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IF Comp 2017: Insignificant Little Vermin
Posted on November 14, 2017	by Sam Kabo Ashwell
Insignificant Little Vermin (Filip Hracek) is a heroic-fantasy dungeon-crawl; the protagonist is an escaped slave in an orcish stronghold, and must fight their way out while doing as much damage as possible.
There are illustrations. These are at their strongest when showing posed figures and details of weapons, less confident when it comes to action scenes, and largely ignore setting.
It's made in a custom choice-based system, designed with mobile devices in mind and centred around combat. There is a simple randomisation system, presented as a five-reel slot machine. I'll be up-front: this is a game with a heavy focus on combat that is not very good at fight scenes, either in terms of writing or choice.
Here's the game's first combat option:
Nobody else is in sight. It's just you, Agruth, and Briana. That's Agruth's first mistake.
Kick Agruth to the ground.
Leap at Agruth.
Punch Agruth.
So those are all basically versions of 'attack'. There is clearly some kind of combat system underlying this, but how it works is not communicated well to the player. Some of this is just about choice of words: 'kick Agruth to the ground' turns out to be some kind of tripping action, 'leap' is tackling to the ground, but both of those phrases are easily read as something else. But in general, you're being asked to make tactical choices based on a very poor idea of what those choices mean. At a certain point you get the choice to 'confuse' opponents: the description when you do it suggests some kind of magical attack, but this ability is never referred to otherwise. After a number of fights – there are a lot of fights in this game, and they blur into one another a bit – I started to get a general sense that some moves do damage while others soften up opponents for future attacks, but by the end of the game I still hadn't really got a sense of tactical choices rather than just picking options and seeing what happened.
This could have been salvaged by the prose. This is a text game, and its central focus is combat. This means that it really needs to be capable of good fight-scene writing. And it really, really isn't.
Describing combat through text is a difficult proposition. Combat is action: it relies on things which are difficult to bring alive in text. Max Gladstone has thought and written about this much more thoroughly than I could hope to, but here are a few very simple thoughts.
You kick Agruth's shin. But he doesn't budge and drops the whip. He draws his sword. "You're dead, slave," he growls at you with hatred. He swings at Briana. She leaps back but she is too slow. Agruth slashes her thigh. She yells in pain.
This is very much a novice writer's approach to a combat scene. It reads like a boring procedural text – like, say, the text output from a roguelike – and this might, in fact, be part of what's going on. This happens, and then this happens, and then this happens. There's little sense of flow or intensity, and even causality gets kind of muddied. Consider this line:
You kick Agruth's shin. But he doesn't budge and drops the whip. He draws his sword.
The structure there is misleading: it implies that both not-budging and whip-dropping are equally consequences of the failed shin-kick. It makes far, far more sense if slightly restructured:
You kick Agruth's shin, but he doesn't budge. He drops the whip and draws his sword.
That's not exactly deathless prose – but it's much clearer about causality, which is pretty damn important for making action flow. Agruth drops the whip in order to draw his sword, not because a shin-kick somehow knocked it out of his hand.
"You pick up the sword and wield it" suggests that combat is being thought of primarily in terms of CRPG mechanics. Outside a game's equipment system, there's no separate action of 'wielding'; it's a general state of action, rather than a specific act. This is a tell that Vermin is taking its cues about combat writing primarily from games writing – but this is a very bad model when you're making IF. Writing about combat in a CRPG or a combat tabletop game can afford to be dry and uninspiring, because those things communicate combat in other ways: through the tension of foregrounded mechanics, the visual action of animation, the tactical considerations of maps. When text has an auxiliary function, it has less pressure on it; writing to those expectations doesn't work when prose forms your central delivery.
Choices aren't a lot more inspiring outside combat, either. At one point you're invited to name a sword – and, like, naming weapons is a pretty cool variety of aesthetic choice, this should be a slam-dunk. The problem is that you can choose between two really boring names, or nothing – and this seems really obviously like a situation that calls for 4-5 high-flavour choices plus a text entry. More than any other, this choice really tanked my confidence in Vermin's design.
Then there's this:
Briana stands towering over Agruth's corpse. She smooths her hair back and looks down into the expanding pool of Agruth's blood, using it as a mirror.
"What?" she says when she notices you're looking.
"We either go now, or die."
This isn't a big thing, but it's significant: it casts the player-character as the Serious Business One and Briana as the silly girl who, in the aftermath of narrowly escaping death, is most concerned with her appearance. This is a dynamic I see a lot in games that have female sidekick characters: interactions that highlight serious, hypercompetent, grimly practical masculinity by contrast to frail, self-absorbed femininity. I don't know that this was consciously intended – Briana is thereafter mostly characterised as a capable fighter, albeit not someone with a whole lot of characteristics other than a desire for revenge – or whether it's that Vermin is taking all of its cues from existing games and not thinking too hard about any of them, and thus has imported their bad patterns of representing femininity. But it's still a bad moment.
And the player-character's line there is immediately undermined: you in fact have time to consider a name for the sword, search Agruth's body, and chat with Briana. You can return to that area throughout the game. That makes the exchange seem more gratuitous.
It's also, I should stress, a very generic piece of fantasy. I said that about Rage Quest, but at least Rage Quest showed some basic interest in exploring the implications of a particular fantasy trope; it didn't take them very far or very interestingly, but it had something. Vermin is just interested in recapitulating tropes, which means that – at best – I would have been unlikely to love this. But it also has some pretty serious issues with execution that are dragging it down. 3.
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The Elwood Talmud Torah Hebrew Congregation (), also known as Elwood Synagogue or Elwood Shule, is a historically significant Orthodox synagogue located in the Melbourne suburb of Elwood, Victoria, Australia. The congregation played an important role in accommodating Melbourne's large population of Jewish Holocaust survivors following World War II.

Overview
The congregation was formed around 1931 or 1932 as a prayer service in a private home. In 1938 the congregation moved to 40 Mitford Street where a part-time Talmud Torah was established in addition to regular prayer services. Prior to World War II the congregation grew with the arrival of Jewish European immigrants. In 1939-40 several members broke away and formed their own congregation, later known as the Adass Israel Congregation.

Elwood Shule flourished during and especially following World War II, as a large number of Jewish Holocaust survivors settled in the area. Services took place in halls in Acland Street, St Kilda, and Hennessy Avenue, Elwood during peak periods, before larger premises were acquired at 26 Avoca Avenue in 1942.

The synagogue's current site, at 39 Dickens Street, was purchased in the early 1950s by synagogue President Abe Sicree. A new synagogue building was constructed and dedicated at the site in 1957, in time for the High Holidays of that year. The following year Rabbi Chaim Gutnick was appointed Chief Minister of the synagogue, a position he held for some forty-five years until his retirement in 2003. A new school campus was constructed on synagogue premises by Moriah College of Elwood in the early 1960s. Following Moriah College's demise the campus was occupied by Mt Scopus Memorial College and later Yesodei HaTorah College. In 1973 the synagogue building was renovated and expanded to a seating capacity of 1068. In 2003 Rabbi Mordechai Gutnick succeeded his father Chaim as Chief Minister of the congregation.

In 2008 the Elwood Talmud Torah Hebrew Congregation celebrated its 75th anniversary. Federal member for Melbourne Ports Michael Danby, an Elwood resident, marked the occasion by paying tribute to the congregation in a speech before Australian Parliament on 16 October 2008, saying,"It is very fortuitous, when viewed in retrospect, that this congregation was formed by central and east European migrants prior to the Second World War. These were the very people who were able to look after the refugees and the displaced persons who came off the ships after the Second World War and who, subsequently, filled the pews at this very special Shule." A similar tribute was delivered before Victorian Parliament by Martin Foley.

See also

 List of synagogues in Australia and New Zealand
 History of the Jews in Australia

References

External links
 Elwood Talmud Torah Hebrew Congregation website

Orthodox synagogues in Australia
Synagogues in Melbourne
Buildings and structures in the City of Port Phillip
1930s establishments in Australia

