text:
One Reviews Thread to Rule them all ... (EMBARGO LIFTS: 9 PM PST)
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Maiarmike
Dec 4 2012, 4:59am
T-2 minutes until elation, or crushing disappointment. // [In reply to] Can't Post
"I warn you, if you bore me, I shall take my revenge"
--J.R.R. Tolkien
BiggieSmaug
HERE WE GOOOOOOO [In reply to] Can't Post
Xanaseb
hmmm interesting review... not very comfortable with the constant comparisons with Lucas and SW tbh... [In reply to] Can't Post
Also, not positive about good ol' Sir Chris Lee :
There's the marrying old characters with new, de-aging certain players and setting events in motion, which unfortunately leads to some particularly clunky, foreboding scenes involving Christopher Lee playing the Middle Earth equivalent of Star Wars' Chancellor Palpatine.
--I'm a victim of Bifurcation--
Join us over at Barliman's chat all day, any day!
Things were so much simpler 11 years ago... [In reply to] Can't Post
As naive as it may have been, I never had any doubt that Fellowship of the Ring would live up to my ridiculously high expectations. I was as excited as anyone possibly could be, and I walked into that theater on cloud nine and walked out the same way. The opinion of the critics never even factored into it.
If only I could have that same mentality now. I think I will refrain from reading the negative ones though. I want to walk in as untainted by negativity as I did 11 years ago.
"All men dream; but not equally. Those who dream by night in the dusty recesses of their minds awake to find that it was vanity; But the dreamers of day are dangerous men. That they may act their dreams with open eyes to make it possible."
- T.E. Lawrence
shadowfaux
NZ Herald Review [In reply to] Can't Post
But more positive concerning a certain creepy thing: [In reply to] Can't Post
The dwarves' quest to reclaim their homeland has been given an extra edge with the addition of the truly nightmare-inducing Azog (Manu Bennett), whose appearance threatens to turn the film into a cross between 300 (he bears more than a passing resemblance to Xerxes) and The Princess Bride (a dwarf with a particularly potent and legitimate grudge Thorrin as a kind of Inigo Montoya).
My Top 5 Wish List for "The Hobbit"
5. Legolas will surf down Smaug's neck
4. Bilbo will be revealed to a Robot
3. Naked PJ cameo as Ghan-Buri-Ghan
2. Use of not only 3D, but smell-o-vision, plus the inclusion of axes coming out of the seats and poking the audience when appropriate
1. Not only keep the claim that Thorin & Co. ran amok in Mirkwood "molesting people", but depict said incident in vivid detail!!!!!
Two More! [In reply to] Can't Post
easterlingchief1
First Rotten Tomatoes review is up! [In reply to] Can't Post
And it's very positive!
http://www.flicks.co.nz/news/review-the-hobbit-an-unexpected-journey/
Wordofmask
alrigh [In reply to] Can't Post
ok first score from NZ herald 4.5/5
we should make the first post of this thread as collecting score post. that is good to all of us.
... and More! [In reply to] Can't Post
indiewire.com
collider.com
Dogsofwar
HitFix gives it a "B" [In reply to] Can't Post
http://www.hitfix.com/...n-unexpected-journey
Huffington Post reviews 48fps [In reply to] Can't Post
48 FPS review
Cinamablend, Movieline, [In reply to] Can't Post
Cinamablend
movieline 'The Hobbit' 3-D Early Review: Back Again, But Not Quite There
Actually several of these first reviews have singled out [In reply to] Can't Post
Azog as a plus for the film.
And sounds as if they enjoyed the film itself as well!
Review from Garfeimeo from TORn [In reply to] Can't Post
TheOneRing.net staffer Garfeimao and long-time TORn friend Nancy Steinman were able to secure tickets to the World Premiere of The Hobbit: An Unexpected Journey. Here are Garfeimao's largely spoiler-free thoughts on what she saw.
I am not here to review the technology, this is just going to be a short and sweet commentary on the movie. Martin Freeman is a joy to watch inhabit the role of Bilbo Baggins: you feel everything he feels, and that is all those emotions you read about in the book. Fear, curiosity, courage, homesickness and that sense that he has bitten off more than he can chew.
There are whole chunks of dialogue lifted directly from the book, you will recognize it throughout the film, I had to restrain myself from cheering during at least one of these lines, won't say which one just yet.
Gandalf is Gandalf, the slightly grumpy, sometimes playful Gandalf the Grey we met 11 years ago is back.
And Richard Armitage as Thorin and Ken Stott as Balin, in particular, shine in their respective roles. You will feel you are in the presence of a King without a Throne. But the kudos really have to go to Andy Serkis as Gollum and Martin's Bilbo.
The Riddles in the Dark will have you on the edge of you seat, even though you know exactly what happens. It is masterfully delivered, you may even cry (yes, one of many spots I teared up at).To be honest, yes there are other bits that are a bit confusing and may feel misplaced, but I will want to watch it a second time before I pass final judgement on this film. I loved the ending, it is rather awesome
resurection [In reply to] Can't Post
So is Azog, who was killed in the Goblin/ Dwarf wars resurrected?
Review from MrCere from TORn [In reply to] Can't Post
At the first public showing of a movie picture depicting a train coming toward the viewers, people scattered in alarm trying to get out of the way. Or so the story goes. Some will do the same with director Peter Jackson�s new film The Hobbit: An Unexpected Journey.
The irony is that The Hobbit's titular character, Bilbo Baggins, doesn�t like to take risks in a film that revels in them.
It's visionary stuff, and Jackson is alone on the frontier with only James Cameron visible on the horizon. The first installment of The Hobbit trilogy is new cinema.
Instead of playing safe and comfortable in lower-risk financial zones, Jackson has pushed the boundaries with 3D, with 48 frames of film per second -� or the digital equivalent -- flashing past viewers' eyes instead of the traditional 24fps. It is definitely different.
The filter of the traditional look of cinema, what we are used to seeing in the dark, is gone. Forever? Consumers will ultimately decide but studios continue to churn out films in 3D despite anything but universal acclaim.
Here, Jackson has gone to such lengths to make the fantastic look real that for many the film will look unreal -� but only if viewers choose to see it in the director�s preferred way. Of the approximately 24,000 theaters that will display The Hobbit: AUJ, around 1,000 will have the technology to display it as it was shot -- HFR 3D. But, for those 1,000, hang onto your arm rests, cinema has just leapt forward.
Bilbo Baggins (masterfully portrayed by Martin Freeman in a performance that will be lost amidst the eye-popping action sequences), as many readers will know, relishes his comfortable life in the cinematically familiar Bag End. He is interrupted by a gaggle of dwarves and a meddlesome wizard who for reasons he cannot himself explain, thinks Bilbo needs to travel with the dwarves.
All of this is familiar ground for readers of Tolkien's original tale. And, as expected, the screenwriters (Fran Walsh, Phillipa Boyens, Guillermo del Toro and Peter Jackson) make great effort to up the stakes and ramp-up the tension.
People who haven�t been paying attention to the promotional lead-in campaign may be surprised to find that joining Freeman and Ian McKellen's Gandalf as lead characters in the film is Richard Armitage as Thorin Oakenshield. It's a role likely to launch Armitage, blessed with an already-fervent fanbase, masculine good looks, and ample ability, into the stratosphere. The character is deeply scarred and tragic. Thorin gives the children�s tale a Shakespearean disaster angle and the film makes the most of it in in ways that may not surprise viewers but will nevertheless delight them.
Andy Serkis returns to work his performance-capture magic as Gollum, and Jackson and his actors reach the peak of performance and technical ... well ... wizardry in a scene that is the film�s best. Despite being dialog-heavy, it is cinematic magic.
So Bilbo, content to play it safe in a movie that most definitely isn�t, is joined by all these dwarves who despite excellent design and characterization, just don�t have enough room in the movie. Considering Jackson�s Middle-earth legacy, many viewers will immediately begin pondering extended editions and hope for more from these characters.
Then -� especially if they see it in 48 fps -� they'll want to schedule another viewing. This writer�s initial reaction to the film was wanting to see it again immediately. As of this writing, that has been impossible. Fans, and those who care about film as an art form (and perhaps as a science) will want to view The Hobbit: An Unexpected Journey in both its incredibly clear new format and as a traditional film.
Others, the Bilbo Baggins of the world, are perhaps better advised to simply see the film as they saw The Lord of the Rings. They will be thrilled with adventure and the interplay of now-familiar movie characters playing again inside the incredible landscapes of Middle-earth. Peering deeper into dwarven culture will come with a (pun-intended) deep satisfaction and they will be thrilled, despite some dark days for fans, back in a delightful rendition of Jackson's Middle-earth..
It goes without saying that not everybody will like everything. Setting aside technology for a moment, sections of the film that delight some will frustrate others.
In one place Gandalf reuses a familiar story-telling device. For some that will hearken back to the familiar. For others it will seem like a rehash. There are a handful of such moments. The 100% CGI orc Azog may feel cartoon despite (or because of) his technical perfection; he is no Gollum. But putting aside the quibbles, Jackson has delivered a tightly paced action flick that is rollicking fun. He has captured again the magic of Tolkien's world with his own stylistic, cinematic stamp.
Radagast the Brown is a delight and injects some of the fringe legends of Tolkien's own hand and some of the lighter tone of the source material into the sometimes dangerously heavy Middle-earth. There is the expanded Goblin Town with its madcap action - more jolly than peilous - which takes license with underground kingdoms. But amidst all this fun we are tossed a dozen decapitations as well creating a puzzle for some parents and sensitive younger viewers. Cate Blanchette casts her graceful spell along with the most welcome Christopher Lee and appropriate the slightly less stressed Elrond from Hugo Weaving, iconic in his own right.
The dead-on casting with exemplary performances from the leads as well as those in small roles make the fantasy feel real more than any technician can -� and the technicians on this film are world-class here. And yes, fantasy films can have exemplary performances and actors selling the fantastic as the everyday, have achieved high-craft in their art. Technically masterful, this adaptation of a classic children�s tale is grand and glorious cinema that is indeed lighter than its Rings cousin but also with nasty battle elements.
What's more, any discussion about the need for three films or the making the decision to do so based on anything but storytelling merits couldn�t be more incinerated if Smaug himself engulfed the conversation in dragon�s fire. The case for three films is settled.
But being first is difficult. Like Ebay selling its first item in 1995 (a laser pointer if you must know) new things can take a while to catch on. The lion's share of reviews are going to address the step forward in display technology (including this one!) but the story will sink or swim because of the characters, their situations and the craft with which they are brought to life.
Viewers need to evaluate: Are they Bilbos? Do they want safe, familiar cinema or do they want to go on an adventure at the theater? The comparison isn't mine originally but I like best the thought that somebody has taken the window out and now we are not looking through the filter of film but looking at the real world. It will take some adjusting.
Will you like it? Well, do you embrace change? Can your entertainment tastes evolve? This is like nothing you have ever seen and it will jar not just your eyes but your brain. The dragon slayers will revel in the new and the adventure. Not every note will ring true for every viewer but whatever else, this movie is about fun and adventure.
And (something that will obviously delight the studio) the real answer to this riddle is that this is a film that cries out to be seen in both formats and compared by the viewer. It is entertaining entertainment but it is also a landmark of technology and a test of crowd behavior. I agree with Jackson and Cameron; this is the future.
Whatever the public and critics conclude, Jackson and his team must be admired for operating on wires high above the crowd with no net or safety line with fearlessness and passion. As for me, I will take that at my cinema experience every time over another action film content to play it safe.
NOTE: Larry D. Curtis spent five weeks on the set during film of The Hobbit and freely acknowledges his opinion of this film is compromised. Further, he is dying to see the film again to have a more informed opinion.
5 more [In reply to] Can't Post
screencrave.com
thewrap.com reviews 48fps
Sydney Morning Herald: Hobbit adaptation a dark, spectacular adventure
heraldsun.com
�The Hobbit�: Prequel Offers Expected � and Excessive � Thrills in an Overstuffed First Film
More, more, more! comingsoon, screencrush, slashfilm, [In reply to] Can't Post
screencrush.com - 5/10???
comingsoon.net 6/10
www.odt.co.nz mostly positive
slashfilm - HFR 3D Review: Peter Jackson�s Latest Is Rousing Yet Repetitive
I might have to wipe my tears of sadness and retire for the night...with my bottle of Jack. // [In reply to] Can't Post
(This post was edited by Maiarmike on Dec 4 2012, 5:40am)
"Martin Freeman's Bilbo is clearly the best screen embodiment of a hobbit yet." [In reply to] Can't Post
Crikey, that's a plaudit and a half.
"Ah, how ironic, the addictive qualities of Sauron�s master weapon led to its own destruction. Which just goes to show, kids - if you want two small and noble souls to succeed on a mission of dire importance... send an evil-minded beggar with them too." - Gandalf's Diaries, final par, by Ufthak.
The Hobbit: An Unexpected Journey' Does What the 'Star Wars' Prequels Failed to Do [In reply to] Can't Post
The Hobbit: An Unexpected Journey' Does What the 'Star Wars' Prequels Failed to Do
chazychris
One thing that every review thus far has in common is unreserved praise for Andy Serkis as gollum. What a treat to look forward to.
"At any rate he is under no enchantment but his own."
-Tolkien, The Hobbit (Gandalf describing Beorn)
not as bad as I thought [In reply to] Can't Post
it's still be good anyway.
Yahoo News! We�ve seen it, and it�s a welcome return to Middle-earthl [In reply to] Can't Post
�The Hobbit� First Look: We�ve seen it, and it�s a welcome return to Middle-earthl


