00a0 \u00bb Technochemistry \u00a0 \u00bb Energetic Materials \u00a0 \u00bb Biochemistry \u00a0 \u00bb Radiochemistry \u00a0 \u00bb Computational Models and Techniques \u00a0 \u00bb Prepublication Non-chemistry \u00a0 \u00bb Forum Matters \u00a0 \u00bb Legal and Societal Issues \u00a0 \u00bb Detritus \u00a0 \u00bb Test Forum\n\nPages: \u00a01 \u00a0 \u00a03\nAuthor: Subject: Detonation Safety Distance\nholmes1880\ndushbag\n\nPosts: 194\nRegistered: 13-12-2010\nLocation: http:\/\/highexplosivesforum.forumotion.com\/\nMember Is Offline\n\nMood: Egregious\n\nDetonation Safety Distance\n\nI am particularly interested in appropriate critical safety distance from the explosive where you will not be harmed externally\/internally from the blast wave pressure\/brisance. *Let us ignore the shrapnel aspect of this relationship, since shrapnel can get you even at 1km.\n\nNobody is planning to have an accident, but for any contingency procedure it is good to know what risk your are taking while handling energetics. Sadly, the scientific research isn't substantial, so many things have to be based on precedents, personal tests.\n\nUsing 1g NG as the standard unit of measure, let's break down potential critical distances assuming extended arms' length measuring from the fingertips:\n\n0.1g.........1cm\n0.5g.........3 cm\n5g............10cm (ear and eye damage is plausible here)\n50g..........50cm (ear and eye damage is very likely)\n\nThis was largely speculations but things we do know from reported accidents:\n\n1. 0.2g ETN has minimal if any brisance at 2cm(I tested it)\n2. 50ml of MEKP will not dismember wrist at 10cm nor take away vision or hearing (I'm not sure about this report because it could have been a deflagration vs. full order)\n3. 5-10g of TATP detonated in a metal container will not cause permanent loss of hearing or vision(not considering shrapnel).\n4. Hitler survived 1kg of C4-type detonation being with 10 feet.\n\nAny contribution to this database is much appreciated.\nThe WiZard is In\nInternational Hazard\n\nPosts: 1617\nRegistered: 3-4-2010\nMember Is Offline\n\nMood: No Mood\n\n Quote: Originally posted by holmes1880 I am particularly interested in appropriate critical safety distance from the explosive where you will not be harmed externally\/internally from the blast wave pressure\/brisance. *Let us ignore the shrapnel aspect of this relationship, since shrapnel can get you even at 1km.\n\nLet me take this where you probably never considered.\n[I have a talent for this.]\n\nYELVERTON, J.T. et al. 1973. Safe Distances from underwater\nExplosions for Mammals and Birds. Lovelace Foundation for\nMedical Educaticn and Research, Project Report:\nDNA-NWED-M-012.\n\nOr how about quail, chickens, geese or pigeons? \u2013\n\nDamon, E.G. The tolerance of Birds to Airblast. Project Report-\n\nIf you are interested in effects ... and have a strong stomach...\neyeball \u2014\n\nJ Rajs & et al\nExplosion-Related Deaths in Sweden \u2014 A Forensic-Pathologic\nand Criminalistic Study\nForensic Science International\n34 (1987) 1-15\n\nThere is a tremendous amount of info available on blast trauma.\n\nVII.10\nUmbrella Effect of a Landmine Blast\nWar Surgery in Afganistan and Iraq : A Series of Cases, 2003-2007\nSC Nessen & et al Editors\nOffice of the Surgeon General\nUS Army 2008\n\nI a good reminder not to step on a landmine!\n\ndjh\n---\nLandmines are relative \u2014\nBetween you and the enemy \u2014 Bad.\nBetween the enemy and you \u2014 Good.\n\nYOU will have to do the leg work.\n\nholmes1880\ndushbag\n\nPosts: 194\nRegistered: 13-12-2010\nLocation: http:\/\/highexplosivesforum.forumotion.com\/\nMember Is Offline\n\nMood: Egregious\n\nWizard, these are awesome sources As I skimmed the abstracts, I think I forgot to include into consideration the damage to the lungs\/hyperventilation death.\n\nI did a 60g ANNM test few weeks ago, and IF something that powerful goes off 100cm away from you, breathing would definitely get constricted for quite some time.....makes perfect sense.\n\n[Edited on 26-12-2010 by holmes1880]\ngrndpndr\nInternational Hazard\n\nPosts: 508\nRegistered: 9-7-2006\nMember Is Offline\n\nMood: No Mood\n\nBlast injuries sound like a gruesome way to go if your not killed outright.Apparently air filled organs are the most subsceptible.Eardrums will be the first to rupture,if thats all your lucky.Then I dont know?Whether damaged lungs that can show up 48hrs later as it begins to get difficult to breath as your lungs fill w\/fluid(white butterfly) or perhaps your intestines are perforated which will lead to sepsis ,blood poisoning..Slop painful death. Gruesome stuff. Lots of factors involved besides overpressure,lenghth of the blast overpressure-FAE,enclosed space such as a cave etc.\n\nLikely why hitler survived,supposedly protected from direct blast by a hardwood divider in the heavy table.Not secluded in a bunker but what appeared to be a home like structure w\/windows etc.(Best of plans..)\nBlasty\nHazard to Others\n\nPosts: 108\nRegistered: 25-7-2008\nMember Is Offline\n\nMood: No Mood\n\nMany years ago I remember reading about a man who survived an explosion at an early dynamite factory. He was the plant's manager and was standing about 4 meters away from a 900 lb batch of nitroglycerin that accidentally went off. All the workers expected that the guy had been killed. He was actually literally \"blown away\" (i.e. he was sent flying through the air and landed on some trees!) by the explosion but survived to tell the tale.