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Bowie, The American Library Association, and The Idiot
Posted on January 2, 2012 January 11, 2012 by Laurie Frost
In 1985 the American Library Association began its READ celebrity poster campaign. Since then, 182 posters have been produced. David Bowie was the eighth to pose for a poster, back in 1987.
It's a strange pose. He looks like he is in mid-air, not floating, though — there's too much tension in his limbs — more like he has jumped very high off a trampoline, bending a leg midair, all the time absorbed in Dostoevsky's The Idiot. The jacket, chosen I suppose for its big R (R is for reading?) is by Roots, which seems a bit like a Canadian LL Bean, but with an emphasis on leather gear. Is the image on the shirt of a glam rock idol?
In 1987, Bowie was 40. He could pass for a high school kid here.
Bowie does read the Russian classics, so choosing The Idiot makes sense for that reason. Of course, Bowie fans know another: Iggy Pop's album The Idiot, produced by Bowie in 1977, which, according to Bowie and Iggy biographer Paul Trykna, was named after, you guessed it, Doetoevsky's novel.
Andrew Kent shot the cover for Iggy Pop's The Idiot, which like the cover for Bowie's "Heroes," (1977) photographed by Masayoshi Sukita, was inspired by the painting Roquairol (1917) by the German Expressionist Erich Heckel.
I haven't been able to find the name of the photographer of the 1987 American Library Association's poster, but it was produced a decade after Bowie was working with Kent and Sukita. Still, its angularity suggests these two artists' works.
Andrew Kent, but the way, is the photographer responsible for what I consider some of the very best images of Bowie from the Thin White Duke/Station to Station era, including this one.
Tags: American Library Association, Andrew Kent, David Bowie, Doestoevsky, Erich Heckel, Iggy Pop, Roquairol, The IdiotCategories: Life, Music
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Polyamory and Love Beyond the Relationship Binary
By Kaia Findlay,
Published December 24, 2021 at 12:02 PM EST
Polyamorous relationships look as different as the people who get into them.
Monogamy is the best relationship structure for some people. But it's not the only way to experience love and partnership, as those in polyamorous relationships explain.
To anyone who thinks being in a relationship is easy: please share your secrets. Relationships take work, and we ask for trust, communication, commitment and fidelity from our partners. But we often understand those concepts based on a standard of monogamy — and that one-partner-fits-all model doesn't work for everyone.
"I am a big believer in being able to make your relationship unique to you." - Natalie Murray
Polyamorous relationships look as different as the people they involve. But they all take some learning and "unlearning" of our standard relationship structures.
Host Anita Rao talks with Rob, a co-organizer of Triangle Polyamory Meetup, and Crystal Byrd Farmer, the website editor for Black & Poly magazine, about how they approach communication and community-building in polyamorous relationships.
And Natalie Murray, a licensed clinical mental health counselor, joins the conversation to talk about how she counsels people exploring polyamory.
Please note: This conversation originally aired on February 12, 2021.
Rob on balancing self-care with multiple relationships:
There's only so many hours in a week that you could fit into putting people in your life. I did stretch myself thin at one point trying to date too many people at the same time. And I don't recommend doing that. You have to keep yourself as your primary focus and try to keep, you know — maintain a safe pace before you're stretched too thin.
Credit Natalie Murray
Natalie Murray, licensed mental mental health counselor, works with couples in polyamorous relationships.
Murray on helping couples find the right relationship structure for them:
One of the things that I try to do is make sure that my couples know they have permission to do what is right for them. So sometimes — when people have done their research, they have decided that this is the right relationship structure for them — they have a hard time dealing with it not working. … And I tell people: You can always change your relationship structure at any time. I have some couples who are open at different times of the year and they're closed the rest, or they're open only to certain people. You are allowed to structure your relationship in a way that works just for you.
Byrd Farmer on understanding jealousy:
Credit Crystal Byrd Farmer
Crystal Byrd Farmer works as an educator and activist in polyamorous communities.
Jealousy is a natural feeling. It's saying that there's something that you want that you don't have, and I think a misconception would be that polyamorous people never get jealous. We do get jealous. We just acknowledge it as a feeling, and it's not something that means we have to suddenly change our relationship or put new rules on it. It's something that we have to talk through and possibly renegotiate.
Please note: This conversation originally aired February 12, 2021.
Embodied Radio Show Polyamorous relationshipsmonogamyTriangle Polyamory MeetupCrystal Byrd FarmerBlack & Poly magazineNatalie MurraypolyamoryEmbodied Radio ShowRelationshipsRomantic Relationships
Kaia Findlay
Kaia Findlay is a producer for Embodied, WUNC's weekly, live talk show on health, sex and relationships. Kaia first joined the WUNC team in 2020 as a producer for The State of Things.
See stories by Kaia Findlay
Anita Rao is an award-winning journalist and the host and creator of "Embodied," a weekly radio show and podcast about sex, relationships & health. She's also the managing editor of WUNC's on-demand content.
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Anita Rao is an award-winning journalist and the host and creator of "Embodied," a live, weekly radio show and seasonal podcast about sex, relationships & health. She's also the managing editor of WUNC's on-demand content.
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Oil set for weekly gain amid virus vaccine progress
'Religious party activist' arrested for vandalising statue of Maharaja Ranjit Singh
Lack of planning badly affected tourism industry: PM
Khyber-Pakhtunkhwa government and World Bank sign an agreement on ecotourism
News Desk December 11, 2020
Prime Minister Imran Khan addresses the launching and partnership signing ceremony of TRAVEL RESPONSIBLY FOR EXPEREINCING ECO TOURISM (TREK) in Islamabad on December 11, 2020.
While underlining the need for protection and preservation of environment in Pakistan, Prime Minister Imran Khan on Friday said that unplanned tourism has badly affected many tourist spots in the country, according to Radio Pakistan.
He said this while addressing a ceremony in Islamabad after witnessing an agreement signed between Khyber-Pakhtunkhwa (K-P) government and World Bank on ecotourism.
The premier emphasised that the model of development of tourist resorts should benefit the local population which would provide opportunities to earn livelihood to them. "It will make our tourism sector sustainable and help in preserving the tourist spots," he added.
Imran Khan lamented that unplanned tourism has badly affected the tourist spots such as Murree, Naraan and Nathia Gali. However, he added that the initiative taken by the K-P government today, involving waste collection and management and recycling, will go a long way in protecting the picturesque sites.
The prime minister expressed satisfaction that domestic tourism has reached record levels in the country.
Experts at a conference last month had underlined the need for a dedicated ecotourism policy for protecting cultural and biological diversity of picturesque tourist attractions across the country.
Massive sensitisation of all the stakeholders on ecotourism was imperative to purge the tourist attractions of haphazard constructions, posing a serious threat to the eco-system of those places, National Tourism Coordination Board (NTCB) Member Aftab Rana stressed at the one-day seminar.
The NTCB member said the country had enormous potential of ecotourism that was needed to be exploited for development of rural economies and uplift of locals' living standards.
"Pakistan is mainly a rural country and tourism can greatly contribute for socio-economic development of those areas which are otherwise neglected," he remarked.
Pakistan's tourism industry made great strides in the preceding years with a wider acknowledgement and interest from all corners.
The PTDC statistics revealed that at least 3.2 million foreigners visited Pakistan in 2019, out of which 17,000 were on tourist visas, Arab News reported.
In January, Forbes listed Pakistan among best 'under-the-radar trips of 2020'. However, tourism around the world including Pakistan took a bad hit after the emergence of Covid-19 pandemic.
#Live_Stream: PM @ImranKhanPTI is addressing a ceremony regrading Eco-Tourism in Islamabad https://t.co/pcNjZT8v3R
— Radio Pakistan (@RadioPakistan) December 11, 2020


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Місто Ресіфі від часу заснування пов'язано із своїм портом, що протягом всієї історії був одним з головних центрів експорту цукру. Великий вплив на історію міста також справило сусіднє місто Олінда, що протягом довгого часу було столицею капітанії та провінції Пернамбуку. Найбільший культурний вплив на місто здійснили португальці, яким місто належало майже весь час до здобуття Бразилією незалежності, та голландці, що володіли ним протягом 24 років у 17 столітті.

Найраніші часи

Перша згадка про місто відноситься до 1537 року, Ресіфі було згадано у форалі Олінди як невелике рибацьке селище. Свою назву місто отримало від португальського слова recife — «риф», через наявність великого числа рифів біля узбережжя.

Протягом наступного періоду Ресіфі виконував функції головного торгового порту капітанії, а потім провінції Пернамбуку. Весь цей час він залишався в тіні Олінди, де мешкала місцева аристократія через стратегічне розташування міста на пагорбі, який полегшував оборону. Укріплення Ресіфі не були настільки надійними через дію моря, а прибережне місто залишалося уразливішим для піратів та іноземних флотів. Так, наприкінці 16 століття Ресіфі атакував і розграбував англійський пірат Джеймс Ланкастер, якому, з трьома кораблями, вдалося робити маленький гарнізон захисників порту.

Між 1620 і 1626 роками губернатор Матіас ді Альбукеркі спробував збудувати у місті нові укріплення (серед них Форт Брум), призначені захистити місто від голландців, що почали військові дій проти Португалії та в 1624 році напали на Баїю.

Нідерландський уряд

Голландське вторгнення почалося 16 лютого 1630 року, коли війська Голландської Вест-Індської компанії у складі 7 тис. солдатів на 67 кораблях висадилися на пляжі Пау-Амарелу. Оскільки забезпечення експорту цукру було важливою задачею Компанії, місто Ресіфі, що мало великий порт, отримало домінуюче значення. До 1636 року поруч зі старим португальським містом було збудовано голландське місто Мауріцстад, що стало столицею Голландської Бразилії. Протягом значної частини періоду голландського панування містом керував граф Йоган Мауріц ван Нассау-Зіген.

Голландці збудували в Бразилії цілий ланцюжок фортець від Сеари біля гирла річки Сан-Франсіску до півдня Алагоасу. Встановивши також контроль над африканським узбережжям від Гвінеї до Анголи, голланці перехопили у португальців транспорт рабів до американських колоній, що почали активно завозитися до Пернамбуку через порт Ресіфі для використання на цукрових плантаціях.

Коли в 1637 році керування колонією отримав грав Нассау, він перевіз до неї велику команду архітекторів та інженерів. Вони збудували в Мауріцстаді численні мости, греблі і канали для покращення герграфічних умов міста. Архітектор Пітер Пост відповідав за проект власне міста і багатьох його споруд, зокрема палацу Фребург, резиденції губернатора Нової Голландії, і будівлі першої в Новому Світі астрономічної обсерваторії.

Голландський уряд проводив політику релігійного співіснування між католиками і кальвіністами. Крім них, в Ресіфі виникла велика юдейська спільнота, також тут в 1642 році була відкрита перша в Америці синагога — Кахал-Цур-Ізраель.

Граф Нассау також покровительствував наукам та витонченим мистецтвам. З ним до Бразилії прибула ціла плеяда натуралістів та живописців для дослідження та зображення нового континенту. Серед них були художники Франс Пост і Альберт Екгоут, що зобразили численні місцеві пейзажі та екзотичні місця, медик Віллем Пісо і натураліст Георг Марґґраф, до досліджували місцеві флору і фауну, лікарські рослини і тропічні хвороби.

Після суперечки з керівництвом Компанії, незадоволеної рівнем доходів з Бразилії, в 1644 році граф Нассау повернувся до Європи. Нові голландські губернатори викликали значне незадоволення вже місцевого населення, що привело до повстання — Пернамбуканського повстання — що в результаті привело до остаточного вибіття голландців з Пернамбуку в 1654 році.

Бродячі торговці

Після вибіття голланців, виробництво цукру в Пернамбуку та його експорт через порт Ресіфі скоротилися через конкуренцію з голландськими володіннями на Антильських островах та у Суринамі, де використовувалися новіші технології. У той же період до Ресіфі почало прибувати багато комерсантів з Португалії, яких зневажливо називали mascates — «бродячі торговці». Вони селилися у місті, просуваючи торгівлю та піднімаючи його добробут. Однак швидкий ріст багатства і значення Ресіфі викликав незадоволення сільськогосподарчої аристократії з Олінди, особливо у світі падіння її доходів через зменшення експорту цукру. Конфлікт економічних та політичних інтересів між старою аристократією та новою буржуазією Пернамбуку привів до збройного конфлікту між ними, «Війни торговців», що тривала у 1710—1711 роках, протягом яких місто стало ареною численних сутичок.