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Regulation of self-driving cars is an increasingly relevant topic in the automotive industry strongly related to the success of the actual technology. Multiple countries have passed legislation and agreed on standards for the introduction of autonomous cars.

Liability

Self-driving car liability is a developing area of law and policy that will determine who is liable when an automated car causes physical damage to persons, or breaks road rules. When automated cars shift the control of driving from humans to automated car technology the driver will need to consent to share operational responsibility which will require a legal framework. There may be a need for existing liability laws to evolve in order to fairly identify the parties responsible for damage and injury, and to address the potential for conflicts of interest between human occupants, system operator, insurers, and the public purse.

Congestion

Authors of the Shared Mobility Principles for Livable Cities recommend that in dense urban areas, governments require that autonomous vehicles must be operated in shared fleets rather than privately owned. They worry the much lower personal time cost of trips (people can do something else instead of driving, or send the car on its own) will create a large amount of  induced demand. They argue a per-trip charge is necessary to discourage low-value trips (such as repeatedly sending cars around town to pick up items).

Conventions on Road Traffic
The Geneva Convention on Road Traffic subscribed to by over 101 countries worldwide, requires the driver to be 18 years old.

The 1968 Vienna Convention on Road Traffic, subscribed to by 83 countries worldwide, establishes principles to govern traffic laws. One of the fundamental principles of the convention had been the concept that a driver is always fully in control and responsible for the  behaviour of a vehicle in traffic.
In 2016, a reform of the convention has opened possibilities for automated features for ratified countries.

In January 2021, proposal of amendment to Article 1 and new Article 34 bis to the 1968 Vienna Convention on Road Traffic was transmitted to one-year period for acceptance.
By 14 January 2022, the amendment to the convention was accepted, for entry into force on 14 July 2022.

UNECE WP.29 GRVA
In February 2018, UNECE's Inland Transport Committee (ITC) acknowledged the importance of WP.29 activities related to automated, autonomous and connected vehicles and requested WP.29 to consider establishing a dedicated Working Party. Following the request, WP.29, at its June 2018 session, decided to convert the Working Party on Brakes and Running Gear (GRRF) into a new Working Party on Automated/Autonomous and Connected Vehicles (GRVA).

In June 2020, WP.29 virtual meeting approved reports from GRVA about its fifth session on "automated/autonomous and connected vehicles" and sixth session on "cyber security and software updates".
The new Regulation on cyber security has been allocated as Regulation 155 and the new Regulation on software updates has been allocated as Regulation 156.
In this way, UN regulation on Level 3 was established.

In March 2021, the following UNECE regulations were published:
 Regulation 155: Cyber security and cyber security management system
 Regulation 156: Software update and software update management system
 Regulation 157: Automated Lane Keeping Systems (ALKS)

On 22 January 2022, these regulations come into effect in some countries.

In June 2022, the 187th session of WP.29 was held, and several amendments on GRVA regulations were agreed based on the continuous leadership of Japan in chairmanship and as proposing national body.
In case of Regulation 157 of Automated Lane Keeping System (ALKS), the speed limit of 60 km/h is to be changed to 130 km/h, and lane changing function is to be added for passenger car.

Legislation and regulation in Japan
Japan is a non-signatory country to the Vienna Convention. In 2019, Japan amended two laws, "Road Traffic Act" and "Road Transport Vehicle Act", and they came into effect in April 2020. In the former act, Level 3 self driving cars became allowed on public roads.
In the latter act, process to designate types for safety certification on Level 3 self driving function of
Autonomous Driving System (ADS) and the certification process for the asserted type were legally defined.
Through the amendment process, the achievements from the national project "SIP-adus" led by Cabinet Office since 2014 were fully considered and accepted.

In May 2020, "The Road Act" was also amended to include definition of automatic operation control equipment in infrastructure, and came into effect.
In July 2020, the next stage national level roadmap plan was officially issued which had considered social deployment and acceptability of Level 4.
At the end of 2020, Ministry of Land, Infrastructure, Transport and Tourism (MLIT) amended its "Safety Regulation for Road Transport Vehicle" to reflect the finalized UNECE WP.29 GRVA's regulations consistently without delay.

In April 2021, National Police Agency (NPA) published its expert committee's report of FY 2020 on summary of issues in research to realize Level 4 mobility services, including required legal amendment issues.
During the summer of 2021, Ministry of Economy, Trade and Industry (METI) prepared with MLIT to launch a project "RoAD to the L4" to cover R&D with social deployment to realize acceptable Level 4 mobility service, and updated its public information in September. As a part of this project, civil law liability problem reflecting changed roles will be clarified.

About misleading representation in marketing, Article 5 of "Act against Unjustifiable Premiums and Misleading Representations" is applied.

At the end of 2021, NPA proposed an amendment bill on "Road Traffic Act" to include approving scheme for Level 4 services.
In March 2022, the Japanese government adopted the bill to amend the act.
In April 2022, the bill was deliberated at the ordinary National Diet session and passed. Under the amended act, a license system was be introduced for operators of transport services using unmanned Level 4 vehicles, which requires no driver in the remotely monitored vehicle within a limited area. Such vehicles are expected to be used for residents in depopulated areas.
In October 2022, NPA unveiled their plan to make the amended "Road Traffic Act" into effect on April 2023, and on 20 December 2022, Japan's cabinet decided to allow Level 4 self-driving cars to be used for transit and delivery services from April 2023.

Legal status in the United States
In the United States, a non-signatory country to the Vienna Convention, state vehicle codes generally do not envisage—but do not necessarily prohibit—highly automated vehicles . To clarify the legal status of and otherwise regulate such vehicles, several states have enacted or are considering specific laws. By 2016, seven states (Nevada, California, Florida, Michigan, Hawaii, Washington, and Tennessee), along with the District of Columbia, have enacted laws for automated vehicles. Incidents such as the first fatal accident by Tesla's Autopilot system have led to discussion about revising laws and standards for automated cars.

In 2017, the Republican-controlled House of Representatives unanimously passed "SELF DRIVE Act" which would speed the adoption of self-driving cars and bar states from setting performance standards. However, a complementary bill in the Senate, "AV START", failed to pass after Democrats raised objections that it didn't do enough to address safety and liability concerns.
A comprehensive regulatory structure has not yet emerged at either the federal or state level in the United States.

, members of the United States House of Representatives are launching a bipartisan effort to help revive legislative efforts to boost self-driving vehicles.

Federal policies
In September 2016, the US National Economic Council and US Department of Transportation (USDOT) released the Federal Automated Vehicles Policy, which are standards that describe how automated vehicles should react if their technology fails, how to protect passenger privacy, and how riders should be protected in the event of an accident. The new federal guidelines are meant to avoid a patchwork of state laws, while avoiding being so overbearing as to stifle innovation. Since then, USDOT has released multiple updates:
 Automated Driving Systems: A Vision for Safety 2.0 (12 September 2017)
 Preparing for the Future of Transportation: Automated Vehicles 3.0 (4 October 2018)
 Ensuring American Leadership in Automated Vehicle Technologies: Automated Vehicles 4.0 (8 January 2020)

The National Highway Traffic Safety Administration (NHTSA) released for public comment the Occupant Protection for Automated Driving System on 30 March 2020, followed by the Framework for Automated Driving System Safety on 3 December 2020. Occupant Protection is intended to modernize the Federal Motor Vehicle Safety Standards considering the removal of manual controls with automated driving systems, while the Framework document is intended to provide an objective way to define and assess automated driving system competence to ensure motor vehicle safety while also remaining flexible to accommodate the development of features to improve safety.

Historically, a vehicle without driving controls such as a steering wheel, accelerator pedal, and brake pedal would not be in compliance with the Federal Motor Vehicle Safety Standards (FMVSS), the minimum safety equipment needed to legally sell a vehicle to the public. On 10 March 2022, NHTSA updated and finalized the rule on safety requirements for the Occupant Protection which now allows a vehicle without driving controls to comply with US regulations.

State policies

, 38 states have laws or executive orders related to autonomous vehicles.

Nevada
In June 2011, the Nevada Legislature passed a law to authorize the use of automated cars. Nevada thus became the first jurisdiction in the world where automated vehicles might be legally operated on public roads. According to the law, the Nevada Department of Motor Vehicles is responsible for setting safety and performance standards and the agency is responsible for designating areas where automated cars may be tested. This legislation was supported by Google in an effort to legally conduct further testing of its Google driver-less car. The Nevada law defines an automated vehicle to be "a motor vehicle that uses artificial intelligence, sensors and global positioning system coordinates to drive itself without the active intervention of a human operator". The law also acknowledges that the operator will not need to pay attention while the car is operating itself. Google had further lobbied for an exemption from a ban on distracted driving to permit occupants to send text messages while sitting behind the wheel, but this did not become law. Furthermore, Nevada's regulations require a person behind the wheel and one in the passenger's seat during tests.

Florida
In April 2012, Florida became the second state to allow the testing of automated cars on public roads.

California
California became the third state to allow automated car testing when Governor Jerry Brown signed SB 1298 into law in September 2012 at Google Headquarters in Mountain View.

On 19 February 2016, California Assembly Bill 2866 was introduced in California. It would allow automated vehicles to operate on public roads, including those without a driver, steering wheel, accelerator pedal, or brake pedal. The bill states that the California Department of Motor Vehicles would need to comply with these regulations by 1 July 2018 for these rules to take effect. , this bill has yet to pass the house of origin. California published discussions on the proposed federal automated vehicles policy in October 2016.