\nquicksilver\nInternational Hazard\n\nPosts: 1820\nRegistered: 7-9-2005\nLocation: Inches from the keyboard....\nMember Is Offline\n\nMood: ~-=SWINGS=-~\n\nMost all the points depicts one extreme lesson: the further the better. Anything you can do to put distance between you and the danger of energetic blast creates a safety element. As distance increases, blast damage diminishes therefore the rules about handling the cap by the leads or the fuse; keeping them away from vital areas of the body, etc. Distance becomes your \"safety zone\" more often than any other single issue.\nThere ARE scales that describe weight levels and proximity but they get back to that vital agenda.\n\nThe WiZard is In\nInternational Hazard\n\nPosts: 1617\nRegistered: 3-4-2010\nMember Is Offline\n\nMood: No Mood\n\n Quote: Originally posted by holmes1880 I am particularly interested in appropriate critical safety distance from the explosive where you will not be harmed externally\/internally from the blast wave pressure\/brisance. *Let us ignore the shrapnel aspect of this relationship, since shrapnel can get you even at 1km.\n\nTo be killed by blast alone requires either a whole bunch of explosives or really bad luck.\n\nEyeballing my ever useful copy of \u2014\n\nWound Ballistics\nOffice of the Surgeon General\nDepartment of the Army 1962\n\n[You can usually find copies for US $25 me thinks.] NB - More than a few stomach turning photos. War is not pretty. p. 105 @ 500 lbs peak pressure sq inch 50% killed 60-100 50% seriously injured. 15 eardrums ruptured \"At the nearest point, peak pressures would be between seven and eight times greater on an object oriented at right angles to the travel of the shock wave; at a distance of 90 feet, the factor would be approximately four; and at 150 feet about three.\" Table 18 Peak pressure in pounds per square inch at varying distances from point of detonation for general-purpose bombs of various weights on a surface parallel to direction of travel of shock wave. At 30 feet 60 &c. &c. 100lb 17 4 500 80 6 1 000 200 20 2 000 400 50 4 000 1 000 170 --------- Deficiencies in the Testing and Classification of Dangerous Materials. J.E. Settles. Annals New York Academy of Sciences Volume 152, Art.1. Pages 199-205. 1968. \"A total of 103 persons suffered injuries in the 81 accidents. Seventy-eight fatalities resulted from these 81 accidents. \"Of the 81 accidents included in this analysis, it was concluded that 23 of them involved only fire, and the principal hazard was radiant heat. It was further concluded that 44 of the accidents involved both fire and explosion. From information available, it seemed justified to assume that no more then 14 of the accidents were characterized by supersonic shock waves that would fall within the accepted definition of \"detonating\" reactions. \"The 14 accidents in which detonating forces were present resulted in injuries to 35 persons and 34 fatalities. It appears from the information available that only one of these 34 deaths resulted from the blast overpressures that are associated with a detonating reaction. However, this one fatality was not the result of blast damage to human tissue. Rather, the blast pressure caused this individual to be propelled as a projectile. The other 33 persons who died in these 14 accidents were located at points where the density of flying fragments, and in some cases, the lethal searing of radiant heat were so great that their deaths were certain, even though there had been no blast effects. \"A serious and disturbing inconsistency is related to the practice of accepting a \"fire hazard only\" label on reactions of such violence and destructive energy as medium-velocity detonation, low-velocity detonations, high-rate explosions, medium-rate explosions, Iow-rate explosions, and even reactions that don't explode at all but kill people by burning them to death.\" djh ---- Don't worry about the bullet with your name on it \u2014 worry about shrapnel labeled occupant. Murhpy's Laws of Combat holmes1880 dushbag Posts: 194 Registered: 13-12-2010 Location: http:\/\/highexplosivesforum.forumotion.com\/ Member Is Offline Mood: Egregious Wizard, Am I understanding that the blast heat generated is more of a lethal factor than the detonation wave? @Blasty, 900lbs of NG at 4m........naw, that's as about impossible to survive as it is a fall from a 20 story building. 900g, I'd go for, but not 900lbs. [Edited on 26-12-2010 by holmes1880] The WiZard is In International Hazard Posts: 1617 Registered: 3-4-2010 Member Is Offline Mood: No Mood Never underestimate bad luck Hirsch, AE & AK Ommaya Lethal Effects on Man of Underwater Detonation of a Firecracker AGARD Conf. Proc. 88:15-1 1971 A firecracker exploded in contact with the skin within six inches of the skull base in a young man while he was swimming underwater. The resulant severe head injury and death appeared to be directly related to this underwater explosion. Reconstruction of the mechanics of his injury indicate that when the head is subjected to impact energies between 440 and 1800 in-lb and impact impulse between 1.8 and 3.5 lb. sec., both skull fracture and brain injury can occur. (Authors abstract) I came upon this in my ever useful copy of \u2014 Shilling and Werts Underwater Medicine and Related Sciences : A Guide to the Literature 2 vols. Plenum Publishing 1973 I have a copy of the paper here somewhere do not be remembering from were\/how I obtained it. Speaking of diving - years ago when I SCUBA'd someone came out with a diver recall device it was a large salute\/firecracker in a 35mm aluminium film can. Underwater it made a crack sounded like a piece of monofilament breaking. Speaking of SCUBA my three best memories are being ... SCUBA diving at Cozumel - swimming behind some sweet young thing... watch her gluteus maximus mussels move her legs. With just a hint of swamp spinach showing from 'round her what Shakespeare called a nest of spices, taint i.e., taint AH - taint pussy. AKA chin rest. Joy, &c., &c. Wall of the Cayman Isl. Amazing down 'bout the 3 ledge like looking down into a vast ink bottle .. jet black. One day the dive boats captain sez \"I was down at 170 feet and I saw you!