A triumvirate is a political regime ruled or dominated by three powerful individuals known as triumvirs. The arrangement can be informal. Though the three are notionally equal, this is the case in reality; the term can be used to describe a state with three different military leaders who all claim to be the sole leader. In the context of the Soviet Union and Russia, the term troika is used for "triumvirate". Another synonym is triarchy. Triumviri were special commissions of three men appointed for specific administrative tasks apart from the regular duties of Roman magistrates; the triumviri capitales, for instance, oversaw prisons and executions, along with other functions that, as Andrew Lintott notes, show them to have been "a mixture of police superintendents and justices of the peace." The capitales were first established around 290–287 BCE. They were supervised by the praetor urbanus; these triumviri, or the tresviri nocturni, may have taken some responsibility for fire control. The triumviri monetalis supervised the issuing of Roman coins.

Three-man commissions were appointed for purposes such as establishing colonies or distributing land. Triumviri mensarii served as public bankers. Another form of three-man commission was the tresviri epulones, who were in charge of organizing public feasts on holidays; this commission was created in 196 BCE by a tribunician law on behalf of the people, their number was increased to seven. The term is most used by historians to refer to the First Triumvirate of Julius Caesar, Marcus Licinius Crassus, Pompey the Great, the Second Triumvirate of Octavianus, Mark Antony, Marcus Aemilius Lepidus. In the Bible triumvirates occurred at some notable events in both the Old Testament and New Testament. In the Book of Exodus Moses, his brother Aaron and, according to some views their nephew or brother-in-law, Hur acted this way during Battle of Rephidim against the Amalekites. In the Gospels as a leading trio among the Twelve Apostles at three particular occasions during public ministry of Jesus acted Peter, son of Zebedee and his brother John.

They were the only apostles present at the Raising of Jairus' daughter, Transfiguration of Jesus and Agony in the Garden of Gethsemane. At the time of the Early Christian Church this triumvirate of the leading apostles changed after the former James’s death, as it became composed of Peter and James, brother of Jesus. One of the most notable triumvirates formed in the history of China was by the Han Dynasty statesmen Huo Guang, Jin Midi, Shangguan Jie 上官桀, following the death of Emperor Wu of Han and the installation of the child emperor Zhao. Despite the Three Excellencies—including the Chancellor, Imperial Secretary, irregularly the Grand Commandant—representing the most senior ministerial positions of state, this triumvirate was supported by the economic technocrat and Imperial Secretary Sang Hongyang, their political lackey; the acting Chancellor Tian Qianqiu was easily swayed by the decisions of the triumvirate. The Three Excellencies existed in Western Han as the Chancellor, Imperial Secretary, Grand Commandant, but the Chancellor was viewed as senior to the Imperial Secretary while the post of Grand Commandant was vacant for most of the dynasty.

After Emperor Guangwu established the Eastern Han, the Grand Commandant was made a permanent official while the Minister over the Masses replaced the Chancellor and the Minister of Works replaced the Imperial Secretary. Unlike the three high officials in Western Han when the Chancellor was senior to all, these new three senior officials had equal censorial and advisory powers; when a young or weak-minded emperor ascended to the throne, these Three Excellencies could dominate the affairs of state. There were other types of triumvirates during the Eastern Han. In Hinduism, the gods Brahma and Shiva form the triumvirate Trimurti "in which the cosmic functions of creation and destruction are personified" by those gods.." Tamil Triumvirate refers to the triumvirate of Chola and Pandya who dominated the politics of the ancient Tamil country. The title was revived a few times for three-headed political'magistratures' in post-feudal times. While French Huguenots had derisively bestowed the name Triumvirate on the alliance formed in 1561 between Catholic Francis, Duke of Guise, Anne de Montmorency, Jacques Dalbon, Seigneur de Saint Andre during the French Wars of Religion, in years the term would be used to describe other arrangements within France.

At the end of the 1700s, when the French revolutionaries turned to several Roman Magistrature names for their new institutions, the three-headed collective Head of State was named Consulat, a term in use for two-headed magistratures since Antiquity.

Gisborne Airport

Gisborne Airport is a regional airport located in the suburb of Elgin 4.2 km from the city centre of Gisborne on the East Coast of the North Island of New Zealand. Gisborne Airport is one of the few airports in the world that has a railway line, the Palmerston North–Gisborne Line, crossing the main runway; the airport has a single terminal with four tarmac gates. Gisborne Airport covers an area of around 160 hectares, it includes a sealed and night-capable runway at 1,310 metres in length, as well as three grass runways suitable for light aircraft. It was announced on September 7, 2018 that the airport will be receiving a $5.5m redevelopment loan to ensure that the airport terminal reflects the unique cultural aspects of the Gisborne region linking with the region’s navigation-themed tourism initiative. On 16 December 2004, Gisborne District Council voted to let Eastland Group manage the airport and lease the assets from 1 April 2005. Gisborne District Council still remains owner of the assets.

The lease runs for 15 years with an option to extend it for a further 15 years. Quote from Eastland Group's website Eastland Group pays Gisborne District Council an annual rental fee indexed to passenger numbers and is accountable for all capital investment, it bears all risk associated with the airports profitability, eliminating the need for ratepayer subsidy. Since taking over, Eastland Group has created a new airport cafe'V2', a new airport logo. List of airports in New Zealand List of airlines of New Zealand Transport in New Zealand NZAIP Volume 4 AD Airport information for NZGS at World Aero Data. Data current as of October 2006. Eastland Group website Gisborne Airport Eastland Group

Classical XY model

The classical XY model is a lattice model of statistical mechanics. It is the special case of the n-vector model for n = 2. Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = The spin configuration, s = j ∈ Λ is an assignment of the angle −π < θj ≤ π for each j ∈ Λ. Given a translation-invariant interaction Jij = J and a point dependent external field h j =, the configuration energy is H = − ∑ i ≠ j J i j s i ⋅ s j − ∑ j h j ⋅ s j = − ∑ i ≠ j J i j cos ⁡ − ∑ j h j cos ⁡ θ j The case in which Jij = 0 except for ij nearest neighbor is called nearest neighbor case; the configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0: P = e − β H Z Z = ∫ Λ ∏ j ∈ Λ d θ j e − β H. where Z is the normalization, or partition function. The notation ⟨ A ⟩ indicates the expectation of the random variable A in the infinite volume limit, after periodic boundary conditions have been imposed; the existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality.

Using the Griffiths inequality in the formulation of Ginibre and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension D, coupling J > 0 and inverse temperature β is dominated by the two point correlation of the ferromagnetic Ising model in dimension D, coupling J > 0 and inverse temperature β/2 ⟨ s i ⋅ s j ⟩ J, 2 β ≤ ⟨ σ i σ j ⟩ J, β Hence the critical β of the XY model cannot be smaller than the double of the critical temperature of the Ising model β c X Y ≥ 2 β c I s In case of long range interaction, J x, y ∼ | x − y | − α, the thermodynamic limit is well defined if α > 1. As in any'nearest-neighbor' n-vector model with free boundary conditions, if the external field is zero, there exists a simple exact solution. In the free boundary conditions case, the Hamiltonian is H = − J therefore the partition function factorizes under the change of coordinates θ j = θ j ′ + θ j − 1 j ≥ 2 That gives Z = ∫ − π π d θ 1 ⋯ d θ L e β J cos ⁡ ( θ 1 − θ