Проте конфлікт не приніс збитків місту і не зупинив росту його багатства. Вже в 1710 році Ресіфі отримав статус незалежного селища (vila independente). У 1711 році його населення становило 16 тис., а у 1745 році зросло до 25 тис. мешканців. Незважаючи на низькі ціни на цукор, тут були збудовані нові величні монастирі й церкви, серед них Монастир і церква Санту-Антоніу (збудовані протягом 18 століття), Позолочена каплиця (відкрита у 1724 році) і Собор Сан-Педру-дус-Клерігус (розпочатий в 1725 році).

Революції

На початку 19 століття історія Ресіфі була відмічена повстаннями, надихнутими ліберальними ідеями Європи: торговці, аристократи і священики разом намагалися отримати більшої автономії колонії від метрополії. Тим часом, домінуючі класи повністю відкидали ідеї скасування рабства і надання влади представникам народу, побоюючись соціальних виступів.

Так, протягом цього періоду прийшов ряд найбільших повстань в історії міста, зокрема Пернамбуканська революція 1817 року, Екваторіальна конфедерація 1824 року і Повстання Праєйра 1848 року. Ресіфі отримав значну фактичну незалежність, тут не визнавали як центральну владу, так і провінціальну владу з сусідньої Олінди. Саме місто продовжило швидко розвиватися, і в 1823 році отримало статус міста (cidade).

20 століття

Від початку 20-го століття в місті тривав період активного культурного розвитку, що почався з доби Прекрасної епохи.

Протягом цього століття місто перетворилася на важливий промисловий центр, численні промислові підприємства міста становлять нині основу його економіки. Також протягом цього періоду в місті розвинулися медицина і туризм.

Література
 Caspar Barlaeus, História dos feitos recentemente praticados durante oito anos no Brasil. Belo Horizonte: Editora Itatiaia; São Paulo: Editora da Universidade de São Paulo, 1974. 418 p. il.
 Gilberto Ferrez, Raras e preciosas vistas e panoramas do Recife (1755—1855). Rio de Janeiro: Fundação Nacional Pró-Memória, 1984.

Посилання

 Recife, a Mauritstad
 História da cidade do Recife
 História do Recife — каталог посилань
 Recife — Minha cidade

Історія Ресіфі


text:
\section{Introduction}
Cooperation and coordination of multi-agent systems has received a great deal of attention due to its widely practical applications in both civilian and military areas \cite{Jorgebook}, \cite{MMbook}. These applications include cooperative control of unmanned aerial vehicles and satellite clusters, flocking, formation control, and control of sensor networks for exploration, surveillance, rescue and missions, video capture in sport events, transportation, etc. Among these tasks of cooperation and coordination control fields,  the search for optimal motion plans can be computationally very expensive, particularly as the number of robots or degrees of freedom of the system gets large. For this reason, we pursue in the work an easily computable motion planning strategy, based on geometric techniques and variational calculus, to avoid collision between agents and static obstacles in the workspace.

Calculus of variations on Riemannian manifolds \cite{Milnor} has been exploited in the past for various applications. In Crouch and Silva Leite \cite{CroSil:91}, \cite{CroSil:95} the authors have used it
to develop a theory of generalized cubic polynomials for
dynamic interpolation problems on Riemannian manifolds, in particular on compact connected Lie groups endowed with a bi-invariant metric.
More recently, Bloch, Camarinha and Colombo \cite{BlCaCoCDC}, \cite{BlCaCoCDC2}, \cite{BCC}  have
used these variational methods to solve obstacle avoidance
problems on Riemannian manifolds and interpolation problems.

 In this paper, as a continuation of the work \cite{Assif}, we address the problem of finding necessary conditions for optimal trajectories of multiple agents on a Riemannian manifold that seek to achieve a specified configuration while avoiding collisions among themselves and static obstacles on the configuration space.

Specifically, the problem studied in this paper consists of finding non-intersecting trajectories of a given number of agents, among a set of admissible curves, to
reach a specified configuration and minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to avoid obstacles and collisions between the agents. To solve the problem, we employ techniques from calculus
of variations on Riemannian manifolds taking into account
that the problem under study can be seen as a higher-order
variational problem \cite{blochcrouch2}, \cite{MargaridaThesis}, \cite{CMdD}, \cite{BlochHussein}.

The structure of the paper is as follow. We start in Section II by
introducing the geometric structures on a Riemannian manifold
that we will use together with admissible variation of curves
and vector fields for the variational problem. Next, in Section III, we introduce variational collision and obstacle avoidance problems on Riemannian
manifolds and first-order derive necessary conditions for optimality. We apply the result to
obstacle and collision avoidance problems for a planar rigid body and agents evolving on a sphere in Section IV. Final comments and ongoing work are discussed at the end of the paper.
\section{Preliminaries on Riemannian Geometry}

Let $M$ be a smooth $ (C^\infty) $ Riemannian manifold with Riemannian metric denoted by $ <\cdot,\cdot>\ : T_{x}M \ \times \ T_{x}M \rightarrow \mathbb{R}$ at each point $x \in M $, where $T_{x}M$ is the tangent space of $M$ at $x$.  The length of a tangent vector is determined by its norm,
$||v_x||=\langle v_x,v_x\rangle^{1/2}$ with $v_x\in T_xM$, for each point $x\in M$.

A \textit{Riemannian affine connection} $\nabla$ on $M$, is a map that assigns to any two smooth vector fields $X$ and $Y$ on $M$, a new vector field, $\nabla_{X}Y$, called the \textit{covariant derivative of $Y$ with respect to $X$}, satisfying $\nabla_{fX}Y=f\nabla_XY,\hbox{ and } \nabla_X(fY)=X(f)Y+f\nabla_XY$, for all vector fields $X,Y\in\mathfrak{X}(M)$ and $f\in\mathcal{C}^{\infty}(M)$, where $\mathfrak{X}(M)$ denotes the set of vector fields on $M$. For the properties of $\nabla$, see for instance \cite{Boothby, bookBullo,Milnor}.

Consider a vector field $W$ along a curve $x(t)$ on $M$. For $n\leq 1$, the $n$-th order covariant derivative of $W$ along $x(t)$ is denoted by $\displaystyle{\frac{D^n W}{dt^n}}$ and we will denote by $\displaystyle{\frac{D^{n+1} x}{dt^{n+1}}}$, the $n$-th order covariant derivative along $x$ of the velocity vector field of $x$.

Given vector fields $X$, $Y$
and $Z$ on $M$, the vector field $R(X,Y)Z$ given by \begin{equation}\label{eq:CurvatureTensorDefinition}
R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z
\end{equation}  defines the \textit{curvature tensor} on $M$, where $[X,Y]$ denotes the \textit{Lie bracket} of the vector fields $X$ and $Y$. $R$ is trilinear in $X$, $Y$ and $Z$ and a tensor of type $(1,3)$. For vector fields $X,Y,Z,W$ on $M$ the curvature tensor satisfies (see \cite{Milnor}, p. 53)
\begin{equation}\label{curvformula}\langle R(X,Y)Z,W\rangle=\langle R(W,Z)Y,X\rangle.\end{equation}

Let $S$ be a submanifold of $M$ and $\Omega$ be the set of all $C^{1}$ piecewise smooth curves $x : [0,T] \ \rightarrow M$ such that $x(0)$, $\frac{dx}{dt}(0)$ and $x(T)$ are fixed, with $x(T)\in S$ and $\frac{dx}{dt}(T) \in T_{x(T)}S$. The set $\Omega$ is called the \textit{admissible set}. For the class of curves in $\Omega$ we introduce the $C^1$ piecewise smooth \textit{one parameter admissible variation} of a curve $x \in \Omega$ by $\alpha : (-\epsilon,\epsilon) \times [0,T] \rightarrow M ;(r,t) \mapsto\alpha(r,t) = \alpha_r(t)$ that satisfy $\alpha_{0} = x$ and $\alpha_r \in \Omega$, for each $r \in (-\epsilon, \epsilon) $.

 The \textit{variational vector field} associated to an admissible variation $\alpha$ is a $C^1$-piecewise smooth vector  field $X$ along $x$ defined by $\displaystyle{X(t) = \frac{D}{\partial r}\Big{|}_{r=0}\alpha(r,t) \in T_{x(t)}\Omega}$ verifying the boundary conditions
\begin{equation*}
X(0)=0,\, X(T) = 0,\,
\frac{DX}{dt}(0)=0, \, \frac{DX}{dt}(T) \in T_{x(T)}S,
\end{equation*} where the tangent space of $\Omega$ at $x$ is the vector space $T_{x}\Omega$ of all $C^1$ piecewise smooth vector fields $X$ along $x$ verifying the former boundary conditions.

\begin{lemma}[\cite{Milnor}, p.$52$]\label{curvature_lemma}
The one parameter variation satisfies
\begin{equation*}
\frac{D}{\partial r}\frac{D^2\alpha}{\partial t^2} = \frac{D^2}{\partial t^2}\frac{\partial \alpha}{\partial r} + R \Big( \frac{\partial \alpha}{\partial r},\frac{\partial \alpha}{\partial t} \Big)\frac{\partial \alpha}{\partial t}.
\end{equation*}
\end{lemma}

\vspace{.2cm}

\section{The variational collision and obstacle avoidance problem on Riemannian manifolds}

\iffalse
\textcolor{red}{Is the graph theory introduced below essential since we do not make use of it later on ?}
Using graph theory, we can model the communication topology among $n$ agents in the formation.  A  graph $\mathcal{G}$ consists of a pair $(\mathcal{V},\mathcal{E})$, where $V=\{1,2,...,n\}$ is a finite nonempty set of nodes describing the kinematics of each single agent and $\mathcal{E}\subset \mathcal{V}\times\mathcal{V}$ denotes the set of edges of the graph, $p:=\frac{n(n-1)}{2}$ symmetric binary relations that link two agents. An edge $e_{(i,j)}$ denotes that node $j$ can obtain information from $i$.
\fi

\label{S:3}
 Let $n$ and $k$ be natural numbers and $T$ be a positive real number. Consider $n$ agents evolving on $M$, a Riemannian manifold with $\dim(M)=m$. Denote by $(p_{0}^{i}, v_{0}^{i})$, for $i=1,2...,n$,  points in $TM$ corresponding to the initial positions and velocities of the agents.