\" True - however, I didn't think it wise to tell him that I was coming up from 2XX feet. I wanted to get the hand on my depth gauge to touch the stop pin... At that depth with every breath you could see a drop in the air pressure gauge on the tank which was empty when I came up. One quick dive! An who .... after lunch having maxed out the tables we dove in shallow waters. Crystal clear water like swimming in a giant bath tub. Lunch upon this day was chili and cheesewiz... Gawd! Swimming along dropping conch shells on sleeping spotted-eagle rays... O' no ... its shit or die. Back to the surface ... no way am I being able to return to the boat. Back to the bottom... undo tank strap, pull down trunks, pull knees up to chest, roll over and BLOWWWW followed by a could of masticated beans floating on the current through the pristine waters. Absolutely the most comfortable dump I have ever taken. Diving with the seal in the Galapagos. Amazing animals underwater! They swim up and look into your mask, if you put your hand out they muzzle it like a dog, which is good as they have canines that would shame a tiger! Burst you bubbles. An underwater pas du deux. Now back on topic... most info on charge size vs. blast over pressure was done in connection with structures not humans. In simulations of special weapon explosions large amounts of ANFO and spheres of pentolite were detonated. The WiZard is In International Hazard Posts: 1617 Registered: 3-4-2010 Member Is Offline Mood: No Mood Noted in passing explosives underwater The use of explosives underwater by diver is covered briefly in \u2014 Ye. P. Shikanov, Editor Spravochnik Vodolaza (Handbook for Divers) Moscow, Voyenizdat 1973 321 pages. English translation by Joint Publications Research Service, JPRS 60691. Sold by the NTIS. My useful copy of the 1970 U.S. Navy Diving Manual NAVSHIPS 0994-001-9010 p.94 change 1. Appx. total force of an underwater blast wave. P= 13,000 (cube root) W ---------------------------- d P = force in ponds per square inch W = weight of explosives in pounds d= distance of explosion from diver (feet) A pressure of 500 psi is sufficient to cause injury to the lungs and intestinal track. Associated details follow. This is repeated in \u2014 the NOAA Diving manual : Diving for Science and Technology Manned Undersea Science and Technology Office US Dept of Commerce 1975 US Gov. Printing Office Not found Handbook U.S. Navy Diving Operations 1 April 1974 NAVSHIPS 0994-009-6010 Nice book the pages are plastic so you can them out diving and not worry 'bout them getting wet. Should take my copy and rinse the saltwater off it! NB \u2014 This search was limited to what I shelve. holmes1880 dushbag Posts: 194 Registered: 13-12-2010 Location: http:\/\/highexplosivesforum.forumotion.com\/ Member Is Offline Mood: Egregious Since a lot of underwater discussion is coming up, I have done a few underwater detonations, and I would NEVER dare to stick my arm in the water when it goes off, let alone jump in for a quick dip. When doing it off the concrete bank, it is absolutely humbling how hard the shockwave hits the reinforced concrete wall......it feels like someone drove a pick-up truck in it. P.S. Wizard, that story about scuba dump was epic! I'd be concerned doing it at a depths.....I'm not sure how the pressure could affect things, The WiZard is In International Hazard Posts: 1617 Registered: 3-4-2010 Member Is Offline Mood: No Mood Quote: Originally posted by holmes1880 P.S. Wizard, that story about scuba dump was epic! I'd be concerned doing it at a depths.....I'm not sure how the pressure could affect things, ---- I have two on land dump stories. I will spare you (all of you) the sorted details! There is a famous workman comp. case. A diver was in la decompression chamber on the throne when someone outside opened the wrong valve at the wrong time! Sucked his guts out through his ..... YAHOOOO-OO! They couldn't remove him from the chamber so they brought in a surgeon who operated on him in situ! I had suggested this to La Myth Busters as a follow up to their airplane toilet myth....! djh ---- Penthouse magazine founder Bob Guccione dies at 79 21x10 DALLAS \u2013 Bob Guccione tried the seminary and spent years trying to make it as an artist before he found the niche that Hugh Hefner left for him in the late 1960s. Where Hefner's Playboy magazine strove to surround its pinups with an upscale image, Guccione aimed for something a little more direct with Penthouse. Something a little more direct. Gynecology through the picture study method comes to mind. \/djh\/ The WiZard is In International Hazard Posts: 1617 Registered: 3-4-2010 Member Is Offline Mood: No Mood Quote: Originally posted by The WiZard is In The use of explosives underwater by diver is covered briefly in \u2014 NB \u2014 This search was limited to what I shelve. Back from the shelves - I had forgotten my favorite diving book it is a fun read. Nicholas B Zinkowski Commercial Oil-Field Diving Cornell Maritime Press 2nd edition 1978 Chapter 8 - Use of explosives. p. 45 Personal Problems Underwater \"I have known many divers who habitually and unabashedly urinate in their dress, considering it one of the evils of the trade.