In December 2016, the California Department of Motor Vehicles ordered Uber to remove its self-driving vehicles from the road in response to two red-light violations. Uber immediately blamed the violations on human error, and has suspended the drivers.

Washington, DC
In Washington, DC's district code:

In the same district code, it is considered that:

Michigan and others
In December 2013, Michigan became the fourth state to allow testing of driver-less cars on public roads. In July 2014, the city of Coeur d'Alene, Idaho adopted a robotics ordinance that includes provisions to allow for self-driving cars.

Legislation in the United Kingdom
In 2013, the government of the United Kingdom permitted the testing of automated cars on public roads. Before this, all testing of robotic vehicles in the UK had been conducted on private property.

In July 2018, "The Automated and Electric Vehicles Act 2018" received royal assent.

In March 2019, the UK became a signatory country to the Vienna Convention.

In 2021, the UK worked on a bill to allow self-driving automated lane keeping systems (ALKS) up to 37 mph (or 60 km/h) after a mixed reaction of experts during the consultation launched in summer 2020. This system would be allowed to give back control to the driver when "unplanned events" such as road construction or inclement weather occurs. The Centre for Connected and Autonomous Vehicles (CCAV) has asked the Law Commission of England and Wales and the Scottish Law Commission to undertake a far-reaching review of the legal framework for "automated" vehicles, and their use as part of public transport networks and on-demand passenger services. The teams developed policy and the full analysis report was published in January 2022.

About misleading representation in marketing, the Society of Motor Manufacturers and Traders (SMMT) published guiding principles as followings:
 An automated driving feature must be described sufficiently clearly so as not to mislead, including setting out the circumstances in which that feature can function.
 An automated driving feature must be described sufficiently clearly so that it is distinguished from an assisted driving feature.
 Where both automated driving and assisted driving features are described, they must be clearly distinguished from each other.
 An assisted driving feature should not be described in a way that could convey the impression that it is an automated driving feature.
 The name of an automated or assisted driving feature must not mislead by conveying that it is the other – ancillary words may be necessary to avoid confusion – for example for an assisted driving feature, by making it clear that the driver must be in control at all times.

In April 2022, UK government confirmed planned changes to "The Highway Code", responding to a public consultation. The changes will clarify drivers' responsibilities in self-driving vehicles, including when a driver must be ready to take back control.
And in August 2022, UK government unveiled a plan to roll out self-driving vehicles on UK roads.

Legislation in Europe
EU
In November 2019, Regulation (EU) 2019/2144 of the European Parliament and of the Council on motor vehicle type approval requirements defined specific requirements relating to automated vehicles and fully automated vehicles. This law is applicable from 2022 and is based on uniform procedures and technical specifications for the systems and other items.

In April 2022, EU released a draft version of its legislation for vehicles with automated driving systems (ADS).

In July 2022, the new "Vehicle General Safety Regulation" come into effect which establishes the legal framework for the approval of automated and fully driverless vehicles (Level 3 and above) in the EU.
And the EU introduced regulations which require all new cars from 6 July 2022 to be fitted with intelligent speed assistance (ISA) (Level 2).

EU Commission delegated regulation contains specific requirements for specific vehicles with regulatory differences between:
 Fully automated vehicles of categories N1, N2 and N3 without driver seat and without occupants
 Fully automated vehicles of categories N1, N2, N3, M1, M2, M3 without driver seat, with occupants
 Dual mode vehicles: vehicles with a driver seat designed and constructed to be driven by the driver in the "manual driving mode" and to be driven by the automated driving system (ADS) without any driver supervision in the "fully automated driving mode"

France
France is a signatory country to the Vienna Convention.
In 2014, the government of France announced that testing of automated cars on public roads would be allowed in 2015. 2000 km of road would be opened through the national territory, especially in Bordeaux, in Isère, Île-de-France and Strasbourg. At the 2015 ITS World Congress, a conference dedicated to intelligent transport systems, the very first demonstration of automated vehicles on open road in France was carried out in Bordeaux in early October 2015.

In May 2018, the government published the first version of the French strategy for the development of automated road mobility to set up the legislative framework, and it brought a result as "The Mobility Orientation Law" in December 2019.

In December 2020, the government updated the strategy to make France the preferred location in Europe for the deployment of automated road mobility services.

The legislative and regulatory framework for the deployment of automated vehicles and transport systems was established through an ordinance in April 2021 and a following decree in June 2021.
The legislative and regulatory framework for the operation of automated vehicles resulting from article 31 of "the Mobility Orientation Law" is scheduled to be finalized in Q1 of 2022.

Germany
Germany is a signatory country to the Vienna Convention.
In July 2021, the "Federal Act Amending the Road Traffic Act and the Compulsory Insurance Act" came into effect.
The Act allows motor vehicles with autonomous driving capabilities, meaning vehicles that can perform driving tasks independently without a person driving, in specified operating areas on public roads. Provisions about autonomous driving in appropriate operating areas correspond to Level 4. Moreover, the new German legislation has major implications for dilemmatic situations. This includes for example the non-discrimination principle that applies to unavoidable crash situations. Moreover, the act elaborates on the technical requirements of autonomous vehicles, including a software system that can operate without permanent supervision of the technical oversight or driver, contains an accident mitigation and reduction system and can initiate a "minimal-risk state."

In February 2022, the Federal Ministry for Digital and Transport (BMDV) submitted the "Ordinance on the Approval and Operation of Motor Vehicles with Autonomous Driving Functions in Specified Operating Areas - Autonomous Vehicles Approval and Operation Ordinance (AFGBV)" to the German Bundesrat for approval.

Legislation in North America

Canada
Canada is a non-signatory country to the Vienna Convention.
At the federal level, "The Motor Vehicle Safety Act" regulates about motor vehicles which was last amended in February 2020.

In August 2021, Transport Canada released the "Guidelines for Testing Automated Driving Systems in Canada" Version 2.0.

Legislation in Asia
China
For historical reason, China is not a signatory country to 1949 Geneva Convention,
although it is a signatory country to 1968 Vienna Convention.
Legislations are conducted at National People's Congress and its Standing Committee which are under control of Chinese Communist Party.

In 2018, China introduced testing regulations to regulate autonomous cars, for conditional automation, high-level automation and full automation (roughly corresponding to Level 3, Level 4 and Level 5).
The rules lay out requirements that vehicles must first be tested in non-public zones, that road tests can only be on designated streets and that a qualified person must always sit in the driver's position, ready to take over control.

In February 2020, eleven constituent departments, represented by National Development and Reform Commission (NDRC), jointly issued the "Strategy for Innovation and Development of Intelligent Vehicles" which describes about roadmap plan until 2025. This plan states about the need to revise the "Road Traffic Safety Law", and surveying and mapping law for intelligent vehicles.

In March 2020, Ministry of Industry and Information Technology (MIIT) published draft GB/T on 6-levels classification framework for driving automation which is basically corresponding to SEA levels.
And in April 2020, MIIT released about the goal of the year which is set to complete the formulation of framework for driving-assist functions and low-level autonomous driving (Level 3).
In January 2021, MIIT planned to add highways to the list of roads where provincial and city-level authorities can authorize automated cars.

In March 2021, Ministry of Public Security (MPS) published draft proposed amendments on the "Road Traffic Safety Law".
In August 2021, The Cyberspace Administration of China (CAC) and MIIT issued "the Provisions on Management of Automotive Data Security (Trial)".
In September 2021, "Data Security Law" came into effect, which broadly expands the extraterritorial reach of China's existing data rules.

In February 2022, MIIT issued the second draft of the "Administrative Measures for Data Security in the Industry and Information Technology Fields".
And in March 2022, MIIT issued "the Guidelines for the Construction of the Internet of Vehicles Cybersecurity and Data Security Standard System".

Legislation in Oceania
Australia
Australia is a non-signatory country to the Vienna Convention.
National Transport Commission (NTC) is in charge of reforming current laws with still achieving national level consistency.
In February 2022, NTC published a policy paper to present proposals on the end-to-end regulatory framework for the commercial deployment of automated
vehicles.

Department of Infrastructure, Transport, Regional Development and Communications (DITRDC) is in charge of developing their policies and bills on first supply and in-service automated vehicle law.

New Zealand
New Zealand is a non-signatory country to the Vienna Convention.
New Zealand legislation does not specifically require a driver to be present for a vehicle to be legally operated on a public road. However, most regulations and relevant international frameworks strongly imply the presence of a driver in the vehicle given that 'automation' was not a consideration at the time of drafting the legislation.

, Ministry of Transport is working on "Autonomous Vehicles Work Programme" with "Long-term Insights Briefing (LTIB)" which will include legislation issues.

Legislation in Middle East
Israel
Israelis a signatory country to the Vienna Convention.
, Israel Innovation Authority (OCS) is working on forming regulatory framework for trials and use of autonomous vehicles with Ministry of Transport and Road Safety (MOT) and Ministry of Justice.

In March 2022, the Knesset passed legislation that will allow companies to pilot autonomous shared transportation with passengers in the vehicle but without a safety driver on Israeli roads. The legislation allows companies and vehicle operators to obtain special licenses from MOT and to conduct trials with autonomous cars including for the purpose of transporting paying passengers and where an independent driving system replaces the driver.

References

Regulation of robots

redpajama_set_name: RedPajamaWikipedia


text:
Q: How to make unclickable footnotes clickable? I have asked all kinds of questions but no solution so far this unclickable footnotes. The only thing I can say is that I'm having this problem since I started using the memoir class.
Here is my code
\documentclass[a4paper,frenchb,12pt,openany,twoside]{memoir}
\let\footruleskip\undefined %undefine footruleskip
\setcounter{tocdepth}{5}
\setsecnumdepth{subsection}
\usepackage{geometry}
\geometry{margin=57pt}
\usepackage{natbib}

%%%%Language
\usepackage[utf8]{inputenc}
\usepackage[francais]{babel}
\usepackage[T1]{fontenc}
\usepackage{charter}
\usepackage[expert]{mathdesign}

%%%%Police
\usepackage{lmodern}% police de caractère
\usepackage{textcomp}% caractères additionnels
\usepackage[normalem]{ulem}
\usepackage{soul}

%%%%%Graphic
\usepackage{graphicx}
\usepackage{xcolor}% gestion des couleurs
\usepackage{fancybox}
\usepackage{lipsum}

%%%%%%%%%gestion de couleur des références;
\usepackage{hyperref}
\definecolor{candyapplered}{rgb}{1.0, 0.03, 0.0}
\usepackage{hyperref}
\hypersetup{hyperfigures = true, colorlinks = true, linkcolor=candyapplered}
\hypersetup{pdfstartview=XYZ}
\usepackage{footnote}

%%%%%Titling
\usepackage{titling}% pour le titre
\usepackage{titlesec}% pour les sections
\usepackage{titletoc}% pour la table des matières
\usepackage{fancyhdr}% pour les en-têtes
\usepackage{enumitem}

%%%%%Math
\usepackage{array}
\usepackage{amsthm,latexsym,amssymb,amsmath, amsfonts}
\usepackage{mathtools}
\usepackage{pstricks}%pour psshadowbox
\usepackage{mathrsfs}
\usepackage{verbatim}
\usepackage{shapepar}
\usepackage{calc}% syntaxe naurelle pour les calculs

%%%%%géstion d'espacement
\usepackage{setspace}
\renewcommand{\baselinestretch}{1.5}

