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Comb

A comb is a tool consisting of a shaft that holds a row of teeth for pulling through the hair to clean, untangle, or style it. Combs have been used since prehistoric times, having been discovered in very refined forms from settlements dating back to 5,000 years ago in Persia.

Weaving combs made of whalebone dating to the middle and late Iron Age have been found on archaeological digs in Orkney and Somerset.

Description

Combs are made of a shaft and teeth that are placed at a perpendicular angle to the shaft. Combs can be made out of a number of materials, most commonly plastic, metal, or wood. In antiquity, horn and whalebone was sometimes used. Combs made from ivory and tortoiseshell were once common but concerns for the animals that produce them have reduced their usage. Wooden combs are largely made of boxwood, cherry wood, or other fine-grained wood. Good quality wooden combs are usually handmade and polished.

Combs come in various shapes and sizes depending on what they are used for. A hairdressing comb may have a thin, tapered handle for parting hair and close teeth. Common hair combs usually have wider teeth halfway and finer teeth for the rest of the comb. Hot combs were used solely for straightening hair during the colonial era in North America.

A hairbrush comes in both manual and electric models. It is larger than a comb, and is also commonly used for shaping, styling, and cleaning hair. A combination comb and hairbrush was patented in the 19th century.

Uses

Combs can be used for many purposes. Historically, their main purpose was securing long hair in place, decorating the hair, matting sections of hair for dreadlocks, or keeping a kippah or skullcap in place. In Spain, a peineta is a large decorative comb used to keep a mantilla in place.

In industry and craft, combs are used in separating cotton fibres from seeds and other debris (the cotton gin, a mechanized version of the comb, is one of the machines that ushered in the Industrial Revolution). A comb is used to distribute colors in paper marbling to make the swirling colour patterns in comb-marbled paper.

Combs are also a tool used by police investigators to collect hair and dandruff samples that can be used in ascertaining dead or living persons' identities, as well as their state of health and toxicological profiles.

Hygiene
Sharing combs is a common cause of parasitic infections much like sharing a hat, as one user can leave a comb with eggs or live parasites, facilitating the transmission of lice, fleas, mites, fungi, and other undesirables. Siblings are also more likely to pass on nits to each other if they share a comb.

Making music
Stringing a plant's leaf or a piece of paper over one side of the comb and humming with cropped lips on the opposite side dramatically increases the high-frequency harmonic content of the hum produced by the human voice box, and the resulting spread sound spectrum can be modulated by changing the resonating frequency of the oral cavity. This was the inspiration for the kazoo, a membranophone.

The comb is also a lamellophone. Comb teeth have harmonic qualities of their own, determined by their shape, length, and material. A comb with teeth of unequal length, capable of producing different notes when picked, eventually evolved into the thumb piano and music box.

Chinese combs
In China, combs are referred to by the generic term  or   and originated about 6000 years ago during the late Neolithic period. Chinese combs are referred as  when referring to thick-tooth comb and   when referred to thin-tooth comb. A form of produced in Changzhou is the Changzhou comb; the Palace Comb Factory, also called Changzhou combs Factory, found in the city of Changzhou started to operate since the 5th century and continues to produce handmade wooden combs up to this day. were also introduced in Japan during the Nara period where they were referred by the generic name.

Japanese combs
In Japan, combs are referred to as. Indigenous Japanese started to be used by Japanese people about 6000 years ago in the Jōmon era. In the Nara period, Chinese combs from the Tang dynasty were introduced in Japan. Another form of comb in Japan is the Satsuma comb, which started to appear around the 17th century and was produced by the samurai warriors of the Satsuma clan as a side job.

Liturgical comb
A liturgical comb is a decorated comb with used ceremonially in both Catholic and Orthodox Christianity during the Middle Ages, and in Byzantine Rite up to this day.

Nit comb
Specialized combs such as "flea combs" or "nit combs" can be used to remove macroscopic parasites and cause them damage by combing. A comb with teeth fine enough to remove nits is sometimes called a "fine-toothed comb", as in the metaphoric usage "go over [something] with a fine-toothed comb", meaning to search closely and in detail. Sometimes in this meaning, "fine-toothed comb" has been reanalysed as "fine toothcomb" and then shortened to "toothcomb", or changed into forms such as "the finest of toothcombs".

Afro pick
An Afro pick is a type of comb having long, thick teeth which is usually used on kinky or Afro-textured hair. It is longer and thinner than the typical comb, and it is sometimes worn in the hair. The history of the Afro pick dates back at least 5,000 years, as a practical tool that may also have cultural and political meaning.

Unbreakable plastic comb
An unbreakable plastic comb is a comb that, despite being made of plastic rather than (more expensive) metal, does not shatter into multiple pieces if dropped on a hard surface such as bathroom tiles, a hardwood floor, or pavement. Such combs were introduced in the mid-twentieth century. Today most plastic combs are unbreakable as technology has reached a point of understanding the causation of brittleness in these products.

Modern artisan combs
Modern artisan combs crafted from a wide variety of new and recycled materials have become popular over recent years. Used skateboard decks, vinyl records, brass, titanium alloy, acrylic, sterling silver, and exotic wood are a few of the materials being used.

title: Comb


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Abbey Park, Leicester

Abbey Park is a public park in Leicester, England, through which the River Soar flows. It is owned and managed by Leicester City Council. It opened in 1882 on the flood plain of the River Soar, and expanded in 1932 to include the area west of the river that had formerly been the medieval St Mary's Abbey, still bounded by large medieval walls. The park includes the archaeological sites of the Abbey and the ruins of Cavendish House, along with a wide range of decorative and recreational parkland features.

It is Grade II* listed in Historic England's Register of Parks and Gardens.

History
In 1876 Leicester town council bought 57 acres of marshy ground between the river and canal from the Earl of Dysart in order to develop flood prevention plans. Planning for this first incarnation of the park was underway by 1879, as part of designs by the borough surveyors for the relief of flooding in the area. However the design for the park itself was opened up to a competition. The winning design, with its bandstand, rustic bridges and planted gardens, was the work of William Barron, a celebrated landscape designer. It was Leicester's first public park of significant size, and was opened on 29 May 1882 by the Prince and Princess of Wales, an event commemorated by an ornate plaque at the Abbey Park Road entrance. The park was created in an area that had previously been described as "marshy ground in a poor district" at a cost of over £40,000. The works included the widening and deepening of the river over a length of around a mile, with the excavated earth used to create mounds within the park, as well as the construction of stone weirs and locks. Three new bridges were constructed crossing the river. An artificial lake was created and over 33,000 trees planted. Excavations as part of the work discovered remains of animals including elephants and rhinoceros. The resulting park occupied 57 acres of land between the River Soar and the Grand Union Canal. Two lodges designed by architect Mr. J. Tait were built at the Abbey Road entrance to the park.

Although the new park was called Abbey Park, the site of the abbey itself, bounded by substantial medieval masonry and brick walls, was not within the park, but was in agricultural land on the other side of the River Soar. The exact location of the Abbey was unknown, with no standing remains other than the boundary walls and the ruins of the sixteenth century Cavendish House. Archaeological investigations of a limited nature began in the 1920s, and popular enthusiasm was fuelled by the 1922 discoveries in Egypt of Tutankhamun's tomb, and a hope that Cardinal Wolsey's tomb might be similarly discovered. After assorted modest excavations, the 32 acre Abbey Grounds site was donated to the city by the Earl of Dysart at the end of 1925. Several years of preliminary works, and sporadic attempts to pin down the site of the Abbey buildings, were followed by more thorough work undertaken in 1929-30. With the rise in unemployment culminating in the Great Depression the city council attempted to alleviate local poverty by employing a team of workmen to clear pernicious weeds from the overgrown site, and the architect W K Bedingfield was able to utilise this process to also search for, and then uncover the buried foundations of the Abbey buildings. A variety of graves were found, but no magnificent tomb of the sort popularly hoped for. Having exposed and mapped the extent of the Abbey Church and a variety of monastic buildings, the workforce built low walls to mark out the foundations and pier bases. They also carried out extensive repairs to the stonework and brickwork of the medieval boundary walls particularly along the north and western sides of the site, traditionally ascribed to Abbot Penny and Abbot Cloune. The north and western wall is now named Abbot Penny's Wall, and is a Grade I listed structure. Paths were laid out across the grounds, tennis courts were built, a large oval was levelled for use as a cricket pitch, and a new stone bridge in a classical style, was built across the River Soar to link the two parts of the park.

Despite the absence of a tomb, the abbey ruins received a memorial slab to Cardinal Thomas Wolsey. He died while en route from York to London on 29 November 1530. A statue of him also stands next to the Park's cafe. Alongside the site of the 12th century Leicester Abbey, which was marked out with low stone walls, are the substantial ruins of Cavendish House (built in the 17th century by William Cavendish, 1st Earl of Devonshire incorporating the Abbey Gatehouse, and by tradition built using stone from the abbey). The house was used by Charles I after the siege of Leicester during the English Civil War in 1645; after he left, his soldiers set fire to it leaving the house gutted. The charred stone window frame is still visible today.

Facilities and events
The park now has an area of 89 acres, and has formal gardens, a sensory garden, a boating lake and model boat lake, a miniature railway operated by Leicester Society of Model Engineers since 1949, a visitor centre, cafe, children's play area with paddling pool, pets corner, tennis courts, a bowling green, and a bandstand. The park has regularly won the Green Flag Award, a national award made annually to parks which reach a high standard.

The park was the site of an annual flower show dating back to the 19th century, which included a swimming gala and evolved into the Abbey Park Show in the 1940s, with the addition a range of entertainment and displays. It continued until 1995, when it was abolished due to falling attendances and rising costs, although a return is periodically discussed. It was also the site of the Pageant of Leicester, held in 1932. The Abbey Park Festival was an annual music festival which was held for over twenty years from 1981. An annual bonfire and fireworks event is held in the park around Guy Fawkes Night.

The Abbey has been the site of training excavations by Archaeology students at Leicester University.

Since 2010 Abbey Park has housed the offices and studios of Takeover Radio within one of its Gatehouse Lodges.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Abbey Park, Leicester


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Googlewhack

A Googlewhack was a contest to find a Google Search query that returns a single result. A Googlewhack must consist of two words found in a dictionary and was only considered legitimate if both of the search terms appear in the result. Published googlewhacks were short-lived since when published to a website, the new number of hits would become at least two: one to the original hit found, and one to the publishing site, unless a screenshot was provided. Googlewhacks generally no longer exist due to changes in Google search indexing.

History
The term googlewhack, coined by Gary Stock, first appeared on the web at UnBlinking on 8 January 2002. Subsequently, Stock created The Whack Stack, at googlewhack.com, to allow the verification and collection of user-submitted Googlewhacks.

Googlewhacks were the basis of British comedian Dave Gorman's comedy tour Dave Gorman's Googlewhack Adventure and book of the same name. In these Gorman tells the true story of how, while attempting to write a novel for his publisher, he became obsessed with Googlewhacks and travelled across the world finding people who had authored them. Although he never completed his original novel, Dave Gorman's Googlewhack Adventure went on to be a  Sunday Times No. 1 best seller in the UK.

