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Trento

Trento is a city on the Adige River in Trentino-Alto Adige/Südtirol in Italy. It is the capital of the autonomous province of Trento. In the 16th century, the city was the location of the Council of Trent. Part of Austria and Austria-Hungary, it was annexed by Italy in 1919. With 120,000 inhabitants, Trento is the third largest city in the Alps and second largest in the Tyrol. Trento is an educational, scientific and political centre in Trentino-Alto Adige/Südtirol, in Tyrol and Northern Italy in general; the University of Trento ranks 2nd among'medium-sized' Universities in the Census ranking and 5th in the Il Sole 24 Ore ranking of Italian universities. The city contains a picturesque Medieval and Renaissance historic centre, with ancient buildings such as Trento Cathedral and the Castello del Buonconsiglio. Together with other Alpine towns Trento engages in the Alpine Town of the Year Association for the implementation of the Alpine Convention to achieve sustainable development in the Alpine Arc.

Trento was awarded the title of Alpine Town of the Year 2004. The city ranks among Italian cities for quality of life, standard of living, business and job opportunities, being ranked 5th in 2017. Trento is one of the nation's wealthiest and most prosperous cities, with its province being one of the richest in Italy, with a GDP per capita of €31,200 and a GDP of €16.563 billion. The township of Trento encompasses the city centre as well as many suburbs of varied geographical and population conditions. Various distinctive suburbs still retain their traditional identity of rural or mountain villages. Trento lies in a wide glacial valley known as the Adige valley, just south of the Dolomite Mountains, where the Fersina River and Avisio rivers join the Adige River. River Adige is one of the three primary south-flowing Alpine rivers; the valley is surrounded by mountains, including Vigolana, Monte Bondone, Paganella and Monte Calisio. Nearby lakes include Lake Levico, Lake Garda and Lake Toblino. Frazioni, or subdivisions of Trento: In 2007, there were 112,637 people residing in Trento, of whom 48% were male and 52% were female.

Minors totalled 18.01 percent of the population compared to pensioners. This compares with the Italian average of 19.94 percent. The average age of Trento residents is 41 compared to the Italian average of 42. In the five years between 2002 and 2007, the population of Trento grew by 5.72 percent, while Italy as a whole grew by 3.56 percent. The current birth rate of Trento is 9.61 births per 1,000 inhabitants compared to the Italian average of 9.45 births. As of 2006, 92.68% of the population was Italian. The largest immigrant group came from other European countries: 4.13%, North Africa: 1.08%, the Americas: 0.85%. Trento Informa reports that in 2011 there were 117,190 people residing in Trento, of whom 48.5% aged between 45 and 65. The average age was 43.1 years. 13,535 were foreigners. The origins of this city on the river track to Bolzano and the low Alpine passes of Brenner and the Reschen Pass over the Alps are disputed; some scholars maintain it was a Rhaetian settlement: the Adige area was however influenced by neighbouring populations, including the Veneti, the Etruscans and the Gauls.

According to other theories, the latter did instead found the city during the 4th century BC. Trento was conquered by the Romans in the late 1st century BC, after several clashes with the Rhaetian tribes. Before the Romans, Trento was a Celtic village. In reality, the name derives from Trent, a tribute to the Celtic god of the waters; the Romans is a tribute to the Roman god Neptune. The Latin name is the source of the adjective "tridentine". On the old city hall, a Latin inscription is still visible: "Montes argentum mihi dant nomenque Tridentum", attributed to Fra' Bartolomeo da Trento. Tridentum became an important stop on the Roman road. After the fall of the Western Roman Empire, the independent bishopric of Trento was conquered by Ostrogoths, Byzantines and Franks becoming part of the Holy Roman Empire. In 1027, Emperor Conrad II created the Prince-Bishops of Trento, who wielded both temporal and religious powers. In the following centuries, the sovereignty was divided between the Bishopric of Trent and the County of Tyrol.

Around 1200, Trento became a mining center of some significance: silver was mined from the Monte Calisio – Khalisperg, Prince-Bishop Federico Wanga issued the first mining code of the alpine region. In the 14th century, the region of Trento was part of Austria; the dukes of Austria were the counts of Tyrol and dominated the region for six centuries. A dark episode in the history of Trento was the Trento blood libel; when a 3-year-old Christian boy, Simonino known as Simon of Trent, disappeared in 1475 on the eve of Good Fri

Stafford Castledine

Stafford William Thomas Castledine was an English cricketer. Castledine was a right-handed batsman, he was born in Nottinghamshire. Castledine made his first-class debut for Nottinghamshire against the touring West Indians in 1933, his debut in the County Championship came when Nottinghamshire played Warwickshire in 1934 County Championship. During the 1934 season, he represented the county in 3 further first-class matches, the last of which came against Sussex. In his 5 first-class matches, he scored 22 runs at a batting average of 3.14, with a high score of 15. In the field he took 6 catches, he died at Nottingham on 17 April 1986. Stafford Castledine at Cricinfo Stafford Castledine at CricketArchive

Convex combination

In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum to 1. More formally, given a finite number of points x 1, x 2, …, x n in a real vector space, a convex combination of these points is a point of the form α 1 x 1 + α 2 x 2 + ⋯ + α n x n where the real numbers α i satisfy α i ≥ 0 and α 1 + α 2 + ⋯ + α n = 1; as a particular example, every convex combination of two points lies on the line segment between the points. A set is convex; the convex hull of a given set of points is identical to the set of all their convex combinations. There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity. A convex combination X of probability distributions Y i is a weighted sum of its component probability distributions called a finite mixture distribution, with probability density function: f X = ∑ i = 1 n α i f Y i A conical combination is a linear combination with nonnegative coefficients.

When a point x is to be used as the reference origin for defining displacement vectors x is a convex combination of n points x 1, x 2, …, x n if and only if the zero displacement is a non-trivial conical combination of their n respective displacement vectors relative to x. Weighted means are functionally the same as convex combinations; the coefficients in a weighted mean are not required to sum to 1. Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field. Affine hull Carathéodory's theorem Simplex Barycentric coordinate system