Consider the set $\Omega_{i}$ of all $C^1$-piecewise smooth curves $x_{i} : [0,T] \rightarrow M$ verifying the boundary conditions
\begin{equation*}
x_i(0) = p_{0}^{i}, \, \frac{dx_i}{dt}(0) = v_{0}^{i},\, x_i(T) = p_T^i \in S, \, \frac{dx_i}{dt}(T) \in T_{x_i(T)}S
\end{equation*} that is, each agent is initially at a fixed position and velocity, and required to reach in a fixed time, a specified position on the submanifold $S$ with its velocity tangent to $S$ at the specified point.

The problem studied in this work consists on designing a trajectory for each agent, satisfying the above boundary conditions and avoiding a set of prescribed static obstacles in the workspace together with ensuring collision avoidance between the agents. The path planning is designed by considering a cost functional which is defined on the set of $C^1$ piecewise smooth trajectories, verifying the above boundary conditions, such that the trajectory that minimizes the defined cost functional will be a feasible solution avoiding collision between the agents and each agent will avoid the obstacles.

Figure $1$ gives a sketch of the situation and the nomenclature used in this paper is given in Table \ref{tab:table1}.

\begin{figure}[h]\label{fig1}
 \includegraphics[scale=0.45]{fig1ecc.eps}
 \caption{Sketch of the situation studied in this work. $x_i(t)$ and $x_r(t)$ denotes trajectories for the $i^{th}$ and $r^{th}$ agents, respectively.}
\end{figure}

\begin{table}[h!]
  \begin{center}
    \caption{Nomenclature}
    \label{tab:table1}
    \begin{tabular}{l|c}
    \hline
      \textbf{Symbol} & \textbf{Description} \\
      \hline
     $M$ & Riemannian manifold. \\
    $S$ & Riemannian submanifold. \\
	$n$ & Number of agents\\
	$T$ & Final time\\
	$J$ & Cost functional\\
	$V$ & Artificial Potential function \\
	$x_i$ & Trajectory of the $i^{th}$ agent\\
	$m$ & Dimension of $M$ \\
	$s$ & Dimension of $S$ \\
	$\frac{D}{dt}$  & Covariant derivative \\
	$\alpha$ & One parameter smooth variation of $x_i$ \\
	$t_{0_i},t_{1_i},...,t_{l_i}$ & Partition of $[0,T]$ for the $i^{th}$ agent \\
	$d_M(x,y)$ & Geodesic distance between $x,y\in M$ \\
	$\alpha$  & Quantity of static obstacles\\
	$B_{\alpha}$ & Smooth real valued function  \\ & describing the $\alpha$ static obstacles
    \end{tabular}
  \end{center}
\end{table}

Define the cost functional $J$ on $\tilde{\Omega} = \Omega_{1}\times$...$\times\Omega_{n}$ as
\begin{align*}
J(x_1, x_2, ..., x_n) =&\frac{1}{2} \int_{0}^{T}\sum_{i=1}^{n} \Big( \Big{\|} \frac{D^2x_i}{dt^2}(t) \Big{\|}^2 + \kappa\Big{\|}\frac{dx_i}{dt}(t)\Big{\|}^2\\&+  V(x_1, x_2, ..., x_n)\Big) dt.
\end{align*}

\vspace{.2cm}

The cost functional $J$  is constructed as a combination of the square magnitudes for covariant acceleration, square magnitudes of velocity regulated by a parameter $\kappa$, both for the individual trajectories of the agents, and an artificial potential function $V$ that penalizes collisions between the agents and also with prescribed static obstacles. The obstacles are described as the zero level surface of a know scalar valued analytic function (see, e.g., \cite{Khatib1986}, \cite{K88}, \cite{Koditschek1990}).

The potential function $V$ is an artificial potential field-based function
represented by a force inducing an artificial repulsion from the surface of the obstacle as. We use the approach introduced by Khatib \cite{Khatib1986}, which consists of using a local inverse potential field going to infinity as the inverse square of a known scalar valued analytic function near the obstacle and collision between the agents, and decaying to zero at some positive level surface far away from the obstacle and when agents arte not close to each other

Let $d_M$ denote the distance function induced by the Riemannain metric on $M$, and $\tau$, $\sigma$ be positive constants. Let the obstacle be the zero sublevel set of a smooth function $B$ in $M$. i.e. $O=\{x \in M | B(x)\leq 0\}$. Then the artificial potential function takes the (non-unique) form

$$V(x_1,..,x_n)=\sum_{i=1}^n\left(\sum_{k=1}^{\alpha}\frac{\tau}{B_{\alpha}(x_i)} + \frac{1}{2}\sum_{j=1}^n\frac{\sigma}{d_M(x_i,x_j)}\right)$$ where $\alpha$ denote the quantity of obstacle. Note that the factor $\frac{1}{2}$ in the expression for $V$ comes from the fact that $d_M(x_i,x_j)=d_M(x_j,x_i)$.

\textbf{Problem:} The variational collision and obstacle avoidance problem consists of minimizing the functional $J$ among $\tilde{\Omega}$.

\vspace{.2cm}

In order to minimize the functional $J$ among the set $\tilde{\Omega}$ we want to find curves $x \in \tilde{\Omega}$ such that $J(x) \leq J(\tilde{x})$, for all admissible curves $\tilde{x}$ in a $C^1$ neighborhood of $x$.
\vspace{.1cm}

\begin{theorem}\label{theorem1}
Let $x_i \in \Omega_i$. If $\alpha$ is an admissible variation of $x_i$ with variational vector field $X_i$, then
\begin{align*}
&\frac{d}{dr}J(\alpha(r))\Big{|}_{r=0}= \int_{0}^{T}\sum_{i=1}^n \Bigg( \Big  \langle X_i,\frac{D^4 x_i}{dt^4}- \kappa\frac{D^2x_i}{dt^2} \\&+ R \Big(\frac{D^2 x_i}{dt^2}, \frac{dx_i}{dt} \Big)\frac{dx_i}{dt}+\mbox{grad}_i V(x_1, x_2, ..., x_n)  \Big \rangle \Bigg) \ dt \\ +&\sum_{i=1}^n \sum_{j_i=1}^{l_i} \Big{[}\Big\langle\frac{DX_{i}}{dt},\frac{D^2x_{i}}{dt^2} \Big \rangle+\Big\langle X_i, \kappa \frac{dx_i}{dt} - \frac{D^3x_i}{dt^3} \Big \rangle\Big{]}^{t_{j_i}^{-}}_{t_{j_{i-1}}^{+}},
\end{align*} where $0=t_{0_i}<....<t_{l_i}=T$ is a partition of $[0,T]$ such that $x_i$ is smooth on each $[t_{j_i},t_{j_i+1}]$, $i=1,\ldots,n$ and $j_i=0,\ldots,l_{i-1}$, and where $\mbox{grad}_i V$ denotes the gradient of $V$ with respect to its $i^{th}$ argument.
\end{theorem}

\vspace{.2cm}

\textit{Proof:} If $\alpha$ is an admissible variation of $x_i \in \Omega_i$ with variational vector field $X_i$, then
\begin{align*}
 \frac{d}{dr}J(\alpha_r) =& \int_0^T \sum_{i=1}^n \Bigg( \Big \langle \frac{D}{\partial r} \frac{D^2 \alpha}{\partial t^2}, \frac{D^2 \alpha}{\partial t^2}\Big \rangle + \kappa\Big \langle\frac{D^2 \alpha}{\partial r \partial t}, \frac{\partial \alpha}{\partial t} \Big \rangle \\&+ \frac{\partial}{\partial r}  V(x_1, x_2, ...,x_{i-1}, \alpha, x_{i-1}, ..., x_n) \Bigg)dt.
\end{align*}

By lemma \ref{curvature_lemma}, the property of curvature tensor (2), and the previous equation, we get

\begin{align*}
\frac{d}{dr}J(\alpha_r) = &\int_0^T\sum_{i=1}^n \Bigg( \Big \langle \frac{D^2}{dt^2}\frac{\partial \alpha}{\partial r}, \frac{D^2 \alpha}{dt^2}\Big \rangle  \\&+ \Big \langle R\Big(\frac{\partial \alpha}{\partial r}, \frac{\partial \alpha}{\partial t}\Big) \frac{\partial \alpha}{\partial t}, \frac{D^2 \alpha}{\partial t^2} \Big \rangle +  \kappa\Big \langle\frac{D^2 \alpha}{\partial r \partial t}, \frac{\partial \alpha}{\partial t} \Big \rangle \\ +& \Big \langle  \frac{\partial\alpha}{\partial r}, \mbox{grad}_i V(x_1, x_2, ..., x_n)  \Big \rangle \Bigg)dt.
\end{align*}

Integrating the first term by parts twice, the third term once, and applying Lemma \ref{curvature_lemma} to the second term, we obtain that
\begin{align*}
\frac{d}{dr}J(\alpha_r) =& \int_0^T \sum_{i=1}^n\Bigg( \Big \langle \frac{\partial \alpha}{\partial r}, \frac{D^4\alpha}{dt^4} + R \Big( \frac{D^2\alpha}{dt^2}, \frac{\partial \alpha}{\partial t} \Big) \frac{\partial \alpha}{\partial t} \\&- \kappa\frac{D^2\alpha}{dt^2} +\mbox{grad}_i V(x_1, x_2, ..., x_n)\Big \rangle \Bigg) dt \\ +&\sum_{i=1}^n \sum_{j_i=1}^{l_i} \Big{[} \Big \langle \frac{D}{dt}\frac{\partial \alpha}{\partial r},\frac{D^2\alpha}{dt^2} \Big \rangle \\&+ \Big \langle \frac{\partial \alpha}{\partial r}, \kappa \frac{\partial \alpha}{\partial t} - \frac{D^3\alpha}{\partial t^3} \Big \rangle\Big{]}^{t_{j_i}^{-}}_{t_{j_i-1}^{+}}
\end{align*}
where the interval $[0,T]$ is partitioned as $ 0 = t_{0_i} < t_{2_i} < ...< t_{l_i} = T $ such that in each subinterval $x_i$ is smooth.