Participants at Googlewhack.com discovered the sporadic "cleaner girl" bug in Google's search algorithm where "results 1–1 of thousands" were returned for two relatively common words such as Anxiousness Scheduler or Italianate Tablesides.

Googlewhack went offline in November 2009 after Google stopped providing definition links. Gary Stock stated on the game's web page soon afterward that he was pursuing solutions for Googlewhack to remain viable.

Score
Some people propose the googlewhack "score", which is the product of the hits of the individual words. Thus a googlewhack score is highest when the individual words produce a large number of hits.

Examples

 * comparative unicyclist
 * maladroit wheezer

Variations
New Scientist has discussed the idea of a Googlewhackblatt, which is similar to a Googlewhack except that it involves finding a single word that produces only one Google result. Lists of these have become available, but as with Googlewhacks, they result in the Googlewhackblatt status of the word being destroyed—unless it is blocked by robots.txt or the word does not produce any Google results before it is added to the list, thus forming the Googlewhackblatt Paradox. Those words that do not produce any Google search results at all are known as Antegooglewhackblatts before they are listed—and subsequently elevated to Googlewhackblatt status if it is not blocked by robots.txt.

Feedback stories are also available on the New Scientist website, thus resulting in the destruction of any existing Googlewhackblatts that are ever printed in the magazine. Antegooglewhackblatts that are posted on the Feedback website become known as Feedbackgooglewhackblatts as their Googlewhackblatt status is created. In addition, New Scientist has more recently discovered another way to obtain a Googlewhackblatt without falling into the Googlewhackblatt Paradox. One can write the Googlewhackblatt on a website, but backward, and then search on elgooG to view the list properly while still keeping the Googlewhackblatt's status as a Googlewhackblatt.

In contrast to Googlewhacks, many Googlewhackblatts and Antegooglewhackblatts are nonsense words or uncommon misspellings that are not in dictionaries and probably never will be.

Practical use of specially constructed Googlewhackblatts was proposed by Leslie Lamport (although he did not use the term).

Research applications
The probabilities of internet search result values for multi-word queries was studied in 2008 with the help of Googlewhacks. Based on data from 351 Googlewhacks from the "WhackStack" a list of previously documented Googlewhacks, the Heaps' law $$\beta$$ coefficient for the indexed World Wide Web (about 8 billion pages in 2008) was measured to be $$\beta=0.52$$. This result is in line with previous studies which used under 20,000 pages. The googlewhacks were a key in calibrating the model so that it could be extended automatically to analyse the relatedness of word pairs.

title: Googlewhack


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Volkswagen Polo

The Volkswagen Polo is a supermini car (B-segment) produced by the German car manufacturer Volkswagen since 1975. It is sold in Europe and other markets worldwide in hatchback, saloon, and estate variants throughout its production run.

History
As of 2018, six separate generations of the Polo have been produced, usually identified by a "Series" or "Mark" number.

Some generations were facelifted midway through production, with the updated versions known unofficially by an addition of the letter F to the mark number, e.g., Mk2F. Some members of the automotive press and some enthusiasts consider the facelifts to be separate models, so have used the unofficial designations Polo Mk1 to Mk7 for previous generations. Each Polo model is also identified by a two- or three-character Volkswagen Group Typ number. Official VW Polo history describes Mark I to Mark IV using either Roman numerals or Arabic numerals, with facelifted variants known as "Phase II" models.

The body style has been varied through the life of the car, originally as a hatchback, which derived from the Audi 50. A saloon version was marketed as the Volkswagen Derby.

Volkswagen vehicles built on different platforms have carried the Polo nameplate. For example, the Volkswagen Polo Playa hatchback sold in Southern Africa in the late 1990s was a rebadged SEAT Ibiza, which has a different body shell from the Polo Mk3 sold in Europe at the same time.

Starting in 1982, Volkswagen sold the Polo in Japan initially through an agreement with Japanese dealership Yanase that specializes in European and North American vehicles. Of all Volkswagens imported into Japan, only the Polo (until 2017) and the Golf (until 1997), complied with Japanese government dimension regulations until the introduction of the VW Up! in 2012.

Related models
The first Polo was effectively a rebadged version of the Audi 50 hatchback launched in August 1974. The Audi 50 was discontinued in 1978, as Audi concentrated on larger luxury models. The Mk1 and Mk2 versions of the Polo were then standalone models in the Volkswagen range.

With the expansion of the Volkswagen Group: Audi (in the 1960s), SEAT (in the 1980s), and Škoda (in the 1990s) were acquired, and the platform used for the Polo was shared with other models.

The Polo shared its platform with the SEAT Ibiza Mk2. The Ibiza was actually launched before the Polo Mk3 and shared essentially all its mechanicals, the dashboard, and other interior components, although no body panels were shared between the two cars. The saloon and estate versions of the Polo Mk3 were actually rebadged SEAT Córdoba models and had no body panels in common with the Polo hatchback. The SEAT Inca and Volkswagen Caddy vans were also based on this model. The Volkswagen Lupo and SEAT Arosa were also based on a shortened version of the Polo Mk3 platform.

The Polo Mk4 continued this trend of platform sharing, with the SEAT Ibiza Mk3 and Škoda Fabia Mk1 and Mk2 both being developed on the same platform and featuring several of the same engines.

The 2009 Polo Mk5 is developed on the latest platform, known as the PQ25, the same platform used in the SEAT Ibiza Mk4 that was launched in 2008 and the Audi A1.

Body styles
The first Polos were hatchbacks, with the saloon being marketed as the Volkswagen Derby.

With the release of the Polo Mk2, the saloon was renamed the Volkswagen Polo Classic' and the "conventional"-styled hatchback (with a sloping tailgate) was renamed as a coupé, the Volkswagen Polo Coupé. Unusually, the third (and new) body that was actually marketed as the Volkswagen Polo Hatchback was closer in concept to a small estate, albeit with exactly the same wheelbase and floor pan as the coupé instead of the (longer) saloon. This latter version, also known as the squareback (in the original German brochures, "steilheck", literally "steep tail"), amongst other nicknames, was the most popular in virtually every country where the Polo was sold. Despite the differences in silhouette and target market segment, all body types were two- or three-door only.

From the Polo Mk3 onwards, the range was more straightforwardly conventional, including unambiguous "saloon", "hatchback", and "estate" models, with only the hatchback offering both five-door and slightly shorter three-door models (both still with quite vertical tailgates, the coupé variation having been retired), the others being four or five-door only and increasing in length from hatch to saloon to estate.

Summary

 * Three-door hatchback (Mk1 to Mk5) – the Mk2 and Mk2F were available in two separate three-door hatchback styles, one of which was badged as a coupé.
 * Two-door saloon (Mk1, Mk1F, Mk2, Mk2F)
 * Four-door saloon (Mk3, Mk3F, Mk4, Mk4F, Mk5, Mk6)
 * Five-door hatchback (Mk3, Mk3F, Mk4, Mk4F, Mk5, Mk6)
 * Five-door estate (Mk3, Mk3F)
 * Five-door crossover SUV-style (two-wheel drive) hatchback (Mk4, Mk4F, Mk5)

Mechanical layout
The Polo is a compact car, with a traditional transversely mounted engine and front-wheel drive. Mk1 Polos only came with four-cylinder petrol engines, but for the Mk2, a diesel engine was offered for the first time, although only in certain markets, others having to wait until the launch of the Mk3. The current range includes a variety of three- and four-cylinder petrol and diesel engines.

Early versions used a four-speed manual transmission, whilst the current car is available with either six-speed manual or seven-speed automatic transmission. The suspension system on all models uses a fully independent MacPherson strut front suspension, and a twist-beam rear suspension. Most models use disc brakes at the front and rear drum brakes, although some recent models have all-round disc brakes.

First generation (86; 1975)
The first-generation Polo, a rebadged version of the Audi 50, was introduced in 1975 and was produced until October 1981. It was also called the Volkswagen Derby. By 1979, 500,000 Polos were produced worldwide. It shared the internal designation Typ 86 with the Audi 50.

The differences between the Audi and Volkswagen models were minor, with the Polo being cheaper and much more basic. The two cars were initially sold alongside each other, but the Audi 50 never sold as well, and was withdrawn in 1978. The Polo was manufactured at the Volkswagen plant in Wolfsburg.

The manufacturer let it be known that Bertone had been involved in the styling of the Polo and its Audi sibling, although the car was essentially an in-house Audi design, with the Italian design studio's contribution restricted to the circular extractor vent cover at the base of the C pillar, and the small "flick-up" at the rear end of the waistline.

In 1977, the Derby saloon was released, which was simply a Polo, identical to the hatchback from the C-pillar forward, with a large boot attached.

The Mark 1 Polo and Derby were facelifted in 1979 (unofficially referred to as the Mark 1F) with plastic bumpers, a different front grille, and a revised dashboard. The round headlights of the Derby were replaced with square ones, bringing it in line with the similar Golf-based Jetta saloon.

Second generation (86C; 1981)
The Polo Mk2 (or Typ 86C) was introduced in October 1981, with the major change being the introduction of a third body style with a steep (almost vertical) rear window, in addition to a version resembling the original Mk1 shape with a diagonal rear window. These two body styles were called the Wagon (in some markets) and Coupé, respectively, although in fact both were three-door hatchbacks, and in some markets the Wagon designation was not used, with that car being simply called the Volkswagen Polo without a suffix. The saloon version received the name Volkswagen Derby. Production was expanded to Spain in the mid-1980s following Volkswagen's takeover of SEAT. By 1983, the one-millionth Polo was produced. The second million were produced by 1986.

The Polo competed in the supermini sector with its new rivals such as the Austin Metro and Ford Fiesta, and it was one of the largest superminis in production at this time, meaning that it was close to larger cars, including the Ford Escort in terms of size, space, and price. Within two years of its launch, however, it was faced with competition from a wave of new rivals, particularly the Fiat Uno and Peugeot 205.

The Polo Mk2 was used extensively by Volkswagen to develop future innovations, for example supercharging with a 40-mm G-Lader supercharger in the GT G40 version. A 60-mm G-Lader would later be used on the larger and more technically challenging G60 engine used in the Golf and Corrado.

A fuel-efficient two-cylinder diesel was prototyped in the mid-1980s with a G40 supercharger to overcome its small capacity, although this did not enter production. A high fuel-efficiency model that did make production was the petrol-engined Formel E (E for economy), introduced at the launch in 1981 with a 1.1-litre engine and from 1983 with a 1.3-litre engine, overdrive top-gear ratio and an early stop-start ignition system (called "SSA"), which would cut the engine when idle for more than two seconds to save fuel whilst temporarily stopped in traffic, and restart the engine on moving the gear lever to the left in neutral.

Similar systems were later used on the Volkswagen Golf Mk3 and various systems from other car manufacturers.

It was a popular import in the UK, competing with the likes of the Peugeot 205, Fiat Uno, and Nissan Micra. While of an outdated design, its reliability and build quality were among the best to be found on a small car of this era. In the years since then, it went on to achieve a higher survival rate than many of its competitors.

However, some engines were only available on certain markets. For example, the British market never received any diesel-engined versions of the Polo, though the diesel engine was already available on many similar cars by the end of the 1980s, namely the Ford Fiesta, Vauxhall Nova, and Peugeot 205.