\" Well sure in a dry suit, however, in a wet suit \u2014 one of life's greatest and free pleasures! Anyone remember the scene with John Glenn while in his Mercury capsule. Movie \"The Right Stuff\"? djh ---- The problem with l'ivresse des grandes profondeurs [rapture of the deep] is that you cannot lay-back and enjoy it for fear of loosing control. I remember one female carbon based unit who overtaken by it dropped her camera. (Cozumel Isl. QR Mexico.) Id following behind (hers) swam down and retrieved it. Then swam up and brought her back to the reality that she was way tooooo deep and should ascend. holmes1880 dushbag Posts: 194 Registered: 13-12-2010 Location: http:\/\/highexplosivesforum.forumotion.com\/ Member Is Offline Mood: Egregious The pressure chamber incident is just....wow. I'm uncomfortable just thinking of what that could feel like. I have a very useful inquiry\/proposition for safety with charges. The particular explosive in mind is KinePak, where we basically add NM after the initiator is already inserted in the charge. Assuming NM does not sensitize NH4NO3 right away (the key assumption), it gives time to safely insert the initiator and then back up from the charge. The only thing to worry about is the small detonator. Any idea whether AN is #6cap sensitive immediately after adding NM, or does it take a few minutes? I'll probably have to test it out, but hate to waste good charge if it fails, even though that's the desired result. I got the idea for this method from this video: http:\/\/www.youtube.com\/watch?v=nkYJ2-eDNEc [Edited on 26-12-2010 by holmes1880] Sickman Hazard to Self Posts: 98 Registered: 9-5-2004 Member Is Offline Mood: Icy and I see! [rquote @Blasty, 900lbs of NG at 4m........naw, that's as about impossible to survive as it is a fall from a 20 story building. 900g, I'd go for, but not 900lbs. [Edited on 26-12-2010 by holmes1880][\/rquote] May have been a low order det. 50 psi seems to be the LD50 for human exposure to blast overpressure: Blast waves and the human body The WiZard is In International Hazard Posts: 1617 Registered: 3-4-2010 Member Is Offline Mood: No Mood Quote: Originally posted by holmes1880 I got the idea for this method from this video: http:\/\/www.youtube.com\/watch?v=nkYJ2-eDNEc Curious most LOX uses an absorbent combustible. See \u2014 US Bureau of Mines RI 3169 Absorbents for Liquid-Oxygen Explosives April, 1932 ---- US Bureau of Mines Technical Paper 294 Progress of Investigations on Liquid-Oxygen Explosives 1923 (Can be DL'd) Number 52A5. 22.1% aluminium dust 77.9% wood pulp Lead block 1.44 that of 40% dynamite. [Straight?] I expect silicon would fail the$ test. I also would like\nto know how they computed the power of their mixture.\n\nFor an interesting report on actual use of LOX in mining I commend \u2014\n\nMH Kurlya and GH Clevenge\nLiquid-oxygen Explosives a Pachuca\nTransactions American Institute of Mining, Metallurgical and\nPetroleum Engineers\nPresented at the New York meeting, February 1923. p. 271-340.\n\nThe on site LO2 plant produced 25 l. (27.5 kg, 60.6 lb per hour.)\n\nThe life of their cartilages max out at 11.5 minutes. As has\nbeen stated - LOX explosives cartridges are the ultimate safety explosive \u2014\nafter a few minutes they are no longer explosive.\n\nI was thinking of magic - once it is explained it is no longer\nmagic. I own this paper because a few years back I found\na bound copy of the complete year for sale at a good price.\nUpon receipt I dis-bound it. Currently I am unable to find\nthe residuum.\nThe WiZard is In\nInternational Hazard\n\nPosts: 1617\nRegistered: 3-4-2010\nMember Is Offline\n\nMood: No Mood\n\nLOX Strength\n\nTransactions American Institute of Mining, Metallurgical and\nPetroleum Engineers\nG St. J. Perrott\nProperties of Liquid-oxygen Explosives 1925\np1248-1275\n\nTable 6.\u2014Propulsive Strength of L.O.X. Cartridges as Measured\nby Ballistic Pendulum\n\nRelative Strength by Volume Compared to Dynamite [40%]\n\n1.16 Gas Black S, cartridge density 0.33\n1.14 Four way tie.\n1.13 One\n\nholmes1880\ndushbag\n\nPosts: 194\nRegistered: 13-12-2010\nLocation: http:\/\/highexplosivesforum.forumotion.com\/\nMember Is Offline\n\nMood: Egregious\n\nSo, no info on how long it takes for NM to sensitize AN? Gerald Hurst's patent is no help, I skimmed it, and ............................nothing regarding duration.\nBlasty\nHazard to Others\n\nPosts: 108\nRegistered: 25-7-2008\nMember Is Offline\n\nMood: No Mood\n\n Quote: Originally posted by holmes1880 @Blasty, 900lbs of NG at 4m........naw, that's as about impossible to survive... 900g, I'd go for, but not 900lbs.