%%%%Page Headings
\addtocounter{chapter}{1}
 \pagestyle{fancy}
 \cfoot{\empty}
 \renewcommand{\chaptermark}[1]{\markboth{#1}{}} %rappel du titre de chapitre
 \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}%rappel du titre de la section
 \lhead[\fancyplain{}{\slshape  \thepage}]
 {\fancyplain{}{\slshape\rightmark} }
 \rhead[\fancyplain{}{\slshape \leftmark}]
 {\fancyplain{}{\slshape \thepage}}
  \addtocounter{chapter}{-1}
\textheight 24cm%
\renewcommand{\chaptermark}[1]{\markboth{#1}{}}
\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
\fancyhf{} % supprime les en-têtes et pieds prédéfinis
\fancyhead[LE,RO]{\sffamily\bfseries\thepage}% Left Even, Right Odd
\fancyhead[LO]{\sffamily\bfseries\rightmark} % Left Odd
\fancyhead[RE]{\sffamily\bfseries\leftmark} % Right Even
\renewcommand{\headrulewidth}{0.85pt}% filet en haut de page
\addtolength{\headheight}{0.5pt} % espace pour le filet
\renewcommand{\footrulewidth}{0pt} % pas de filet en bas
\fancypagestyle{plain}{ % pages de tetes de chapitre
\fancyhead{} % supprime lentete
\renewcommand{\headrulewidth}{0pt} % et le filet
}

%%%%%%%%%gestion des chapitre,section;
\renewcommand{\thechapter}{\Roman{chapter}}

\makeatletter
\newcommand\thickhrulefill{\leavevmode \leaders \hrule height 1ex \hfill \kern \z@}
\setlength\midchapskip{10pt}
\makechapterstyle{VZ14}{
\renewcommand\chapternamenum{}
\renewcommand\printchaptername{}
\renewcommand\chapnamefont{\normalfont\sffamily\large\scshape}
\renewcommand\printchapternum{%
\chapnamefont\null\thickhrulefill\quad
\@chapapp\space\thechapter\quad\thickhrulefill}
\renewcommand\printchapternonum{%
\par\thickhrulefill\par\vskip\midchapskip
\hrule\vskip\midchapskip
}
\renewcommand\chaptitlefont{\normalfont\sffamily\huge\bfseries\centering}
\renewcommand\afterchapternum{%
\par\nobreak\vskip\midchapskip\hrule\vskip\midchapskip}
\renewcommand\afterchaptertitle{%
\par\vskip\midchapskip\hrule\nobreak\vskip\afterchapskip}
}
\makeatother
\chapterstyle{VZ14}

\titleformat{\section}
[hang]% style : hang, display, runin, leftmargin, ...
{\filright\sffamily\Large\bfseries}% fonte numéro + titre
{\thesection}% numéro
{0.5em}% espace entre le numéro et le titre
{}% fonte titre

\titleformat{\subsection}
[hang]% style : hang, display, runin, leftmargin, ...
{\filright\sffamily\large\bfseries}% fonte numéro + titre
{\thesubsection}% numéro
{0.5em}% espace entre le numéro et le titre
{}% fonte titre

%%%%%%%%%gestion des theoreme,lemme,preuve,remarque;
\newtheoremstyle{mes_theoremes}{0.45em}{0em}{}{}{\normalfont\filright\sffamily\bfseries}{\newline}{\parindent}{\thmname{#1}\thmnumber{ #2}\thmnote{ (#3)}}

\theoremstyle{mes_theoremes}

\newtheorem{pro}{Proposition}[section]
\newtheorem{res}{Remarques}[section]
\newtheorem{exs}{Exemples}[section]
\newtheorem{ex}{Exemple}[section]
\newtheorem{Th}{Théorème}[section]
\newtheorem{de}{Définition}[section]
\newtheorem{Le}{Lemme}[section]
\newtheorem{Not}{Notation}[section]
\newtheorem{re}{Remarque}[section]
\newtheorem{prts}{Propriétés}[section]
\newtheorem{prt}{Propriété}[section]
\newtheorem{co}{Corollaire}[section]
\newtheorem{con}{Contre Exemple}[section]
\newtheorem{col}{Conclusion}[section]
\newtheorem*{sol}{Solution.}
%\newtheorem*{pre}{Preuve.}
\newtheorem*{hyp}{Hypothèses :}
\newenvironment{pre}[1][]%
{\par\parindent0pt\medskip\normalfont{\filright\sffamily\bfseries Preuve\ifx&#1&\else\space\no#1\fi.} \par}{\par\medskip}%

\begin{document}
\chapter{Préliminaires}
\section{Notations et définitions}
Dans cette section, nous rappelons quelques définitions, notations et résultats d'analyse fonctionnelle (espaces, opérateur,$\ldots$) et quelques théorèmes qui nous seront utiles dans notre étude.\\

\noindent Soit $J=[a,b]$ un intervalle fini ou infinie de $\mathbb{R}$ où $a,b \in \mathbb{R}$ et $a<b$ et soit $(X,|~.~|)$ un espace de Banach réel.\\
$\blacktriangleright$ $C(J,X)$ l'espace de Banach des fonctions continues $y:J \rightarrow X$, muni de la norme
$$\Vert y\Vert_{\infty} = \sup \lbrace |y(t)|~:~t\in J\rbrace.$$
$\blacktriangleright$ $L^{1}(J,X)$ l'espace de Banach des fonctions mesurables $y:J \rightarrow X$ qui sont Lebesgue intégrable, muni de la norme
$$\Vert y\Vert_{L^{1}} = \int_{a}^{b} |y(s)|~ds.$$
$\blacktriangleright$ $L^{\infty}(J,X)$ l'espace de Banach des fonctions mesurables $y:J \rightarrow X$ qui sont essentiellement bornées, muni de la norme
$$\Vert y\Vert_{\infty} = \inf \lbrace c>0 ~:~ |y(t)|\leq c~~p.p.\footnote{p.p: presque partout}~t\in J\rbrace.$$
$\blacktriangleright$ $AC^{1}(J,X)$ l'espace de Banach des fonctions dérivables $y:J \rightarrow X$ dont la  première dérivée est absolument continue.
\end{document}

These are the results after using the solution in the accepted answer below,

A: The reason for this issue is that package hyperref should be in your case the last called package!
Another issue here is that you should french as class option and option for babel to get rid of the relevant warning.
You should rework the packages you call and the order of calling ... Do you really need all that packages? Do you understand what they do? Please use only packages you know or you have read the documentation (for example with texdoc hyperref in your console/terminal).
Please see the following code
\documentclass[a4paper,french,12pt,openany,twoside]{memoir} % <=========
\let\footruleskip\undefined %undefine footruleskip
\setcounter{tocdepth}{5}
\setsecnumdepth{subsection}
\usepackage{geometry}
\geometry{margin=57pt}
\usepackage{natbib}

%%%%Language
\usepackage[utf8]{inputenc}
\usepackage[french]{babel} % <==========================================
\usepackage[T1]{fontenc}
\usepackage{charter}
\usepackage[expert]{mathdesign}

%%%%Police
\usepackage{lmodern}% police de caractère
\usepackage{textcomp}% caractères additionnels
\usepackage[normalem]{ulem}
\usepackage{soul}

%%%%%Graphic
\usepackage{graphicx}
\usepackage{xcolor}% gestion des couleurs
\usepackage{fancybox}
\usepackage{lipsum}

\definecolor{candyapplered}{rgb}{1.0, 0.03, 0.0}

\usepackage{footnote}

%%%%%Titling
\usepackage{titling}% pour le titre
\usepackage{titlesec}% pour les sections
\usepackage{titletoc}% pour la table des matières
\usepackage{fancyhdr}% pour les en-têtes
\usepackage{enumitem}

%%%%%Math
\usepackage{array}
\usepackage{amsthm,latexsym,amssymb,amsmath, amsfonts}
\usepackage{mathtools}
\usepackage{pstricks}%pour psshadowbox
\usepackage{mathrsfs}
\usepackage{verbatim}
\usepackage{shapepar}
\usepackage{calc}% syntaxe naurelle pour les calculs

%%%%%géstion d'espacement
\usepackage{setspace}
\renewcommand{\baselinestretch}{1.5}

\usepackage[hyperfigures=true]{hyperref} % <================================ last called package
\hypersetup{colorlinks = true, linkcolor=candyapplered}
\hypersetup{pdfstartview=XYZ}

%%%%Page Headings
\addtocounter{chapter}{1}
 \pagestyle{fancy}
 \cfoot{\empty}
 \renewcommand{\chaptermark}[1]{\markboth{#1}{}} %rappel du titre de chapitre
 \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}%rappel du titre de la section
 \lhead[\fancyplain{}{\slshape  \thepage}]
 {\fancyplain{}{\slshape\rightmark} }
 \rhead[\fancyplain{}{\slshape \leftmark}]
 {\fancyplain{}{\slshape \thepage}}
  \addtocounter{chapter}{-1}
\textheight 24cm%
\renewcommand{\chaptermark}[1]{\markboth{#1}{}}
\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
\fancyhf{} % supprime les en-têtes et pieds prédéfinis
\fancyhead[LE,RO]{\sffamily\bfseries\thepage}% Left Even, Right Odd
\fancyhead[LO]{\sffamily\bfseries\rightmark} % Left Odd
\fancyhead[RE]{\sffamily\bfseries\leftmark} % Right Even
\renewcommand{\headrulewidth}{0.85pt}% filet en haut de page
\addtolength{\headheight}{0.5pt} % espace pour le filet
\renewcommand{\footrulewidth}{0pt} % pas de filet en bas
\fancypagestyle{plain}{ % pages de tetes de chapitre
\fancyhead{} % supprime lentete
\renewcommand{\headrulewidth}{0pt} % et le filet
}

%%%%%%%%%gestion des chapitre,section;
\renewcommand{\thechapter}{\Roman{chapter}}

\makeatletter
\newcommand\thickhrulefill{\leavevmode \leaders \hrule height 1ex \hfill \kern \z@}
\setlength\midchapskip{10pt}
\makechapterstyle{VZ14}{
\renewcommand\chapternamenum{}
\renewcommand\printchaptername{}
\renewcommand\chapnamefont{\normalfont\sffamily\large\scshape}
\renewcommand\printchapternum{%
\chapnamefont\null\thickhrulefill\quad
\@chapapp\space\thechapter\quad\thickhrulefill}
\renewcommand\printchapternonum{%
\par\thickhrulefill\par\vskip\midchapskip
\hrule\vskip\midchapskip
}
\renewcommand\chaptitlefont{\normalfont\sffamily\huge\bfseries\centering}
\renewcommand\afterchapternum{%
\par\nobreak\vskip\midchapskip\hrule\vskip\midchapskip}
\renewcommand\afterchaptertitle{%
\par\vskip\midchapskip\hrule\nobreak\vskip\afterchapskip}
}
\makeatother
\chapterstyle{VZ14}

\titleformat{\section}
[hang]% style : hang, display, runin, leftmargin, ...
{\filright\sffamily\Large\bfseries}% fonte numéro + titre
{\thesection}% numéro
{0.5em}% espace entre le numéro et le titre
{}% fonte titre

\titleformat{\subsection}
[hang]% style : hang, display, runin, leftmargin, ...
{\filright\sffamily\large\bfseries}% fonte numéro + titre
{\thesubsection}% numéro
{0.5em}% espace entre le numéro et le titre
{}% fonte titre

%%%%%%%%%gestion des theoreme,lemme,preuve,remarque;
\newtheoremstyle{mes_theoremes}{0.45em}{0em}{}{}{\normalfont\filright\sffamily\bfseries}{\newline}{\parindent}{\thmname{#1}\thmnumber{ #2}\thmnote{ (#3)}}

\theoremstyle{mes_theoremes}

\newtheorem{pro}{Proposition}[section]
\newtheorem{res}{Remarques}[section]
\newtheorem{exs}{Exemples}[section]
\newtheorem{ex}{Exemple}[section]
\newtheorem{Th}{Théorème}[section]
\newtheorem{de}{Définition}[section]
\newtheorem{Le}{Lemme}[section]
\newtheorem{Not}{Notation}[section]
\newtheorem{re}{Remarque}[section]
\newtheorem{prts}{Propriétés}[section]
\newtheorem{prt}{Propriété}[section]
\newtheorem{co}{Corollaire}[section]
\newtheorem{con}{Contre Exemple}[section]
\newtheorem{col}{Conclusion}[section]
\newtheorem*{sol}{Solution.}
%\newtheorem*{pre}{Preuve.}
\newtheorem*{hyp}{Hypothèses :}
\newenvironment{pre}[1][]%
{\par\parindent0pt\medskip\normalfont{\filright\sffamily\bfseries Preuve\ifx&#1&\else\space\no#1\fi.} \par}{\par\medskip}%

\begin{document}

\chapter{Préliminaires}
\section{Notations et définitions}
Dans cette section, nous rappelons quelques définitions, notations et
résultats d'analyse fonctionnelle (espaces, opérateur,$\ldots$) et
quelques théorèmes qui nous seront utiles dans notre étude.
\url{https://tex.stackexchange.com/questions/484190}

\noindent Soit $J=[a,b]$ un intervalle fini ou infinie de $\mathbb{R}$
où $a,b \in \mathbb{R}$ et $a<b$ et soit $(X,|~.~|)$ un espace de Banach
réel.