Facelift (1990–1994)

The Mark II Facelift (referred to as the Mark IIF, also erroneously known as the "Mark 3") was a far-reaching facelift of the MkII, including a reskin of the bodywork, launched in the autumn of 1990. The new look had square headlights, enlarged and reshaped taillights, bigger bumpers, and a new interior (dashboard and door trim). The three different body styles were maintained. Under the skin, the car received modifications to the chassis, suspension, and brakes, as well as the cosmetic differences. The new Polo retained the previous four-cylinder engines, but now as well as the carburetted 1.0-litre, a fuel-injected model was available with single-point injection, and all engines came with a catalytic converter as standard to combat tightening European emissions regulations. The saloon was only produced in Spain, and production ceased in 1992 due to disappointing sales.

At the time of launch of the Mark IIF Polo, the highest-performance model was the Polo GT. This featured a multipoint fuel-injected version of the 1272-cc engine, and a top speed more than 100 mph. This produced 75 PS and had a quoted top speed of 107 mi/h. Times for 0–60 stood at 11.1 seconds. The defining features of the GT include red piping in the bumpers, black overhead cloth, a tachometer, and a red "GT" badge in the grille. This was succeeded by the launch of the G40 in May 1991, displacing the GT as the most powerful Polo at the time, with a top speed around 120 mph – rivalling the likes of the Ford Fiesta XR2i, Peugeot 205 GTI, and Renault Clio 16v.

Soon after the launch of the Mark IIF, another sporting model was added to the range — a new version of the supercharged G40, now as a full production model in all markets rather than the limited batch of Mark II G40s. As with the previous model, Volkswagen Motorsport modified G40 Cup cars were sold for racing in a one-make series, the Volkswagen Polo G40 Cup. Features that define the G40 from other Polo models at the time (on top of the GT) include a bee-sting aerial, BBS cross-spoke alloy wheels, Le Mans interior trim, and front and rear red "G40" badges.

Third generation (6N/6KV; 1994)
The Mark III Polo or Typ 6N, (sometimes referred to as the "Mark 4" by enthusiasts as it is the Polo's fourth guise) appeared in 1994, and was a completely new model (on a new chassis), available as three- and five-door hatchback versions, the latter making VW the last major European manufacturer to finally offer rear side doors in this class. It shared its platform with the SEAT Ibiza Mark 2. This platform actually used the floorpan of the Volkswagen Golf Mk3 (a multitude of mechanical parts and all of the suspension components were interchangeable among the three models). Although the dashboard and a number of mechanical components, including engines, were shared with the Ibiza, outwardly the two cars were different, with no shared body panels.

An all-new 1.0-litre petrol engine was added to the range from launch, while the 1.3-litre petrol engine from the MK2 was briefly carried over. Also new to the range were 1.4- (8-valve or 16-valve) and 1.6-litre petrol engines, as well as a 1.9-litre diesel (with or without a turbocharger).

The saloon and estate versions of the Mark III, which debuted in 1995, were essentially rebadged version of the SEAT Córdoba, which were launched early in 1994. They were referred to internally by Volkswagen as the Typ 6KV, and shared body panels with the SEAT model rather than the Polo hatchback models, with some cosmetic alterations such as new rear and front bumpers and headlights. The Volkswagen Caddy 9K van also shares the same platform and front-end styling as the 6KV models.

The later Volkswagen Lupo and SEAT Arosa city cars were based on a shortened version of the Typ 6N platform and shared many components.

Polo Playa (1996–2003)

The Volkswagen Polo Playa was a model for the South African market. It was sold instead of the European Polo Mark III from 1996 until 2002 and was effectively a rebadged SEAT Ibiza Mark II. In 2002, the Mark IV Polo was adopted in South Africa, and the separate Polo Playa model was dropped.

Fourth generation (6Q/9N/9N3; 2002)
Unveiled in September 2001, the all-new Mark IV (or Typ 9N, sometimes erroneously referred to as the "Mark 4") model was put on sale in early 2002. It shares its platform with the SEAT Ibiza Mk3, Škoda Fabia Mk1, and Škoda Fabia Mk2. The car is all new compared to the Mark III/F and bears structural resemblance to Golf MK4 (1J). The rear taillights resemble those of the B5.5 Passat. Outwardly, the most recognisable change is the use of quadruple round headlights similar to the Lupo's.

Volkswagen Racing rallied a Polo S1600 in the 2003 Junior World Rally Championships, winning the Turkish round. The Super 1600 developed 165 kW/215 to its front wheels. This version of the Polo was a mixed success in the United Kingdom. It sold reasonably well (though not as well as some earlier Polos), but several customer-satisfaction surveys by high-profile motoring magazines such as Top Gear gave the Polo a very low rating.

Top Gear's 2005 survey rated the Polo as the third-least satisfying supermini to own, with only the Fiat Punto and Rover 25 receiving worse ratings. It fell behind most of its crucial rivals, namely the Ford Fiesta, Vauxhall Corsa, Citroën C3, and Peugeot 206.

A saloon version of the Mk4 Polo was produced for markets outside Europe, including most Latin American countries, South Africa and China.

Fifth generation (6R/6C/61; 2009)
Volkswagen launched the fifth generation Polo (internal designation Typ 6R) at the Geneva Motor Show in March 2009. For the first time in Polo's history, the car was declared European Car of the Year, for 2010. The Polo was also declared 2010 World Car of the Year at the New York International Auto Show in April 2010. It also won What Car? Supermini of the Year 2010, as well as being awarded Japan Import Car of the Year for 2010–2011.

It shares its platform with the SEAT Ibiza Mk4 and the Audi A1. Production for the UK market started in August 2009, with first deliveries beginning in October 2009. The Polo Mark V is 44 mm longer, 32 mm wider, and sits 13 mm lower to the road than the previous generation Polo. Boot capacity is increased by 10 L to 280 L of storage space with 952 L with the seats folded down. The car is 7.5% lighter than its predecessor. The Polo has thoracic airbags and has been awarded a five-star Euro NCAP crash impact rating.

Volkswagen Vento/Polo Sedan
In 2010, a saloon based on the Polo Mk5 platform was developed for launch in India and Russia. This saloon is known as the Vento or Polo Sedan, depending on the country. It has a length of 4,384 mm, increased wheelbase (2,552 mm) and ground clearance (168–170 mm), and had one petrol engine (1.6-litre, four-cylinder, 105 PS; coupled with either five-speed manual or six-speed automatic gearbox) and one diesel engine (1.6-litre, turbocharged four-cylinder common-rail, 105 PS; only five-speed manual gearbox) options available.

In several markets, it is succeeded by the Virtus, while some markets such as Mexico offered the Virtus alongside the Vento for the purpose of keeping a budget-friendly option. In Russia, the Polo Mk5-based Polo Saloon is succeeded by the Škoda Rapid-based Polo liftback.

Volkswagen Ameo
The Ameo is another saloon based on the Polo Mk5 which was released in June 2016. It was specifically developed for the Indian market which offers lower excise tax for vehicles shorter than 4 meter in length. The Ameo is more heavily derived from the hatchback body of the Polo, retaining its shorter wheelbase and the rear doors, unlike the larger Vento/Polo Saloon. It was discontinued in 2020 due to low demand.

Sixth generation (AW/BZ; 2017)
The sixth-generation Polo was first unveiled in Berlin on 16 June 2017, and launched in late 2017. Based on the Volkswagen Group MQB A0 platform, it is claimed to carry improvements in cabin space, engines, and interior technology. It is the first Polo generation that is not available with a 3-door body style.

The Mk6 Polo is 81 mm longer, 63 mm wider and 7 mm lower than its predecessor. It has a wheelbase of 2564 mm, which is 92 mm longer than the previous model. The adoption of the MQB platform brought an increase in rigidity and stiffness over the previous generation, where the stiffness has increased from 14,000 Nm per degree to more than 18,000 Nm per degree according to Volkswagen.

Boot space has increased by about 25 percent from 280 to 351 L. Optionally, it now features the second-generation version of the 11.7-inch Active Info Digital Display Cockpit, a first in its class. As standard, the car comes with front collision detection, blind-spot assist, and emergency stopping. The car is said to be extremely customizable, available in 14 exterior colours and 17 dashboard colours.

Production for all right-hand drive markets are allocated to the Volkswagen plant in Uitenhage/Kariega, South Africa. The South African plant is also the sole producer of the Polo GTI.

Europe
In Europe, three trim levels were announced during its debut in 2017, which are Trendline, Comfortline and Highline, the ‘Beats’ special edition, and the performance-oriented Polo GTI. In UK the trim levels in 2017 were SE, SEL, R-Line, GTI and GTI+.

European models initially were available with a range of 1.0-litre, three-cylinder engines with various outputs; 1.0-litre 65 PS and 75 PS naturally aspirated versions, a natural gas-powered 1.0-litre TGI producing 89 PS - as well as 95 PS TSI. The 1.6-litre TDI engine was also available with 80 PS and 95 PS outputs. The GTI model gets a 200 PS 2.0-litre TSI engine.

Latin America
The Latin American-spec Polo debuted in Brazil in September 2017. At the time of its introduction, the Polo was positioned between the Fox and Golf in the Brazilian market, and marks the reintroduction of the nameplate in the country since the discontinuation of the fourth-generation Polo in 2015. Some changes were made compared to the European version, including a reworked suspension to provide higher ground clearance, a different headlight assembly and a revised front bumper design.

Engine options available are the basic 1.0-litre MPI petrol engine producing 85 PS, 1.6-litre MSI rated at 119 PS, and the top-spec 1.0-litre TSI with 130 PS marketed as '200 TSI', referencing its 200 Nm maximum torque.

In early 2020, the GTS variant was launched in Brazil, featuring a 150 PS 1.4-litre turbocharged petrol engine and cosmetic additions.

In Mexico, the Brazilian-imported, facelifted Polo was released in 2023, six years after the Brazilian model's launch, sharing the same engines and specification with the Virtus saloon. It replaced the Indian-made Polo Mk5.

Polo Track
Announced in November 2022 as a replacement for the Gol, the Polo Track was released in February 2023 as a low-cost version of the Mk6 Polo. Aimed at fleet buyers, the Track version has a shorter list of features and a simpler exterior appearance. It is powered by a 1.0-litre MPI (1.6-litre 16V MSI 105 hp for Colombia and other countries) petrol flex engine producing 85 PS.

Safety
The Polo in its most basic version for Latin America received 5 stars for adult occupants, 5 stars for toddlers, and Advanced Award from Latin NCAP in 2017.

The facelifted Polo was reassessed by Latin NCAP in 2022 and achieved a three-star safety rating against more demanding Latin NCAP assessment protocols than the original 2017 test. Latin NCAP noted that since the original test, the car had lost a manual passenger airbag disconnection switch that allows for safe installation of a rearward-facing child restraint on the front passenger seat.

Facelift
In April 2021, the restyled version of the sixth-generation Polo was revealed for the European market. It gained a redesigned front end, featuring reshaped headlights and an LED light bar on the grille. A matrix LED setup is optional for the first time in a Polo. The rear end now adopts wider LED taillights extending onto the tailgate. With the facelifted version, the Polo comes with a fully digital instrument cluster as standard. The dashboard also received a redesigned touch climate control.

