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In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean and mode are all known as measures of central tendency, in colloquial usage any of these might be called an average value; the arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means. The most common type of average is the arithmetic mean. If n numbers are given, each number denoted by ai, the arithmetic mean is the sum of the as divided by n or AM = 1 n ∑ i = 1 n a i = a 1 + a 2 + ⋯ + a n n The arithmetic mean simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = /2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A; the mean 5 is not less than the minimum 2 nor greater than the maximum 8.

If we increase the number of terms in the list to 2, 8, 11, the arithmetic mean is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = /3 = 7; the geometric mean of n positive numbers is obtained by multiplying them all together and taking the nth root. In algebraic terms, the geometric mean of a1, a2... an is defined as GM = ∏ i = 1 n a i n = a 1 a 2 ⋯ a n n Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers. Example: Geometric mean of 2 and 8 is GM = 2 ⋅ 8 = 4 Harmonic mean for a non-empty collection of numbers a1, a2... an, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the ai's: HM = 1 1 n ∑ i = 1 n 1 a i = n 1 a 1 + 1 a 2 + ⋯ + 1 a n One example where the harmonic mean is useful is when examining the speed for a number of fixed-distance trips. For example, if the speed for going from point A to B was 60 km/h, the speed for returning from B to A was 40 km/h the harmonic mean speed is given by 2 1 60 + 1 40 = 48 A well known inequality concerning arithmetic and harmonic means for any set of positive numbers is AM ≥ GM ≥ HM See Inequality of arithmetic and geometric means.

Thus for the above harmonic mean example: AM = 50, GM ≈ 49, HM = 48 km/h. The mode, the median, the mid-range are used in addition to the mean as estimates of central tendency in descriptive statistics; these can all be seen as minimizing variation by some measure. The most occurring number in a list is called the mode. For example, the mode of the list is 3, it may happen that there are two or more numbers which occur often and more than any other number. In this case there is no agreed definition of mode; some authors say they are all modes and some say there is no mode. The median is the middle number of the group, thus to find the median, order the list according to its elements' magnitude and repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If one value is left, it is the median; this method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. The 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, /2 = 5.

The mid-range is the arithmetic mean of lowest values of a set. The table of mathematical symbols explains the symbols used below. Other more sophisticated averages are: t

The Associates (band)

The Associates were a Scottish post-punk and new wave band, formed in Dundee in 1979 by singer Billy MacKenzie and guitarist Alan Rankine. The group first gained recognition after releasing an unauthorized cover of David Bowie's "Boys Keep Swinging" as their debut single in 1979, which landed them a contract with Fiction Records, they followed with their debut album The Affectionate Punch in 1980 and the singles collection Fourth Drawer Down in 1981, both to critical praise. They achieved commercial success in 1982 with the UK Top 10 album Sulk and UK Top 20 singles "Party Fears Two" and "Club Country", during which time they were associated with the New Pop movement. Rankine left the group that year, leaving MacKenzie to record under the Associates name until 1990, they reunited in 1993. MacKenzie committed suicide in 1997. Billy MacKenzie and guitarist Alan Rankine met in Edinburgh, Scotland in 1976 and formed the cabaret duo the Ascorbic Ones, although Rankine claimed that this was "a fantasy band that Bill and I dreamt up to give ourselves a past".

In 1978, they recorded songs as Mental Torture before changing the name to the Associates. Disappointed that their early recordings were not getting picked up, MacKenzie concocted the stunt of doing a cover of David Bowie's "Boys Keep Swinging", without copyright permission, just six weeks after Bowie's version hit the UK Top 10. Released in June 1979, this debut Associates single reached No. 15 in Record Mirror's Scottish chart and gained them airplay on John Peel's Radio One show. MacKenzie said that the band recorded the Bowie song "to prove the point, it was a strange way of proving it. People said,'That is awful. How dare they!'" The ensuing attention earned them a contract with Fiction Records, their debut album, The Affectionate Punch, followed on 1 August 1980. A string of 1981 non-album singles on the label Situation Two were compiled as Fourth Drawer Down, released that October; these releases saw the band develop an interest in experimenting with unorthodox instrumentation and recording techniques, including sounds being amplified through the tube of a vacuum cleaner on the track "Kitchen Person".