$\blacktriangleright$ $C(J,X)$ l'espace de Banach des fonctions continues
$y:J \rightarrow X$, muni de la norme
$$\Vert y\Vert_{\infty} = \sup \lbrace |y(t)|~:~t\in J\rbrace.$$
$\blacktriangleright$ $L^{1}(J,X)$ l'espace de Banach des fonctions
mesurables $y:J \rightarrow X$ qui sont Lebesgue intégrable, muni de la
norme
$$\Vert y\Vert_{L^{1}} = \int_{a}^{b} |y(s)|~ds.$$
$\blacktriangleright$ $L^{\infty}(J,X)$ l'espace de Banach des fonctions
mesurables $y:J \rightarrow X$ qui sont essentiellement bornées, muni de
la norme
$$\Vert y\Vert_{\infty} = \inf \lbrace c>0 ~:~ |y(t)|\leq c~~p.p.\footnote{p.p: presque partout}~t\in J\rbrace.$$
$\blacktriangleright$ $AC^{1}(J,X)$ l'espace de Banach des fonctions
dérivables $y:J \rightarrow X$ dont la  première dérivée est absolument
continue.
\end{document}

and the result:

Edit:
Your used MiKTeX is too old, please update your system!

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}\label{sec.intro}
For any positive integer $n$ the higher cohomology of
the line bundle $\mathcal O(n)$ on $\mathbb P^{m-1}$ vanishes. The dimension
of the space of global sections of this bundle is easily
calculated to be ${n+m \choose n}$ via the identification
of $H^0(\mathbb P^{m-1},\mathcal O(n))$ with the vector space of homogeneous
forms of degree $n$ in $m$ variables.

If we view $\mathbb P^{m-1}$ as the space of lines in an $m$-dimensional
vector space $V$, then the line bundle $\mathcal O(n)$ is the $n$-th tensor
power of the dual of the tautological line subbundle $\mathcal O(-1)$.
Generalizing to the
Grassmannian of $k$-planes we are led to a number of questions about
the cohomology of vector bundles on Grassmannians.

The most obvious question
is the following: Let $V$ be a vector space of dimension $m$. Compute the dimension of the space of sections $H^0(\Gr(k,V),
\mathcal V^{\otimes n})$ as a function of $k,m,n$, where $\mathcal V$ is the
dual of the tautological rank $k$ subbundle on the Grassmannian of $k$-planes in
$V$. Likewise, if we are interested in the dimension of linear systems
we could ask for the dimension of the linear system $H^0(\Gr(k,V),
(\text{det } \mathcal V)^{\otimes n})$.

In this note we use Bott-Borel-Weil theory to answer these
questions. In particular we will compute for any (irreducible)
representation $W$ of $\GL_k$ the dimension $H^0(\Gr(k,m),{\mathcal W})$
where ${\mathcal W}$ is the vector bundle on $\Gr(k,m)$ whose
fiber at a point corresponding to a $k$-dimensional linear subspace $L$
is the dual vector space $W^*$ viewed as a $\GL(L)$-module. In addition
we explain when the higher cohomology vanishes.

Our motivation came from trying to understand
cohomology of vector bundles on classifying spaces.
It is also closely related to the study of representations
of $ind$-groups (\cite{twoivansandthewolf}).
The classifying space, $B\mathbb G_m$,
for the multiplicative group $\mathbb G_m$,  is the infinite projective
space $\mathbb P^\infty$.
The sequence of bundles $\{\mathcal O_{\mathbb P^m}(n)\}_{m=1}^\infty$ defines
a line bundle on $B\mathbb G_m$ whose
global sections is the limit of the countable sequence
of finite dimensional vector spaces $H^0(\mathbb P^m,\mathcal O(n))$.

In a similar way the classifying space $B\GL_k$ may be identified
with the infinite Grassmannian $\Gr(k,\infty)$ which is the limit
of the finite-dimensional Grassmannians $\Gr(k,m)$.
For a given $\GL_k$-module $W$, the associated
sequence of vector bundles on $\Gr(k,m)$ defines a vector
bundle on the classifying space $B\GL_k$. As we show below, the global
sections of these bundles on $Gr(k,m)$ have natural
$\GL_m$-module structures and the sequence of global sections
$\{H^0(Gr(k,m),{\mathcal W})\}_{m=k}^\infty$ defines
a representation of
the $ind$-group $\GL_\infty$. When $W$ is irreducible this
is an irreducible representation of $\GL_\infty$ in the sense
of \cite{twoivansandthewolf}.

\section{Definitions and statement of results}
We work over an arbitrary algebraically closed field $K$.

For any $k$ and $m \geq k$ we may construct
the Grassmannian $\Gr(k,m)$ as follows: Let
$U_{k,m}\subset \mathbb A^{mk}$ be the open set parametrizing $m \times k$
matrices of rank $k$. There is a free action of $\GL_k$
on $U_{k,m}$ given by right matrix multiplication and
$\Gr(k,m) = U_{k,m}/\GL_k$.
The principal bundle $U_{k,m} \to \Gr(k,m)$ is the {\it frame
bundle}; i.e., the fiber over a point of $\Gr(k,m)$ corresponding
to a linear subspace $L$ is the set of all possible bases for $L$
in the vector space $K^m$. When $k =1$, this gives the familiar
construction of $\mathbb P^{m-1}$ as the quotient of $\mathbb A^m -\{0\}$ by
$\mathbb G_m$.

If $W$ is a (left) $\GL_k$-module then we obtain a vector bundle
on $\Gr(k,m)$ by taking the quotient $\mathcal W = U_{k,m}  \times_{\GL_k} W$
where $\GL_k$ acts diagonally on the product $U_{k,m} \times W$.
When $W$ is the defining
representation of $\GL_k$, then ${\mathcal W}$ is the dual of the
tautological
rank $k$ subbundle of $\Gr(k,m)$. Likewise if $W$ is the determinant
character, then $\mathcal W$ is the line bundle which gives the Pl\"ucker
embedding.

Since $\GL_m$ acts on $U_{k,m}$ by left matrix multiplication
the quotient bundle $\mathcal W = U_{k,m} \times_{\GL_k} W$
is $\GL_m$-equivariant, so the cohomology of $\mathcal W$ is a (left) $\GL_m$-module.
The main result of this paper is the determination
of these cohomology modules.

To state our theorem we recall some notation about $\GL_k$-modules.
Let $T_k \subset \GL_k$ be the group of diagonal matrices and let
$B_k$ the group of upper triangular matrices.  Any nonincreasing
sequence of integers $\lambda = (\lambda_1, \lambda_2,\ldots
\lambda_k)$ determines a $\GL_k$-module $V^{(k)}_\lambda$ which
is defined as follows:
The sequence of integers $\lambda$
determines a character of $T_k$ which we also call $\lambda$.
This character extends to a
representation of $B_k$. Let $L_\lambda$ be the
line bundle $\GL_k \times_{B_k} \lambda$, where
$B_k$ acts
diagonally,\footnote{In \cite[p.393]{Fulton-Harris}
the line bundle whose global sections is the highest
weight module $V_\lambda$ is denoted $L_{-\lambda}$.}
and set $V_\lambda^{(k)} =
H^0(\GL_k/B_k,L_\lambda)$.
When the characteristic of $K$ is 0 then
$V^{(k)}_\lambda$ is irreducible. In any characteristic,
the groups $H^i(GL_k/B_k, L_{\lambda})$ vanish for
$i > 0$ \cite{Demazure}.

\begin{remark} With this notation $V_\lambda$ and $V_{\lambda'}$ restrict
to the same representation of $\SL_k$ if and only if
$\lambda_l - \lambda'_l$ is constant (i.e. independent of $l$).
In this case $V_{\lambda} = V_{\lambda'} \otimes D^{\otimes r}$
where $r = \lambda_l - \lambda'_l$ and $D$ is the determinant
character of $\GL_k$.
\end{remark}

Let $\lambda = (\lambda_1, \ldots , \lambda_k)$ be a sequence
of non-increasing integers and let $V_\lambda^{(k)}$ be the associated
representation of $\GL_k$.
Let $\mathcal V_\lambda$ be the corresponding
vector bundle on $\Gr(k,m)$.

\begin{theorem} \label{thm.main}
(a) If $\lambda_k \geq 0$, then
$H^0(\Gr(k,m), \mathcal V_\lambda) = V_\lambda^{(m)}$,
and $H^i(\Gr(k,m), \mathcal V_\lambda) = 0$ for $i>0$.
Here $V_\lambda^{(m)}$ denotes the $\GL_m$-module
with highest weight $(\lambda_1, \ldots , \ldots \lambda_k, 0, \ldots , 0)$.\\

(b) If $\lambda_k < 0$, then for sufficiently large $m$,
$H^i(\Gr(k,m), \mathcal V_\lambda) = 0$ for all $i$.
\end{theorem}

\section{Bott-Borel-Weil theory for parabolics}
The material here is well known, but we do not know a reference with
algebraic proofs.

Let $H$ be an affine algebraic group and let $\pi\colon X \to Y$ be an
$H$-principal bundle.\footnote{This means \cite[Definition
0.10]{GIT} that $\pi$ is flat, $X \to Y$ is a geometric quotient and
that $X \times_Y X$ is $H$-equivariantly isomorphic to $H \times
X$. This definition implies that the action of $H$ on $X$ is free.}
If $\mathcal F$ is an $H$-equivariant $\mathcal O_X$-module, set
$\mathcal F^H$ to be subsheaf of invariant sections of $\pi_*\mathcal F$.
Given an $\mathcal O_Y$-module ${\mathcal G}$, $\pi^*{\mathcal G} =
\pi^{-1}{\mathcal G}
\otimes_{\pi^{-1}\mathcal O_Y} \mathcal O_X$ has a natural $H$-action given
by the action of $H$ on $\mathcal O_X$. Because
$H$ acts freely, the functors ${\mathcal G} \mapsto \pi^*{\mathcal G}$
and $\mathcal F \mapsto \mathcal F^H$ are inverse to each other.