\n\n\"Compare our next nitroglycerine incident. In 1886 Mr. Wilson P. Foss was manager of the Clinton Dynamite Company's plant at Plattsburgh, New York. He was inside a building and standing four yards from a wash tank that contained 900 lbs of nitroglycerine when an accidental gush of live steam from an open valve detonated the nitroglycerine, destroying the building and leaving a crater 30 ft deep. Workers at the plant hurried to the site, expecting to find Mr. Foss's remains among the wreckage. He had, however, been blown by the force of the explosion out of sight round the bend of a nearby frozen river, and to the startled incredulity of the workforce he reappeared to them striding on the ice around the river bend.\" - John Bond, \"The hazards of life and all that: a look at some accidents and safety curiosities, past and present\", CRC Press, 1996. Page 63.\n\n Quote: as it is a fall from a 20 story building.\n\nhttp:\/\/untoldvalor.blogspot.com\/2007\/07\/alan-mcgee-luckiest-...\n\nDon't underestimate the quite frankly bizarre survival luck some people have had.\n\n[Edited on 27-12-2010 by Blasty]\ncrazedguy\nHazard to Others\n\nPosts: 143\nRegistered: 12-11-2010\nMember Is Offline\n\nMood: You can't fix stupid\n\n@ Blasty: The guy surviving 900 lbs of NG seems believable to be so close you are pushed with the pressure wave rather than crushed by it, but I have heard that story of the 20,000 foot fall before and find a piece of glass slowing the impact enough to save someones life hard to believe.\n\nWarning: i do stupid things\nsimply RED\nHazard to Others\n\nPosts: 182\nRegistered: 18-8-2005\nLocation: Stelianovsk\nMember Is Offline\n\nMood: booM\n\n\"50g..........50cm\"\n\nNo gun - No fun!\nThe WiZard is In\nInternational Hazard\n\nPosts: 1617\nRegistered: 3-4-2010\nMember Is Offline\n\nMood: No Mood\n\n Quote: Originally posted by crazedguy @ Blasty: The guy surviving 900 lbs of NG seems believable to be so close you are pushed with the pressure wave rather than crushed by it, but I have heard that story of the 20,000 foot fall before and find a piece of glass slowing the impact enough to save someones life hard to believe.\n\n------\nWell the 900 NG story is A- true and B- has nuances. There being a\ngood description in Vangleder and Schlatter, I will scan it after\nI finish reading the news on line and going out in the 6.3o F\ncold to move some of the snow around.\nbbartlog\nInternational Hazard\n\nPosts: 1139\nRegistered: 27-8-2009\nLocation: Unmoored in time\nMember Is Offline\n\nMood: No Mood\n\nThing with 900 lbs of NG, one could readily believe that there might be some initial deflagration that would push the victim\/survivor fast and far enough that once the DDT occurred he was already underway, so to speak, and thus at a distance where the real shockwave only accelerated him rather than crushing him to death. In cases like these our theoretical models about what should happen must give way to actual reports.\nThe WiZard is In\nInternational Hazard\n\nPosts: 1617\nRegistered: 3-4-2010\nMember Is Offline\n\nMood: No Mood\n\nQuote: Originally posted by The WiZard is In\n Quote: Originally posted by crazedguy @ Blasty: The guy surviving 900 lbs of NG seems believable to be so close you are pushed with the pressure wave rather than crushed by it, but I have heard that story of the 20,000 foot fall before and find a piece of glass slowing the impact enough to save someones life hard to believe.\n\n------\nWell the 900 NG story is A- true and B- has nuances. There being a\ngood description in Vangleder and Schlatter, I will scan it after\nI finish reading the news on line and going out in the 6.3o F\ncold to move some of the snow around.\n\nFinished moving the snow around, you should have seen the\nsmile on the Snow Man's face when he heard the snow blower\ncoming!\n\nBroke the plow blade off my Gator, snapped 1\/2\" bolt, large\nspring flew off into the snow, hope to find it comes April of May.\n\n\\\n\nI shelve a 1998 Ayer Company aka Arno Press reprint of G & S 1927.\nJimbo Jones\nHazard to Others\n\nPosts: 102\nRegistered: 15-10-2009\nMember Is Offline\n\nMood: No Mood\n\n Quote: Originally posted by simply RED \"50g..........50cm\" Hahahahaha, that made my day....\n\nNice way to clean your ear wax.\n\nBy the way, Hitler \u201csurvived, as did everyone else who was shielded from the blast by the conference table leg.\u201d\n\nThe lightweight, wooden structure of the building loses even more from the performance of the used \u201cplastic explosive\u201d. If this 1 kilo was placed in bunker\u2026..\nPages: \u00a01 \u00a0 \u00a03\n\n Sciencemadness Discussion Board \u00bb Special topics \u00bb Energetic Materials \u00bb Detonation Safety Distance Select A Forum Fundamentals \u00a0 \u00bb Chemistry in General \u00a0 \u00bb Organic Chemistry \u00a0 \u00bb Reagents and Apparatus Acquisition \u00a0 \u00bb Beginnings \u00a0 \u00bb Responsible Practices \u00a0 \u00bb Miscellaneous \u00a0 \u00bb The Wiki Special topics \u00a0 \u00bb Technochemistry \u00a0 \u00bb Energetic Materials \u00a0 \u00bb Biochemistry \u00a0 \u00bb Radiochemistry \u00a0 \u00bb Computational Models and Techniques \u00a0 \u00bb Prepublication Non-chemistry \u00a0 \u00bb Forum Matters \u00a0 \u00bb Legal and Societal Issues \u00a0 \u00bb Detritus \u00a0 \u00bb Test Forum","date":"2021-02-28 22:11:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22868143022060394, \"perplexity\": 11928.97222376346}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178361776.13\/warc\/CC-MAIN-20210228205741-20210228235741-00137.warc.gz\"}"} | null | null |
Fabricated the control stick and torque tube. The tube mounts via rod end bearings to the fuse crossmembers.
