Career.

Costa was born in Toulon, France, the son of Brazilian parents. His father Joaquim Ribeiro da Costa, from Salvador, Bahia, was a naval engineer, and his mother Alina Ferreira da Costa, was from Manaus, Amazonas. He was educated at the Royal Grammar School, Newcastle upon Tyne, England, and at the "Collège National" in Montreux, Switzerland, until 1916, he graduated as an architect in 1924 from the National School of Fine Arts in Rio de Janeiro. After some early works in the eclectic manner, he adopted Modernism in 1929.  In 1930 Costa established a partnership with Russian-born Brazilian architect Gregori Warchavchik, and also became the Director of the National School of Fine Arts where he had studied. Even though he found students eager to be taught in the "new style," his ruthless administration won him the opposition of the faculty and student body, and Costa eventually had to resign after a year in office. He joined the newly created SPHAN ("Servico do Patrimônio Histórico e Artístico Nacional" – National Service of Historic and Artistic Heritage) in 1937 under Rodrigo Melo Franco de Andrade. He remained at the National Heritage Service until retirement, acceding to the top post of director, where he was followed by his granddaughter Maria Elisa Costa. During his tenure as regional and then national director, he became involved in numerous controversial decisions (see Controversies).  Costa became a figure associated with reconciling traditional Brazilian forms and construction techniques with international modernism, particularly the work of Le Corbusier. His works include the Brazilian pavilion at the New York World's Fair of 1939 (designed with Oscar Niemeyer), the Parque Guinle residential complex in Rio of 1948, and the Hotel do Park São Clemente in Nova Friburgo of 1948. In the 1950s, Costa was invited to advise on the UNESCO building in Paris and to lecture at the Conference of Venice.  Among his major works are also the Ministry of Education and Health, in Rio (1936–43), designed with Niemeyer, Roberto Burle Marx, among others, and consulted by Le Corbusier, and the Pilot Plan of Brasília, a competition winner designed in 1957 and mostly built in 1958–1960.  Costa taught geometry and drawing at the Liceu de Artes e Ofícios of Rio from 1938–1954. The Liceu was affiliated with the Associação Académica de Coimbra where Costa also taught until 1966, and received a Medal of Merit from the Portuguese government.

Controversies.

During his long tenure as regional, later national chief of the Brazilian Historic and Artistic Heritage Institute ("Instituto do Patrimônio Histórico e Artístico Nacional" – IPHAN), Costa pushed for systematic documentation of existing architectural and urbanistic heritage, but his critics allege that he let his personal preferences and political opinions interfere with the bases of his decisions. In 1975, he created a public controversy by refusing to sign the landmarking act of Palácio Monroe, the former seat of the Brazilian Senate, built in 1906. The building was slated for demolition because of the construction of the subway but in the face of public and media outcry, the construction company shifted the line to preserve the building. This effort, however, was in vain since landmark status was denied and a developer razed the building shortly thereafter.  Costa preferred the heritage of the Portuguese colonization over that of any other time or ethnic group (except for Brazilian Modernism). Because of that attitude, inculcated also on younger preservationists thanks to Costa's influence in the architecture schools, much of 19th- and early 20th-century architecture, including the architecture of German, Japanese and Italian immigrants, was lost to urban renewal in the 1960s and 1970s.  In 1936, when the competition to design the new Ministry of Education and Health was held, the winner was an eclectic design by architect Arquimedes Memória. Costa used his political connections within the government to scrap the competition result and instead form a new design team headed by himself, the Roberto Brothers and a young architect who had been Costa's intern, Niemeyer.

Design of Brasilia.

Summary.

Costa is best known for his urban plan for the city of Brasília, located in Brazil's hinterland. Costa won the job in a 1957 public competition in order to replace Rio de Janeiro as the capital of Brazil. His "Plano Piloto" (Pilot Plan) for Brasília, is in the shape of an irregular cross, suggesting an airplane or dragonfly. While the majority of the project's architecture was designed by Oscar Niemeyer, Costa's own Parque Guinle project was the model for Brasília's many residential tower-in-a-park superblocks. The new city was inaugurated on 21 April 1960 and represents one of the largest adoptions of Modernism in a singular project to the present day.  Although named as an UNESCO World Heritage Site in 1987, the city is notorious for its windswept emptiness and anti-pedestrian layout. Some streets are badly lit because the height and the spacing of light standards were not changed with the advent of mercury-vapor lamps, and World Heritage Site designation has prevented remediation. Overall, despite the city's positive features, the concept and execution have sparked controversy which is explored in "The Modernist City". Brasilia has been expressed as an attempt at a utopian city or in the nickname "ilha da fantasia" (fantasy island), indicating the sharp contrast between the city and the surrounding regions, marked by poverty and disorganization.

Background and Concept.

The middle of the 20th century saw urban struggles for Brazil. Brazilian cities, particularly Rio de Janeiro and São Paulo had seen an increase of problems regarding transportation, sufficient housing, public utilities, and distribution of essential goods such as food. By the 1950s, the frustration of the upper class residents of these locations had convinced the political elite that a long term solution was necessary, making the idea of constructing a new capital city a compelling idea.  There was no consensus on the project from the Brazilian people. Many thought that the new capital would be a "monumental urbanistic and social disaster", believing that a project of that magnitude and cost would cause massive inflation in the Brazilian economy. The critics believed that the current state of Brazil, defined by poverty, corruption, and disease, would not be addressed by the building of a new capital city; it would merely distract from the nation's real issues while giving the populist politicians, such as Juscelino Kubitschek, a way of enriching their personal legacies.  Despite the criticism, the idea of a new capital received widespread backing. To the supporters, Brasília would symbolize Brazil's coming of age. The Brazilian coastal cities were remnants of the colonial era and the center for agricultural trade. Despite Brazil's imposing geographic presence in South America, the country was largely a coastal country. There had been several publicists calling for a "marcha para oeste", or westward march, in order to utilize the bountiful excess of land to the west. Also, it was felt that due to the current capital's location at Rio, the government's views of the nation was distorted. To be placed in a central location for these reasons, Brasília could become a new source for new growth for the country, a springboard for modernization and industrial production.  The new capital would be the diverging point for a series of new infrastructure development, such as long-range highways. It would be an optimistic step towards a unified, modern, and prosperous Brazil; towards fulfilling Brazil's "continental destiny" while also relieving the pressure on the existing urban centers. As put by Brazilian writer and diplomat José Osvaldo de Meira Penna in 1958: "A centrally located capital...might make Brazil more conscious of her role in the Americas, her terrestrial frontiers with Paraguay, Bolivia, Peru, Colombia, and Venezuela. When the attention of the elites turns from their nearly exclusive interest in Europea, the'splendid isolation' will be broken which has until now separated, in spirit and in fact, Brazil from continental neighbors".  The idea to establish a new Brazilian capital was adopted, and the location for Brasília would be within the Brazilian State of Goiás. The search for the site had been calculated, based on factors such as the favorability of the climate, water supply, land quality for agriculture, a suitable power source within 100 kilometers, and access to ground and air transportation.

Costa's Design.

Once it had been determined that there would be a new capital and the location had been established, the project entered into the design process. The plan was to be selected among competing plans from various Brazilian architects and city planners by an international jury. The participating designers were given the necessary information to make suitable plans for the space. Each participant was to submit two pieces in their proposal, a basic layout of the city and a supporting report that exhibited logistical details.  The design of Lucio Costa was selected for Brasilia. Although the design had not been as detailed as some of the other plans submitted, the jurors found it favorable due its features that would complement future population growth. The plan, called Costa's "Plano Piloto", conceptually demanded four components: (1) the government buildings, (2) the residential zones or superblocks, (3) the vehicular circulation and transportation infrastructure, (4) and the city center. In all, this original design was intended to hold a population of 500,000 people. In order to fulfill this, Costa's design called for an urban plan in the form of a cross, often described as an airplane, bird, or dragonfly. Some scholars have found symbolism in this shape of plan, a reference to the cross of the early Portuguese conquerors, the bow and arrow of the native populations of Brazil, or the jet shape as a symbol for future innovation.  The plan essentially divided the new capital into two parts, the Monumental Axis and the Residential Axis. The Monumental Axis, which points east to west, would serve as the home for the new civic buildings. This wide mall was intended to create a feeling of grandeur and significance, and is the location of the ministries, national congress, supreme court and other administrative buildings. The two "wings" of the Residential Axis, which runs north to south, would contain the residential and commercial developments. Costa envisioned these as the most important aspect of the design, as they would be home to a majority of the city's daily operations, such as local commerce, schools, recreation and churches. The Residential Axis consisted of 96 superblocks, which were limited to six-storey buildings, and 12 additional superblocks limited to three-storey buildings.  Each of the individual super blocks was supposed to have a distinct style for each superblock and a uniform paint scheme that set it apart from the surrounding superblocks. Costa's goal for the superblocks was to create neighborhoods and communities that were small self-contained and self-sufficient. He envisioned that they would have apartment buildings of a consistent modern style that housed both upper and middle classes, making integration a key theme of the city's design. The Residential Axis's two wings were meant to have a character that was intimate and peaceful, rather than the Monumental Axis that was designed to be imposing.  One of the unique aspects about the plan was how it was designed to handle expansion and growth. Unlike most cities, which add to a grid or expand the urban plan with growth, Costa's design provided the complete and large-scale plan. The original plan included paving streets in areas of the city that would not be put to immediate use, making the landscape of the city defined and hard to change during future construction. It was already "born with an adult skeleton". However, this did mean that large portions of the city would not see high levels of activity in the early years. Another contributing factor in the thorough plan was an emphasis on establishing an organized road system that would best serve the expanding use of automobiles. Costa's design of auto-routes was meant to create a system where traffic would be free flowing. He hoped that by establishing this grid at the city's founding that future construction projects regarding traffic improvements could be avoided.