Let $X' \stackrel{\pi'} \to Y'$ and $X \to Y$ be principal
$H$-bundles. Let $q \colon X' \to X$ be a $H$-equivariant.
Then there is an induced map $p\colon Y' \to Y$ such that the
diagram
\begin{equation} \label{diag.cartesian}
\begin{array}{ccc}
X' & \stackrel{q} \to & X\\
\pi' \downarrow  & & \pi \downarrow\\
Y' & \stackrel{p} \to & Y
\end{array}
\end{equation}
commutes.

\begin{lemma} \label{lem.commute}
Diagram \eqref{diag.cartesian} is cartesian.
\end{lemma}
\begin{proof}
Since the diagram commutes there is an $H$-equivariant map
$X' \to Y' \times_Y X$. Now $X' \to Y'$ and $Y' \times_Y X \to Y'$
are both principal bundles over $Y$.  After base change by
a flat surjective morphism $X' \to Y'$ both bundles
are trivialized. Thus the map of $Y'$-schemes,
$X' \to Y' \times_Y X$ is an isomorphism after flat surjective
base change. Therefore it is an isomorphism.
\end{proof}

\begin{lemma} \label{lem.invcommute}
Let
$\begin{array}{ccc}
X' & \stackrel{q} \to & X\\
\pi' \downarrow  & & \pi \downarrow\\
Y' & \stackrel{p} \to & Y
\end{array}$
be a commutative (hence cartesian) diagram of principal
$H$-bundles as above. Then for any $H$-equivariant $\mathcal O_{X'}$-module $\mathcal F$
there are natural isomorphism of $\mathcal O_Y$-modules
$(R^iq_* \mathcal F)^H = R^ip_*(\mathcal F^H)$ for any $i\geq 0$.
\end{lemma}
\begin{proof}

Because $\mathcal F = \pi^{\prime *}\mathcal F^H$, and
cohomology commutes with base change \cite[Proposition 9.3]{Hartshorne},
there are natural isomorphisms
$R^i q_* \mathcal F = \pi^* R^ip_*\mathcal F^H$. Pushing forward by
$\pi$ and taking $H$-invariants yields the desired isomorphism.
\end{proof}

Let $G$ be an algebraic group, $B \subset G$ a Borel
subgroup and $P \supset B$ a parabolic subgroup.
Let $p \colon G/B \to G/P$ be the projection.
\begin{prop} \label{prop.parabolic}
If $\lambda$ is a $B$-module, then $G \times_P H^i(P/B, P \times_B \lambda)
= R^ip_*(G \times_B \lambda)$ as $G$-equivariant bundles on $G/P$.
\end{prop}

\begin{proof}
The map $G \times P/B \to G/B$ given by $(g,pB) \mapsto gpB$
is a $P$-principal bundle, where $P$ acts freely on $G \times P/B$
by the formula $q\cdot(g,pB) = (gq, q^{-1}pB)$ where $q \in P$.

Thus we have a commutative diagram with the vertical
arrows proper and the horizontal arrows quotient maps by the free action
of $B$.
$$\begin{array}{ccc}
G \times P/B & \stackrel{q} \to & G\\
\downarrow & & \downarrow\\
G/B &\stackrel{p} \to & G/P
\end{array}$$
Consider the $P$-equivariant line bundle $L =G \times (P \times_B \lambda)$
on $G \times P/B$.
Since $q$ is a projection,
$R^iq_*L = G \times H^i(P/B, P \times_B \lambda)$.
On the other hand $L^P = G \times_B \lambda$ so
so by
Lemma \ref{lem.invcommute},
$G \times_P H^i(P/B, P \times_B \lambda)$ is naturally isomorphic
to $R^ip_*(G \times_B \lambda)$. Since this
isomorphism is natural, it is equivariant for the $G$-actions
on these bundles.
\end{proof}
\section{Proof of Theorem \ref{thm.main}}
Let $G = \GL_m$, and let $B_m$ be the Borel subgroup
of upper triangular matrices. Let $P$ be the parabolic
subgroup of matrices of the form
$\left(\begin{array}{cc}
A_k & U \\
0 & A_{m-k}
\end{array}\right)$ with $A_k \in \GL_{k}$, $A_{m-k} \in \GL_{m -k}$ and $U$ an
arbitrary $k \times (m-k)$ matrix.  The subgroup $N$ of matrices of
the form $\left(\begin{array}{cc} I & U \\ 0 &
A_{m-k}\end{array}\right)$ is normal and $P/N = \GL_k$.

\begin{lemma} \label{lem.framebundle}
$U_{k,m}$ is isomorphic to the homogeneous space $G/N$.
\end{lemma}
\begin{proof}
Any $m \times k$ matrix of maximal rank can be viewed as the first $k$
columns of a nonsingular $m \times m$ matrix.  Since $\GL_m$ acts
transitively on itself by left multiplication, it also acts
transitively on the space $U_{k,m}$ of $m \times k$ matrices of
maximal rank. The stabilizer of the $m \times k$ matrix $\left(\begin{array}{c} I_k \\ 0\end{array} \right)$ is
the subgroup $N \subset \GL_k$.
\end{proof}

Using Lemma \ref{lem.framebundle} we obtain the well
known identification of $\Gr(k,m)$ $= U_{k,m}/\GL_k = G/P$. Likewise for
any $\GL_k$-module,
the vector bundle $U_{k,m} \times_{\GL_k} V$ is identified
with $G \times_P V$ where $V$ is made into a $P$-module via
the surjective map $P \to P/N = \GL_k$.

Since $P$ is a parabolic subgroup containing $B$
we have a proper map $G/B \stackrel{p} \to G/P$.
Let $L_\lambda$ be the line bundle on $G/B$
corresponding to the weight $\lambda = (\lambda_1, \ldots , \lambda_k,
0, \ldots , 0)$. Then by Bott-Borel-Weil $H^0(G/B, L_\lambda) =
V^{(m)}_\lambda$.

\begin{lemma} \label{lem.pushforward}
$p_*(L_\lambda)= \mathcal V_\lambda$ as $\GL_m$ equivariant bundles
on $\Gr(k,m)$.
\end{lemma}
\begin{proof}
By Proposition \ref{prop.parabolic}, $p_*(L_\lambda) = G \times_P H^0(P/B,
P \times_B L_\lambda)$. Thus to prove the Lemma
we must compute $H^0(P/B, P \times_B L_\lambda)$.

Since $P$ is a unipotent extension of $GL_k \times GL_{m-k}$,
the homogeneous space $P/B$ is isomorphic to the
product $GL_k/B_k \times GL_{m-k}/B_{m-k}$,
where $B_k = B \cap (\GL_k \times I_{m-k})$ and
$B_{m-k} = B \cap (I_k \times \GL_{m-k})$. Thus
\begin{eqnarray}
H^0(P/B, P \times_B \lambda) & = & H^0(\GL_k/B_k, \GL_k \times_{B_k}
(\lambda|B_k)) \otimes \nonumber \\
& & H^0(\GL_{m-k}/B_{m-k}, \GL_{m-k} \times_{B_{m-k}} (\lambda|B_{m-k})). \nonumber
\end{eqnarray}

Since $\lambda = (\lambda_1, \ldots , \lambda_k, 0, \ldots ,0)$,
its restriction $B_k$ is the highest weight vector $(\lambda_1, \ldots
, \lambda_k)$, and its restriction to $B_{m-k}$ is trivial.
\end{proof}

Part (a) of the theorem now follows easily from Lemma \ref{lem.pushforward}.
First observe that $H^0(\Gr(k,m),\mathcal V_\lambda) = H^0(G/P,p_*L_\lambda)
= H^0(G/B, L_\lambda)$.
If all of the $\lambda_k$'s are nonnegative then
$H^0(G/B, L_\lambda) = V^{(m)}_{\lambda}$ and
$H^i(G/B, L_\lambda) = 0$ for $i > 0$.
It follows from the Leray spectral
sequence that
that $H^i(G/P, p_*L_\lambda) = 0$ as well.

We now prove (b). Suppose that $\lambda_k < 0$. It suffices
to show that, for $m$ sufficiently large, that $H^i(G/B, G \times_B
\lambda) = 0$ where is the character of the maximal torus
with weight $\lambda = (\lambda_1,\ldots , \lambda_k, 0, \ldots , 0)$.

Let $\{0\}=V_0 \subset V_1 \ldots \subset V_m$ be the
flag fixed by the Borel subgroup of upper triangular
matrices $B_m \subset \GL_m$. For any $l < m$
let $P_l$ be the parabolic subgroup fixing the subflag
$V_0 \subset V_1 \subset \ldots \subset V_l$.  Then $P_{l} \subset
P_{l-1}$ and the fibers of the map $p_l\colon G/P_{l} \to G/P_{l-1}$
are isomorphic to the projective space $\mathbb P^{m-l}$.
In this way the flag variety $G/B = G/P_{m-1}$ is realized
as being at the top of the tower of projective bundles
$$G/B \stackrel{p_{m-1}} \to G/P_{m-2} \to \ldots \to G/P_2 \stackrel{p_2}
\to G/P_1 = \mathbb P^{m-1}.$$

The group $P_l$ is isomorphic to the product $B_l \times \GL_{m-l}$
where $B_l$ is the Borel group of upper triangular matrices in
$\GL_l$. Thus, irreducible representations of $P_l$ are of
the form $\chi \times W$ where $\chi$ is a character of
the maximal torus in $B_l$ and $W$ is an irreducible representation
of $GL_{m-l}$.
If $\chi = (a_1, \ldots , a_l)$ is a character of the maximal
torus of $B_l$
then the same argument used in the proof Proposition \ref{prop.parabolic}
shows that $R^ip_{l*} (G \times_{P_l} \chi) =
G \times_{P_{l-1}} H^i(P_{l-1}/P_{l}, P_{l-1} \times_{P_l} \chi)$.
Under the identification $P_{l-1}/P_{l} = \mathbb P^{m-l +1}$ the
line bundle $P_{l-1} \times_{P_l} \chi$ corresponds to
$\mathcal O_{\mathbb P^{m-l+1}}(a_l)$.

If $\lambda$ is a character of $T_m$ with
$\lambda_{k+1} = \ldots \lambda_m = 0$ then
we may view $\lambda$ as $P_l$-module for $l \geq k$.
The argument of the previous paragraph
implies that  $H^i(G/B, G \times_B \lambda) = H^i(G/P_k, G \times_{P_k}
\lambda)$. If $\lambda_k < 0$ then, for all $j$,
$H^j(\mathbb P^{m-k}, \mathcal O_{\mathbb P^{m-k}}(\lambda_k)) = 0$ whenever $m \geq -\lambda_k +k$.
Thus $R^jp_{k*}(G/\times_{P_k} \lambda) = 0$ for all $j$ as long
as $m$ is sufficiently large.
Therefore, by the Leray spectral sequence, it follows that
$H^i(G/B, L_\lambda) = 0$ for all $i$.

\section{Some explicit dimension computations}\label{sec.formulas}
We conclude by using Theorem~\ref{thm.main} to compute the dimensions
of $H^0(\Gr(k,m),\mathcal V_\lambda)$ for some particular $\mathcal V_\lambda$
involving symmetric powers and the determinant. To do this, we combine
Theorem~\ref{thm.main} with \cite[Theorem 6.3]{Fulton-Harris} to
obtain a convenient formula for the dimensions.