Taking $r = 0$ in the last equation,
\begin{align*}
 \frac{d}{dr}J(\alpha_r)\Big{|}_{r=0} =& \int_0^T \sum_{i=1}^n\Bigg( \Big \langle X_i, \frac{D^4x_i}{dt^4} + R \Big( \frac{D^2x_i}{dt^2}, \frac{\partial x_i}{\partial t} \Big) \frac{\partial x_i}{\partial t}  \\&- \kappa\frac{D^2x_i}{dt^2} +\mbox{grad}_i V(x_1, x_2, ..., x_n) \Big \rangle \Bigg) dt \\&+\sum_{i=1}^n\sum_{j_i=1}^{l_i} \Big{[} \Big \langle \frac{DX_i}{dt},\frac{D^2x_i}{dt^2} \Big \rangle \\&\hspace{1.5cm}+ \Big \langle X_i, \kappa \frac{dx_i}{dt} - \frac{D^3x_i}{dt^3} \Big \rangle\Big{]}^{t_{j_i}^{-}}_{t_{j_i-1}^{+}}.\hfill\square
\end{align*}

As we are interested in variations that satisfy the boundary conditions above, the corresponding variational vector fields $X_i$ should satisfy $X_i(0)=X_i(T)=0$,  $\frac{DX_i}{dt}(0)=0$, $\frac{DX_i}{dt}(T) \in T_{x_i(T)}S$.

The next result characterizes necessary conditions for optimality in the variational collision and obstacle problem and restriction on the boundary conditions tom solve the corresponding boundary value problem.

\begin{theorem}\label{mainTh}
If $\tilde{x} \in \tilde{\Omega}$ is a local minimizer of $J$, then $ \forall i \in \{1,2...,n\}$, it satisfies
\begin{enumerate}
\item \begin{align*}0=&\frac{D^4 x_i}{dt^4} + R \Big(\frac{D^2 x_i}{dt^2}, \frac{dx_i}{dt} \Big)\frac{dx_i}{dt}- \kappa\frac{D^2x_i}{dt^2}\\&+\mbox{grad}_i V(x_1, x_2, ..., x_n),\end{align*}
\item $x_i$ is smooth on $[0,T]$,
\item $\frac{D^2x_i}{dt^2}(T) \perp T_{x_{i}(T)}S$,
\item If the final point $x_i(T)$ is not fixed but just constrained to lie anywhere in the submanifold $S$, then  $$\kappa \frac{dx_i}{dt} - \frac{D^3x_i}{dt^3} \perp T_{x_i(T)}S.$$
\end{enumerate}
\end{theorem}
\proof
Assume $\tilde x \in \tilde\Omega$ is a local minimizer of $J$. Consider a variation of $\tilde{x}$, $\tilde{\alpha}_{r,i}(t) := (x_1(t),.. ,\alpha_{r,i}(t),... ,x_n(t))$, where $\alpha_{r,i}(t)$ is an admissible variation of $\Omega_i$ with variational vector field $X_i$. Then $\frac{d}{dr}J(\alpha_{r,i})\Big{|}_{r=0} = 0 \ \forall i \in \{1,2,...n\}$.
This is because given that the variations are independent, they have to vanish individually for each agent.

 Let us consider a variation of the $i^{th}$ agent with its variational vector field $X_i$ defined as
\begin{equation*}
f(t) \Big[ \frac{D^4x_i}{dt^4} + R\Big(\frac{D^2x}{dt^2},\frac{dx}{dt}\Big)\frac{dx}{dt} - \kappa\frac{D^2x}{dt^2}
+\mbox{grad}_i V \Big)  \Big]
\end{equation*} \\where $f:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth real valued function on $[0,T]$ such that $f(t_{j_i}) = f'(t_{j_i}) = 0$ and $ f(t) > 0, t \neq t_{j_i}, j_i = 1,...,l_i$.
So, we have
\begin{align*}
0=&\frac{d}{dr}J(\alpha_r)\Big{|}_{r=0}\\=& \int_0^T\sum_{i=1}^n  \Bigg(f(t) \Big{|}\Big{|} \frac{D^4x_i}{dt^4}+ R \Big( \frac{D^2x_i}{dt^2}, \frac{d x_i}{d t} \Big) \frac{dx_i}{dt}\\&\hspace{1.0cm}- \kappa\frac{D^2x_i}{dt^2}
+\mbox{grad}_i V(x_1, x_2, ..., x_n) \Big\|^2 \Bigg) dt \\ &\hspace{1.2cm}+ \sum_{j_i=1}^{l_i}\Big \langle f'(t)[\cdot]+f(t)\frac{D}{dt}[\cdot],\frac{D^2x_i}{dt^2} \Big \rangle \Big{]}^{t_{j_i}^{-}}_{t_{j_i-1}^{+}} \\&\hspace{1.4cm}+\sum_{j_i=1}^{l_i}\Big \langle f(t)[\cdot], \kappa \frac{dx_i}{dt} - \frac{D^3x_i}{dt^3} \Big \rangle\Big{]}^{t_{j_i}^{-}}_{t_{j_i-1}^{+}}
\end{align*}
where $[\cdot]= \frac{D^4x_i}{dt^4}+ R \Big( \frac{D^2x_i}{dt^2}, \frac{d x_i}{d t} \Big) \frac{dx_i}{dt}- \kappa\frac{D^2x_i}{dt^2}
+\mbox{grad}_i V(x_1, x_2, ..., x_n)$.

Now, since $f(t_{j_i})=f'(t_{j_i})=0$, the second and third term in the above summation vanish.
Since $f(t)$ is greater then zero outside a set of measure zero, the first integrand has to identically vanish which establishes the first statement of the theorem, i.e.
\begin{equation*}
\begin{split}
\Big\| \frac{D^4x_i}{dt^4} +& R \Big( \frac{D^2x_i}{dt^2}, \frac{d x_i}{d t} \Big) \frac{d x_i}{d t} - \kappa\frac{D^2x_i}{dt^2}  +\mbox{grad}_i V \Big\| = 0
\end{split}
\end{equation*}
from which statement 1 follows.

Having made the first term vanish, now, choose $X_i \in T_{x_i}\Omega_i$ such that
\begin{align}
X_i(t_{j_i}) &= \frac{D^3x_i}{dt^3}(t_{j_i}^+) - \frac{D^3x_i}{dt^3}(t_{j_i}^-), \quad \forall j_i = 1,..,l_i-1, \nonumber\\
\frac{DX_i}{dt}(t_{j_i}) &= \frac{D^2x_i}{dt^2}(t_{j_i}^-) - \frac{D^2x_i}{dt^2}(t_{j_i}^+), \quad \forall j_i = 1,..,l_i-1,\nonumber \\
X_i(T) &= \frac{DX_i}{dt}(T) = X_i(0) = \frac{DX_i}{dt}(0)= 0.\label{constraint}
\end{align}
Therefore,
\begin{align*}
0=&\frac{d}{dr}J(\alpha_r)\Big{|}_{r=0}=\sum_{j_i=1}^{l_i-1} \Big \| \frac{D^2x_i}{dt^2}(t_{j_i}^-) - \frac{D^2x_i}{dt^2}(t_{j_i}^+) \Big \|^2  \\+& \Big \| \frac{D^3x_i}{dt^3}(t_{j_i}^+) - \frac{D^3x_i}{dt^3}(t_{j_i}^-) \Big \|^2
\end{align*}
which implies that
\begin{equation*}
\frac{D^2x_i}{dt^2}(t_{j_i}^-) = \frac{D^2x_i}{dt^2}(t_{j_i}^+) \hbox{ and } \frac{D^3x_i}{dt^3}(t_{j_i}^+) = \frac{D^3x_i}{dt^3}(t_{j_i}^-).
\end{equation*}

 Since $x_i$ is a $C^1$ curve with continuous covariant derivatives  up to order 3, $x_i$ is $C^3$ on $[0,T]$. But, we have shown that $x_i$ is the solution of a fourth order smooth ODE, which means the fourth derivative can be expressed as a smooth function of derivatives up to order 3. The $k^{th}$ order derivative can be expressed as a smooth function of derivatives upto order $(k-1)$, and so by induction, $x_i$ is smooth on $[0,T]$. Hence, statement 2 follows.

Now, the only non zero terms left in the cost function are given as
\begin{align*}
   - &\Big \langle X(0),\kappa \frac {dx_i}{dt}(0)-\frac{D^3x_i}{dt^3}(0) \Big \rangle + \Big \langle \frac{DX_i}{dt}(T),\frac{D^2x_i}{dt^2}(T) \Big \rangle \\
   -&\Big \langle \frac{DX_i}{dt}(0),\frac{D^2x_i}{dt^2}(0) \Big \rangle +\Big \langle X(T),\kappa \frac {dx_i}{dt}(T)-\frac{D^3x_i}{dt^3}(T) \Big \rangle
\end{align*}

Due to the constraints \eqref{constraint}, we are always forced to choose $X_i \in T_{x_i}\Omega_i$ such that
\begin{equation*}
\begin{split}
X_i(0)=X_i(T) =\frac{DX_i}{dt}(0)= 0.
\end{split}
\end{equation*}
Now, we choose a variational vector field $X_i$ such that we also have $\displaystyle{\frac{DX_i}{dt}(T) = \Pi_{T_{x_i(T)}S}\left(\frac{D^2x_i}{d t^2}(T)\right)}$
where $ \Pi_{T_{x_i(T)}S}V$ is the orthogonal projection onto $T_{x_i(T)}S$ of $V \in T_{x_i(T)}M$. Since
 $X_i(T) = 0$, $\displaystyle{\frac{dX_i}{dt}(T) = \frac{DX_i}{\partial t}(T) \in T_{x_i(T)}S}$. Since $X_i$ is the variational vector field of an admissible variation,
\begin{equation*}
\begin{split}
\frac{d}{dr}J(\alpha_r)\big{|}{r=0} & = \Big \langle  \Pi_{T_{x_i(T)}S}\left(\frac{D^2x_i}{\partial t^2}(T)\right), \frac{D^2x_i}{\partial t^2}(T) \Big \rangle = 0 \\
&\implies \Pi_{T_{x_i(T)}S}\left(\frac{D^2x_i}{\partial t^2}(T)\right) = 0.
\end{split}
\end{equation*} and hence statement 3 holds.