Construction and Establishment.

The President of Brazil, Juscelino Kubitschek (1956-1961), approved Costa's plan and construction of Brasília, fulfilling the promise of the Brazilian constitution and some of his own personal political campaign promises. Construction started in 1956, using the urban design of Costa and the services of chief architect Oscar Niemeyer. The building materials for the large project were primarily from local sources. The residential buildings in the superblocks were all constructed with local concrete. The initial phase of Brasília was completed on 21 April 1960, in only four years. By the time of the inauguration on this date, while the city was not fully complete, Costa's vision had taken its physical form; the main buildings on the Monumental Axis, such as the Congress and Ministries had been built, along with the main transit center, the majority of the highway system, and several of the superblocks. In order to complete the large project in a short window of time, funding had to be organized swiftly, sufficient labor had to be acquired, and equipment and material had to be bought and transported onto the site.  Labor for the project required the migration of workers into the area and construction camps were built to house workers with the permission of the authorities. These temporary settlements, also called "Cidade Livre", were eventually slated for removal. However, these efforts were ineffective, and the settlements grew into self-sustaining communities, and were eventually adopted as satellite cities (suburbs). Gradually over the decades to come, the satellite communities continued to form on the outskirts of Brasilia, serving a valuable function towards battling low-cost housing issues that arose in the city. While alterations have been made to the design since the city's construction, the original plan by Costa prominently survives.  Following the opening of Brasilia, the modern design attracted the interest of artists, theater and musical groups, and younger generations of Brazilian citizens. One reason for this particular crowd was a coordinated effort to make Brasilia the new cultural center of the country. There were venues during the city's first years to help encourage artistic activities, such as the International Congress of Art Critics in 1959, an annual National Salon for several years in the early 1960s, and the establishment of several musical and theatrical groups.  Since its initial completion, Brasilia has gone through several stages of population changes. After initial interest following the opening, population numbers decreased before recovering in the late 1960s. Brasilia is now estimated to be Brazil's 3rd most populous city.

Lúcio Marçal Ferreira Ribeiro Lima Costa (27 February 1902 – 13 June 1998) was a Brazilian architect and urban planner, best known for his plan for Brasília.

Lúcio Costa (27. února 1902 Toulon, Francie – 13. června 1998 Rio de Janeiro, Brazílie) byl brazilský architekt a urbanista. Vytvořil plán výstavby nového brazilského hlavního města Brazílie.


Usage.

Like the "gladius", the "pugio" was probably a stabbing weapon, the type of weapon said to have been preferred by the Romans. Of them, late Roman writer Vegetius, says: Vegetius may be overstating this point however, as there are a number of surviving Roman depictions of soldiers slashing with their weapons in addition to stabbing with them. This is shown best on the Adamklissi metopes.

History.

Etymology.

The word "pūgiō" possibly descends from the Proto-Indo-European root "*peuĝ"-, "stab, stick". The root is the same as in English "pugilist", "boxer". It is still possible to use punch and stab synonymously in many Indo-European languages; hence, Latin "pugnus" and Greek πυγμή "pygmḗ" mean "fist". The Smith article cited below proposes that the "pugio" was the weapon grasped by the fist; however, the Latin word for swordplay was "pugna", an exchange of thrusts without the intermediary of fists, although it could also be a fistfight.

Early 1st century AD.

Ultimately the "pugio" was descended from Spanish originals of a variety of types, but by the early 1st century AD the Roman dagger typically had a large blade, which could be 'leaf-shaped' (type 'A') or which might alternatively have narrowed from the shoulders to run parallel to about half the blade's length, before narrowing to a sharp point (type 'B'). In size they ranged from 18 cm to 28 cm (7 in to 11 in) long and 5 cm (2 in) or more in width. A midrib ran close to the length of each side, either standing out from the face (mainly on type 'A' blades) or sunken and defined by grooves on either side (mainly on type 'B' blades). The tang was wide and flat initially, and the grip was riveted through it, as well as through the shoulders of the blade. The pommel expansion was originally round but by the early 1st century AD this was being replaced by a pommel expansion typically of a bulbous, roughly trapezial shape, often topped by three decorative rivets.  The "pugio" was accommodated in its own sheath. By the second quarter of the 1st century AD three types of sheath were in use. All of these had four suspension rings and a bulbous terminal expansion which was pierced by a large rivet. Both depictional evidence and the evidence of wear patterns on surviving examples show that the two lower rings were unused. The first type (type 'A') was made with a curving metal (normally iron) plate at both front and back, surrounding a wooden 'lining'. The front plate was usually heavily decorated with inlaid brass, silver, niello and red, yellow or green enamel. These sheaths featured round free-running suspension rings, attached by bifurgated mountings which were riveted on. Modern reconstructions of these sheaths which feature applied brass plates attached by rivets are incorrect and nothing of this type has ever been found. This common mistake is derived from a misinterpretation of a line drawing in an archaeological report of an iron type 'A' sheath decorated with silver inlay and decorative rivets. The second type (type 'B') was a wooden sheath, probably covered with leather to the front of which a metal (almost always iron) plate had been attached. This plate was fairly flat and was heavily decorated with inlaid silver (or occasionally tin) and enamel. The suspension rings resembled small Roman military buckles and were hinged to the sides of the sheath. The third type ('frame type') was made of iron and consisted of a pair of curved channels which ran together at the lower end of the sheath, where they were normally worked into a flattened round terminal expansion and pierced with rivet, although there is an example from Titelberg whose channels have been inserted into a bulbous terminal formed with a ferule to accept the channels before a decorated rivet has been used to fix all three elements together. The channels were joined by two horizontal bands at the top and middle of the sheath and these bands also retained the suspension rings, which were round like those of type 'A' sheaths. These sheaths would have been built around a wooden core, which does not survive in the archaeological record.  According to the Latin dictionaries, such as Lewis & Short, the term first appears in the late republican author, Marcus Tullius Cicero, with reference to the dagger used by Marcus Brutus to stab Julius Caesar. Suetonius confirms that all the conspirators used "pugiones" on that occasion and some later killed themselves with it. In the pages of Cicero, the "pugio" appears to have been the favoured weapon for high-ranking assassination and suicide because it was so easily concealed in the folds of flowing Roman garments.

Later 1st century and after.

Like other items of legionary equipment, the dagger underwent some changes during the 1st century AD. At some time in the first half of the 1st century AD a rod tang was introduced, and the hilt was no longer riveted through the tang and was instead secured only at the shoulders of the blade. This in itself caused no great change to the "pugio's" appearance, although the archaeological evidence strongly suggests that the rod tang was less secure and that handles attached in this way could become detached, a possibility that may be proved by the existence of two surviving "pugiones" from different sites which both retain replacement handles, one of which is a recycled sword grip. Some of the blades associated with rod tangs were narrower (under 4.5 cm (1.75 in) wide), with little or no waisting, or reduced or virtually non-existent midribs (type 'C' blades).  Throughout the period the outline of the hilt remained essentially the same. Like the earlier period it was made with two layers of horn, wood or bone sandwiching the tang, each overlaid with a thin iron plate, which could either be solid, becoming thinner at both the guard and the pommel expansion, which was roughly trapezial in shape, or made of thin metal embossed to this shape. The hilt was often decorated with silver inlay. The hilt is 10 cm to 13 cm (4 in - 5 in) long overall and that the grip is quite narrow, the presence of an expansion in the centre of the grip provides for a very secure grip.  References to the "pugiones" are more common in the literature of the empire, especially in Tacitus and Suetonius. Tacitus reports that Gnaeus Domitius Corbulo had a soldier executed for not wearing a sword while digging a trench and another for wearing only a "pugio" in the same activity. This does not mean however, that the "pugio" was carried universally and a study of 1st century AD figural tombstones reveals that there were certainly soldiers who did not carry "pugiones". The evidence of a preserved 1st century AD writing tablet also reveals that some cavalrymen also carried "pugiones".  The "pugio" seems to have been carried much less during the 2nd century AD and when it did reappear in large numbers early in the 3rd century AD, the blades had become much larger and the pommel expansion on the handle had become crescent shaped. These later daggers seem always to have been carried in 'frame type' sheaths.  "Pugiones" became ornate sidearms of officers and dignitaries as well, a custom reminiscent of the knives after which the Saxons were named. These Germanic mercenaries served in the Roman army. The emperors came to wear daggers to symbolize the power of life and death. The emperor Vitellius attempted to resign his position and offered his dagger to the consul, but it was refused and Vitellius was forced to stay by popular acclaim and the Praetorian guard. Tacitus also relates that a centurion, Sempronius Densus, of the Praetorian guard, drew a dagger to save Lucius Calpurnius Piso Licinianus momentarily.