For the Latin American market, the facelifted 2023 Polo does not receive the taillight shape redesign.

Virtus
Volkswagen developed the sixth-generation Polo in a saloon body style badged as the Volkswagen Virtus, which is produced in Brazil and exported to most Latin American countries. It is also produced in India since 2022. It is marketed as the Polo Sedan in South Africa since September 2022.

Polo liftback (Russia)
In Russia and the neighbouring CIS countries, a separate Polo model was released in 2020 to replace the Polo Mk5 saloon. Unrelated to the Mk6 Polo, the model heavily based on the global Škoda Rapid liftback to cut costs instead of having to produce the MQB A0-based Volkswagen Virtus. The Russian-market Polo features a Jetta-like front fascia, and a unique rear fascia to differentiate it from the Rapid, while sharing the front bumper and hood panel with the Jetta VA3. As it is based on the Rapid, it is based the A05+ (PQ25) platform and has a liftback tailgate instead of a conventional saloon trunk opening.

It is offered with a 1.6-litre MPI four-cylinder, naturally aspirated engine with 89 hp with a five-speed automatic gearbox, and 109 hp also with a six-speed automatic transmission, and a 1.4-litre TSI four-cylinder with 123 hp, only with a seven-speed DSG dual-clutch transmission.

In early 2022, Volkswagen ceased production and new car sales of the global Skoda Rapid-based Volkswagen Polo for the CIS markets, due to international sanctions being imposed on Russia as a result of the 2022 Russian invasion of Ukraine, as Volkswagen withdrew entirely from the Russian auto market as a response to the war.

GTI
The Volkswagen Polo GTI is a hot hatch version of the Polo.

Mk2 Polo GT G40 and Mk2F Polo G40
The Volkswagen Polo GTI can trace its roots back to the original hot Polo, the supercharged 1.3-litre 115 PS G40. It was sold in the United Kingdom between 1990 and 1994 in the form of the Mk2F Polo but was available during the late 1980s in limited numbers in Europe in the form of the Mk2 Polo GT G40. The car was expensive compared to its rivals of the time, so did not sell in vast numbers. This makes it highly collectible today, and according to Top Gear, "a potential future classic."

Mk3 Polo GTI
After production of the G40 ended in 1994, Volkswagen decided to release the first GTI-branded Polo in a limited batch of just 3000. It was released in 1995 and available only in left-hand drive. It featured a 1.6-litre, 16-valve 120 PS engine. The GTI was not available in the UK at any stage during its limited production, so as of 1994, the UK only had the 100 bhp 1.4-litre, 16-valve as a model with any sporting intent.

Facelift (6N2)
Between 1999 and 2002, Volkswagen offered two sporting models — the 16V and GTI. The 16V came with the 1.4-litre, 16-valve (V), 100 PS engine, and had options such as 15-in Spa alloys and air conditioning. However, the GTI – available only in three- or five-door hatchback body styles in three colours being red, silver, and black, used a 1.6-litre 16-V 125 PS engine with variable valve timing. External changes included a deeper front splitter with honeycomb mesh grilles, lowered sports suspension (by 10 mm), a subtle rear spoiler, deeper side skirts, fog lights, and 15" BBS RXII split rims for the wheels, bearing 195/45/15 tyres. There were also standard extras such as Climatronic fully automatic air conditioning, xenon headlights with a headlight washer system, a six-disc CD autochanger with a GAMMA head unit, exclusive sports interior with leather steering wheel, handbrake and gearstick, chrome inserts, and driver aids such as ABS with EBD and an EDL (a system to aid traction). Leather and satellite navigation were also optional extras. The Polo GTI Mk3F, however, never got the new six-speed gearbox that was introduced in the smaller Volkswagen Lupo GTI, which shared the same engine.

Mk4 Polo GT
With the introduction of the 2002 Polo, the GTI model was discontinued and was given no direct replacement. A GT model, though, was produced, featuring the same 1.9-litre TDI, 130 PS engine found in the popular Škoda Fabia vRS and SEAT Ibiza FR TDI, as well as a six-speed gearbox. Although this model had a relatively slow 0–100 km/h (62 mph) time over 9 seconds, it did have impressive midrange clout with torque figures of 310 Nm (228 lb-ft).

Mk4 Polo GTI
The GTI was not released until late 2005, after yet another Polo revamp was reintroduced. This time it boasted a 1.8-litre, 150 PS engine, which had been used in models from the Mk4 Volkswagen Golf GTI to the Audi A6. Despite the impressive figures, this new model lacked the standard features of the Polo GTI Mk3, with xenon headlights not even on the options list, and fully digital climate control only being an expensive option. Although faster than the 2000–2002 Polo GTI, the newer model was also seen as being off the pace when compared to its rivals, most of which are now nudging 200 PS. In Europe, this led to VW quickly beefing up the Polo further to create the Polo GTI Cup Edition, which was tuned to around 180 PS and featured more aggressive styling, 17" wheels and larger diameter 312mm front disc brakes. The Polo GTI Cup Edition has 177 bhp, 29 bhp more than the standard Polo GTI. The standard Polo GTI model completes 0–100 km/h in 8.2 seconds, but in the Cup, it completes 0–100 km/h in 7.5 seconds.

Mk5 Polo GTI
The Mk5 Polo GTI was launched at the 2010 Geneva Motor Show. The new GTI is powered by VW's award-winning 180 PS 1.4-litre TSI engine (adapted from that used in the current Scirocco), which uses both a supercharger and turbocharger to provide torque throughout the revolution range. The Mk5 Polo is also 7.5% lighter than its predecessor and with a 30 PS increase in power over the previous generation Polo GTI it accelerates from 0 to 100 km/h (62 mph) in 6.9 seconds. The Mk5 Polo also includes features not found on previous generation Polos, such as touch-screen satellite navigation and a seven-speed version of VW's DSG gearbox as standard.

Facelift
The Mk5 Polo GTI facelift was launched in 2015. It featured a new 1.8-litre turbocharged engine, which had been developed by Audi. It produced 190 bhp and was capable of sprinting from 0-60 mph in 6.4 seconds (real world test). It was now also available with a manual transmission.

Mk6 Polo GTI
The Mk6 Polo GTI was launched in 2017. It features a slightly detuned version of the 2.0-litre TSI petrol engine from the Mk7 Golf GTI. It produced 200 PS, an 8 PS increase over the previous 1.8-litre engine, and was capable of getting to 100 km/h in under 6.7 seconds. Available transmissions are a 6-speed manual or a 7-speed DSG. Along with the engine, the GTI version came standard with lowered suspension, GTI badges, GTI bumpers, 17-in wheels, sill extensions, bigger brakes, red brake calipers, twin exhausts, tartan seats, sport steering, red stitching, and a roof spoiler. It is equipped with a standard ‘sport’ chassis and optional ‘Sport Select’ chassis.

Performance versions and motorsport
Volkswagen helped consolidate the pre-eminence of the so-called hot hatch genre of high-performance hatchbacks with their Golf GTI in 1975 and has produced a number of performance versions of the Polo. The first of these was the Polo GT version of the Polo Mk1F.

The Polo Mk2 and Mk2F were available as supercharged G40 models. The GT G40 with its 1.3-litre 85 kW engine could reach 100 km/h in 8.1 seconds from standstill and had a maximum speed of 196 km/h. It was used by Volkswagen to set a number of world endurance speed records, such as the 1.3-litre class records for speed over 24 hours and speed over a distance of 5000 km.

The fastest version of the Polo Mk3 on the United Kingdom market was the 1.6-litre, 16-V, 92 kW model. A 88 kW Polo GTI model was also produced, but only in a limited edition in Germany, and this was the first time the GTI label had been used for a Polo. A GTI version of the Polo Mk3F, with a 92 kW 1.6-litre petrol engine was also produced.

In 2004, Volkswagen Individual, a specialist division of Volkswagen, produced a limited number of (Polo Mk4) Club Sports with a 1.8-litre, turbocharged engine producing 132 kW. Available only in Germany, this was based on the one-make racing series Polo Cup Racer hatchback. The Club Sport came with a roll cage inside the vehicle and Recaro racing seats as standard.

A GTI version of the Polo Mk4F was launched in 2006. These feature styling similar to that of the contemporary Golf GTI and a turbocharged 20-V, 110 kW, 1.8-litre petrol engine. It has a 0–100 km/h time of 8.2 seconds and a top speed of 134 mi/h.

Volkswagen Individual have also engineered an even faster Polo called the Polo GTI Cup Edition. Available with the same 1.8-litre, turbocharged engine, albeit with 180 hp, its claimed 0–100 km/h is 7.5 seconds, and it has a claimed top speed of 225 km/h.

Volkswagen Racing in South Africa rallied a four-wheel drive Polo Mk4F that shared some components with its sister World Rally Championship Škoda Fabia; the S2000 has a 2.0-litre, 191 kW engine. Mk4 Polos have been entered into the Junior World Rally Championship. The Polo also competed in the Russian Touring Car Championship.

A number of one-make race series were made of the Polo, starting with the G40 Cup for Polo Mk2 and Mk2F G40 versions. The current Polo Cup championship for 105 hp cars is a support race at rounds of the Deutsche Tourenwagen Masters.

Sébastien Ogier won the FIA World Rally Championship for Drivers in 2013, 2014, 2015 and 2016 driving a Volkswagen Polo R WRC.

Sales
The sales of Volkswagen Polo set a benchmark for Volkswagen, which sold more than 12 million cars globally so far; in February 2010 Volkswagen produced the 11,111,111th Polo worldwide at the celebration event of its first production anniversary in Pune, India. In 2010, its first full year on sale in the United Kingdom, more than 45,000 units were sold. It was the UK's sixth-best selling new car.

11 years later, in 2021, the Polo has maintained consistent sales figures with it being the fifth best-selling car in the UK in 2021 having had 30,634 new registrations throughout the year.

In South Africa, the previous model Polo is produced and branded as the Polo Vivo, and is consistently the top-selling passenger car.

title: Volkswagen Polo


text:
Turtle (disambiguation)

A turtle is a reptile, most of whose body is shielded by a special bony or cartilaginous shell.