In 1981, Rankine and MacKenzie released a version of "Kites" under the name 39 Lyon Street, with Christine Beveridge on lead vocals. The B-side, was credited to the Associates; the band's breakthrough came in 1982 with the release of the single "Party Fears Two". Buoyed along by the popularity of synthpop at the time, the song reached No. 9 on the UK Singles Chart. Two other hits followed, "Club Country" and "18 Carat Love Affair". On 14 May 1982, the band released Sulk. Martha Ladly, of Martha and the Muffins, contributed backing keyboards to this album. Rankine left the band in 1982 just before the Sulk tour; this proved disastrous for the band's career. MacKenzie continued to write and record music under the name Associates until 1990; the albums Perhaps, The Glamour Chase and Wild and Lonely were recorded during this period. However, recordings were sporadic and subsequent Associates records failed to reach the charts in the UK and sold far fewer than their early albums; the Associates name was put to rest, MacKenzie released an electronica-influenced solo album Outernational in 1992 with limited success.

In 1993, MacKenzie and Rankine began working on new material together. News of an Associates revival generated hype and speculation of a tour, the demos recorded by the two were promising. However, MacKenzie was not committed to the reunion and touring with it, so Associates split for a final time. MacKenzie went back to his solo work, signing a deal with Nude Records and finding a new collaborative partner in Steve Aungle. Between 1987 and 1992, Billy worked with Swiss avant-garde outfit Yello. MacKenzie wrote the lyrics of the song "The Rhythm Divine" performed by Shirley Bassey on the album One Second, with MacKenzie singing backing vocals. MacKenzie contributed to three Yello albums: One Second and Baby; some tracks for The Glamour Chase and Outernational were recorded with Boris Blank at Yello's recording studio. Rankine became a lecturer in music at Stow College in Glasgow, worked with Belle and Sebastian on their 1996 debut album, Tigermilk. MacKenzie committed suicide in 1997 at age 39, shortly after the death of his mother.

He had been suffering from clinical depression. He was contemplating a comeback at the time with material co-written with Aungle; the albums Beyond the Sun and Eurocentric were released posthumously and, in 2004, reconstructed and expanded with new unreleased songs into the two albums Auchtermatic and Transmission Impossible. Before MacKenzie's death all Associates records had been deleted. Former band member Michael Dempsey and the MacKenzie estate began a reissue programme to make sure the band's legacy continued, reissuing every Associates album, including a 25th anniversary edition of The Affectionate Punch in 2005. In addition to the original albums, two compilation albums were released: Double Hipness, a collection of early tracks with the 1993 reunion demos. In 2002, The Glamour Chase was released as a set titled The

Donald Robinson (bishop)

Donald William Bradley Robinson was an Australian Anglican Archbishop of Sydney from 1982 to 1992. Robinson was born in Lithgow, New South Wales on 9 November 1922, the son of Richard Bradley Robinson, some time Archdeacon of North Sydney, his first year of secondary school studies was at North Sydney Boys High School. His undergraduate studies were interrupted by service in World War 2, he was ordained in 1950 by Howard Mowll, Archbishop of Sydney, began his ministry with curacies at St Matthew’s, Manly and St Philip's Church, Sydney. He was a lecturer and Vice-Principal at Moore College and at Sydney University until 1973 when he became the Bishop of Parramatta, he was consecrated a bishop on 25 January 1973 at Sydney. Nine years he was elected as Archbishop of Sydney and the Metropolitan of New South Wales. Michael Jensen argues that Robinson's work "has been crucial for shaping how Sydney Anglicans think about and preach from the Bible."