The pugio (plural: "pugiones") was a dagger used by Roman soldiers as a sidearm. It seems likely that the "pugio" was intended as an auxiliary weapon, but its exact purpose to the soldier remains unknown. Officials of the empire took to wearing ornate daggers in the performance of their offices, and some would wear concealed daggers as a defense against contingencies. The dagger was a common weapon of assassination and suicide; for example, the conspirators who stabbed Julius Caesar used "pugiones".

Pugio byla římská dýka a součást římské uniformy. Používala se od 2. stol. před Kr. - 3. stol. po Kr.


History.

First formation: 1942–1949.

The British Army first took to the sky during the 19th century with the use of observation balloons. In 1911 the Air Battalion of the Royal Engineers was the first heavier-than-air British military aviation unit. The following year, the battalion was expanded into the Military Wing of the Royal Flying Corps which saw action throughout most of the First World War until 1 April 1918, when it was merged with the Royal Naval Air Service to form the Royal Air Force.  Between the wars, the army used RAF co-operation squadrons.  At the beginning of the Second World War, Royal Artillery officers, with the assistance of RAF technicians, flew Auster observation aircraft under RAF-owned Air Observation Post (AOP) Squadrons. Twelve such squadrons were raised —three of which belonged to the RCAF— and each performed vital duties in a wide array of missions in many theatres.  Early in the war, Winston Churchill announced the establishment of a new branch of army aviation, the Army Air Corps, formed in 1942. The corps initially comprised the Glider Pilot Regiment and the Parachute Battalions (subsequently the Parachute Regiment), Air Landing Regiments, and the Air Observation Post Squadrons. In March 1944, the SAS Regiment was added to the corps.  One of their most successful exploits during the war was Operation Deadstick, including the attack on Pegasus Bridge, which occurred on 6 June 1944, prior to the landings on Normandy. Once the three gliders landed, some roughly which incurred casualties, the pilots joined the glider-borne troops (Ox & Bucks Light Infantry) to act as infantry. The bridge was taken within ten minutes of the battle commencing and the men there withstood numerous attempts by the Germans to re-capture the location. They were soon reinforced and relieved by soldiers from Lord Lovat's 1 Special Service Brigade.  The AAC was broken up in 1949, with the SAS returning to its independent status, while the Parachute Regiment and Glider Pilot Regiment came under the umbrella of the Glider Pilot and Parachute Corps.

Second formation: 1957–present.

In 1957 the Glider Pilot and Parachute Corps was split, with the Parachute Regiment becoming an independent formation, while the Glider Pilot Regiment was merged with the Air Observation Squadrons of the Royal Artillery into a new unit, the Army Air Corps.  From 1970, nearly every army brigade had at least one Aviation Squadron that usually numbered twelve aircraft. The main rotor aircraft during the 1970s were the Westland Scout and Bell Sioux general purpose helicopters. The Sioux was replaced from 1973 by the Westland Gazelle used for Airborne reconnaissance; initially unarmed, they were converted to carry 68mm SNEB rocket pods in 1982, during the Falklands War. The Scout was replaced from 1978 by the Westland Lynx, which was capable of carrying additional firepower in the form of door gunners.  Basic rotary flying training was carried out on the Sioux in the 1970s, on the Gazelle in the 1980s and 1990s, and is currently conducted on the Eurocopter H145 through the Defence Helicopter Flying School.  Fixed-wing types in AAC service have included the Auster AOP.6 and AOP.9 and DHC-2 Beaver AL.1 in observation and liaison roles. In 1989, the AAC commenced operating a number of Britten-Norman Islander aircraft for surveillance and light transport duties. The corps operated the DHC-1 Chipmunk T.10 in a training role until its replacement by the Slingsby T67 Firefly in the 1990s. The Firefly was replaced by the Grob Tutor in 2010.

Cold War.

During the Cold War the majority of Army Air Corps units were based in Germany and part of the British Army of the Rhine. At the beginning of 1989 the Army Air Corps structure was as follows:

War on Terror.

A further boost in the Army Air Corps' capability came in the form of the Westland Apache AH.1 attack helicopter. In 2006, British Apaches deployed to Afghanistan as part of the NATO International Security Assistance Force. In 2004, Britten-Norman Defender fixed wing aircraft were purchased for Afghanistan and Iraq.

Current structure and deployment.

Mascot.

The Army Air Corps adopted their first Corps Mascot: Zephyr, a bald eagle, in October 2011

Training.

The training of future Army Air Corps aircrew is delivered by the joint service UK Military Flying Training System. Elementary Flying Training is delivered at RAF Barkston Heath, alongside Royal Navy students.  Training Units, Middle Wallop

Personnel.

The strength of the Army Air Corps is about 2,000 Regular personnel, of which 500 are officers. However, the AAC draws an additional 2,600 personnel from the Royal Logistic Corps, the Royal Electrical and Mechanical Engineers and the Adjutant General Corps. Therefore, total related Army Air Corps personnel is around 4,600.

Aircraft.

As of 2019, the AAC solely operates rotary-wing aircraft in the operational environment. The AAC uses the same designation system for aircraft as the Royal Air Force and the Fleet Air Arm. Fixed-wing training aircraft include the Grob Tutor for Army Flying Grading and the Grob Prefect (for Elementary Flying Training by No. 674 Squadron Army Air Corps).  Today AAC aviators fly four types of helicopter, and within each type there are usually several marks/variants which carry out different roles. Pilots train with No. 1 Flying Training School at RAF Shawbury. The School is a tri-Service organisation consisting of civilian and military instructors that take the student from basic flying through to more advanced flying such as instrument flying, navigation, formation flying and captaincy. In service aircraft include: the Bell 212HP AH1, the Eurocopter AS365N3 Dauphin II, the Airbus Helicopters H135 Juno, the Westland Gazelle AH1, the Westland Wildcat AH.1 and the AgustaWestland Apache AH1.  Transfer of Fixed-Wing assets  In April 2019, the Army transferred the AAC’s Islander and Defender aircraft operated by 651 Squadron from 5 Regiment to No. 1 Group Royal Air Force. The aircraft will be operated by RAF personnel, replacing Army personnel through attrition.

Command & units.

The AAC regiments will be consolidated into the following structure: Aviation Reconnaissance Force  Wattisham Flying Station Headquarters (WFS HQ), formerly the Attack Helicopter Force (AHF) (Supporting 3rd (UK) Division, 16 Air Assault Brigade and Royal Marines) Wattisham | Apache AH.1  Army Reserves  Joint Special Forces Aviation Wing  1 and 9 Regt AAC will merge under one headquarters (1 Regt AAC) and re-locate to RNAS Yeovilton to form a large regiment equipped with the new AgustaWestland Wildcat helicopter. The Regular component of Army Air Corps capability will consist of two regular aviation regiments equipped with Apache, one large regular aviation regiment equipped with Wildcat, and one regular manned aerial surveillance regiment, although the Gazelle out-of-service date is currently 2018 and it has not yet been confirmed whether or how the capability will be replaced. All five squadrons from 1 and 9 AAC will remain. Four squadrons will be the front line Lynx Wildcat squadrons, one (652 Squadron) will become the Wildcat Operational Conversion Squadron.  There will be two frontline squadrons of Apache helicopters in both 3 and 4 Regiment AAC, with 3 Regiment also parenting the Apache OCU, 653 Squadron. One Regiment will be at high readiness at any one time. One of the squadrons will be attached to and/or the new aircraft carriers for expeditionary operations. Another will be attached to the lead armoured battlegroup. However, under Army 2020 Refine, 3 Regiment AAC will be tasked to support the armoured infantry, and strike brigades of 3 (UK) Division and 4 Regiment AAC will support the army's 16 Air Assault Brigade and 3 Commando Brigade of the Royal Marines.  5 Regiment may be disbanded following the future retirement of the Gazelle fleet operated by 665 Squadron and also as a consequence of the transfer in 2019 of the Defender and Islander operated by 651 Squadron to the RAF.

Independent units.

No. 651 Squadron personnel part of No. 1 Group RAF based at JHC FS Aldergrove.

Battle honours.

The Army Air Corps is classed, in UK military parlance, as a "Combat Arm". It, therefore, carries its own guidon and is awarded battle honours. The honours awarded to the AAC are:

The Army Air Corps (AAC) is a component of the British Army, first formed in 1942 during the Second World War by grouping the various airborne units of the British Army (which are no longer part of the AAC). Today, there are eight regiments (seven Regular Army and one Reserve) of the AAC as well as four Independent Flights and two Independent Squadrons deployed in support of British Army operations across the world. They are located in Britain, Brunei, Canada, and Germany. Some AAC squadrons provide the air assault elements of 16 Air Assault Brigade through Joint Helicopter Command.

Army Air Corps (Armádní vzdušný sbor) je letecká složka Britské armády. Skládá se z osmi pluků (sedmi v rámci pravidelné armády, a jeden v teritoriální), a dále dvou samostatných squadron a čtyř letek, poskytujících podporu Britské armádě v místech její dislokace. Jeho základny se nacházejí v Británii, Bruneji, Kanadě, a Německu.


Geography.

