British Empire

The British Empire comprised the dominions, colonies, protectorates, mandates and other territories ruled or administered by the United Kingdom. It originated with the overseas possessions and trading posts established by England between the late 16th and early 18th centuries. At its height, it was the largest empire in history and, for over a century, was the foremost global power. By 1922 the British Empire held sway over about 458 million people, one-fifth of the world's population at the time, and covered more than 13,000,000 sq mi (33,670,000 km²), almost a quarter of the Earth's total land area. As a result, its political, legal, linguistic and cultural legacy is widespread. At the peak of its power, the phrase "the empire on which the sun never sets" was often used to describe the British Empire, because its expanse around the globe meant that the sun was always shining on at least one of its territories.

During the Age of Discovery in the 15th and 16th centuries, Portugal and Spain pioneered European exploration of the globe, and in the process established large overseas empires. Envious of the great wealth these empires generated, England, France, and the Netherlands began to establish colonies and trade networks of their own in the Americas and Asia. A series of wars in the 17th and 18th centuries with the Netherlands and France left England (and then, following union between England and Scotland in 1707, Great Britain) the dominant colonial power in North America and India.

Order of the British Empire

The Most Excellent Order of the British Empire is the "order of chivalry of British constitutional monarchy", rewarding contributions to the arts and sciences, work with charitable and welfare organisations and public service outside the Civil Service. It was established on 4 June 1917 by King George V, and comprises five classes, in civil and military divisions, the most senior two of which make the recipient either a knight if male, or dame if female. There is also the related British Empire Medal, whose recipients are affiliated with, but not members of, the order.

Appointments to the Order of the British Empire were at first made on the nomination of the self-governing Dominions of the Empire, the Viceroy of India, and the colonial governors, as well as on nominations from within the United Kingdom. As the Empire evolved into the Commonwealth, nominations continued to come from the Commonwealth realms, in which the monarch remained head of state. These overseas nominations have been discontinued in realms that have established their own Orders—such as the Order of Australia, the Order of Canada, and the New Zealand Order of Merit—but members of the Order are still appointed in the British Overseas Territories.

Big O notation

In mathematics, big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation (after Edmund Landau and Paul Bachmann), or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size. In analytic number theory, it is used to estimate the "error committed" while replacing the asymptotic size, or asymptotic mean size, of an arithmetical function, by the value, or mean value, it takes at a large finite argument. A famous example is the problem of estimating the remainder term in the prime number theorem.

Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation.

Empire

An empire is defined as "an aggregate of nations or people ruled over by an emperor or other powerful sovereign or government, usually a territory of greater extent than a kingdom, as the former British Empire, French Empire, Spanish Empire, Russian Empire, Byzantine Empire or Roman Empire." An empire can be made solely of contiguous territories such as the Austro-Hungarian Empire, or of territories far remote from the homeland, such as a colonial empire.

Aside from the more formal usage, the term "empire" can also be used to refer to a large-scale business enterprise (e.g. a transnational corporation), a political organisation controlled by a single individual (a political boss) or a group (political bosses). The term "empire" is associated with other words such as imperialism, colonialism, and globalization. Empire is often used to describe a displeasure to overpowering situations. The effects of imperialism exist throughout the world today.

An imperial political structure can be established and maintained in two ways: (i) as a territorial empire of direct conquest and control with force or (ii) as a coercive, hegemonic empire of indirect conquest and control with power. The former method provides greater tribute and direct political control, yet limits further expansion because it absorbs military forces to fixed garrisons. The latter method provides less tribute and indirect control, but avails military forces for further expansion. Territorial empires (e.g., the Mongol Empire and Median Empire) tend to be contiguous areas. The term, on occasion, has been applied to maritime empires or thalassocracies, (e.g., the Athenian and British empires) with looser structures and more scattered territories. Empires are usually larger than kingdoms.

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