Now we consider a scenario where $x(T)$ is not fixed but is only constrained to lie at an arbitrary point in $S$. In that case $X(T)$ is not always zero. So the term in the cost function

\begin{equation}\label{condition4}\Big \langle X_i, \kappa \frac{d x_i}{dt} - \frac{D^3\alpha}{\partial t^3} \Big \rangle(T)\end{equation} in the expression for $\frac{dJ}{dr}$ and it does not vanish because $X_i(T)=0$.
Since $x_i(T)\in S$, we must now have $X_i(T)\in T_{x_i(T)}S$. By choosing a variation such that $$X_i(T)=\Pi_{T_{x_i(T)}S}\left(\kappa \frac{dx_i}{dt} - \frac{D^3 x_i}{dt^3} \right)(T),$$ it follows that $$\kappa \frac{dx_i}{dt} - \frac{D^3x_i}{dt^3} \perp T_{x_i(T)}S$$ because if otherwise, it will contribute a strictly positive value for $\frac{d}{dr}J(\alpha_r)|_{r=0}$ when $X_i$ is chosen such that Hence statement $4$ holds. \hfill$\square$

\iffalse
If $\Gamma^{k}_{ij}(x) = 0 \ \forall x \in S$, then $\frac{DX_i}{\partial t}(T) = \frac{dX_i}{dt}(T)$. Choose $X_i \in T_{x_i}\Omega_i$ such that \\
\begin{equation*}
\begin{split}
X_i(T) = \Pi_{T_{x_i(T)}S}(\frac{D^3x_i}{dt^3} - k\frac{dx_i}{dt}) \qquad  \frac{DX_i}{dt}(T) = \Pi_{T_{x_i(T)}S}(\frac{D^2x_i}{\partial t^2}(T))
\end{split}
\end{equation*}
Therefore
\begin{equation*}
\begin{split}
\at{\frac{d}{dr}J(\alpha_r)}{r=0} & = \Big \langle  \Pi_{T_{x_i(T)}S}(\frac{D^3x_i}{dt^3} - k\frac{dx_i}{dt}), \frac{D^3x_i}{dt^3} - k\frac{dx_i}{dt} \Big \rangle = 0 \\
&\implies \Pi_{T_{x_i(T)}S}(\frac{D^3x_i}{dt^3} - k\frac{dx_i}{dt}) = 0
\end{split}
\end{equation*}
Hence, 3.2 (iv) follows.
\fi

\vspace{.5cm}

\textbf{Boundary conditions and well-posedness:}
Note that condition (1) in Theorem 3.2 is a fourth order ODE in each of the $n$ $x_i$ variables which are $s$ dimensional. So, the total order of the system of ODEs is $4sn$, and hence, for well posedness, $4sn$ conditions are required in the form of initial and/or boundary conditions which is verified for each case in the tables given below. Recall again that $S$ is a $m$ dimensional manifold.
\begin{itemize}
    \item Case 1: Final position fixed, final velocity in $TS$ \\ \\
        \begin{tabular}{l|c}
        \hline
      \textbf{Condition} & \textbf{$\#$ of constraints}\\
        \hline
        $x_i(0)=p_0^i$ & $sn$\\

        $\frac{dx_i}{dt}(0)=v_0^i$ & $sn$ \\

        $x_i(T)=p_T^i\in S$ & $sn$ \\

        $\frac{dx_i}{dt}(T)\in T_{x_i(T)}S$ & $n(s-m)$ \\

        Condition (3) in Theorem 3.2 & $mn$ \\
        \hline
        TOTAL &    $4sn$

	\end{tabular} \\
	\item Case 2: Final velocity also fixed \\ \\
	 \begin{tabular}{l|c}
        \hline
      \textbf{Condition} & \textbf{$\#$ of constraints}\\
        \hline
        $x_i(0)=p_0^i$ & $sn$\\

        $\frac{dx_i}{dt}(0)=v_0^i$ & $sn$ \\

        $x_i(T)=p_T^i$ & $sn$ \\

        $\frac{dx_i}{dt}(T)=v_T^i$ & $sn$ \\

        Condition (3) in Theorem 3.2 & Vacuous \\
      \hline
        TOTAL &    $4sn$\\

	\end{tabular} \\
	\item Case 3: Final position and velocity not fixed but in $TS$\\ \\
	 \begin{tabular}{l|c}
        \hline
      \textbf{Condition} & \textbf{$\#$ of constraints}\\
        \hline
        $x_i(0)=p_0^i$ & $sn$\\

        $\frac{dx_i}{dt}(0)=v_0^i$ & $sn$ \\

        $x_i(T)\in S$ & $s(n-m)$ \\

        $\frac{dx_i}{dt}(T)\in T_{x_i(T)}S$ & $s(n-m)$ \\

       Condition (3) in Theorem 3.2 & $mn$ \\

        Condition (4) in Theorem 3.2 & $mn$ \\
        \hline
        TOTAL &    $4sn$\\

	\end{tabular}
\end{itemize}

\section{Examples}
\subsection{Application to variational obstacle avoidance problem for a planar rigid body on $SE(2)$.}
The  special euclidean Lie group $SE(2)$ consists of all the transformations of $\mathbb{R}^2$ of the form $z \mapsto Rz+v$, where $v\in \mathbb{R}^2$ and $R\in SO(2)$. This Lie group is, as a smooth manifold, diffeomorphic to $\mathbb{R}^{2}\times S^{1}$.

The Riemannian metric on $SE(2)\simeq\mathbb{R}^{2}\times \mathbb{S}^{1}$, locally parametrized by $\gamma=(q,\theta)=(x,y,\theta)$, is determined by a diagonal matrix with $\mbox{diag =}(m,m,J)$. The reason stems from the fact that it is the same matrix that yields the kinetic energy of a planar rigid body with $m$ being the mass of the body and $J$ the moment of inertia of the body. The curvature tensor is zero under this metric.

We denote by $\gamma_i=(q_i,\theta_i)=(x_i, y_i,\theta_i)$ the trajectory of the $i$-th agent, $i=1,2,..., n$, and represent by $p^k=(p_x^k,p_y^k)$ the center of an obstacle with circular shape in the $xy$-plane, with radius $r^k$. The submanifold $S$ of $SE(2)$ given by $\mathbb{S}^1\times S(p^0,l)$, where $S(p^0,l)$ is the circle with center $p^0\in\mathbb{R}^2$ and radius $l$.
We consider the artificial potential function $V$ given by
\begin{align*}V(\gamma) =& \frac{1}{2}\sum_{i=1}^n\sum_{j=1}^{n}\frac{\sigma}{\|q_i - q_j\|^2-4d^2}\\&+\sum_{k=1}^{\alpha}\sum_{i=1}^n\frac{\tau}{\|q_i - p^k\|^2-(d+r^k)^2},\end{align*} where $d$ is the radius of the smallest ball containing each agent and $\tau,\sigma\in\mathbb{R}^{+}$.
Note that, for $i=1,2,..,n$,
\begin{align*}\frac{\partial V}{\partial x_i} (\gamma)=&\sum_{j=1}^{n}\sum_{j=1}^{n}\frac{\sigma}{(x_i-x_j)^2+(y_i-y_j)^2-4d^2}\\&\times((x_j-x_i)\frac{\partial}{\partial x}+(y_j-y_i)\frac{\partial}{\partial y})\\
&+\sum_{i=1}^{n}\sum_{k=1}^{\alpha}\frac{2\tau}{(x_i-p_x^k)^2+(y_i-p_y^k)^2-(d+r^k)^2}\\&\times((p_x^k-x_i)\frac{\partial}{\partial x}+(p_y^k-y_i)\frac{\partial}{\partial y}).
\end{align*}

By Theorem \ref{mainTh}, the equations determining necessary conditions for the extrema in the variational collision and obstacle avoidance problem are\begin{align*}

                   \theta_i^{(4)}=&\kappa\theta_i'',\\
                  x_i^{(4)}=&\kappa x_i''+\sum_{i=1}^{n}\sum_{j=1}^n\frac{\sigma(x_j-x_i)}{(x_i-x_j)^2+(y_i-y_j)^2-4d^2}\\+&\sum_{i=1}^{n}\sum_{k=1}^{\alpha}\frac{2\tau (p_x^k-x_i)}{(x_i-p_x^k)^2+(y_i-p_y^k)^2-(d+r^k)^2},\\
                  y_i^{(4)}=&\kappa y_i''+\sum_{i=1}^n,\sum_{j=1}^n\frac{\sigma(y_j-y_i)}{(x_i-x_j)^2+(y_i-y_j)^2-4d^2}\\&+\sum_{i=1}^{n}\sum_{k=1}^{\alpha}\frac{2\tau (p_y^k-y_i)}{(x_i-p_x^k)^2+(y_i-p_y^k)^2-(d+r^k)^2},
                 \end{align*}  with given beginning position and velocity
             $(x(0),y(0),\theta(0))$ and   $(x^{\prime}(0),y^{\prime}(0),\theta^{\prime}(0))$ and satisfying the end-boundary conditions $x_i''(T)(x_i(T)-p_x^0)+y_i''(T)(y_i(T)-p_y^0)=0$ and $(\kappa x_i'(T) - x_i''''(T))((x_i(T)-p_x^0)+(\kappa y_i'(T) - y_i''''(T))((y_i(T)-p_y^0)=0$.

\subsection{Application to variational collision and obstacle avoidance problem on $S^2$.}
Next, we study the variational collision and obstacle avoidance problem on the sphere. This problem has applications in, for instance, trajectory planning for aircrafts in the presence of unsafe regions of flight (which can be considered as obstacles). Here, $S^2=\{x \in \mathbb{R}^3, ||x||_2=1\}$ is considered as a Riemannian submanifold of $\mathbb{R}^3$ and hence, its Riemannian metric is inherited from the standard metric on $\mathbb{R}^3$ as an inner product space. $S^2$ can be parametrised in spherical polar coordinates (or longitudes and latitudes) as  $(x_1,x_2,x_3)(\theta,\phi)=(\sin\theta \sin\phi, \sin\theta \cos\phi, \cos\theta)$ and this serves as a coordinate system for $S^2$ except at a great circle passing through $(0,0,1)$ where the parametrization becomes singular and/or non-injective.
With this choice of parametrization, if $x(t)=(\theta(t),\phi(t))$ is the representation of a parametrized curve (in this chart), then it can be shown that for the actual curve $x(t)$,
\begin{align*}
    \frac{D^4x}{Dt^4}&=(\theta''''  +5\sin(2\theta)\theta'^2\phi'^2+(1-7\cos^2\theta)\theta''\phi'^2\\&+(5-17\cos^2\theta \theta'\phi'\phi''-3\sin\theta \cos\theta \phi''^2\\&-2\sin(2\theta)\phi'\phi'''+\sin\theta \cos^3\theta \phi'^4)\frac{\partial}{\partial \theta}+\\&(\phi''''-7\theta'\theta''\phi'-5\theta'^2\phi'')+4cot\theta \theta''' \phi' + (6cot\theta \theta'' \phi'' \\&+4\cot\theta \phi'''\theta'+(\sin(2\theta)-\cot\theta(5\cos^2\theta-1))\theta'\phi'^3)\\&-6\cos^2\theta\phi'^2\phi''-2\cot\theta\phi'\theta'^3\frac{\partial}{\partial \phi}
\end{align*}
 and
 $R(\frac{D^2x}{Dt^2},\frac{dx}{dt})\frac{dx}{dt}=0$.