The actual Land’s End, or Peal Point, is a modest headland compared with nearby headlands such as Pedn-men-dhu overlooking Sennen Cove and Pordenack, to the south. The present hotel and tourist complex is at Carn Kez, 200 m south of the actual Land’s End. Land's End has a particular resonance because it is often used to suggest distance. Land's End to John o' Groats in Scotland is a distance of by road and this "Land's End to John o' Groats" distance is often used to define charitable events such as end-to-end walks and races in the UK. Land's End to the northernmost point of England is a distance of by road.  The westernmost promontory at Land's End is known as Dr Syntax's Head. The character Dr Syntax was invented by the writer William Combe in his 1809 comic verse "The Tour of Dr Syntax in Search of the Picturesque", which satirised the work of seekers of the "picturesque" such as William Gilpin. A nearby promontory is called Dr Johnson's Head after Samuel Johnson, who referred to a hypothetical Cornish declaration of independence in his 1775 essay "Taxation no Tyranny".  The area around Land's End has been designated part of an Important Plant Area, by the organisation Plantlife, for rare species of flora.  Land's End is a popular venue for rock climbers.  The Longships, a group of rocky islets are just over offshore, and together with the Seven Stones Reef and the Isles of Scilly which lie about southwest — are part of the mythical lost land of Lyonesse, referred to in Arthurian literature.

Geology.

The cliffs are made of granite, an igneous rock, which means they are resistant to weathering, and have steep cliff faces. There are two varieties of granite represented at Land's End. Adjacent to the hotel the granite is coarse-grained with large phenocrysts of orthoclase, sometimes more than in length. To the north, at the First and Last House, there is a finer-grained granite with fewer and smaller phenocrysts, and the different granites can be seen from a distance by the smoother weathering of the finer variety. The granite dates to 268–275 million years ago of the Permian period. The contact zone between the Land's End granite pluton and the altered ′′country rocks′′ is nearby and the Longships Lighthouse, offshore, is built on the country rock.

History.

In 1769, the antiquarian William Borlase wrote: Of this time we are to understand what Edward I. says (Sheringham. p. 129.) that Britain, Wales, and Cornwall, were the portion of Belinus, elder son of Dunwallo, and that that part of the Island, afterwards called England, was divided in three shares, viz. Britain, which reached from the Tweed, Westward, as far as the river Ex; Wales inclosed by the rivers Severn, and Dee; and Cornwall from the river Ex to the Land's-End.  Tourists have been visiting Land’s End for over three hundred years. In 1649, an early visitor was the poet John Taylor, who was hoping to find subscribers for his new book "Wanderings to see the Wonders of the West". In 1878 people left Penzance by horse-drawn vehicles from outside the Queens and Union hotels and travelled via St Buryan and Treen, to see the Logan Rock. There was a short stop to look at Porthcurno and the Eastern Telegraph Company followed by refreshments at the First and Last Inn in Sennen. They then headed for Land’s End, often on foot or horse, because of the uneven and muddy lanes. Over one hundred people could be at Land's End at any one time. At Carn Kez, the First and Last Inn owned a small house which looked after the horses while visitors roamed the cliffs. The house at Carn Kez developed into the present hotel. The earliest part of the house was damaged by the Luftwaffe when a plane returning from a raid on Cardiff jettisoned its remaining bombs. 53 fisherman were injured or killed. In the build-up to D-Day American troops were billeted in the hotel leaving the building in a bad state. Land's End was owned by a Cornish family until 1982, when it was sold to David Goldstone. In 1987, Peter de Savary outbid the National Trust to purchase Land’s End for almost £7 million from David Goldstone. He had two new buildings erected and much of the present theme park development was instigated by him. He sold both Land's End and John o' Groats to businessman Graham Ferguson Lacey in 1991. The current owners purchased Land's End in 1996 and formed a company named Heritage Great Britain PLC. Attractions at the theme park include children's playgrounds and recorded music. Twice a week in August, Land's End hosts 'Magic in the Skies', a night-time firework spectacular with music by British composer Christopher Bond and narrated by actress Miriam Margolyes. Within the complex is the Land's End Hotel.  In May 2012, Land's End received worldwide publicity as the starting point of the 2012 Summer Olympics torch relay.

End to end.

Land's End is either the start or finishing point of end to end journeys with John o'Groats in Scotland. One of the earliest was by Carlisle who left Land's End on 23 September 1879, went to John O'Groats House and arrived back at Land's End on 15 December; taking 72 days (exclusive of Sundays); covering. To prove his journey, he kept a log book which was stamped at any Post Office he passed. An early end to end on bicycle was completed by Messrs Blackwell and Harman of Canonbury Bicycle Club. Starting at Land's End they covered in thirteen days in July/August 1880. Nearly two years later the Hon I Keith-Falconer travelled. from Land's End, in twelve days, 231⁄4 hours, on a bicycle.

Greeb.

On the south side of Carn Kez the land slopes away to a shallow valley containing a small stream and the former Greeb Farm. In 1879 a derrick was used for hauling seaweed from the beach 40 feet below for use as a soil improver.

Land's End ( or "Pedn an Wlas") is a headland and holiday complex in western Cornwall, England, situated within the Penwith peninsula about west-south-west of Penzance at the western end of the A30 road. To the east of it is the English Channel, and to the west the Celtic Sea.

Land's End ( "Penn an Wlas" či "Pedn an Wlas") je mys a rekreační komplex na západním výběžku poloostrova Cornwall. Je to nejzápadnější bod ostrova Velká Británie, a tedy i hrabství Cornwall a pevninské Anglie. Nachází se asi 13 kilometrů jihozápadně od města Penzance na západním konci silnice A30.


History.

Interwar period.

The National Library was founded on 29 August 1919, one year after independence, as the State Library ("Valsts Bibliotēka"). Its first chief librarian and bibliographer was Jānis Misiņš (1862-1944), who made his immense private collection the basis of the new library. Within a year, until 1920, the stocks had grown to 250,000 volumes. Starting in the same year, all publishers were obliged to hand in a deposit copy of their works. Since 1927, the Library has published the "National Bibliography of Latvia".  There were significant additions in 1939 and 1940, when the State Library took over many of the libraries and collections of the Baltic Germans. Most resettled to the Nazi German "Reich." Among these collections was a large part of the collection of the "Society for History and Archaeology of Russia's Baltic Provinces", est. 1834, the primary historical society of the Baltic Germans. In 1940, holdings encompassed 1.7 million volumes, so that they had to be stored in two different locations in the Old Town (Jēkaba iela 6/8 and Anglikāņu iela 5).

German and Soviet occupation.

During World War II, Germany invaded Latvia and occupied Riga (1941-1944). At that time, German authorities renamed the State Library as Country Library ("Zemes bibliotēka"), eliminating reference to a sovereign Latvian state).  Following the war, Latvia was under Soviet rule, and the institution was known as State Library of the Latvian SSR ("Latvijas PSR Valsts bibliotēka"). According to Soviet customs, in 1966 the library received an honorary name, commemorating Vilis Lācis, a writer and the late prime minister of Soviet Latvia. From 1946, literature deemed 'dangerous' from the Soviet perspective was withdrawn from the shelves and until 1988, when the Soviet Union gave up control, could be accessed only with a special permit. In 1956, the State Library moved into a new building at Krišjāņa Barona iela.

Renewed independence.

Since the reestablishment of Lativan national independence in 1991, the institution has been called National Library of Latvia. In 1995, it received as a permanent loan the "Baltic Central Library" of Otto Bong (1918-2006), a collection pertaining to the history, regional studies and languages of the Baltic countries. In 2006, the National Library joined the European Library online service.

Collections.

The Library's holdings today encompass more than 5 million titles, incl. about 18,000 manuscripts from the 14th century up to modern times. One of the characteristic cornerstones of the NLL, which characterizes every national library, is the formation of the collection of national literature, its eternal storage and long-term access.  The NLL is a centre of theoretical research and practical analyses of the activities of Latvian libraries. The Library carries out the functions of the centre of Latvia Interlibrary Loan, ensures the library and information service to the Parliament of the Republic of Latvia – the Saeima, implements the standardisation of the branch. Since the very outset, its main concern has been the national bibliography. The massive union catalogue "Seniespiedumi latviešu valodā" (Ancient Prints in Latvian 1525–1855, published in Riga, 1999) received the Spīdola Prize in 2000 and was awarded "The Beautiful Book of the Year 99". In 2005, the "Letonikas grāmatu autoru rādītājs (1523-1919)" (Index of the Authors of Lettonica Books (1523–1919)) was published, providing information about versatile branches of science and representatives of various nations, Latvia being the main focus of their publications.  The NLL includes several collections of posters (artists Oskars Šteinbergs (1882–1937), Sigismunds Vidbergs (1890–1970), Raoul Dufy (1877–1953), Bernhard Borchert (1863–1945), Niklāvs Strunke (1894–1966) and others).

Lettonica.

Digitising collections at the NLL started in 1999. At present the Latvian National Digital Library Letonica, which was formed in 2006, holds digitized collections of newspapers, pictures, maps, books, sheet-music and audio recordings. In 2008 NLL launched two major digital projects. Periodika.lv is the NLL's collection of digitized historical periodicals in Latvian with the possibility to read full texts and search page by page. Latvia has a tradition of Song and Dance Festivals organized every four years. The historical materials from the first Song Festival in 1864 till the Latgale Song Festival in 1940 can be explored in another digital collection of the National Library of Latvia.

New building.