This is what builders of plans-built aircraft spend a lot of time doing....staring at the task at hand and mentally running through options. It doesn't take much time to come up with a solution, but it may take much more time to resolve the simplest, most minimal design which is nearly always the best solution.
There isn't much guidance on fuel tank mounting in the D.VII docs, just the suggestion to use foam pipe insulation as shock absorbers. The trick is to come up with a simple design that allows the tank to be easily removed but still safely supports eleven gallons of gas without compromising the tank.
Here is the forward tank floor support I decided upon. While rummaging through my box of scraps, I found a piece of left-over 1.25" 6061 with a part number from my first homebuilt project over twenty years ago, a Fish Super Koala. I chuckled at the irony of flying a part of that long-gone aircraft in the Fokker.
The tube was coped for a tight fight to the fuse upright, riveted with a couple of .187" steel rivets, then a gusset added for additional security. This tube can be permanently fixed because the tank can be slid back enough to clear this tube.
The other tank support is a left-over length of 1/2" 4130 from previous projects (never throw anything away!). Because this support must be removed in order for the tank to drop out, it is secured with AN-3 bolts. Final version will have a full length piece of automotive rubber tubing as a shock mount. I'll also weld on tabs that will capture the rear corner of the tank to prevent it from sliding backward. The steel tube is heavier than aluminum but allows me to use smaller diameter to prevent interference with the pilot's toes.
In totality, the tank installation looks pretty good. The pipe insulation wedges the tank into place and my bottom supports hold it tight up against the fuse members. Removing the side panels will result in good access to the tank if it needs to be removed.
Earlier in the day I fabricated the instrument panel and firewall. I'm waiting on new turtledeck and side panel sheets to replace those damaged in shipping.
Rear tank support with straps welded in place to prevent tank from sliding back. They will be padded during final installation.
After establishing hard mounting points for the forward fuse sheet metal by installing a ton of K1000-8 nutplates, the firewall, side panels and belly pan are in place.
Aluminum angles were riveted to various fuse tubes to carry the nutplates. The nutplates were installed with squeezed flush-head rivets so the panels will lie flat against the angles. I installed a belly pan so the fuse can be opened up for service--it should also stand up better to engine residue than fabric.
Aluminum angle was cut with saw kerfs so it would smoothly follow the contour of the instrument panel and carry nutplates to attach the turtledeck.
Same thing with the firewall. Close to 50 nutplates have been installed so far but they will pay off with ease of service access when the time comes for the condition inspection.
The firewall shipped with the kit is 6061 aluminum which offers practically no protection for the fuel tank if there is an engine fire. After pondering this for many weeks I decided I needed a real firewall.
A 24" x 24" piece of 26g (0.022") galvanized steel was purchased locally and a new firewall constructed. I retained the upper portion of the original firewall because I had already fabricated the bowed angle with platenuts. The steel extends above the tank and satisfies my concerns. We picked up a pound or two of weight but this won't matter if the firewall is ever called upon to do its job.
Looking more like a D.VII with the sheet metal in place. Figuring out how to sequence all the panels so they can be removed from the finished aircraft is an interesting mental exercise. It is necessary to think from the inside to the outside to avoid boxing in something that won't come out later.
Getting the deck panel measured, cut, and fitted is a major milestone. Here is the scrap sheet of aluminum used for trial fitting. It then was refined with paper masks to make a pattern. The challenge is getting a close fit around the cabanes and over the curved firewall and instrument panel....and putting the bends in exactly the right place.
These angles were riveted to the fuse side panels so wing root fairings could be attached. There is a gap in the angles so the side panel can be easily removed for service. The rear angles will be permanently attached after the fuse is covered.
A finished root fairing. A pair of 1.5" holes allow the lift wires to attached to the lift tangs.
The D.VII didn't have a windscreen but I want one to prevent collisions with hard-shell bugs. A couple sheets of Lexan were mocked up for a simple windscreen with careful measurements and a cardboard template.
Aluminum angles and brackets were fabricated and all the pieces attached with 1/8" rivets. This screen is inexpensive and easily modified or replaced if flight test prove this one is ineffective.