Let $\lambda =
(\lambda_1, \ldots , \lambda_k)$ be a sequence of nonincreasing
integers, and let $V_\lambda^{(k)}$ be the associated representation
of $\GL_k$ with $\mathcal V_\lambda$ the corresponding vector bundle on
$\Gr(k,m)$.

\begin{lemma} \label{lem.dimensions}
If all $\lambda_i \ge 0$, then \[ \dim H^0(\Gr(k,m),\mathcal V_\lambda) = \prod_{i < j} \frac{\lambda_i - \lambda_j +j-i}{j-i},\] where $j$ runs from 2 to $m$, and $\lambda_{k+1}= \cdots = \lambda_m = 0$.
\end{lemma}

The first proposition gives a symmetry result about the dimension of a
power of the determinant. Throughout the rest of this section, let $V$
denote the standard $k$-dimensional representation of $\GL_k$ and $\mathcal V$
the corresponding tautological rank $k$ vector bundle on $\Gr(k,m)$.

\begin{prop} \label{prop.symmetry}
For all $0 < k \le m$ and $l > 0$,
\[ \dim H^0(\Gr(k,m),(\text{det } \mathcal V)^l) = \dim H^0(\Gr(k,k+l),(\text{det } \mathcal V)^{m-k})\]
\[ = \prod_{j=k+1}^m \frac{{l+j-1 \choose l}}{{l+j-k-1 \choose l}}. \]
\end{prop}

\begin{proof}
The partition $\lambda$ associated to $(\text{det } \mathcal V)^l$ is $\lambda=(l,\dots,l,0,\dots,0)$. Hence by Lemma~\ref{lem.dimensions}, \[ \dim H^0(\Gr(k,m),(\text{det } \mathcal V)^l) = \prod_{j=k+1}^m \prod_{i=1}^k \frac{l+j-i}{j-i} \]
\[ = \prod_{j=k+1}^m \frac{l!(l+j-1)!(j-k-1)!}{l!(j-1)!(l+j-k-1)!} = \prod_{j=k+1}^m \frac{{l+j-1 \choose l}}{{l+j-k-1 \choose l}} =  \prod_{j=1}^{m-k} \frac{{l+j+k-1 \choose l}}{{l+j-1 \choose l}}, \]
\[ \dim H^0(\Gr(k,k+l),(\text{det } \mathcal V)^{m-k}) = \prod_{j=k+1}^{k+l} \prod_{i=1}^k \frac{m-k+j-i}{j-i} \]
\[ = \prod_{j=k+1}^m \frac{(m-k)!(m-k+j-1)!(j-k-1)!}{(m-k)!(m+j-1)!(j-1)!} = \prod_{j=k+1}^{k+l} \frac{{m-k+j-1 \choose m-k}}{{m-k+j-k-1 \choose m-k}} \]
\[ = \prod_{j=1}^l \frac{{m+j-1 \choose m-k}}{{m+j-k-1 \choose m-k}}.\]

To show that the two dimensions are equal, we induct on $l$. When $l=1$, it is easy to compute that both dimensions are ${m \choose k}$. Suppose now that $s > 1$, and we have proven that the dimensions are the same for $l=s-1$. The induction hypothesis gives us that
\[ \prod_{j=1}^{s-1} \frac{{m+j-1 \choose m-k}}{{m+j-k-1 \choose m-k}} = \prod_{j=1}^{m-k} \frac{{s+j+k-2 \choose s-1}}{{s+j-2 \choose s-1}}.\]
Hence, letting $l=s$, \[ \prod_{j=1}^{s} \frac{{m+j-1 \choose m-k}}{{m+j-k-1 \choose m-k}} = \frac{{m+s-1 \choose m-k}}{{m+s-k-1 \choose m-k}} \prod_{j=1}^{m-k} \frac{{s+j+k-2 \choose s-1}}{{s+j-2 \choose s-1}} \]
\[ = \frac{{m+s-1 \choose m-k}}{{m+s-k-1 \choose m-k}} \prod_{j=1}^{m-k} \frac{s+j-1}{s+j+k-1} \prod_{j=1}^{m-k} \frac{{s+j+k-1 \choose s}}{{s+j-1 \choose s}}.\]

The rightmost product is what we want, so it suffices to show that the
product of the first two factors is one. Note that
\[ \prod_{j=1}^{m-k} \frac{s+j-1}{s+j+k-1} = \frac{\frac{(s+m-k-1)!}{(s-1)!}}{\frac{(s+m-1)!}{(s+k-1)!}} = \frac{{m+s-k+1 \choose m-k}}{{m+s-1 \choose m-k}},\] which completes the proof.
\end{proof}

\begin{remark} Assume that the characteristic of the ground
field is $0$.  If we fix $k$ and $l$ and let $m$ go to infinity, then
sequence of irreducible $\GL_m$ representations $\{V^{(m)}_{(l,\ldots
, l)}\}$ determines an irreducible representation of the $ind$-group
$\GL_\infty$. Proposition \ref{prop.symmetry} implies that the
dimensions of the finite dimensional representations $V^{(m)}_{(l, l,
\ldots ,l)}$ are the same as the dimensions of the irreducible
$\GL_{k+l}$-modules $V^{(l+k)}_{(m-k,\ldots , m-k)}$.
\end{remark}

\begin{remark}
As noted in \cite{twoivansandthewolf} the infinite Grassmannian
$\Gr(k,\infty)$ is an $ind$-projective variety.  For fixed $l$, the
sequence of $\GL_m$-modules $\{\Sym^l(\wedge^k V_m)/V^{(m)}_{(l,\ldots
, l)}\}$ is the $ind$-representation of $\GL_\infty$ corresponding to
the degree $l$ component of the ideal of Pl\"ucker relations for the
embedding of $\Gr(k,\infty)$ into the infinite projective space given
by the sequence of projective spaces $\{\mathbb P(\wedge^k V_m)\}$. (Here
$V_m$ denotes the standard representation of $\GL_m$.)

Proposition \ref{prop.symmetry} allows us to compute the
sequence of dimensions of the Pl\"ucker relations in degree
$l$. In particular the dimension of the $\GL_m$-module of
Pl\"ucker relations in degree $l$ is
$${ {m \choose k} + l \choose l} - \prod_{j=k+1}^m \frac{{l+j-1
\choose l}}{{l+j-k-1 \choose l}}.$$
\end{remark}

\begin{prop} \label{prop.sym}
For all $r \ge 0$ and $0 < k \le m$, \[ \dim H^0(\Gr(k,m),\Sym^r \mathcal V) =
{r+m-1 \choose r}. \]
\end{prop}

\begin{proof}
Use Lemma~\ref{lem.dimensions} and the partition $(r,0,\dots,0)$.
\end{proof}

\begin{prop} \label{prop.symdet}
For all $r \ge 0$, $l \ge 0$, and $0 < k \le m$, \[ \dim H^0(\Gr(k,m),\Sym^r \mathcal V \otimes (\text{det } \mathcal V)^l) \]
\[= \frac{{r+l+m-1 \choose r}}{{r+l+k-1 \choose l}} {l+m-1 \choose l} \prod_{j=k+1}^m \frac{{l+j-2 \choose l}}{{l+j-k-1 \choose l}}. \]
\end{prop}

As we would expect, when $l=0$, we get $\dim H^0(\Gr(k,m),\Sym^r \mathcal V)$, and when $r=0$, we have (after a bit of manipulation) the dimension of $H^0(\Gr(k,m),(\text{det } \mathcal V)^l)$. Also, the product over $j$ is exactly $\dim H^0(\Gr(k-1,m-1),(\text{det } \mathcal V)^{l})$ (again easy to see after a little algebra).

\begin{proof}
We use the partition $\lambda=(r+l,l,\dots,l,0,\dots,0)$, where $\lambda_{j} = 0$ for all $k+1 \le j \le m$. By Lemma~\ref{lem.dimensions}, the dimension is \[ \prod_{j=2}^k \frac{r+j-1}{j-1} \prod_{j=k+1}^m \frac{r+l+j-1}{j-1} \prod_{j=k+1}^m \prod_{i=2}^k \frac{l+j-i}{j-i}.\]

The first product is ${r+k-1 \choose r}$. Consider the second
product. We have
\[ \prod_{j=k+1}^m \frac{r+l+j-1}{j-1} = \frac{\frac{(r+l+m-1)!}{(r+l+k-1)!}}{\frac{(m-1)!}{(k-1)!}} = \frac{(r+l+m-1)!(k-1)!}{(r+l+k-1)!(m-1)!} \]
\[ = \frac{(r+l+m-1)!}{r!(l+m-1)!} \frac{r!(l+m-1)!(k-1)!}{(r+l+k-1)!(m-1)!} \frac{l!(r+k-1)!}{l!(r+k-1)!} \]
\[ = {r+l+m-1 \choose r} {r+l+k-1 \choose l}^{-1} {l+m-1 \choose l} {r+k-1 \choose r}^{-1}. \]

The last factor is \[ \prod_{j=k+1}^m \prod_{i=2}^k \frac{l+j-i}{j-i} = \prod_{j=k+1}^m \frac{\frac{(r+l+m-1)!}{(r+l+k-1)!}}{\frac{(m-1)!}{(k-1)!}} = \prod_{j=k+1}^m \frac{(l+j-2)!(j-k-1)!}{(l+j-k-1)!(j-2)!}.\] Multiplying the top and bottom by $l!$ yields \[ \prod_{j=k+1}^m \frac{{l+j-2 \choose l}}{{l+j-k-1 \choose l}}. \]

Finally, multiplying all three products together completes the proof.
\end{proof}

\begin{remark}
Using Theorem~\ref{thm.main} it is easy to show that if $d \leq k$ then
$H^0(\Gr(k,m), \mathcal V^{\otimes d}) = V_m^{\otimes d}$ where
$V_m$ is the standard representation of $\GL_m$.
In particular $\dim H^0(\Gr(k,m), \mathcal V^{\otimes d}) = m^d$.
An interesting question is to determine whether
there are relatively simple formulas
for the dimension of
$H^0(\Gr(k,m),\mathcal V^{\otimes d} \otimes (\text{det } \mathcal V)^l)$.
If $l \geq  0$ then Theorem \ref{thm.main} gives the summation formula
$$\dim H^0(\Gr(k,m),\mathcal V^{\otimes d} \otimes (\text{det } \mathcal V)^l) =
\sum_{\lambda} m_\lambda\; \dim V^{(m)}_{\lambda +l}$$
where the sum is over all $k$-element partitions $\lambda$ of $d$,
$m_\lambda$ is the product of the hook lengths
in the tableau corresponding to the partition
$(\lambda_1, \ldots \lambda_k)$, and
$V^{(m)}_{\lambda +l}$ refers to the $\GL_m$-module with
highest weight $(\lambda_1 + l, \ldots \lambda_k +l, 0, \ldots 0)$.
Unfortunately, we have do not know how to simplify this sum.
Other summation formulas can also be obtained using the Riemann-Roch
or localization theorems.
When $-d < l < 0$,  there is the added difficulty
that some, but not all, of the sequences $(\lambda_1 + l, \lambda_2
+l, \ldots , \lambda_k +l, 0, \ldots 0)$ have negative entries,
so the corresponding line bundles on $\GL_m/B_m$ have no
cohomology for sufficiently large $m$.
\end{remark}