The first discussions about the need for a new National Library began as early as 1928, and the significance of the project of this century was confirmed by high-level international recognition of the value of its collections. In 1999 almost all 170 UNESCO member states adopted a resolution during its General Conference, calling on member states and the international community to ensure all possible support for the implementation of the NLL project.  The continuous growth of the Library had made it necessary to transfer parts of the stocks into other buildings. By NLL had its holdings distributed among five locations in Riga. Furthermore, since 1998, some stocks had to be stored in a depot in Silakrogs outside the capital.  The Parliament finally authorized a new building to be constructed on the left bank of the Daugava River. On 15 May 2008, after discussions lasting for many years, the state agency "Three New Brothers" and the "Union of National Construction Companies" signed the contract on the construction of the new "National Library of Latvia". On 18 May 2014, the main facility of the Library at Krišjāņa Barona iela was closed for the move.  In 2008, construction started according to the design of noted Latvian-American architect Gunnar Birkerts, who had been based in the Detroit, Michigan area since the early 1950s. He had been commissioned to design the building in 1989. He was inspired by the Glass Mountain of Latvian mythology.  The new building has 13 floors and is 68 m high. Construction costs were given as 193 million euros. 480 people work there.  As part of Riga's programme for its title as European Capital of Culture, selected holdings were symbolically carried from the old to the new building by a human chain on 18 January 2014. The new building was finally opened on 29 August that year, the Library's 95th anniversary.  Today the NLL building is a dominant landmark on the Riga cityscape. It has space for conferences and conventions, and other community events. Among others, it hosted the 4th summit of the EU's Eastern Partnership programme in May 2015, and a debate chaired by the BBC's Jonathan Dimbleby on 14 March 2016.

The National Library of Latvia (), also known as Castle of Light ("Gaismas pils"), is a national cultural institution under the supervision of the Ministry of Culture of Latvia. The National Library of Latvia was formed in 1919 after the independent Republic of Latvia was proclaimed in 1918. The first supervisor of the Library was Jānis Misiņš, a librarian and the founder of the Latvian scientific bibliography (1862–1945).

Lotyšská národní knihovna (, zkratka LNB) je národní knihovna podle zákona a největší knihovna v Lotyšsku vůbec. Sídlí v hlavním městě Rize, v nové budově zvané "Zámek světla" od srpna 2014. Instituce byla založena 29. srpna 1919.


History.

Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold. This new derivative – the Levi-Civita connection – was "covariant" in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.  It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries.  In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified "ad hoc" by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.

Motivation.

The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, formula_1, which takes as its inputs: (1) a vector, u, defined at a point "P", and (2) a vector field, v, defined in a neighborhood of "P". The output is the vector formula_2, also at the point "P". The primary difference from the usual directional derivative is that formula_1 must, in a certain precise sense, be "independent" of the manner in which it is expressed in a coordinate system.  A vector may be "described" as a list of numbers in terms of a basis, but as a geometrical object a vector retains its own identity regardless of how one chooses to describe it in a basis. This persistence of identity is reflected in the fact that when a vector is written in one basis, and then the basis is changed, the components of the vector transform according to a change of basis formula. Such a transformation law is known as a covariant transformation. The covariant derivative is required to transform, under a change in coordinates, in the same way as a basis does: the covariant derivative must change by a covariant transformation (hence the name).  In the case of Euclidean space, one tends to define the derivative of a vector field in terms of the difference between two vectors at two nearby points. In such a system one translates one of the vectors to the origin of the other, keeping it parallel. With a Cartesian (fixed orthonormal) coordinate system "keeping it parallel" amounts to keeping the components constant. Euclidean space provides the simplest example: a covariant derivative which is obtained by taking the ordinary directional derivative of the components in the direction of the displacement vector between the two nearby points.  In the general case, however, one must take into account the change of the coordinate system. For example, if the same covariant derivative is written in polar coordinates in a two dimensional Euclidean plane, then it contains extra terms that describe how the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate grid expands, contracts, twists, interweaves, etc. In this case "keeping it parallel" does "not" amount to keeping components constant under translation.  Consider the example of moving along a curve "γ"("t") in the Euclidean plane. In polar coordinates, γ may be written in terms of its radial and angular coordinates by "γ"("t") = ("r"("t"), "θ"("t")). A vector at a particular time "t" (for instance, the acceleration of the curve) is expressed in terms of formula_4, where formula_5 and formula_6 are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change. In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transport, depends on the path along which the vector is translated.  A vector e on a globe on the equator at point Q is directed to the north. Suppose we parallel transport the vector first along the equator until at point P and then (keeping it parallel to itself) drag it along a meridian to the pole N and (keeping the direction there) subsequently transport it along another meridian back to Q. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the "curvature" of the surface of the globe. The same effect can be noticed if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. The infinitesimal change of the vector is a measure of the curvature.

Informal definition using an embedding into Euclidean space.

Suppose a (pseudo) Riemannian manifold formula_8, is embedded into Euclidean space formula_9 via a twice continuously-differentiable (C) mapping formula_10 such that the tangent space at formula_11 is spanned by the vectors  and the scalar product on formula_13 is compatible with the metric on "M":  For a tangent vector field, formula_15, one has  The last term is not tangential to "M", but can be expressed as a linear combination of the tangent space base vectors using the Christoffel symbols as linear factors plus a vector orthogonal to the tangent space:  In the case of the Levi-Civita connection, the covariant derivative formula_18, also written formula_19, is defined as the orthogonal projection of the usual derivative onto tangent space:  Since formula_21 is orthogonal to tangent space, one can solve the normal equations:  On the other hand,  implies  and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:  For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.

Formal definition.

A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the duals of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).

Functions.

Given a point "p" of the manifold, a real function "f" on the manifold, and a tangent vector v at "p", the covariant derivative of "f" at "p" along v is the scalar at "p", denoted formula_27, that represents the principal part of the change in the value of "f" when the argument of f is changed by the infinitesimal displacement vector v. (This is the differential of "f" evaluated against the vector v.) Formally, there is a differentiable curve formula_28 such that formula_29 and formula_30, and the covariant derivative of "f" at "p" is defined by  When v is a vector field, the covariant derivative formula_32 is the function that associates with each point "p" in the common domain of "f" and v the scalar formula_33. This coincides with the usual Lie derivative of "f" along the vector field v.

Vector fields.

A covariant derivative formula_34 at a point "p" in a smooth manifold assigns a tangent vector formula_35 to each pair formula_36, consisting of a tangent vector v at "p" and vector field u defined in a neighborhood of "p", such that the following properties hold (for any vectors v, x and y at "p", vector fields u and w defined in a neighborhood of "p", scalar values "g" and "h" at "p", and scalar function "f" defined in a neighborhood of "p"):  If u and v are both vector fields defined over a common domain, then formula_46 denotes the vector field whose value at each point "p" of the domain is the tangent vector formula_47. Note that formula_37 depends not only on the value of v at "p" but also on values of u in an infinitesimal neighbourhood of "p" because of the last property, the product rule.

Covector fields.

Given a field of covectors (or one-form) formula_49 defined in a neighborhood of "p", its covariant derivative formula_50 is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, formula_50 is defined as the unique one-form at "p" such that the following identity is satisfied for all vector fields u in a neighborhood of "p"  The covariant derivative of a covector field along a vector field v is again a covector field.

Tensor fields.

Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields formula_53 and formula_54 in a neighborhood of the point "p": and for formula_53 and formula_57 of the same valence The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.  Explicitly, let "T" be a tensor field of type. Consider "T" to be a differentiable multilinear map of smooth sections "α", "α",..., "α" of the cotangent bundle "T""M" and of sections "X", "X",... "X" of the tangent bundle "TM", written "T"(α, α,..., "X", "X",...) into R. The covariant derivative of "T" along "Y" is given by the formula

Coordinate description.

Given coordinate functions  any tangent vector can be described by its components in the basis  The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination formula_62. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field formula_63 along formula_64.  the coefficients formula_66 are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols.  Then using the rules in the definition, we find that for general vector fields formula_67 and formula_68 we get  so  The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field "u". In particular  In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.  For covectors similarly we have  where formula_73.  The covariant derivative of a type tensor field along formula_74 is given by the expression:  Or, in words: take the partial derivative of the tensor and add: formula_76 for every upper index formula_77, and formula_78 for every lower index formula_79.  If instead of a tensor, one is trying to differentiate a "tensor density" (of weight +1), then you also add a term If it is a tensor density of weight "W", then multiply that term by "W". For example, formula_81 is a scalar density (of weight +1), so we get:  where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.

Examples.

For a scalar field formula_83, covariant differentiation is simply partial differentiation:  For a contravariant vector field formula_85, we have:  For a covariant vector field formula_87, we have:  For a type (2,0) tensor field formula_89, we have:  For a type (0,2) tensor field formula_91, we have:  For a type (1,1) tensor field formula_93, we have:  The notation above is meant in the sense  Covariant derivatives do not commute; i.e. formula_96. It can be shown that:  where formula_98 is the Riemann tensor. Similarly,  and  The latter can be shown by taking (without loss of generality) that formula_101.

Notation.

In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.  Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this notation we write the same as: Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates formula_103, but also depends on the vector v itself through formula_104.  In some older texts (notably Adler, Bazin & Schiffer, "Introduction to General Relativity"), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe:

Derivative along curve.

Since the covariant derivative formula_106 of a tensor field formula_107 at a point formula_108 depends only on the value of the vector field formula_109 at formula_108 one can define the covariant derivative along a smooth curve formula_111 in a manifold: Note that the tensor field formula_107 only needs to be defined on the curve formula_111 for this definition to make sense.  In particular, formula_115 is a vector field along the curve formula_116 itself. If formula_117 vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by arc length.  The derivative along a curve is also used to define the parallel transport along the curve.  Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.