Let us assume $\kappa=0$ and there are only two agents indexed by $i\in\{1,2\}$.
 Next, we need a potential function for each agent to capture the obstacle avoidance terms.

 Let the obstacle be described by  $\{x\in S^2 | f(x)\leq0\}$ where $f$ is a smooth real valued function on $S^2$. Assume thatn the obstacle is a circle centered at the north pole in $S^2$ of geodesic radius $\frac{\pi}{4}$, then we have in local coordinates, $f(\theta,\phi)=\theta-\frac{\pi}{4}$. Then we choose a smooth potential function that is non-negative for $f>0$ which approaches to infinity as $f\rightarrow 0^+$. It should also become constant when $f$ is sufficiently large and positive. For the above situation, a  candidate $V$ would be $$V_i(\theta_i,\phi_i)=\frac{1}{(\theta_i-\frac{\pi}{4})^2}.$$

 Then, using the Riemannian metric of $S^2$ in the above chart, we can find the gradient of $V_i$ which is required in Theorem 3.2. It is given by $$\hbox{grad}(V_i(\theta_i,\phi_i))=\Big{[}0,...,\frac{-2}{(\theta-\frac{\pi}{4})^3},0,...\Big{]}^T$$ where the non-zero pair is the $i^{th}$ pair.

 Now, we need another potential function to capture the collision avoidance terms. As far as the collision term is concerned, if $d_M(x,y)$ denotes the Riemannian distance between two points $x,y$, for the sphere it may be defined by using the exponential function as $$d_{S^1}(x,y)=||\exp_x^{-1}y||,$$ and it can be shown that $$\frac{\partial}{\partial r}d_{S^1}(x,y(r))\big|_{r=0}=-\Big{<}\frac{\partial y}{\partial r}\big|_{r=0}, \exp_{y(0)}^{-1}x\Big{>}.$$
  Then we need a potential function that goes to infinity as the inter-agent distance approaches to zero and it is close to zero when the inter-agent distances are large enough. A candidate function would be
 $\displaystyle{F(d^2_S)=\frac{1}{2}\frac{1}{d_S^2}}$.
 Then, it can be shown as in [20] that for the potential function $V_{12}=F(d_S^2(x_1,x_2))$, $$(\hbox{grad} V)^i=(-F'(d_S^2)\exp_{x_1}^{-1}x_2)$$ where$(\hbox{grad} V)^i$ denotes the $i^{th}$ component of gradient.

 Now for two points $x,y\in S^2$, we have
\begin{align*}
    \exp_x^{-1}(y)=\cos^{-1}\bigg(\big<x,y\big>\frac{y-\big<x,y\big>x}{\sqrt{1-\big<x,y\big>^2}} \bigg)
\end{align*} and hence $\displaystyle{d_S(x,y)=\bigg|\bigg| \exp_x^{-1}(y)\bigg|\bigg|=\cos^{-1}(\big<x,y\big>)}$.
So the final potential function is constructed by combining the obstacle and collision terms as $V=V_1+V_2+V_{12}$, and therefore substituting the respective gradients. \\ \\
 So, equipped with a parametric representation of the curve, the conditions of Theorem 3.2 give a two point boundary value problem for the parametrized curve $(\theta_i(t),\phi_i(t))$ may be solved numerically to determine the optimal trajectories.

\section{Conclusions and Future work}

We discussed the problem of collision and obstacle avoidance of multi-agent systems on a  Riemannian manifold and
derived, from the point of view of calculus of variations,
first order necessary conditions for optimality in the
problem. We have shown how the main result can be applied
for the particular cases of planar rigid bodies and agents evolving on a sphere. In future work we intend to extend the results presented in this paper to the Lie group case and sub-Riemannian problems which allows to consider non-holonomic vehicles.
as well as to explore numerical results.

redpajama_set_name: RedPajamaArXiv


It's all about the food consumed and the commercials aired during the Super Bowl, a $10 billion industry, which, ultimately, is good for the economy. According to a survey from the Retail Advertising and Marketing Association, consumers who watched the game spent approximately $11 billion on food and other goodies (I'm thinking Rolaids should be a sponsor). 1.25 billion chicken wings and 4.4 million pizzas were eaten while watching grown men run after a football. Add the chips, dips and beers to everything else eaten that day makes the average fan add 2,000+ calories to his intake. Wow! Good luck on that treadmill this week.
The cost of commercials airing during the 2013 Super Bowl went for a whopping $3.8 million for 30 seconds. Up from 2012's cost of $3.5 million. That's a far cry from the $42,000 charged for spots during the first Super Bowl in 1967. All the commercials were cute, as usual, but that Budweiser commercial showing the Clydesdale with a personality was my winner.
And finally, there was the half-time show with, who else, but Beyoncé, I really have to admit that she has great . . . hair. And it wasn't affected by all that fire, ice and smoke. But, was she really singing?
I'm sure you know by now that the Ravens are the 2013 Super Bowl Champions. It was a good game, although I know nothing about football. Ahh, can't wait for next year!!!


My father lived an active life of 76 years. There are days when I think of him deeply and feel as though I've 'touched' him.
His life can be seen in a collection of pictures in a box: a young, single man sporting a shotgun; a stern looking father standing behind his children; a rugged outdoorsman proudly displaying a day's catch; an adored husband; an aging, pensive man, appearing to be planning his next project or adventure. I shuffle through these pictures and for a moment, 'touch my father's hand'.
I'm surrounded by reminders of my father, keeping alive my memory of him and the times we shared. At my workbench, I hold the tools he left behind and sense the touch of his strong, calloused hand, a hand toughened on the instruments of his trade and his handyman skills: a hammer and hand crafted wooden box, still full of his selection of nails; old hand saws in need of sharpening; chisels and broad knives; a duster that outlived its usefulness as a pure bristle paintbrush.
In a storage room, a wallpaper board he gave me warps with age. Always the teacher, I'm sure he was hoping that I was watching and learning some of his decorating skills. I was. I move my hand slowly across its paste stained surface and 'touch my father's hand'.
Under my desk lies his rifle that we sighted at the range each year in preparation for hunting season. I never saw him shoot a deer. Unlucky, a poor shot, or just a man who enjoyed the woods without the gaming, Now, I feel the cold steel of the tarnished barrel, the etched wood of the worn stock, and the smooth, shiny finish of the trigger, and I 'touch my father's hand'.
A once classic set of golf clubs stands alone in the corner. We woke early on Saturdays to meet my brothers for a morning of serious, but fun, golf. Good shots were applauded, poor shots were cursed, and loose change went to the winners. Once again, the time together presented him with great teaching moments about golf, life and friendship. Today, I pull a club from the worn golf bag that has stiffened with age, grip it and give it a waggle. Almost seems like Saturday morning.
An array of fishing rods lay against the cellar wall. Old reels, pitted by ocean salt, collect dust and spider webs while looking down at me from an overhead shelf. Homemade lures hang by their treble hooks on a nail. I brush away the webs and give one of the reels a crank or two. For a fleeting moment, I see him casting in the foamy, white surf along a sandy Massachusetts beach or rocky Rhode Island shore, places where we fished, side by side. I feel the stiff wire leader run between my fingers, as I close my eyes to a carefree time, a 'once upon a time'.
l have many wonderful memories of my father. I have things of his that depict him and his life. Yet, I don't have him, anymore. It's the natural progression of life, isn't it? But today, when I reached out and held those things that once were his, I 'touched my father's hand'…..and it was wonderful, again.
This is beautiful – thank you for sharing. It brings back many memories of my father who lived to be 91, rode his bike until he was 90 and tended his vegetable and flower garden until the same age.
The tittle is very appropriate.
it was very touching.I have some thoughts on my DAD.He worked as a coal miner in Pa.In his spare time he was a cobbler. Fixed many of my shoes.I have the Shoe tree that he worked on. Every now & then I go to the back of the basement to remember him.
Thanks, Steve. Sounds like both our dads were hard working men. Thank you for your feedback, Steve. I'm glad you liked it.
My father was an artist, and I have many of his pieces that I cherish. I also have his art box with paint brushes and dried up oil paints. Just as you said, holding them is like touching his hand. Thank you for evoking sweet memories.
Thank you for enjoying my essay, Ida. I'm so glad that it moved you. While time marches on , it's still pleasant to hold onto certain memories, especially of folks so close and important to us. My dad was that kind of man and it sounds like yours was, too.
So beautifully put and so nice that even after all these years, you remember him so well and so fondly. I'm sure he would be nothing but honored.
This is a very poignant and beautiful tribute to your father, Steve. He sounds like an amazing man.
A wonderful tribute to your father, Steve.


text:
Valerie Jacqueline Candida "Val" Ross (17 October 1950 – 17 February 2008) was a Canadian writer and winner of the 2004 Norma Fleck Award for Canadian children's non-fiction. She was also a journalist for the newspaper The Globe and Mail, and won a National Newspaper Award in 1992 for critical writing.

Biography
Val Ross was born in Toronto in 1950, to Jack and Erma Ross, and had one younger brother, Philip ("Pip"). Val attended the Institute of Child Study, a progressive primary school operated by the University of Toronto.  After completing high school at Jarvis Collegiate Institute, she considered becoming a visual artist and studied at St Martin's School of Art in London, England. Eventually she graduated with a bachelor's degree from the University of Toronto.

Before establishing her career as a freelance writer, she worked in urban planning, tourism writing and as a broadcaster at CBC.  She traveled extensively, developing her interests in community, culture and media.