redpajama_set_name: RedPajamaArXiv


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Dr. ANGELA A RHOADES , – INTERNAL MEDICINE, Is Female and specialized in INTERNAL MEDICINE, Dr. RHOADES complete education, website and Fee details are given here. Dr. RHOADES is one of most famous doctor in the CLARION, RHOADES is practicing in medical field for 10 Years. Doctor ANGELA completed its graduation from LAKE ERIE COLLEGE OF OSTEOPATHIC MEDICINE in the year 2006 and was awarded with degree by the medical institute LAKE ERIE COLLEGE OF OSTEOPATHIC MEDICINE. From 10 years Dr. ANGELA A RHOADES had been awarded by local authorities in the state PA, USA. Currently Dr. RHOADES is Working in CLARION HOSPITAL hospital.
All doctors in united states have unique identification and thus Dr. RHOADES. Three kind of Identities are assigned to Dr. ANGELA A RHOADES , .
Dr. ANGELA RHOADES was one of the brilliant student in the medical institute LAKE ERIE COLLEGE OF OSTEOPATHIC MEDICINE, RHOADES was awarded medical credential – by the Institute during the year 2006.
Dr. RHOADES graduated in 2006 since then is practicing in medical field for about 10 years. Since 2006 no malpractice has been reported for this physician. Dr. ANGELA RHOADES is practicing in HEALTH SERVICES OF CLARION, INC. with other 25 doctors. Doctor and his group mates are now practicing in CLARION HOSPITAL, CLARION – PA and other hospitals lited here.
Medicare is social insurance program in USA and Dr. ANGELA RHOADES accepts Medicare medical insurance approved amounts as May or May not. If physician Dr. ANGELA A RHOADES takes Medicare approved payments then 30 insurance companies under Medicare are accepted by Dr. RHOADES.
Dr. ANGELA A RHOADES , Works with HEALTH SERVICES OF CLARION, INC., due to some privacy issue we can not provide the direct email contacts here online.
Phone number of doctor ANGELA A RHOADES is available at the official website of the organisation HEALTH SERVICES OF CLARION, INC., also you may contact the office of doctor at CLARION HOSPITAL and other hospital can also give phone number of Doctor RHOADES. However due to privacy issue we can not provide phone number of RHOADES here.
Dr. ANGELA A RHOADES , – INTERNAL MEDICINE is one of the most famous doctor in CLARION, PA, having specialization in INTERNAL MEDICINE. Here is reviews and rating Based on various surveys.
[wp_charts title="Doctor Rating" type="line" align="alignright" margin="5px 20px" datasets="4.1,3.1,4.9,4.5,3.1,4.9,3.7″ scaleoverride="false" scalesteps="8″ scalestepwidth="8″ scalestartvalue="0″ scaleFontSize="20″ canvaswidth="920px" canvasheight="460px" relativewidth="2″ width="100%" height="100%" labels="Emergency,Office,Staff,Waiting Time,Explanation,Answers,Time Spent"].
Dr. RHOADES worked in various hospitals in practice life under the organisation HEALTH SERVICES OF CLARION, INC.. Office of Dr. ANGELA RHOADES is in the corresponding Hospitals. Dr. RHOADES working office hours are from 10:00 to 16:30 during this time you may visit the Docotr's office at HEALTH SERVICES OF CLARION, INC., However emergency support by Dr. ANGELA is not surely known.
Name Of Hospitals Where Dr. ANGELA A RHOADES , – INTERNAL MEDICINE provides service.
WEST PENN ALLEGHENY HEALTH SYSTEM, INC.
Getting appointment with Dr. ANGELA RHOADES is very easy by visiting the hospitals where doctor provides services, however emergency appointment with Dr. ANGELA RHOADES depends upon time available to doctor. User rating are 4.1 out of five, which is very good and is above average.
You may also contact the HEALTH SERVICES OF CLARION, INC. for getting General appointment or emergency appointment as Dr. RHOADES Provides all its services under HEALTH SERVICES OF CLARION, INC..
Dr. ANGELA A RHOADES , is primarily specialized in INTERNAL MEDICINE, and thus provides best treatments in its area of interest INTERNAL MEDICINE. However Doctor ANGELA RHOADES also have some other specializations.
Dr. ANGELA A RHOADES , – INTERNAL MEDICINE is highly famous and intelligent doctor in CLARION, passed out from LAKE ERIE COLLEGE OF OSTEOPATHIC MEDICINE in 2006, and have a very tough experience of 10 years thus providing best medical treatments.
0 – Malpractice reports by the CLARION Authorities.
0 – Malpractice by Dr. RHOADES in USA.


Published online December 22, 2015.
Jason S. Sexton is a systematic theologian with research interests at the crossroads of theology and culture. He is a lecturer in the honors program of California State University, Fullerton, and a visiting fellow at University of California, Riverside's Center for Ideas and Society.


text:
Best Practices, Incidents, Leadership	 30 Oct 2018 30 Oct 2018
Captain, I have a concern…
Last year, I was inbound to a port-that-will-remain-nameless, with the local pilot at the conn. As we maneuvered up a winding channel, the channel curved to port, yet the pilot ordered starboard rudder as we approached the next turn. Our 3rd officer, who had been onboard for over two months already and well-drilled in bridge resource management theory piped up and said, "Mr. Pilot – the channel goes to port, why are you using starboard rudder?" The pilot responded by glancing at the rudder angle indicator and out the window. He then turned and told the helmsman, "Midships," followed shortly by, "Port 20." The pilot then turned, smiled at the 3rd officer, and said, "Thanks."
The 3rd officer had effectively implemented the first step of the PACE model for "graded assertiveness." Originating in the medical field, graded assertiveness and the PACE (Probe-Alert-Challenge-Emergency) model were necessary to overcome the power dynamic between nurses and doctors. Much like the power dynamic between the pilot and the 3rd officer, the power dynamic between doctors and nurses is such that nurses are frequently hesitant to question a doctor, even when a patient is at risk of harm.
Probe – "Mr. Pilot – the channel goes to port, why are you using starboard rudder?"
Alert – " Mr. Pilot – the channel goes to port. We will ground if we continue to turn to starboard.
Challenge – "Mr. Pilot – we will ground if we turn to starboard. I recommend turning to port immediately.
Emergency – "Hard to port!" (or other appropriate order)
Despite the experience and knowledge of maritime pilots, captains and officers on vessels, they are not infallible. The bridge team (including the helmsman, lookout, cadet, deck officers and captain) are all there to assist in error trapping. Unfortunately, sometimes this safety net fails entirely, such as in the "heavy contact" of CMA CGM Centaurus with the berth (and gantry cranes!) in Jebel Ali in May 2017.
The report from the UK Marine Accident Investigation Board (MAIB) detailed the failures in bridge resource management that contributed to this incident. The vessel's parent company, CMA CGM, issued guidance to their fleet shortly after that included the instruction, "Once pilot decision looks unsafe to you, challenge and be ready to take over command." At some point in the maneuver to the berth, utilizing the PACE model might have avoided or mitigated this incident.
The Five Step Assertive Statement Process
The PACE model is effective in acute situations where immediate actions need to be taken to correct the situation, such as the rudder order or speed directed by the pilot. When a concern needs to be raised in a less immediate situation, a less aggressive, but equally effective method is the Five Step Assertive Statement Process.
The Five Step Assertive Statement Process comes out of the aviation industry. Leading the transportation industry in human factors studies, the aviation industry had multiple serious incidents where a problem had been identified by the co-pilot or first officer. Yet, these concerns had not been properly communicated to or been ignored by the pilot due to the power dynamic between pilot and co-pilot. Much like the aviation industry, the maritime industry has its fair share of egos, where the senior officers or advisors (marine pilots) feel themselves to be above reproach (or questioning!).
The five steps are :
Start with the person's formal title (i.e. Captain / Mr. Pilot). Starting with anything else can diminish the importance of the message.
State, "I have a concern." This is a trigger statement. In the aviation industry, by policy, this is statement that requires the captain acknowledge and consider the concerns of the crew member. Shipping companies might consider adding such a policy to their safety management system (SMS).
State your concern and provide details.
Suggest an alternate plan.
Seek permission to implement the alternate plan.
In practice, this statement might sound something like :
"Captain, I have a concern. There appears to be a crack between 1 port bunker tank and 3 port water ballast tank. The level on 1 port has gone down while 3 port has gone up with no ballast or fuel transfer operations taking place. I recommend we treat 3 port ballast tank as contaminated and do not discharge it as planned in the next port. Does that sound like a plan?
In reality, neither of these systems will be effective without buy-in from all parties. There are plenty of captains, chief engineers, pilots and other senior officers still out there that will reprimand a subordinate for questioning their decisions. There are probably an equal number of junior officers and ratings who are unwilling to voice their concerns due to lack of training or prior negative responses. The combination of these two groups often prevents adequate communication which then becomes a casual factor in incident investigations.
The solution? A strong first step is the institution of a "just culture" within an organization and on its vessels. A just culture provides the safe space within which concerns and safety issues can be discussed without fear of repercussions. The implementation of such a culture requires policies first, followed by actions that support those policies. Additionally, policies and training that support the concept of graded assertiveness at all levels is critical. If junior officers and ratings are never encouraged to voice their concerns, will they? Even worse, what about the captain that won't question the pilot when he is concerned? What can YOU improve on your vessel?
Filed under: graded assertiveness, human factors, PACE
The Effective Crew Project : Solent University
The London P&I Club : Captain's Rounds

redpajama_set_name: RedPajamaCommonCrawl


The summer's hotly-anticipated European Championships in France are looming rapidly, and we're getting into the party spirit with a series of stonking price boosts.
So be sure to stay tuned every day in the run-up to the tournament, which kicks off in Paris on June 10 when France take on Romania at the Stade de France.
Today we're boosting two teams vying to oust Germany and Northern Ireland in Group C plus we are offering an enhancement for one of the tournament's top strikers.
Keep your eyes peeled and head to Ladbrokes.com, online, on mobile and on tablet throughout the day.


* To identify these factors through a survey instrument which will assist in the dissemination of information to students utilizing factors that they have chosen.
1. How significant are factors of the environment in making career choices? 2. How significant are factors of opportunity in making career choices? 3. How significant are factors of personality in making career choices? 4. Which factor, environment, opportunity, or personality, is most significant to students?
1. Some students do not begin to explore 'real' career possibilities until after graduation. Academic colleges, technical colleges, industry, and armed forces can provide students with relevant information earlier in their schooling. They can be more aggressive, giving students information they could test and use in their daily studies and apply to their career choice.
2. Before graduating, some students do not consider enough alternative choices in career selection to justify making an informed decision. Sources of influence such as parents or mentors could be brought into a circle of counseling and discussion, to help the student form a comprehensive career plan or outline.


text:
Now online: The Signal of Liberty newspaper
Sat, 10/17/2009 - 11:05am by amy
Take a step back to the 1840s when a cord of hickory wood or "produce of every description" assured Ann Arborites a year's subscription to The Signal of Liberty--the voice of the Liberty Party in Michigan and the state's most prominent antislavery newspaper. Published in Ann Arbor above a shop on Broadway by Theodore Foster and Reverend Guy Beckley, whose home on Pontiac Trail was an important stop on the Underground Railroad, The Signal of Liberty brought Ann Arborites seven years of emotionally charged news from the many stories of slavery's outrages to the speeches of great statesmen and important national issues of the day, to more sobering local news, notices, and advertisements for dubious health cures. You can now browse The Signal of Liberty or search the full text online.
Submitted by camelsamba on Sat, 08/06/2011 - 11:20pm
The links to "a cord of
The links to "a cord of hickory wood" and "advertisements for dubious health cures" don't work. :^( Oh well, perhaps it's better that we can't learn about those dubious health cures! :^)
Submitted by 1funone on Sun, 08/14/2011 - 12:44pm
this has a game code


text:
Oliarus modica är en insektsart som först beskrevs av Walker 1857.  Oliarus modica ingår i släktet Oliarus och familjen kilstritar. Inga underarter finns listade i Catalogue of Life.

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Kilstritar
modica