Relation to Lie derivative.

A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.  There is however another generalization of directional derivatives which "is" canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over "C"("M")) in the direction argument, while the Lie derivative is linear in neither argument.  Note that the antisymmetrized covariant derivative, and the Lie derivative "L""v" differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization "is" the Lie derivative.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.

Kovariantní derivace je geometrický způsob derivování vektorových a tenzorových polí, při němž se využívá jejich paralelní přenos pomocí tzv. konexe. Základní vlastností kovariantního derivování je, že převádí tenzorová pole opět v tenzorová pole, což při obvyklém způsobu derivování neplatí.


Overview.

Arising alongside video game development in the 1980s, the term "gameplay" was used solely within the context of video games, though now its popularity has begun to see use in the description of other, more traditional, game forms. Generally, gameplay is considered to be the overall experience of playing a video game excluding factors like graphics and sound. Game mechanics, on the other hand, is the sets of rules in a game that are intended to produce an enjoyable gaming experience. Academic discussions tend to favor terms like "game mechanics" specifically to avoid "gameplay" since the latter term is too vague.

Types.

There are three components to gameplay: "Manipulation rules", defining what the player can do in the game, "Goal Rules", defining the goal of the game, and "Metarules", defining how a game can be tuned or modified. In video games gameplay can be divided into several types. For example, "cooperative gameplay" involves two or more players playing on a team. Another example is "twitch gameplay" which is based around testing a player's reaction times and precision, maybe in rhythm games or first-person shooters. Various gameplay types are listed below.

Ambiguity in definition.

The term "gameplay" can be quite ambiguous to define, thus it has been differently defined by different authors.  For instance:

Playability.

Playability is the ease by which the game can be played or the quantity or duration that a game can be played and is a common measure of the quality of gameplay. Playability evaluative methods target games to improve design while player experience evaluative methods target players to improve gaming." This is not to be confused with the ability to control (or play) characters in multi-character games such as role playing games or fighting games, or factions in real-time strategy games.  Playability is defined as: a set of properties that describe the Player Experience using a specific game system whose main objective is to provide enjoyment and entertainment, by being credible and satisfying, when the player plays alone or in company. Playability is characterized by different attributes and properties to measure the video game player experience.

Playability's facets.

The playability analysis is a very complex process due to the different point of view to analyze the different part of video game architecture. Each facet allows us to identify the different playability's attributes and properties affected by the different elements of video game architecture. The playability's facets are:  Finally, the "global" playability of a video game will be deduced through each attribute value in the different playability's facets. It is crucial to improve the playability in the different facets to guarantee the best player experience when the player plays the video game.

Gameplay is the specific way in which players interact with a game, and in particular with video games. Gameplay is the pattern defined through the game rules, connection between player and the game, challenges and overcoming them, plot and player's connection with it. Video game gameplay is distinct from graphics and audio elements.

Hratelnost je pojem užívaný především v oblasti počítačových her, který vyjadřuje, jak je hra zábavná a jak vysoký stupeň komplexního pocitu štěstí, uspokojení a napětí může v člověku navodit během jejího hraní, což ovlivňuje i dobu, po kterou je hráč ochoten trávit svůj čas u hry. Tato vlastnost hry je závislá na tom, jak hra dokáže zaujmout, rozvíjet mysl či reflexy a přinést člověku příjemné pocity. Pojem hratelnost je tak značně subjektivní a závisí na preferencích každého člověka.


Life.

Early years  Kostenko was born in a family of teachers. In 1936, she moved from Rzhyshchiv to Kyiv, where she finished her secondary education. From 1937 to 1941 she studied at the Kyiv school #100, which was located on Trukhanov island, where her family lived. The school and the whole village was burned in 1943. The poem "I grew up in Kievan Venice" is devoted to these events.  After graduating from high school, she studied at the Kiev Pedagogical Institute, and later - at the Maxim Gorky Literature Institute in Moscow, from where she graduated with distinction in 1956.  Sixtiers  She was one of the first and most noteworthy in the band of young Ukrainian poets, acting at the turn of the 1950s-1960s. The period of the so-called "sixtiers" created the newest styles in the Ukrainian literature, made it create something new, atypical, then avant-garde, but, as always, ruthless and as critical as possible to the authorities and the existing regime.  In the early 1960s, she took part in the literary evenings of the Kiev Creative Youth Club. Following her graduation she published three collections of poetry in 1957 ("The Rays of the Earth"), 1958 ("Sails") and 1961("Wandering of the Heart"). These books became immensely popular among her Ukrainian readers, however they also forced her into publication silence as she was unwilling to submit to Soviet authorities.  Since 1961, she was criticized for "apoliticism." In 1963, the collection of poems by Lina Kostenko "The Star Integral" was removed from print and another collection of poems "The Prince's Mountain" was removed from typography. During these years, Kostenko's poems were published in Czechoslovakia's magazines, Poland's newspapers, and only occasionally in Ukraine's one, mostly in "samizdat."  In 1965 Lina Kostenko signed a letter of protest against arrests of the Ukrainian intelligentsia. She was present at the trial of Mykhailo Osadchyi and Myroslava Zvarychevska in Lviv. During the trial of the Horyn brothers, she threw them flowers. Together with Ivan Drach she appealed to the editorial office of the magazine "Zhovten" (now "Dzvin") and to the Lviv writers with a proposal to speak out in defense of the arrested. The writers did not dare to protest, but filed a lawsuit with the request to admit Bohdan Horyn on bail as the youngest of arrested. These efforts did not influence the trials, although they made a huge moral significance.  In May 1966 in the National Writers' Union of Ukraine, where the "nationalist outlaws" were labeled, a part of the youth held the ovation of L. Kostenko, who defended his position and defended Ivan Svitlichny, Opanas Zalyvaha, Myhajlo Kosiv and Bohdan Horyn.  In 1968 she wrote letters in defense of Viacheslav Chornovil in response to the defamation against him in the newspaper "Literary Ukraine." After that, the name of Lina Kostenko was not mentioned in the Soviet press for many years. She worked "in the drawer", knowing that her works were not going to be published.  In 1973 Lina Kostenko got on the "black lists," compiled by the secretary of the Central Committee on ideology of the Communist Party of Ukraine Valentyn Malanchuk. Only in 1977, after the departure of V. Malanchuk, was her collection of poems "On the shore of the eternal river" published, and in 1979, under a special decree of the Presidium of the Socialist-Revolutionary Guard, one of her greatest works was published, a historical novel in the verses "Marusia Churai" (about at 17th century Ukrainian folksinger) which had stagnated with recognition for 6 years. She was awarded the Taras Shevchenko National Prize of the Ukrainian SSR for in 1987.  Kostenko also wrote collections of poems "Originality" (1980) and "Garden of Unthawed Sculptures" (1987), collection of poems for children "The Lilac King" (1987). Her most recent collection is Berestechko, a book length historical poem.

Lina Vasylivna Kostenko (, born 19 March 1930 in Rzhyshchiv, Kiev Oblast, in the Ukrainian SSR of the Soviet Union) is a Ukrainian poet and writer, recipient of the "Shevchenko Award" (1987).

Lina Vasylivna Kostenková, ukrajinsky "Ліна Василівна Костенко" (* 19. březen 1930, Ržišiv u Kijeva) je ukrajinská spisovatelka. Jedna z klíčových představitelů generace ukrajinských spisovatelů nazývaných "Šedesátníci".


Life.

Early life.

Lewis Brumfield was born in Mansfield, Ohio, in 1896 to Charles Brumfield, a bank cashier and real estate speculator, and Annette Marie Coulter Brumfield, the daughter of an Ohio farmer. (Brumfield later changed the spelling of his name to "Louis Bromfield" because he thought it looked more distinguished.)  As a boy, Bromfield loved working on his grandfather’s farm. In 1914, he enrolled in Cornell University to study agriculture. Yet his family’s deteriorating financial situation forced him to drop out after only one semester. Deeply in debt, his parents sold their house in central Mansfield and moved to Bromfield’s grandfather’s farm on the outskirts of town. From 1915 to 1916, Bromfield struggled to revive the unproductive family farm, an experience he later wrote about bitterly in his autobiographical novel "The Farm". In 1916, he enrolled in Columbia University to study journalism, where he was initiated into the fraternal organization Phi Delta Theta. His time at Columbia would be brief; he left after less than a year to volunteer in World War I with the American Field Service.  Bromfield served in Section 577 of the US Army Ambulance Corps and was attached to the French infantry. He saw major action during the Ludendorff Offensive and the 100 Days Offensive and was briefly captured by the German army in the summer of 1918. Though he later claimed to have been awarded the Croix de Guerre, there is no evidence of this decoration in French or American military records.

New York.

Bromfield was discharged from the army in 1919. He found work in New York City as a journalist, critic and publicity manager, among other jobs. In 1921, he married the socialite Mary Appleton Wood during a small ceremony near her family home in Ipswich, Massachusetts. They would have three daughters, Ann Bromfield (1925-2001), Hope Bromfield (1927-2016) and Ellen Bromfield (1932-2019).  In 1924, Bromfield published his first novel, "The Green Bay Tree", which featured a headstrong, independent female protagonist — a feature that would recur in many of his later books. A second novel, "Possession", was published in 1925. Stuart Sherman, John Farrar and other leading critics of the day praised the quality of his early fiction.