Turtle(s) may also refer to:

Bird

 * European turtle dove (archaic name: turtle)

Computing

 * Turtle (robot), a class of educational robots used most prominently in the 1970s and 1980s
 * Turtle (syntax), a Terse RDF Triple language
 * Turtle F2F, a tool for exchanging content in an anonymous and secure way over a friend-to-friend (F2F) network
 * Turtle graphics, using a relative cursor, the "turtle"
 * Turtles, small values near the end of the list in sorting; See Comb sort

Fictional characters

 * Turtle (comics), in DC Comics, two antagonists of The Flash
 * Turtle (Entourage), a character in the HBO television series
 * The Great and Powerful Turtle, or "The Turtle", a character in the Wild Cards novels
 * Turtle, a character in Barbara Kingsolver's novels, The Bean Trees and Pigs in Heaven

Gaming

 * Turtle (game term), a defensive gaming strategy
 * Turtles (video game), a 1981 arcade game by Konami
 * Turtle Entertainment, a German esports organiser

Music

 * The Turtles, an American folk-rock band, known for their 1967 song "Happy Together"
 * Turtles (South Korean band), a K-Hip-Hop band
 * "Turtle", a 2017 song by Twice from the album Twicetagram

Other media

 * Turtle (magazine), a 1979–2014 American bi-monthly for preschoolers
 * Turtle: The Incredible Journey, a 2009 German–British–Austrian documentary
 * Teenage Mutant Ninja Turtles, a superhero franchise

Places

 * Turtle Mountain (Alberta), Canada

United States

 * Turtle, Missouri, a ghost town
 * Turtle, Wisconsin, a town
 * Turtle Mountains (California)
 * Turtle Mound, a prehistoric archaeological site in Florida
 * Turtle Pond, Central Park, Manhattan, New York
 * Rio Turtle or Turtle Rock, West Virginia

Retail brands

 * Turtle's Records & Tapes, a defunct American retail chain
 * Turtles (chocolate), a type of candy trademarked by DeMet's Candy Company

Watercraft

 * Turtle (submersible), the first military diving bell, built during the American Revolutionary War
 * DSV Turtle, a retired ocean research submersible of the US Navy

Other uses

 * Turtle beans
 * Botts' dots, often called turtles in Washington and Oregon
 * Turtle, a member of the Ancient Order of Turtles, or Turtle Club
 * Turtle, a member of NASA Astronaut Group 22
 * Turtle, a type of spinning float, a breakdancing move

People named Turtle

 * John Turtle (1937–2024), Australian academic and endocrinologist
 * Roger Turtle (fl. 1326–1344), English politician
 * Tommy Turtle (1950–2020), a British Army officer from Ireland
 * Turtle Bunbury (born 1972), Irish historian and author

title: Turtle (disambiguation)


text: User:Zulugrid

title: User:Zulugrid


text:
Democracy Now!

Democracy Now! is an hour-long TV, radio, and Internet news program based in Manhattan and hosted by journalists Amy Goodman (who also acts as the show's executive producer), Juan González, and Nermeen Shaikh. The show, which airs live each weekday at 8 a.m. Eastern Time, is broadcast on the Internet and via more than 1,400 radio and television stations worldwide.

The program combines news reporting, interviews, investigative journalism and political commentary from a progressive perspective. It documents social movements, struggles for justice, activism challenging corporate power and operates as a watchdog outfit regarding the effects of American foreign policy. Democracy Now! views as its aim to give activists and the citizenry a platform to debate people from "The Establishment". The show is described as progressive by fans as well as critics, but Goodman rejects that label, calling the program a global newscast that has "people speaking for themselves". Democracy Now! describes its staff as "includ[ing] some of this country's leading progressive journalists."

Democracy Now Productions, the independent media nonprofit organization that produces Democracy Now!, is funded entirely through contributions from listeners, viewers, and foundations such as the Park Foundation, Ford Foundation, Lannan Foundation, and the J.M. Kaplan Fund. It has over $36 million in assets and about a $10 million annual budget. Democracy Now! does not accept advertisers, corporate underwriting or government funding. The show has become popular on the internet, and from the late 2010s onward, has been involved in pioneering extensive media cooperation in the public sphere across the US.

Background

Democracy Now!, also called ''Democracy Now! The War and Peace Report, Democracy Now Independent Global News, or Democracy News'', was founded on February 19, 1996, at WBAI in New York City by journalists Amy Goodman, Juan González, Larry Bensky, Salim Muwakkil, and Julie Drizin. It originally aired on five Pacifica Radio stations. Goodman is the program's principal host, with Juan González and Nermeen Shaikh as frequent co-hosts. Jeremy Scahill, an investigative reporter and co-founding editor for The Intercept, has been a frequent contributor since 1997.

The show covered the Seattle protests (1999) targeting the World Trade Organization.

Democracy Now! partnered with Free Speech TV (FSTV) and Deep Dish Television to cover the 2000 Democratic National Convention. The event marked a turning point for Democracy Now!, as in addition to its presence on radio, it became a television show. From then onward, Democracy Now! has had their content promoted and broadcast on FSTV.

Democracy Now! began broadcasting on television every weekday shortly after September 11, 2001, and is the only public media in the U.S. that airs simultaneously on satellite and cable television, radio, and the internet.

Democracy Now! has been critical of the Trans-Pacific Partnership (TPP) trade deal. After data disclosures by the whistleblower organisation WikiLeaks regarding the TPP in 2010, Democracy Now! has given a significant media platform and extensively covered them since, and like some other news networks cooperated with its leader Julian Assange. Coverage of WikiLeaks by Democracy Now! was sympathetic.

In 2011, reporter Sharif Abdel Kouddous covered the Egyptian revolution for Democracy Now!.

On February 19, 2016, Democracy Now! marked 20 years on the air with an hour-long retrospective look back at "two decades of independent, unembedded news", with highlights chosen from over 5,000 episodes. Amy Goodman also published a book entitled Democracy Now!: 20 Years Covering the Movements Changing America, and launched a 100-city tour across the United States to mark the 20th anniversary of Democracy Now!, with scheduled broadcasts of the show recorded during her travels.

Studios
Democracy Now! began as a radio program broadcast from the studios of WBAI, a local Pacifica Radio station in New York City. In early September 2001, amid a months-long debate over the mission and management of Pacifica, Democracy Now! was forced out of the WBAI studios. Goodman took the program to the Downtown Community Television Center located in a converted firehouse building in New York City's Chinatown, where the program began to be televised. Only a few days later on September 11, 2001 Democracy Now! was the closest national broadcast to Ground Zero. On that day Goodman and colleagues continued reporting beyond their scheduled hour-long time slot in what became an eight-hour marathon broadcast. Following 9/11, in addition to radio and television, Democracy Now! expanded their multimedia reach to include cable, satellite radio, Internet, and podcasts.

In November 2009, Democracy Now! left their broadcast studio in the converted DCTV firehouse, where they had broadcast for eight years. The studio subsequently moved to a repurposed graphic arts building in the Chelsea District of Manhattan. In 2010, the new 8,500-square-foot Democracy Now! studio became the first radio or television studio in the nation to receive LEED Platinum certification, the highest rating awarded by the U.S. Green Building Council.

Syndication
Democracy Now! is the flagship program of the Pacifica Radio network. It also airs on several NPR member stations. The television simulcast airs on public-access television and several PBS stations; by satellite on Free Speech TV and Link TV, and free-to-air on C Band. Democracy Now! is also available on the Internet as downloadable and streaming audio and video. In total, nearly 1,400 television and radio stations broadcast Democracy Now! worldwide.

Awards and reception
Democracy Now! and its staff have received several journalism awards, including the Gracie Award from American Women in Radio & Television; the George Polk Award for its 1998 radio documentary Drilling and Killing: Chevron and Nigeria's Oil Dictatorship, on the Chevron Corporation and the deaths of two Nigerian villagers protesting an oil spill; and Goodman with Allan Nairn won Robert F. Kennedy Memorial's First Prize in International Radio for their 1993 report, Massacre: The Story of East Timor, which involved first-hand coverage of genocide during the Indonesian occupation of East Timor.

On October 1, 2008, Goodman was named as a recipient of the 2008 Right Livelihood Award, in connection with her years of work establishing Democracy Now! and in 2009, she, like her frequent guest Glenn Greenwald, was awarded the first annual Izzy Award (named after journalist I. F. "Izzy" Stone) for "special achievement in independent media". Her co-host Juan González was inducted into the New York chapter of the Society of Professional Journalists' Hall of Fame on November 19, 2015.

2008 Republican National Convention arrests
Three journalists with Democracy Now!—including principal host Amy Goodman, and news producers Nicole Salazar and Sharif Abdel Kouddous—were detained by police during their reporting on the 2008 Republican National Convention protests in Saint Paul, Minnesota. Salazar was filming as officers in full riot gear charged her area. As she yelled "Press!" she was knocked down and told to put her face in the ground while another officer dragged her backward by her leg across the pavement. The video footage of the incident was immediately posted on the Internet, leading to a large public outcry against her arrest. When a second producer, Kouddous, approached, he too was arrested, and charged with a felony. According to a press release by Democracy Now!, Goodman herself was arrested after confronting officers regarding the arrest of her colleagues. The officers had established a line of "crowd control", and ordered Goodman to move back. Goodman claims she was arrested after being pulled through the police line by an officer, and subsequently (as well as Kouddous) had her press credentials for the convention physically stripped from her by a Secret Service agent. All were held on charges of "probable cause for riot". A statement was later released by the city announcing that all "misdemeanor charges for presence at an unlawful assembly for journalists" would be dropped. The felony charges against Salazar and Kouddous were also dropped.

Goodman, Salazar, and Kouddous subsequently filed a lawsuit against the cities of St. Paul and Minneapolis as well as other defendants. According to Baher Asmy of the Center for Constitutional Rights, "[a]ll three plaintiffs that are journalists with Democracy Now reached a final settlement with the city of Minneapolis and St. Paul, and the United States Secret Service, that will resolve the claims that they had against them from unlawful and quite violent arrests." The settlement includes $100,000 in compensation and a promise of police training.

2016 North Dakota access pipeline protests
In September 2016, an arrest warrant for criminal trespass was issued for Amy Goodman after covering for Democracy Now! the Dakota Access Pipeline protests during which guards unleashed dogs and pepper spray on protesters in Morton County, North Dakota. An arrest warrant was reportedly also issued for Green Party presidential candidate Jill Stein and her running mate, Ajamu Baraka.

Goodman elected to turn herself in. Three days before the court date, the charges were increased to engaging in a riot, which carried a penalty of up to 30 days in jail and a $1,500 fine. On October 17, 2016, the judge quickly dismissed the charges, but Morton County prosecutors insisted the case is still open and that they may pursue further charges in the future. *
 * Goodman asserted the importance of freedom of the press and said that Democracy Now! would continue covering the developing situation in North Dakota.
 * Goodman asserted the importance of freedom of the press and said that Democracy Now! would continue covering the developing situation in North Dakota.
 * Goodman asserted the importance of freedom of the press and said that Democracy Now! would continue covering the developing situation in North Dakota.

Notable guests, interviews, and on-air debates
Democracy Now! has featured appearances from Green Party candidate Jill Stein during the 2016 United States presidential election.

Listenership
According to a 2016–17 Quantcast survey, "democracynow.org reaches over 395K U.S. monthly people".

title: Democracy Now!


text:
Grassmannian

In mathematics, the Grassmannian $$\mathbf{Gr}_k(V)$$ (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all $$k$$-dimensional linear subspaces of an $$n$$-dimensional vector space $$V$$ over a field $$K$$. For example, the Grassmannian $$\mathbf{Gr}_1(V)$$ is the space of lines through the origin in $$V$$, so it is the same as the projective space $$\mathbf{P}(V)$$ of one dimension lower than $$V$$. When $$V$$ is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension $$k(n-k)$$. In general they have the structure of a nonsingular projective algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to $$\mathbf{Gr}_2(\mathbf{R}^4)$$, parameterizing them by what are now called Plücker coordinates. (See below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors, and include $$\mathbf{Gr}_k(V)$$, $$\mathbf{Gr}(k,V)$$,$$\mathbf{Gr}_k(n)$$, $$\mathbf{Gr}(k,n)$$ to denote the Grassmannian of $$k$$-dimensional subspaces of an $$n$$-dimensional vector space $$V$$.