Here are exits for the rudder cables. This will provide a place to anchor fabric.
Similar exit for the elevator pushrod. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,209 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/469330\/doubt-on-derivation-of-ols-estimators-as-unbiased-estimators-of-optimal-linear-p","text":"# Doubt on derivation of OLS estimators as unbiased estimators of Optimal Linear Predictors\n\nI'm studying from C. Shalizi's lecture notes https:\/\/www.stat.cmu.edu\/~cshalizi\/ADAfaEPoV\/ .\n\nIn the third chapter he introduces the optimal linear estimator of a random variable $$Y$$ conditioned to another (possibly vector) $$X$$: $$f(X)=\\beta X,\\qquad \\beta = \\frac{1}{\\text {Cov}(X,X)}\\text {Cov}(X,Y).$$\n\nDefining the error $$Y-f(X)=\\epsilon$$ he states that, in general, $$\\mathbb E(\\epsilon|X)\\neq 0$$, which I understand.\n\nHowever, at page 45 he is proving that the Ordinary Least Squares estimators $$\\hat \\beta$$ give unbiased estimates of $$\\beta$$ (as far as I understand, without any assumption about the actual correctness of the linear model). Here's the derivation.\n\nMy confusion concerns the step from Eq. (2.24) to (2.25), i.e. the second $$+0$$. Isn't he assuming here that the conditional expectation is $$\\mathbb E (\\epsilon \\vert X)=0$$? And, relatedly, why in Eq. (2.24) has the $$\\mathbb E(|\\boldsymbol X = \\boldsymbol x)$$ for $$\\mathbb \\epsilon$$ been replaced by an apparently unconditional expectation mean?\n\nAfter some thought I realized this is probably just an error\/typo from the author, which really meant that the unconditional expectation (averaged over the data set $$\\boldsymbol X=\\boldsymbol x$$) of the $$\\hat \\beta$$ estimator is equal to $$\\beta$$. Indeed, it doesn't make much sense to think of being able to estimate the full regression line by making repeated measurements of $$Y$$ for few fixed values of $$X$$... unless the truth is a linear model, for sure :-)\n\nIf nobody comes up with corrections or anything to add, I will add the above as answer.\n\n\u2022 yes. what you say is correct. formally, it seems that the author is using what is referred to as the tower property: $E(E(\\epsilon | X = x)) = E(\\epsilon) = 0$. May 29 '20 at 22:36\n\nThe linear coefficient estimator(s) $$\\hat \\beta$$ is conditionally (over the input data) unbiased only if the underlying process is truly linear.\nOn the other hand, $$\\hat \\beta$$ is an unconditionally unbiased estimator of the optimal linear predictor. This can be formally proved by integrating Eq. (2.24) over the marginal $$X$$ distribution and using the tower property as pointed out by @mlofton.","date":"2021-10-27 12:52:19","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\": 19, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.93205726146698, \"perplexity\": 355.88538657595245}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588153.7\/warc\/CC-MAIN-20211027115745-20211027145745-00506.warc.gz\"}"} | null | null |
Sporting Club Grants Program Open
Sporting groups and organisations in the Murray Plains Electorate can benefit from the $4.6 million Sporting Club Grants Program which is now open.
Leader of The Nationals and Member for Murray Plains Peter Walsh said the program is designed to give clubs the tools and resources they need to keep them viable and competitive.
"Our sporting clubs can put some new life into their club with new uniforms modern equipment increased knowledge and stronger operational capacity " Mr Walsh said.
"Many of our local sporting groups and organisations have stretched budgets and this program assists those involved with sporting groups to represent their club and region to the best of their ability " he said.
"It is hoped the grants will also help to attract and accommodate new clubs members nurture talent and overcome financial barriers that might prevent some people from being more active more often " Mr Walsh said.
The program has three funding categories:
Category 1: Uniforms or Equipment
Grants up to $1000 to purchase uniforms or other equipment essential for participation.
Category 2: Skill Development
Grants up to $2500 to improve the skills of club members by providing training for coaches
officials administration staff and management committees.
Category 3: Club Operational Capacity
Grants up to $5000 to improve the operational effectiveness and efficiency of clubs through strategic planning or to increase community participation through accessible sport and active recreation opportunities.
Applications are now open and will close on 3rd March 2016.
For more information or to apply for a grant please http://sport.vic.gov.au/grants-and-programs/sporting-club-grants-program
Category: MediaBy Peter Walsh MP 21 January 2016
PreviousPrevious post:Bridge – Get On With ItNextNext post:Free Travel Doesn't Buy Country People | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 621 |
Q: Flask POST sign up form retrieve value of dropdown box I am using this HTML script for a sign up form in flask and the code will not let me retrieve the data in the same way I retrieve the username and password using request.form['']. I would like to pass the value of the option a user selects from a dropdown to my python program. Is there any way to solve this issue? Thanks in advance for any responses!