Paris and Hollywood.

In November 1925, Bromfield moved to Paris, where he became associated with many of the central figures of the Lost Generation, especially Gertrude Stein and Ernest Hemingway. His third novel, "Early Autumn", a harsh portrait of his wife’s Puritan New England background, won the 1927 Pulitzer Prize. “He is, of all the young American novelists, pre-eminently the best and most vital,” John Carter wrote that year in the "New York Times".  Bromfield continued to write best-selling novels in the late 1920s and early 1930s, including "A Good Woman", "The Strange Case of Miss Annie Spraag" and "The Farm", an autobiographical novel that romanticized his family’s agrarian past. He also worked briefly in Hollywood as a contract screenwriter for Samuel Goldwyn, Jr.

Senlis and India.

In 1930, he moved into a renovated 16th-century rectory, the Presbytère St-Etienne, in Senlis, north of Paris. There he built an elaborate garden on the banks of the River Nonette, where he hosted parties that were well-known among artists, writers and socialites of the period. Regular guests included Gertrude Stein, Alice B. Toklas, Elsa Schiaparelli, Dolly Wilde, Leslie Howard, Noël Haskins Murphy, Douglas Fairbanks, Sir Francis Rose, F. Scott and Zelda Fitzgerald. Janet Flanner, who was a frequent witness to the weekly gatherings at Bromfield's Senlis estate, once said that Bromfield "collected people (and noted their value) the way some men do stamps."  Bromfield’s passion for horticulture increased over the course of the 1930s. He learned techniques of intensive gardening from his peasant neighbors in Senlis and formed a close bond with Edith Wharton, who designed the formal gardens at the Pavillon Colombe, her estate in nearby Saint-Brice-Sous-Fôret.  During this period, Bromfield also made two long trips to India. He visited Sir Albert Howard’s soil institute in the state of Indore (where Bromfield was exposed to early organic farming methods) and spent time in Baroda City (present-day Vadodara) as a guest of Sayajirao Gaekwad III, the Maharajah of Baroda. His travels informed one of his most critically acclaimed bestsellers, "The Rains Came" (1937), which was adapted into a popular 1939 film starring Myrna Loy and Tyrone Power. He later used proceeds from this book to finance Malabar Farm, saying that “nothing could be more appropriate than giving the farm an Indian name because India made it possible.”  At the end of the Spanish Civil War, Bromfield served as the chairman of the Paris-based Emergency Committee for American Wounded, which helped repatriate volunteers who had fought in the Abraham Lincoln Brigades. He later received the French Legion of Honor for this effort. An outspoken critic of Neville Chamberlain’s policy of appeasement (most notably in the 1939 book "England, Dying Oligarchy"), he left Europe shortly after the Munich Agreement with a hazy plan to move to Ohio and raise his children on an “honest-to-God farm.”

Malabar Farm and The Friends of the Land.

In December 1938, Bromfield purchased 600 acres of worn-out farmland near the town of Lucas in Pleasant Valley, Richland County, Ohio. He built a 19-room Greek Revival-style farmhouse that he dubbed the Big House. Using expertise and labor from New Deal agencies like the Soil Conservation Service and Civilian Conservation Corps, Bromfield rehabilitated his land and in the process learned the principles of soil conservation. He later turned Malabar into a showcase for what he called the “New Agriculture.” Among the novel farming techniques that he promoted at Malabar were the use of green manures, contour plowing, “trash farming,” sheet composting and strip cropping.  In 1941, Bromfield became first vice president of the Friends of the Land, a new national volunteer organization allied with the U.S. Soil Conservation Service, that sought to correct the ruinous farming practices that had culminated in the Dust Bowl and other incidents of widespread soil erosion in the 1930s. The organization brought together many prominent voices in 20th century ecology and agriculture, including Paul B. Sears, Hugh Hammond Bennett and Aldo Leopold. Bromfield used his celebrity to promote the work of agricultural reformers, including Edward Faulkner, whose 1943 book "Plowman’s Folly" criticized the moldboard plow and advocated “trash farming” (a forerunner to no-till agriculture) to avoid erosion and maintain soil fertility. Bromfield also helped popularize the organization’s journal, "The Land", which featured contributions from E.B. White, John Dos Passos, Henry A. Wallace, Aldo Leopold and Rachel Carson, among many others. Bromfield established Malabar’s national reputation in 1945 by hosting the wedding of his good friend Humphrey Bogart to Lauren Bacall. Bromfield served as best man. Malabar was often visited by celebrities, including Kay Francis, Joan Fontaine, Ina Claire, Mayo Methot and James Cagney. E.B. White captured the atmosphere of the farm in a 1948 poem in the "New Yorker": Strangers arriving by every train,  Bromfield terracing against the rain,  Catamounts crying, mowers mowing,  Guest rooms full to overflowing,  Boxers in every room of the house,  Cows being milked to Brahms and Strauss,  Kids arriving by van and pung,  Bromfield up to his eyes in dung,  Sailors, trumpeters, mystics, actors,  All of them wanting to drive the tractors,  All of them eager to husk the corn,  Some of them sipping their drinks till morn [...]

Decline and death.

Bromfield’s newfound interest in agriculture and environmentalism coincided with a collapse of his literary reputation. Critics like Malcolm Cowley, Orville Prescott and Edmund Wilson dismissed his later fiction as contrived and superficial. Yet Bromfield’s books continued to be popular with readers; his 1947 novel "Colorado" sold more than 1 million copies. He also began writing a series of memoirs about agriculture and the environment, beginning with the best-selling "Pleasant Valley" (1945).  As Bromfield’s literary career faltered, he began to run into major financial difficulties, compounded by the high cost of maintaining his experimental farm and his lavish lifestyle. Among many failed business schemes, he tried to raise capital by creating satellite versions of Malabar in Wichita Falls, Texas and Itatiba, Brazil. After the death of his wife Mary in 1952, he began a relationship with the billionaire heiress Doris Duke, who shared his interest in horticulture and conservation. Bromfield told a newspaper reporter early in 1956 that he and Duke “might get married.” But their romance was cut short because of his deteriorating health. He died of multiple myeloma on March 18 at the University Hospital in Columbus.

Influence and legacy.

After Bromfield’s death, Malabar Farm was eventually turned into a state park and tourist attraction. Malabar Farm State Park hosts thousands of annual visitors and maintains some aspects of Bromfield's management philosophy. One of the park's notable features is the Doris Duke Woods, named for Doris Duke, whose donation helped rescue Malabar from development after Bromfield’s death.  Many of Bromfield’s agricultural writings remain in print. Farmers and environmentalists such as Wendell Berry and Joel Salatin have cited Bromfield as an important influence. In 1989, Louis Bromfield was posthumously elected to the Ohio Agricultural Hall of Fame, and in December 1996, the centennial of his birth, the Ohio Department of Agriculture placed a bust of him in the lobby named for him at the department's new headquarters in Reynoldsburg, Ohio.  Bromfield’s youngest daughter Ellen Bromfield Geld continued her father’s work in Brazil, where she and her husband Carson Geld moved in 1952. They built a farm, Fazenda Pau d’Alho, and Ellen became a well-known newspaper columnist and author. She died in 2019.

Louis Bromfield (December 27, 1896 – March 18, 1956) was an American author and conservationist. A bestselling novelist in the 1920s, he reinvented himself as a farmer in the late 1930s and became one of the earliest proponents of sustainable and organic agriculture in the United States. He won the Pulitzer Prize, founded the experimental Malabar Farm in Ohio, and played an important role in the early environmental movement.

Louis Bromfield, rodným jménem "Lewis Brumfield" (* 27. prosince 1896, Ohio — 18. března 1956, Columbus), byl americký spisovatel a publicista. Proslavil se ale i jako ochránce přírody.


General information.

NGC 1097 is also a Seyfert galaxy. Deep photographs revealed four narrow optical jets that appear to emanate from the nucleus. These have been interpreted as manifestations of the (currently weak) active nucleus. Subsequent analysis of the brightest jet's radio-to-X-ray spectral energy distribution were able to rule out synchrotron and thermal free-free emission. The optical jets are in fact composed of stars. The failure to detect atomic hydrogen gas in the jets (under the assumption that they were an example of tidal tails) using deep 21 cm HI imaging with the Very Large Array radio telescope and numerical simulations led to the current interpretation that the jets are actually the shattered remains of a cannibalized dwarf galaxy.  NGC 1097 has a supermassive black hole at its center, which is 140 million times the mass of the Sun. Around the central black hole is a glowing ring of star-forming regions with a network of gas and dust that spirals from the ring to the black hole. An inflow of material toward the central bar of the galaxy causes new stars to be created in the ring. The ring is approximately 5,000 light-years in diameter, the spiral arms of the galaxy extend tens of thousands of light-years beyond the ring.  NGC 1097 has two satellite galaxies, NGC 1097A and NGC 1097B. Dwarf elliptical galaxy NGC 1097A is the larger of the two. It is a peculiar elliptical galaxy that orbits 42,000 light-years from the center of NGC 1097. Dwarf galaxy NGC 1097B (5 x 10 solar masses), the outermost one, was discovered by its HI emission, and appears to be a typical dwarf irregular. Little else is known about it.