In the early 1970s she began studying Washin Ryu, a form of karate taught by Sensei Burt Konzak, attaining a brown belt. Val was also highly involved with several groups of women in the Toronto area that met frequently for recreational walks (The "Walking Women").

She wrote for Chatelaine, Saturday Night, and Toronto Life, and then was hired as a staff writer and editor at Maclean's, where she worked in the late 1970s and early 1980s. She reported on the conflict in El Salvador, and became an important voice for human rights and freedom of expression.  She spent the remainder of her career at The Globe and Mail, as a publishing reporter, as deputy editor of the Comment section, and as an arts reporter.

She wrote two children's books, 2003's The Road to There and 2006's You Can't Read This. The former, a history of cartography, won the $10,000 Norma Fleck Award in 2004. You Can't Read This is a history of banned literature.

Her final book, the posthumous Robertson Davies: A Portrait in Mosaic, is an oral history of Canadian writer Robertson Davies. She continued to work on the book even after being diagnosed with brain cancer on the day after her 57th birthday in 2007.

She was married to Morton Ritts, and had three children Max Ritts, Madeleine Ritts, and Zoe Ritts.

She died in Toronto on 17 February 2008 of brain cancer.

Selected works

 2003: The Road to There (Norma Fleck Award winner) (Gelett Burgess Children's Book Award Honors)
 2006: You Can't Read This
 2008: Robertson Davies: A Portrait in Mosaic (posthumous)

References
 Martin, Sandra. "Raw courage, strong convictions: She was a reporter's reporter", The Globe and Mail, 2008-02-18, p. S11.

Deaths from brain tumor
Canadian children's writers
Canadian women non-fiction writers
1950 births
2008 deaths
Deaths from cancer in Ontario
Canadian women children's writers
20th-century Canadian journalists
20th-century Canadian women writers


text:
Thirteen Reasons Why Audiobook Free Download7
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13 Reasons is an American Netflix collection based on the 2007 unique Thirteen Reasons Why by Jay Asher and also adjusted by Brian Yorkey for Netflix. Diana Son and also Brian Yorkey are co-showrunners on the series.
Initially developed as a film set to be released by Universal Photo with Selena Gomez ahead function, the adaptation was grabbed as a television collection by Netflix in late 2015. Gomez functioned as an exec producer. The collection premiered on March 31, 2017, bring in 6.08 million U.S. viewers in the very first three days and also its 2nd season released on May 18, 2018, (Although the second season is not associated with the original publication, unlike the first period). In 2018, 13 Reasons was the most viewed Netflix reveal. Jay Asher – Thirteen Reasons Why Audiobook Free. The series was renewed on June 6, 2018, for a third season, which premiered on August 23, 2019. On August 1, 2019, 13 Reasons that was restored for a 4th and final season, which premiered on June 5, 2020.
The first tape is dealt with to Justin Foley, Hannah's very first kiss. She attends to the very first time they kissed after meeting in a playground in their community, and also describes that it behaved and entirely innocent. However, she also discusses that Justin later on boasted to his close friends that even more had actually taken place in the park, and Hannah was called a slut for the remainder of senior high school. At this moment, Clay pauses the tapes and goes to his pal Tony's home, where he obtains his mobile Walkman to make sure that he can listen to the tapes while adhering to the places of each of Hannah's stories. The places are designated on a map that she slipped into the lockers of each recipient before her death.
The second tape is dealt with to Alex Standall, who assembled a checklist during Hannah's freshman year that stated she had the most effective butt in the freshman course. Hannah discusses that the title led to a boy assaulting her in the local sweet-shop, just adding to her feelings of instability after being labeled a slut.
The 3rd tape is addressed to Jessica Davis, that was a new pupil during Hannah's freshman year, the year that Hannah likewise relocated to the community. The two were presented by their support counselor, who wished to give them a support network while they adjusted to senior high school. Hannah never ever considered her or Alex Standall (that was also a new pupil that year) her friends, although they hung around typically at a neighborhood coffee residence. Jessica as well as Alex dated and afterwards broke up. After the best ass occurrence, Jessica was upset that Alex "selected" Hannah over her, as well as both stopped speaking. Jessica continued to raise the rumors that Hannah was a slut.
The 4th tape is dealt with to Tyler Down, that stalked Hannah outside of her bedroom window and took photos of her. Along with Tyler, however, the tape also resolves an unnamed girl that Hannah recognized rather well, and also whose aid she enlisted to help capture the Peeping Tom. The unnamed lady made a decision to offer Tyler a program, initial asking Hannah to offer her a back massage therapy and then exclaiming over missing sex playthings that she "discovered" in Hannah's cabinet. Hannah felt broken by Tyler as her privacy had actually been infringed on, yet she carries on to resolve the unrevealed woman next.
The story of Courtney Crimsen is informed in the fifth tape. She is known as a wonderful, well-liked girl, a persona which Hannah discusses that she fell for. Following the occasion with Tyler, nevertheless, Hannah began to feel overlooked by Courtney, and also when Courtney welcomed her to a party she still really felt questionable. Courtney deserted her at the event, as well as later on Hannah found out that Courtney had been spreading out exists about the "surprises" she 'd found in Hannah's cabinet.
The 6th tape is resolved to Marcus Cooley, who Hannah matched with in a collection of school Valentines. He showed up late to a day for gelato that she arranged, and then molested her in a cubicle at the restaurant where they had actually fulfilled.
The 7th tape is addressed to Zach Dempsey. Hannah clarifies that in her Peer Communications class, there was a collection of bags at the back of the space where students can leave praises for each various other. After her assault in the diner, Zach comforted her. However, she later found that he had been removing notes from her bag. By this factor, Hannah discusses, she was currently dispirited, and that activity only intensified her condition. She additionally discusses that she wrote an anonymous note to her Peer Communications course, in which she shared that she had been feeling self-destructive. No one in the course took the note seriously in any way, rather assuming that it was made up for attention.
The eighth tape is dealt with to Ryan Razor, who participated in a verse class with Hannah while she was spiraling additionally right into depression. They shared their verse with each other, including a rhyme that Hannah discussed not having the ability to accept herself. Ryan later on anonymously released the poem in his magazine, The Lost as well as Found Gazette, betraying Hannah's trust.
The ninth tape is dealt with to Clay, that Hannah especially says does not deserve to be on the list. She says that he is a truly nice child. She discusses that they were at a celebration with each other, extremely close to the end of her life, when every little thing was spiraling out of control. They had a very long discussion as well as discovered a whole lot about each other, and Hannah clarifies that she wanted she had learnt more about him earlier. They kissed, and she felt overwhelmed by remembering what had actually happened in between her as well as Justin.
The tenth tape is addressed to Justin once more. Hannah clarifies that on the night of the party, he enabled his pal to enter a room where an intoxicated, subconscious girl was laying and also rape her. Hannah does not call the kid that devoted the rape; she explains that if she named him, he would certainly skip town and also stop the chain of tapes. Nevertheless, she blames Justin, to a certain extent, for permitting the young boy right into the room to begin with.
The eleventh tape is dealt with to Jenny Kurtz, that offered Hannah a trip residence the night of the event and also struck a quit sign. Later on, a car crash occurred at the crossway, as a result of the quit sign not existing, and also a senior at Hannah's institution was eliminated. Jay Asher – 13 Reasons Why Audiobook Online. The day of his funeral led Hannah to start thinking of her very own funeral service.
The twelfth tape is dealt with to Bryce Pedestrian, who sexually attacked Hannah in a hot tub at the party prior to she left. It is on this tape that Hannah begins talking about how she prepared to kill herself.
The thirteenth tape is dealt with to Mr. Porter, an English educator that completed as an advice counselor for an amount of time at Hannah's school. She mosted likely to him with her feelings of depression and seclusion, and he did nothing to aid her.
As the tapes are played, Clay strolls to every area that Hannah talks about, really feeling more and more troubled as he does so. By the end of the tapes, he really feels transformed, and also the unique ends with him resolving a pupil named Skye, who he sees acting like Hannah, potentially protecting against an additional suicide.
The initial season received universal praise from doubters and also audiences, that commended its topic as well as spreading. Nevertheless, the second, third, as well as fourth seasons' favorable reception began to decrease.
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text:
// Licensed to the .NET Foundation under one or more agreements.
// The .NET Foundation licenses this file to you under the MIT license.
// See the LICENSE file in the project root for more information.

using System.Runtime.Serialization;
using Microsoft.CodeAnalysis.Remote;

namespace Microsoft.CodeAnalysis.ExternalAccess.Razor
{
    [DataContract]
    internal readonly struct RazorPinnedSolutionInfoWrapper
    {
        [DataMember(Order = 0)]
        internal readonly Checksum UnderlyingObject;

        public RazorPinnedSolutionInfoWrapper(Checksum underlyingObject)
            => UnderlyingObject = underlyingObject;

        public static implicit operator RazorPinnedSolutionInfoWrapper(Checksum info)
            => new(info);
    }
}

redpajama_set_name: RedPajamaGithub


Controlling the running state of the 1 ph motor by PLC ; pressing START and STOP pushbuttons i.e. motor should remain in ON state after START pushbutton is pressed and should OFF when STOP pushbutton is pressed. Checking if the Motor is running normally by pressing TEST pushbutton.
X2 – Error signal from Motor to PLC.
Press START button lightly and X0 = ON. The Motor will keep running if no error occurred (X3 = OFF). The action can be practiced by a latching circuit which takes output Y0 as one of the input condition to keep the motor running even if the START button is not pressed(See normally open contact Y0 below X0).
When STOP button is pressed, X1 = ON and Y0 = OFF. The motor will stop running.
If error occur (X3 = ON), Y1 will be OFF and the motor will stop running.
When TEST is pressed (X2 = ON), Y0 = ON. The motor will start running if no error occurred (X3 = OFF). On the contrary, when TEST is released, the motor will stop running. The testing function is performed by this process.


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The laptop is very poorly, and that makes posting a little difficult. It's all going down on paper: pencil and the Moleskine, the red hot blogging tools. Hope to find computer doctor shortly.
Is the Earth More Free?


Mary comes to River Architects with an extensive background that seamlessly melds finance and construction. While working 36 years in local government, she oversaw the financial aspect of many large construction projects. In that capacity, she worked closely with the architectural firms and contractors hired to complete the work. This experience has solidly prepared her to manage and understand the workings of our architectural firm, and her financial and accounting skills are a valuable asset to the River Architects� team.