Motivation
By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differential manifold, one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Suppose we have a manifold $$M$$ of dimension $$k$$ embedded in $$\mathbf{R}^n$$. At each point $$ x\in M$$, the tangent space to $$M$$ can be considered as a subspace of the tangent space of $$\mathbf{R}^n$$, which is also just $$\mathbf{R}^n$$. The map assigning to $$ x$$ its tangent space defines a map from $M$ to $$\mathbf{Gr}_k(\mathbf{R}^n)$$. (In order to do this, we have to translate the tangent space at each $$x \in M$$ so that it passes through the origin rather than $$x$$, and hence defines a $$k$$-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This can with some effort be extended to all vector bundles over a manifold $$M$$, so that every vector bundle generates a continuous map from $$M$$ to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

Low dimensions
For $k = 1$, the Grassmannian $Gr(1, n)$ is the space of lines through the origin in $n$-space, so it is the same as the projective space $$\mathbf{P}^{n-1}$$of $n − 1$ dimensions.

For $k = 2$, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces $Gr(2, 3)$, $Gr(1, 3)$, and $P^{2}$ (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is $Gr(2, 4)$.

The Grassmannian as a differentiable manifold
To endow $$\mathbf{Gr}_k(V)$$ with the structure of a differentiable manifold, choose a basis for $$V$$. This is equivalent to identifying $$V$$ with $$K^n$$, with the standard basis denoted $$(e_1, \dots, e_n)$$, viewed as column vectors. Then for any $$k$$-dimensional subspace $$w\subset V$$, viewed as an element of $$\mathbf{Gr}_k(V)$$, we may choose a basis consisting of $$k$$ linearly independent column vectors $$(W_1, \dots, W_k)$$. The homogeneous coordinates of the element $$ w \in \mathbf{Gr}_k(V)$$ consist of the elements of the $$n\times k$$ maximal rank rectangular matrix $$W$$ whose $$i$$-th column vector is $$W_i$$, $$i = 1, \dots, k $$. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices $$W$$ and $$\tilde{W}$$ represent the same element $$w \in \mathbf{Gr}_k(V)$$ if and only if
 * $$\tilde{W} = W g$$

for some element $$g \in GL(k, K)$$ of the general linear group of invertible $$k\times k $$ matrices with entries in $$K$$. This defines an equivalence relation between $$n\times k$$ matrices $$W$$ of rank $$k$$, for which the equivalence classes are denoted $$[W]$$.

We now define a coordinate atlas. For any $$n \times k$$ homogeneous coordinate matrix $$W$$, we can apply elementary column operations (which amounts to multiplying $$W$$ by a sequence of elements $$g \in GL(k, K)$$) to obtain its reduced column echelon form. If the first $$k$$ rows of $$W$$ are linearly independent, the result will have the form
 * $$\begin{bmatrix} 1 \\ & 1 \\ & & \ddots \\ & & & 1 \\ a_{1,1} & \cdots & \cdots & a_{1,k} \\ \vdots & & & \vdots \\ a_{n-k,1} & \cdots & \cdots & a_{n-k,k} \end{bmatrix}$$

and the $$(n-k)\times k$$ affine coordinate matrix $$A$$ with entries $$(a_{ij})$$ determines $$w$$. In general, the first $$k$$ rows need not be independent, but since $$W$$ has maximal rank $$k$$,  there exists an ordered set of integers $$1 \le i_1 < \cdots < i_k \le n$$ such that the $$k \times k$$ submatrix $$W_{i_1, \dots, i_k}$$ whose rows are  the $$(i_1, \ldots, i_k)$$-th rows of $$W$$ is nonsingular. We may apply column operations to reduce this submatrix to the identity matrix, and the remaining entries uniquely determine $$w$$. Hence we have the following definition:

For each ordered set of integers $$1 \le i_1 < \cdots < i_k \le n$$, let $$U_{i_1, \dots, i_k}$$ be the set of elements $$w\in \mathbf{Gr}_k(V)$$ for which, for any choice of homogeneous coordinate matrix $$W$$, the $$k\times k$$ submatrix $$W_{i_1, \dots, i_k}$$ whose $$j$$-th row is the $$i_j$$-th row of $$W$$ is nonsingular. The affine coordinate functions on $$U_{i_1, \dots, i_k}$$ are then defined as the entries of the $$(n-k)\times k $$ matrix $$A^{i_1, \dots, i_k}$$ whose rows are those of the matrix $$W W^{-1}_{i_1, \dots, i_k}$$ complementary to $$ (i_1, \dots, i_k)$$, written in the same order. The choice of homogeneous $$n \times k$$ coordinate matrix $$W$$ in $$[W]$$ representing the element $$w\in \mathbf{Gr}_k(V)$$ does not affect the values of the affine coordinate matrix $$A^{i_1, \dots, i_k}$$ representing $w$ on the coordinate neighbourhood $$U_{i_1, \dots, i_k}$$. Moreover, the coordinate matrices $$A^{i_1, \dots, i_k}$$ may take arbitrary values, and they define a diffeomorphism from $$U_{i_1, \dots, i_k}$$ to the space of $$K$$-valued $$(n-k)\times k $$ matrices. Denote by

 * $$\hat{A}^{i_1, \dots, i_k} := W (W_{i_1, \dots, i_k})^{-1}$$

the homogeneous coordinate matrix having the identity matrix as the $$k \times k$$ submatrix with rows $$(i_1, \dots, i_k)$$ and the affine coordinate matrix $$A^{i_1, \dots, i_k}$$ in the consecutive complementary rows. On the overlap $$ U_{i_1, \dots, i_k} \cap U_{j_1, \dots, j_k}$$ between any two such coordinate neighborhoods, the affine coordinate matrix values $$A^{i_1, \dots, i_k}$$ and $$A^{j_1, \dots, j_k}$$ are related by the transition relations

where both $$W_{i_1, \dots, i_k}$$ and $$W_{j_1, \dots, j_k}$$ are invertible. This may equivalently be written as

where $$ \hat{A}^{i_1, \dots, i_k}_{j_1, \dots, j_k} $$ is the invertible $$k \times k $$ matrix whose $$l$$th row is the $$j_l$$th row of $$\hat{A}^{i_1, \dots, i_k}$$. The transition functions are therefore rational in the matrix elements of $$A^{i_1, \dots, i_k} $$, and $$\{U_{i_1, \dots, i_k}, A^{i_1, \dots, i_k}\}$$ gives an atlas for $$\mathbf{Gr}_k(V)$$ as a differentiable manifold and also as an algebraic variety.

The Grassmannian as a set of orthogonal projections
An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators ( problem 5-C). For this, choose a positive definite real or Hermitian inner product $$\langle \cdot, \cdot \rangle$$ on $$V$$, depending on whether $$V$$ is real or complex. A $$k$$-dimensional subspace $$w$$ determines a unique orthogonal projection operator $$P_w:V\rightarrow V$$ whose image is $$w\subset V$$ by splitting $$V$$ into the orthogonal direct sum
 * $$V = w \oplus w^\perp $$

of $$w$$ and its orthogonal complement $$w^\perp$$ and defining
 * $$ P_w(v) =\begin{cases} v \quad \text{ if } v \in w \\

0 \quad \text{ if } v\in w^\perp. \end{cases} $$ Conversely, every projection operator $$P$$ of rank $$k$$ defines a subspace $$w_P := \mathrm{Im}(P)$$ as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold $$ \mathbf{Gr}(k, V)$$ with the set of rank $$k$$ orthogonal projection operators $$P$$:
 * $$ \mathbf{Gr}(k, V) \sim \left\{ P \in \mathrm{End}(V) \mid P = P^2 = P^\dagger,\, \mathrm{tr}(P) = k \right\}.$$

In particular, taking $$V = \mathbf{R}^n$$ or $$V = \mathbf{C}^n$$ this gives completely explicit equations for embedding the Grassmannians $$\mathbf{Gr}(k, \mathbf{R}^N) $$, $$\mathbf{Gr}(k, \mathbf{C}^N) $$ in the space of real or complex $$n\times n $$ matrices $$\mathbf{R}^{n \times n}$$,  $$\mathbf{C}^{n \times n}$$, respectively.

Since this defines the Grassmannian as a closed subset of the sphere $$\{X \in \mathrm{End}(V) \mid \mathrm{tr}(XX^\dagger) = k\}$$ this is one way to see that the Grassmannian is a compact Hausdorff space. This construction also turns the Grassmannian $$\mathbf{Gr}(k, V)$$ into a metric space with metric
 * $$d(w, w') := \lVert P_w - P_{w'} \rVert,$$

for any pair $$ w, w' \subset V$$ of $$k$$-dimensional subspaces, where $\|⋅\|$ denotes the operator norm. The exact inner product used does not matter, because a different inner product will give an equivalent norm on $$V$$, and hence an equivalent metric.

For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.

Grassmannians Gr(k,Rn) and Gr(k,Cn) as affine algebraic varieties
Let $$M(n, \mathbf{R})$$ denote the space of real $$n \times n $$ matrices and the subset $$ P(k, n, \mathbf{R})\subset M(n, \mathbf{R})$$ of matrices $$ P \in M(n, \mathbf{R}) $$ that satisfy the three conditions:

 * $$P$$ is a projection operator: $$P^2=P$$.
 * $$P$$ is symmetric: $$P^T=P$$.
 * $$P$$ has trace $$\operatorname{tr}(P)=k$$.

There is a bijective correspondence between $$ P(k, n, \mathbf{R})$$ and the Grassmannian $$\mathbf{Gr}(k, \mathbf{R}^n)$$ of $$k$$-dimensional subspaces of $$\mathbf{R}^n$$ given by sending $$P\in P(k, n, \mathbf{R}) $$ to the $$k$$-dimensional subspace of $$\mathbf{R}^n$$ spanned by its columns and, conversely, sending any element $$w\in\mathbf{Gr}(k, \mathbf{R}^n)$$ to the projection matrix
 * $$P_w:= \sum_{i=1}^k w_i w_i^T,$$

where $$(w_1, \cdots, w_k)$$ is any orthonormal basis for $$w\subset\mathbf{R}^n$$, viewed as real $$n$$ component column vectors.