NGC 1097 (also known as Caldwell 67) is a barred spiral galaxy about 45 million light years away in the constellation Fornax. It was discovered by William Herschel on 9 October 1790. It is a severely interacting galaxy with obvious tidal debris and distortions caused by interaction with the companion galaxy NGC 1097A. Three supernovae (SN 1992bd, SN 1999eu, and SN 2003B) have been observed in NGC 1097 since 1992.

NGC 1097 (také známá jako Caldwell 67) je spirální galaxie s příčkou v souhvězdí Pece vzdálená přibližně 64 milionů světelných let. Objevil ji britský astronom William Herschel v roce 1790. Je to galaxie výrazně interagující s její sdruženou galaxií NGC 1097A a je zjevně narušená vzájemným působením slapových sil.


History.

The dedication of this church to Saint Stephen, the first patron saint of the city, follows that of a preexisting church with that name, now covered by the Rozafa Castle, and on whose remains a mosque was built. Marin Barleti, a notable 16th-century Albanian writer, mentions the former church and the saint patron of the city in his "Historia de vita et gestis Scanderbegi Epirotarum principis" ("The story of life and deeds of Skanderbeg, the prince of Epirotes"), when he describes the Siege of Shkodra which lasted from 1478 to 25 April 1479 and saw the League of Lezhe helplessly protect the castle from the Ottoman Empire army, until the castle eventually fell to Turkish hands.  During the first three centuries of the Ottoman occupation, the bishops of the city were constrained to live outside the city. In 1762, 5 years after the formation Pashalik of Scutari, the first Albanian Pashalik under the Bushati family, Monsignor Pal Pjeter Kamsi was finally allowed again to reside in the city.  In 1851, after the faithful had continuously applied for the permission from the Sultan for the building of the church, Abdülmecid I issued the decree.  The dome, firstly built in wood, almost fell off 35 years after construction, so in 1897 it was replaced which other materials which last to today. In 1909, Albanian painter Kolë Idromeno coffered the vault. His paintings in the vault can still be seen today, especially that of "Lady of Shkodër" followed by two angels who are dressed in Shkodër folk costumes. In the background is the city of Shkodër, surrounded byRozafa castle.  An earthquake in 1905 and the bombing of 12 March 1913 during the Siege of Scutari damaged the bell tower and the clock, which had been given by the Kakarriqi brothers. In 1925, the Kakarriqi family purchased another clock which worked till 1967, when the church was transformed into a "Palace of Sports" following the Cultural Revolution. Ironically, even the Congress of the Communist Women of Albania was held in the cathedral in 1973. The towers were destroyed in 1967 and the portal was boarded up.  In 1990 the cathedral was reopened and on November 11, 1990, the first Mass in the country since 1967 was held in the cathedral. A symbolic Mass was held in the cathedral on March 21, 1991. In 1993 it was visited by Pope John Paul II.

Structure and construction.

Background.

Legend has it that the discussion on the size of the cathedral was decided upon the agreement that the size should be as long the throwing of a string from the stripes of one skinned cow. "Gjergj Nikë Sheldija", who was present in that discussion, has told that the string was firstly taken into the hands of the Scutari's Wāli, who put it into his little finger and threw it. After his throwing the consul of United Kingdom threw it much further, the consul of France even further, and the consuls of Russia, Greece, and even the Austro-Hungarian Empire went much further than that of the Wāli. This one, smiling under his moustache, is said to have murmured: "You probably want to put here all the Catholics of Albania".

Construction.

The land was given by mecenate "Palok Krandja". Work commenced on 7 April 1858 after a major delay. There was a pompous ceremony on that day for the construction of this cathedral which would be one of the biggest Catholic cathedrals in the Balkans at that time, and many local and diplomatic personalities participated at the inauguration of the construction work. Besides giving his permission for the building through a decree, Sultan Abdülmecid I had also sent 700 Turkish liras, whereas Pope Pius IX had provided 1,000 golden scudi. The first phase of construction finished in 1867, 9 years later.  The cathedral was 74 meters long, 50 meters wide and 23.5 meters tall. Initially the main hall had a 6,000 person capacity (with people standing up).  There was a marathon for fund raising at that time: besides the contribution, which included even proceeds from sold jewelries, each Catholic family gave a worker to contribute to the construction for a period of at least 6 months. "Gurash Shënkolli", a Shkodër citizen, is said to have been in Vienna during the time of the marriage of Franz Joseph I of Austria, and to have managed to enter the Emperor's office. Gurash asked the Emperor for financial aid for the construction of the church. Not only was Gurash given 150 napoleons, but he was also congratulated by the Austrian Emperor for his "typically Albanian" audacity in asking that way and to an Emperor for that type of money.  During the construction, the big arch fell twice on the ground, and the fix was made through the help of an Austrian engineer, whose name is unknown to us today. As the construction work was ending, a bell tower was still missing. Archbishop Guerini gathered the representatives of the wealthiest Catholic families. It was decided that three categories, depending on the level of wealth, would provide each 20, 15 and 10 Napoleon gold. This resulted in a dispute because many people didn't want to be ranked in lower wealth categories. At that point, Gjon Nushi, one of the wealthiest people of the community, openly proposed to Filip Parruca, another wealthy man: "Let's finish the bell tower together, and on its wall we'll have a commemorative stone to remind posterity that it was our money to build it". This discussion was enough for the community to accept rather the bishop's proposal. "Filip Parruca", "Mark Pema" and "Pjetër Pema" purchased in Venice the bell, and had it put on the tower.

Shkodër Cathedral (, also known as "St Stephen's Catholic Cathedral" or "Kisha e Madhe - the Great Church") is a cathedral in the city of Shkodër in northwestern Albania. The Roman Catholic cathedral is dedicated to Saint Stephen.

Katedrála svatého Štěpána ve Skadaru (albánsky Katedralja e Shën Shtjefnit) je římskokatolická katedrála ve severoalbánském městě Skadar. Je duchovním centrem římskokatolických věřících v Albánii. Albánská římskokatolická komunita představuje více než 10 % obyvatel země a je soustředěna právě v severní části Albánie.


Definitions.

There are three conceptually different but equivalent approaches to defining the Lie bracket:

Vector fields as derivations.

Each smooth vector field "X" on a manifold "M" may be regarded as a differential operator acting on smooth functions "C"("M"). Indeed, each smooth vector field "X" becomes a derivation on "C"("M") when we define "X"("f") to be a function whose value at a point "p" is the directional derivative of "f" at "p" in the direction "X"("p"). Furthermore, any derivation on "C"("M") arises from a unique smooth vector field "X".  In general, the commutator formula_2 of any two derivations formula_3 and formula_4 is again a derivation, where formula_5 denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

Flows and limits.

Let formula_7 be the flow associated with the vector field "X", and let D denote the tangent map derivative operator. Then the Lie bracket of "X" and "Y" at the point can be defined as the Lie derivative:  This also measures the failure of the flow in the successive directions formula_9 to return to the point "x":

In coordinates.

Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold "M"), in practice one often wants to compute the bracket in terms of a specific coordinate system formula_11. We write formula_12 for the associated local basis of the tangent bundle, so that general vector fields can be written formula_13and formula_14for smooth functions formula_15. Then the Lie bracket can be computed as:  If "M" is (an open subset of) R, then the vector fields "X" and "Y" can be written as smooth maps of the form formula_17 and formula_18, and the Lie bracket formula_19 is given by:  where formula_21 and formula_22 are "n×n" Jacobian matrices multiplying the "n×"1 column vectors "X" and "Y".

Properties.

The Lie bracket of vector fields equips the real vector space formula_23 of all vector fields on "M" (i.e., smooth sections of the tangent bundle formula_24) with the structure of a Lie algebra, which means [ •, • ] is a map formula_25 with:  An immediate consequence of the second property is that formula_28 for any formula_29.  Furthermore, there is a "product rule" for Lie brackets. Given a smooth (scalar-valued) function "f" on "M" and a vector field "Y" on "M", we get a new vector field "fY" by multiplying the vector "Y" by the scalar "f"("x") at each point. Then:  where we multiply the scalar function "X"("f") with the vector field "Y", and the scalar function "f" with the vector field. This turns the vector fields with the Lie bracket into a Lie algebroid.  Vanishing of the Lie bracket of "X" and "Y" means that following the flows in these directions defines a surface embedded in "M", with "X" and "Y" as coordinate vector fields:  Theorem: formula_31 iff the flows of "X" and "Y" commute locally, meaning formula_32 for all and sufficiently small "s", "t".  This is a special case of the Frobenius integrability theorem.

Examples.

For a Lie group "G", the corresponding Lie algebra formula_33 is the tangent space at the identity formula_34, which can be identified with the vector space of left invariant vector fields on "G". The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation formula_35.  For a matrix Lie group, whose elements are matrices formula_36, each tangent space can be represented as matrices: formula_37, where formula_38 means matrix multiplication and "I" is the identity matrix. The invariant vector field corresponding to formula_39is given by formula_40, and a computation shows the Lie bracket on formula_41 corresponds to the usual commutator of matrices:

Applications.

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

Generalizations.

As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields "X" and "Y" on a smooth manifold "M" a third vector field denoted.

Lieova závorka je operátor, který přiřazuje kterýmkoliv dvěma vektorovým polím "X" a "Y" na hladké varietě "M", třetí vektorové pole označované "[X, Y]". Lieova závorka vystihuje nekomutativitu toků generovanými těmito poli.