<select id="cell" name="carrier" data-name="cellcarrier">
<option value="verizon">Verizon</option>
<option value="att">At&t</option>
<option value="tmobile">T-Mobile</option>
<option value="sprint">Sprint</option>
<option value="alltel">AllTel</option>
<option value="boost">Boost Mobile</option>
<option value="cricket">Cricket</option>
<option value="metropcs">MetroPCS</option>
<option value="uscellular">US Cellular</option>
<option value="virginmobile">Virgin Mobile</option>
<option value="republicwireless">Republic Wireless</option>
</select>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,601 |
package techreborn.client.render.entitys;
import net.minecraft.client.render.VertexConsumerProvider;
import net.minecraft.client.render.block.BlockRenderManager;
import net.minecraft.client.render.entity.EntityRenderer;
import net.minecraft.client.render.entity.EntityRendererFactory;
import net.minecraft.client.render.entity.TntMinecartEntityRenderer;
import net.minecraft.client.util.math.MatrixStack;
import net.minecraft.screen.PlayerScreenHandler;
import net.minecraft.util.Identifier;
import net.minecraft.util.math.MathHelper;
import net.minecraft.util.math.Vec3f;
import org.jetbrains.annotations.Nullable;
import techreborn.entities.EntityNukePrimed;
import techreborn.init.TRContent;
/**
* Created by Mark on 13/03/2016.
*/
public class NukeRenderer extends EntityRenderer<EntityNukePrimed> {
private final BlockRenderManager blockRenderManager;
public NukeRenderer(EntityRendererFactory.Context ctx) {
super(ctx);
this.shadowRadius = 0.5F;
this.blockRenderManager = ctx.getBlockRenderManager();
}
@Nullable
@Override
public Identifier getTexture(EntityNukePrimed entityNukePrimed) {
return PlayerScreenHandler.BLOCK_ATLAS_TEXTURE;
}
@Override
public void render(EntityNukePrimed entity, float f, float g, MatrixStack matrixStack, VertexConsumerProvider vertexConsumerProvider, int i) {
matrixStack.push();
matrixStack.translate(0.0D, 0.5D, 0.0D);
if ((float) entity.getFuse() - g + 1.0F < 10.0F) {
float h = 1.0F - ((float) entity.getFuse() - g + 1.0F) / 10.0F;
h = MathHelper.clamp(h, 0.0F, 1.0F);
h *= h;
h *= h;
float j = 1.0F + h * 0.3F;
matrixStack.scale(j, j, j);
}
matrixStack.multiply(Vec3f.POSITIVE_Y.getDegreesQuaternion(-90.0F));
matrixStack.translate(-0.5D, -0.5D, 0.5D);
TntMinecartEntityRenderer.renderFlashingBlock(blockRenderManager, TRContent.NUKE.getDefaultState(), matrixStack, vertexConsumerProvider, i, entity.getFuse() / 5 % 2 == 0);
matrixStack.pop();
super.render(entity, f, g, matrixStack, vertexConsumerProvider, i);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,910 |
{"url":"http:\/\/mathhelpforum.com\/math-topics\/207213-mechanics-new-question-just-need-slight-help.html","text":"# Math Help - Mechanics new question just need a slight help\n\n1. ## Mechanics new question just need a slight help\n\ni couldnt solve the last part where u have to find the resultant i know that the forumla to find the resultant but yet cant find it. I know that horizantal forces is 192cos50 but what would be the sum of vertical forces help\n\n2. ## Re: Mechanics new question just need a slight help\n\n$T_1 \\cos(40) = T_2 \\cos(50)$\n\n$T_1 = \\frac{T_2 \\cos(50)}{\\cos(40)}$\n\n$T_1 \\sin(40) + T_2 \\sin(50) + 150 = 400$\n\n$T_1 \\sin(40) + T_2 \\sin(50) = 250$\n\n$\\frac{T_2 \\cos(50)\\sin(40)}{\\cos(40)} + T_2 \\sin(50) = 250$\n\n$T_2\\left[\\cos(50)\\tan(40) + \\sin(50)] = 250$\n\n$T_2 = 192 \\, N$\n\n$T_1 = 161 \\, N$\n\nforces on the pulley in the x-direction (leftward) ...\n\n$T_2 \\cos(50)$\n\nforces on the pulley in the y-direction (downward) ...\n\n$T_2 \\sin(50) + T_2$\n\nResolving forces acting on the pulley ...\n\n$|F| = \\sqrt{[T_2 \\cos(50)]^2 + [T_2 \\sin(50) + T_2]^2} = 360 \\, N$\n\n$\\theta = \\arctan\\left[\\frac{ T_2 \\sin(50) + T_2}{T_2 \\cos(50)}\\right] = 70^\\circ$ relative to the horizontal ... same as $20^\\circ$ relative to the vertical\n\n3. ## Re: Mechanics new question just need a slight help\n\nu deserve a million thanks well i just wanted to know what would be the vertical forces well thanks anyway u did a too much writing\n\n4. ## Re: Mechanics new question just need a slight help\n\nOriginally Posted by abdulrehmanshah\nu deserve a million thanks well i just wanted to know what would be the vertical forces well thanks anyway u did a too much writing\nNote that I have to do the problem from the start so that I understand how the entire problem develops.\n\nIt's like coming into a movie after it has been running a while ... you really do not understand what is going on.","date":"2015-05-22 15:40:01","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 13, \"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\": 0.5783059597015381, \"perplexity\": 1018.8635514443425}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-22\/segments\/1432207925274.34\/warc\/CC-MAIN-20150521113205-00162-ip-10-180-206-219.ec2.internal.warc.gz\"}"} | null | null |
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