An analogous construction applies to the complex Grassmannian $$\mathbf{Gr}(k, \mathbf{C}^n)$$, identifying it bijectively with the subset $$P(k, n, \mathbf{C})\subset M(n,\mathbf{C})$$ of complex $$n \times n $$ matrices $$ P\in M(n,\mathbf{C}) $$ satisfying where the self-adjointness is with respect to the Hermitian inner product $$ \langle \, \cdot, \cdot \, \rangle $$ in which the standard basis vectors $$(e_1, \cdots, e_n) $$ are orthonomal. The formula for the orthogonal projection matrix $$P_w$$ onto the complex $$k$$-dimensional subspace $$w\subset \mathbf{C}^n$$ spanned by the orthonormal (unitary) basis vectors $$(w_1, \cdots, w_k)$$ is
 * $$P$$ is a projection operator: $$P^2=P$$.
 * $$P$$ is self-adjoint (Hermitian): $$P^\dagger=P$$.
 * $$P$$ has trace $$\rm{tr}(P)=k$$,
 * $$P_w:= \sum_{i=1}^k w_i w_i^\dagger.$$

The Grassmannian as a homogeneous space
The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group $$\mathrm{GL}(V)$$ acts transitively on the $$k$$-dimensional subspaces of $$V$$. Therefore, if we choose a subspace $$w_0 \subset V$$ of dimension $$k$$, any element $$w\in\mathbf{Gr}(k, V)$$ can be expressed as
 * $$ w = g (w_0) $$

for some group element $$g \in \mathrm{GL}(V) $$, where $$g$$ is determined only up to right multiplication by elements $$\{h \in H\}$$ of the stabilizer of $$w_0$$:
 * $$ H:=\mathrm{stab}(w_0):=\{h\in \mathrm{GL}(V) \,|\, h(w_0)=w_0 \} \subset \mathrm{GL}(V) $$

under the $$\mathrm{GL}(V)$$-action.

We may therefore identify $$\mathbf{Gr}(k, V)$$ with the quotient space
 * $$\mathbf{Gr}(k, V) = \mathrm{GL}(V)/H$$

of left cosets of $$H$$.

If the underlying field is $$\mathbf{R}$$ or $$\mathbf{C}$$ and $$\mathrm{GL}(V)$$ is considered as a Lie group,  this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field $$K$$, the group $$\mathrm{GL}(V)$$ is an algebraic group, and this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, $$H$$ is a parabolic subgroup of $$\mathrm{GL}(V)$$.

Over $$\mathbf{R}$$ or $$\mathbf{C}$$ it also becomes possible to use smaller groups in this construction. To do this over $$\mathbf{R}$$, fix a Euclidean inner product $$q$$ on $$V$$. The real orthogonal group $$O(V, q)$$ acts transitively on the set of $$k$$-dimensional subspaces $$\mathbf{Gr}(k, V)$$ and the stabiliser of a $$k$$-space $$w_0\subset V$$ is
 * $$O(w_0, q|_{w_0})\times O(w^\perp_0, q|_{w^\perp_0})$$,

where $$w_0^\perp$$ is the orthogonal complement of $$w_0$$ in $$V$$. This gives an identification as the homogeneous space
 * $$\mathbf{Gr}(k, V) = O(V, q)/\left(O(w, q|_w)\times O(w^\perp, q|_{w^\perp})\right)$$.

If we take $$V = \mathbf{R}^n$$ and $$w_0 = \mathbf{R}^k \subset \mathbf{R}^n$$ (the first $$k$$ components) we get the isomorphism
 * $$\mathbf{Gr}(k,\mathbf{R}^n) = O(n)/\left(O(k) \times O(n - k)\right).$$

Over $C$, if we choose an Hermitian inner product $$h$$, the unitary group $$U(V, h)$$ acts transitively, and we find analogously
 * $$\mathbf{Gr}(k, V) = U(V, h)/\left(U(w_0, h|_{w_0}) \times U(w_0^\perp|, h_{w_0^\perp})\right),$$

or, for $$V = \mathbf{C}^n$$ and $$w_0 = \mathbf{C}^k \subset \mathbf{C}^n$$,
 * $$\mathbf{Gr}(k, \mathbf{C}^n) = U(n)/\left(U(k) \times U(n-k)\right).$$

In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension $k(n − k)$.

The Grassmannian as a scheme
In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.

Representable functor
Let $$\mathcal E$$ be a quasi-coherent sheaf on a scheme $$S$$. Fix a positive integer $$k$$. Then to each $$S$$-scheme $$T$$, the Grassmannian functor associates the set of quotient modules of
 * $$\mathcal{E}_T := \mathcal E \otimes_{O_S} O_T$$

locally free of rank $$k$$ on $$T$$. We denote this set by $$\mathbf{Gr}(k, \mathcal{E}_T)$$.

This functor is representable by a separated $$S$$-scheme $$\mathbf{Gr}(k, \mathcal{E})$$. The latter is projective if $$\mathcal {E}$$ is finitely generated. When $$S$$ is the spectrum of a field $$K$$, then the sheaf $$\mathcal{E}$$ is given by a vector space $$V$$ and we recover the usual Grassmannian variety of the dual space of $$V$$, namely: $$\mathbf{Gr}(k, V)$$. By construction, the Grassmannian scheme is compatible with base changes: for any $$S$$-scheme $$S'$$, we have a canonical isomorphism
 * $$\mathbf{Gr}(k, \mathcal{E} ) \times_S S' \simeq \mathbf{Gr}(k, \mathcal{E}_{S'})$$

In particular, for any point $$s$$ of $$S$$, the canonical morphism $$\{s\} = \text{Spec}K(s) \rightarrow S$$ induces an isomorphism from the fiber $$\mathbf{Gr}(k, \mathcal {E})_s$$ to the usual Grassmannian $$\mathbf{Gr}(k, \mathcal{E} \otimes_{O_S} K(s))$$ over the residue field $$K(s)$$.

Universal family
Since the Grassmannian scheme represents a functor, it comes with a universal object, $$\mathcal G$$, which is an object of $$\mathbf{Gr} \left (k, \mathcal{E}_{\mathbf {Gr}(k, \mathcal E)} \right),$$ and therefore a quotient module $$\mathcal G$$ of $$\mathcal E_{\mathbf {Gr}(k, \mathcal E)}$$, locally free of rank $$k$$ over $$\mathbf{Gr}(k, \mathcal{E})$$. The quotient homomorphism induces a closed immersion from the projective bundle:
 * $$\mathbf{P}(\mathcal G) \to \mathbf{P} \left (\mathcal E_{\mathbf{Gr}(k, \mathcal E)} \right) = \mathbf P({\mathcal E}) \times_S \mathbf{Gr}(k, \mathcal E). $$

For any morphism of $S$-schemes:

this closed immersion induces a closed immersion
 * $$ \mathbf{P}(\mathcal G_T) \to \mathbf{P} (\mathcal{E}) \times_S T.$$

Conversely, any such closed immersion comes from a surjective homomorphism of $$O_T$$-modules from $$\mathcal E_T$$ to a locally free module of rank $$k$$. Therefore, the elements of $$\mathbf{Gr}(k, \mathcal E)(T)$$ are exactly the projective subbundles of rank $$k$$ in

Under this identification, when $$T=S$$ is the spectrum of a field $$ K$$ and $$\mathcal E$$ is given by a vector space $$V$$, the set of rational points $$\mathbf{Gr}(k, \mathcal{E})(K)$$ correspond to the projective linear subspaces of dimension $$k-1$$ in $$\mathbf{P}(V)$$, and the image of $$\mathbf{P}(\mathcal G)(K)$$ in

is the set
 * $$\left\{ (x, v) \in \mathbf{P}(V)(K) \times \mathbf{Gr}(k, \mathcal E)(K) \mid x\in v \right\}.$$

The Plücker embedding
The Plücker embedding is a natural embedding of the Grassmannian $$\mathbf{Gr}(k, V)$$ into the projectivization of the $$k$$th Exterior power $$\Lambda^k V$$ of $$V$$.

Suppose that $$w\subset V$$ is a $$k$$-dimensional subspace of the $$n$$-dimensional vector space $$V$$. To define $$\iota(w)$$, choose a basis $$ (w_1, \cdots, w_k)$$ for $$w$$, and let $$\iota(w)$$ be the projectivization of the wedge product of these basis elements: $$\iota(w) = [w_1 \wedge \cdots \wedge w_k],$$ where $$ [ \, \cdot \, ]$$ denotes the projective equivalence class.

A different basis for $$w$$ will give a different wedge product, but the two will differ only by a non-zero scalar multiple (the determinant of the change of basis matrix). Since the right-hand side takes values in the projectivized space, $$\iota$$ is well-defined. To see that it is an embedding, notice that it is possible to recover $$w$$ from $$\iota(w)$$ as the span of the set of all vectors $$v\in V$$ such that
 * $$ v \wedge \iota (w) = 0$$.

Plücker coordinates and Plücker relations
The Plücker embedding of the Grassmannian satisfies a set of simple quadratic relations called the Plücker relations. These show that the Grassmannian $$\mathbf{Gr}_k(V)$$ embeds as a nonsingular projective algebraic subvariety of the projectivization $$\mathbf{P}(\Lambda^k V)$$ of the $$k$$th exterior power of $$V$$ and give another method for constructing the Grassmannian. To state the Plücker relations, fix a basis $$(e_1, \cdots, e_n)$$ for $$V$$, and let $$w\subset V$$ be a $$k$$-dimensional subspace of $$V$$ with basis $$(w_1, \cdots, w_k)$$. Let $$(w_{i1}, \cdots, w_{in})$$ be the components of $$w_i$$ with respect to the chosen basis of $$V$$, and $$(W^1, \dots, W^n)$$ the $$k$$-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:
 * $$ W^T = [W^1\, \cdots W^n]= \begin{bmatrix} w_{11} &\cdots & w_{1n}\\ \vdots & \ddots & \vdots\\ w_{k1} & \cdots & w_{kn}

\end{bmatrix} ,$$

For any ordered sequence $$1\le i_1 < \cdots < i_k \le n$$ of $$k$$ positive integers, let $$w_{i_1, \dots, i_k}$$ be the determinant of the $$k \times k$$ matrix with columns $$[W^{i_1}, \dots , W^{i_k}]$$. The elements $$\{w_{i_1, \dots, i_k} \, \vert \, 1 \leq i_1 < \cdots < i_k \leq n\}$$ are called the Plücker coordinates of the element $$w \in \mathbf{Gr}_k(V)$$ of the Grassmannian (with respect to the basis $$(e_1, \cdots, e_n)$$ of $$V$$). These are the linear coordinates of the image $$\iota(w)$$ of $$w$$ under the Plücker map, relative to the basis of the exterior power $$ \Lambda^k V$$ space generated by the basis $$(e_1, \cdots, e_n)$$ of $$V$$. Since a change of basis for $$w$$ gives rise to multiplication of the Plücker coordinates by a nonzero constant (the determinant of the change of basis matrix), these are only defined up to projective equivalence, and hence determine a point in $$ \mathbf{P}(\Lambda^k V)$$.

For any two ordered sequences $$1 \leq i_1 < i_2 \cdots < i_{k-1} \leq n$$ and $$1 \leq j_1 < j_2 \cdots < j_{k+1} \leq n$$ of $$k-1$$ and $$k+1$$ positive integers, respectively, the following homogeneous quadratic equations, known as the Plücker relations, or the Plücker-Grassmann relations, are valid and determine the image $$\iota(\mathbf{Gr}_k(V))$$ of $$\mathbf{Gr}_k(V)$$ under the Plücker map embedding:
 * $$\sum_{l=1}^{k+1} (-1)^\ell w_{i_1, \dots, i_{k-1}, j_l} w_{j_1, \dots , \widehat{j_l}, \dots j_{k+1}} = 0,$$

where $$j_1, \ldots, \widehat{j_l}, \ldots j_{k+1}$$ denotes the sequence $$j_1, \ldots, j_{k+1}$$ with the term $$j_l$$ omitted. These are consistent, determining a nonsingular projective algebraic variety, but they are not algebraically independent. They are equivalent to the statement that $$\iota(w)$$ is the projectivization of a completely decomposable element of $$\Lambda^k V$$.