White Horse Tavern (Newport, Rhode Island)

The White Horse Tavern was constructed before 1673 and is believed to be the oldest tavern building in the United States. It is located on the corner of Farewell and Marlborough streets in Rhode Island. Francis Brinley, an English immigrant, constructed the original building on the site in 1652. In 1673, he sold the lot to William Mayes; the building was used for large meetings, including use as a Rhode Island General Assembly meeting place, a court house, a city hall. William Mayes obtained a tavern license in 1687, his son William Mayes, Jr. operated it through the early eighteenth century. The operation was named "The White Horse Tavern" in 1730 by owner Jonathan Nichols. Tories and British troops were quartered there during the British occupation of Newport in the American Revolution, around the time of the Battle of Rhode Island. Newport's Van Bueren family donated money to the private Preservation Society of Newport to restore the building in 1952, after years of neglect as a boarding house.

After the restoration, the building was sold and once again operated as a private tavern and restaurant, it remains a popular drinking and dining location today. List of the oldest restaurants in the United States Oldest buildings in America National Register of Historic Places listings in Newport County, Rhode Island List of the oldest buildings in Rhode Island List of oldest companies List of oldest companies in the United States


Teprotide is nonapeptide, isolated from the snake Bothrops jararaca. It is an angiotensin converting enzyme inhibitor, which inhibits the conversion of angiotensin I to angiotensin II and may potentiate some of the pharmacological actions of bradykinin, it has been looked at as an antihypertension agent. The antihypertensive effects of teprotide were first observed by Sergio Ferreira in 1965 and it was first isolated by Ferreira et al. along with eight other peptides in 1970. Teprotide was synthesized in 1970 by Ondetti et al. and from there its antihypertensive properties were studied more closely. Teprotide was chosen as a lead because of its long-lasting in vivo activity; this was demonstrated by Bianchi et al. by administering teprotide to dogs and rats and observing that it inhibited the vasopressor response induced by angiotensin I. Teprotide was shown to be an effective antihyperension agent but it had limited use because of its expense and lack of oral activity, it was found that teprotide inhibits the enzyme that converts angiotensin I to angiotensin II.

From this researchers conducted structure-activity studies which allowed them to identify the active binding site of the ACE which allowed for the development of antihypertension drugs to be developed. Captopril was the first antihypertension drug developed by Cushman. Many ACE inhibitors have been developed since this time but this was the start of them

Left-leaning red–black tree

A left-leaning red–black tree is a type of self-balancing binary search tree. It is a variant of the red–black tree and guarantees the same asymptotic complexity for operations, but is designed to be easier to implement. All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of log N in a tree of N keys, the behavior observed in practice is that same multiple faster than the worst-case bound, close to the optimal log N nodes examined that would be observed in a balanced tree. In a left-leaning red-black 2-3 tree built from N random keys: A random successful search examines log2 N − 0.5 nodes. The average tree height is about 2 log2 N The average size of left subtree exhibits log-oscillating behavior. Robert Sedgewick. Left-leaning Red–Black Trees. Direct link to PDF. Robert Sedgewick. Left-Leaning Red–Black Trees. Two versions: Robert Sedgewick. Left-Leaning Red–Black Trees, from seminar at Dagstuhl in February 2008.

Outdated. Robert Sedgewick. Left-Leaning Red–Black Trees, from April 2008. May 19, 2009. Formalizing Arne Andersson trees and Left-leaning Red–Black trees in Agda Julien Oster. March 22, 2011. An Agda implementation of deletion in Left-leaning Red–Black trees Kazu Yamamoto. 2011.10.19. Purely Functional Left-Leaning Red–Black Trees Robert Sedgewick. 20 Apr 2008. Animations of LLRB operations Open Data Structures - Section 9.2.2 - Left-Leaning Red–Black Trees, Pat Morin Left-Leaning Red-Black Trees Considered Harmful