When $$\dim(V) =4$$, and $$ k=2$$ (the simplest Grassmannian that is not a projective space), the above reduces to a single equation. Denoting the homogeneous coordinates of the image $$\iota(\mathbf{Gr}_2(V) \subset \mathbf{P}(\Lambda^2 V)$$ under the Plücker map as $$(w_{12}, w_{13}, w_{14}, w_{23}, w_{24}, w_{34})$$, this single Plücker relation is
 * $$w_{12}w_{34} - w_{13}w_{24} + w_{14}w_{23} = 0. $$

In general, many more equations are needed to define the image $$\iota(\mathbf{Gr}_k(V))$$ of the Grassmannian in $$ \mathbf{P}(\Lambda^k V)$$ under the Plücker embedding.

Duality
Every $$k$$-dimensional subspace $$W \subset V$$ determines an $$ (n-k)$$-dimensional quotient space $$V/W$$ of $$V$$. This gives the natural short exact sequence:
 * $$ 0 \rightarrow W \rightarrow V \rightarrow V/W \rightarrow 0.$$

Taking the dual to each of these three spaces and the dual linear transformations yields an inclusion of $$(V/W)^*$$ in $$V^*$$ with quotient $$W^*$$
 * $$ 0 \rightarrow (V/W)^* \rightarrow V^* \rightarrow W^* \rightarrow 0.$$

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between $$k$$-dimensional subspaces of $$V$$ and $$(n-k)$$-dimensional subspaces of $$V^*$$. In terms of the Grassmannian, this gives a canonical isomorphism
 * $$ \mathbf{Gr}_k(V) \leftrightarrow \mathbf{Gr}{(n-k}, V^*)$$

that associates to each subspace $$W \subset V$$ its annihilator $$W^0\subset V^*$$. Choosing an isomorphism of $$V$$ with $$V^*$$ therefore determines a (non-canonical) isomorphism between $$\mathbf{Gr}_k( V)$$ and $$\mathbf{Gr}_{n-k}(V)$$. An isomorphism of $$V$$ with $$V^*$$ is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of Grassmannians sends any $$k$$-dimensional subspace into its $$(n-k)$$}-dimensional orthogonal complement.

Schubert cells
The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for $$\mathbf{Gr}_k(V)$$ are defined in terms of a specified complete flag of subspaces $$V_1 \subset V_2 \subset \cdots \subset V_n=V$$ of dimension $$\mathrm{dim}(V_i) = i$$. For any integer partition
 * $$\lambda =(\lambda_1, \cdots, \lambda_k)$$

of weight
 * $$|\lambda|=\sum_{i=1}^k\lambda_i$$

consisting of weakly decreasing non-negative integers
 * $$\lambda_1 \geq \cdots \geq \lambda_k \geq 0, $$

whose Young diagram fits within the rectangular one $$(n-k)^k$$, the Schubert cell $$X_\lambda(k,n)\subset \mathbf{Gr}_k(V)$$ consists of those elements $$W \in \mathbf{Gr}_k(V)$$ whose intersections with the subspaces $$\{V_i\}$$ have the following dimensions
 * $$ X_\lambda(k,n) = \{W \in \mathbf{Gr}_k(V)\, | \, \dim(W \cap V_{n-k+j-\lambda_j}) = j\}. $$

These are affine spaces, and their closures (within the Zariski topology) are known as Schubert varieties.

As an example of the technique, consider the problem of determining the Euler characteristic $$\chi_{k,n} $$ of the Grassmannian $$\mathbf{Gr}_k(\mathbf{R}^n)$$ of $k$-dimensional subspaces of $R^{n}$. Fix a $$1$$-dimensional subspace $$\mathbf{R}\subset \mathbf{R}^n$$ and consider the partition of $$\mathbf{Gr}_k(\mathbf{R}^n)$$ into those $k$-dimensional subspaces of $R^{n}$ that contain $R$ and those that do not. The former is $$\mathbf{Gr}_{k-1}(\mathbf{R}^{n-1})$$ and the latter is a rank $$k$$ vector bundle over $$\mathbf{Gr}_k(\mathbf{R}^{n-1})$$. This gives recursive formulae:
 * $$ \chi_{k,n} = \chi_{k-1,n-1} + (-1)^k \chi_{k, n-1}, \qquad \chi_{0,n} = \chi_{n,n} = 1.$$

Solving these recursion relations gives the formula: $$\chi_{k,n}=0$$ if $$n$$ is even and $$k$$ is odd and
 * $$\chi_{k, n} = \begin{pmatrix}\left\lfloor \frac{n}{2} \right\rfloor \\ \left\lfloor \frac{k}{2} \right\rfloor

\end{pmatrix} $$ otherwise.

Cohomology ring of the complex Grassmannian
Every point in the complex Grassmann manifold $$\mathbf{Gr}_k(\mathbf{C}^n)$$ defines a $$k$$-plane in $$n$$-space. Fibering these planes over the Grassmannian one arrives at the vector bundle $$E$$ which generalizes the tautological bundle of a projective space. Similarly the $$(n-k)$$-dimensional orthogonal complements of these planes yield an orthogonal vector bundle $$F$$. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of $$E$$. In particular, all of the integral cohomology is at even degree as in the case of a projective space.

These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of $$E$$ and $$F$$. Then the relations merely state that the direct sum of the bundles $$E$$ and $$F$$ is trivial. Functoriality of the total Chern classes allows one to write this relation as
 * $$c(E) c(F) = 1.$$

The quantum cohomology ring was calculated by Edward Witten. The generators are identical to those of the classical cohomology ring, but the top relation is changed to

reflecting the existence in the corresponding quantum field theory of an instanton with $$2n$$ fermionic zero-modes which violates the degree of the cohomology corresponding to a state by $$2n$$ units.

Associated measure
When $$V$$ is an $$n$$-dimensional Euclidean space, we may define a uniform measure on $$\mathbf{Gr}_k(V)$$ in the following way. Let $$\theta_n$$ be the unit Haar measure on the orthogonal group $$ O(n)$$ and fix $$w\in \mathbf{Gr}_k(V)$$. Then for a set $$ A \subset\mathbf{Gr}_k(V)$$, define
 * $$ \gamma_{k, n}(A) = \theta_n\{g \in \operatorname{O}(n) : gw \in A\}.$$

This measure is invariant under the action of the group $$ O(n)$$; that is,
 * $$\gamma_{k,n}(gA)= \gamma_{k,n}(A)$$

for all $$g \in O(n)$$. Since $$\theta_n(O(n))=1$$, we have $$\gamma_{k,n}(\mathbf{Gr}_k(V))= 1 $$. Moreover, $$\gamma_{k,n}$$ is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

Oriented Grassmannian
This is the manifold consisting of all oriented $$k$$-dimensional subspaces of $$\mathbf{R}^n$$. It is a double cover of $$\mathbf{Gr}_k(\mathbf{R}^n)$$ and is denoted by $$\widetilde{\mathbf{Gr}}_k(\mathbf{R}^n)$$.

As a homogeneous space it can be expressed as:
 * $$\widetilde{\mathbf{Gr}}_k(\mathbf{R}^n)=\operatorname{SO}(n) / (\operatorname{SO}(k) \times \operatorname{SO}(n-k)).$$

Orthogonal isotropic Grassmannians
Given a real or complex nondegenerate symmetric bilinear form $$Q$$ on the $$n$$-dimensional space $$V$$ (i.e., a scalar product), the totally isotropic Grassmannian $$\mathbf{Gr}^0_k(V, Q)$$ is defined as the subvariety $$\mathbf{Gr}^0_k(V, Q) \subset \mathbf{Gr}_k(V) $$ consisting of all $$k$$-dimensional subspaces $$w\subset V$$ for which
 * $$ Q(u, v)=0, \, \forall \, u, v \in w. $$

Maximal isotropic Grassmannians with respect to a real or complex scalar product are closely related to Cartan's theory of spinors. Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit, under the spin representation, the so-called projective pure spinor variety which, similarly to the image of the Plücker map embedding, is cut out as the intersection of a number of quadrics, the Cartan quadrics.

Applications
A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.

Another important application is Schubert calculus, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc. in a projective space that intersect a given set of points, lines, etc., using the intersection theory of Schubert varieties. Subvarieties of Schubert cells can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the Gaudin model, using the Bethe ansatz method.

A further application is to the solution of hierarchies of classical completely integrable systems of partial differential equations, such as the Kadomtsev–Petviashvili equation and the associated KP hierarchy. These can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold. The KP equations, expressed in Hirota bilinear form in terms of the KP Tau function are equivalent to the Plücker relations. A similar construction holds for solutions of the BKP integrable hierarchy, in terms of abelian group flows on an infinite dimensional maximal isotropic Grassmann manifold.

Finite dimensional positive Grassmann manifolds can be used to express soliton solutions of KP equations which are nonsingular for real values of the KP flow parameters.

The scattering amplitudes of subatomic particles in maximally supersymmetric super Yang-Mills theory may be calculated in the planar limit via a positive Grassmannian construct called the amplituhedron.

Grassmann manifolds have also found applications in computer vision tasks of video-based face recognition and shape recognition, and are used in the data-visualization technique known as the grand tour.

title: Grassmannian


text:
Joseph Adhémar

Joseph Alphonse Adhémar (1797–1862) was a French mathematician. He was the first to suggest that ice ages were controlled by astronomical forces in his 1842 book Revolutions of the Sea.

The Earth's orbit is elliptical, with the Sun at one focus; lines drawn through the summer and winter solstice; and the spring and autumn equinox; intersect with the Sun at right angles. The Earth is closest to the Sun (perihelion) near the northern hemisphere winter solstice. The Earth moves faster through its orbit when closer to the Sun. Hence, the period from the northern hemisphere's autumn equinox to winter and spring is shorter by around seven days than the period from spring to summer to autumn; the reverse is true in the southern hemisphere. Hence, northern hemisphere winter is shorter.

Because of this, Adhemar reasoned that because the southern hemisphere had more hours of darkness in winter, it must be cooling, and attributed the Antarctic ice sheet to this. Adhemar knew of the 22,000 year cycle of precession of the equinoxes, and theorised that the ice ages occurred in this cycle.

One immediate objection to the theory was that the total insolation during a year does not vary at all during the precessional cycle, only its seasonal distribution. Another was that the timing was wrong; however this could not be tested by observations available at the time.

Adhemar's theory was further developed, first by James Croll and later by Milutin Milanković.

Adhemar predicted the Antarctic ice sheet and theorised about its thickness by comparing the depths of the Arctic and circum-Antarctic oceans. Finding the Antarctic oceans deeper (the measurements he used may not have been fully representative) and attributing this to the gravitational attraction of the Antarctic ice sheet, he postulated a truly enormous ice sheet approximately 90 km thick.

title: Joseph Adhémar


text: User talk:209.104.141.158

title: User talk:209.104.141.158