Year 1056 was a leap year starting on Monday of the Julian calendar. August 31 – Empress Theodora dies after a 18-month reign by a sudden illness at Constantinople, she is succeeded by Michael VI who has served as military finance minister under the former Emperor Romanos III. Michael is appointed through the influence of Theodora's most trusted adviser; this ended the Macedonian Dynasty. Theodosius, a nephew of the former Emperor Constantine IX, tries to usurp the Byzantine throne and liberates all the prisoners who flocks his banner. With their support he marches through the streets of Constantinople to the Palace. There, the Varangian Guard forms outside to stop him. Theodosius heads for Hagia Sophia, he is captured and exiled to Pergamum. October 5 – Emperor Henry III dies after a 10-year reign at Bodfeld, an imperial hunting lodge in the Harz Mountains, he is succeeded and enthroned by his 5-year-old only son Henry IV as "king of the Germans" by Pope Victor II at Aachen – while his mother, Empress Agnes of Poitou, becomes co-regent.

Ottokar I, count of Steyr, becomes margrave of the Karantanian March. June 16 – In response to the attack on Hereford Cathedral, Leofgar the bishop of Hereford takes an army into Wales to deal with the Welsh prince Gruffydd ap Llywelyn, he along with a large number of English troops is killed in battle at Glasbury-on-Wye by the Welsh. Earl Harold Godwinson comes to peaceful terms with Gruffydd; the Pagoda of Fogong Temple in Shanxi in northern China is built during the Liao Dynasty. Work begins on the Pizhi Pagoda of Lingyan Temple in Shandong under the opposing Song Dynasty. Dromtön, an Atiśa chief disciple, founds Reting Monastery in the Reting Tsangpo Valley as the seat of Kadam lineage of Tibetan Buddhism; the Muslims expel 300 Christians from Jerusalem, European Christians are forbidden to enter the Church of the Holy Sepulcher. Abdallah ibn Buluggin, emir of Granada Al-Muqtadi, caliph of the Abbasid Caliphate Baldwin II of Mons, count of Hainaut Ermengol IV, Spanish nobleman Fujiwara no Kiyohira, Japanese nobleman and samurai Hildegarde of Burgundy, French noblewoman Ibn Tahir of Caesarea, Arab scholar and historian Nestor the Chronicler, Russian monk and historian Sæmundur Sigfússon, Icelandic priest and scholar William II, king of England Zhou Bangyan, Chinese bureaucrat and ci poet February 10 – Æthelstan, bishop of Hereford February 11 – Herman II, archbishop of Cologne June 16 – Leofgar, bishop of Hereford August 31 Odda of Deerhurst, English nobleman Theodora, empress of the Byzantine Empire September 10 – William, margrave of the Nordmark October 5 – Henry III, Holy Roman Emperor November 25 – Flann Mainistreach, Irish poet and historian Áed Ua Forréid, bishop of Armagh Anselm of Liège, French chronicler Benedict IX, pope of the Catholic Church Ekkehard IV, Swiss monk and chronicler Hilal al-Sabi', Buyid historian and writer Leo of Ohrid, Byzantine archbishop and theologian Yahya ibn Umar al-Lamtuni, Almoravid chieftain

View factor

In radiative heat transfer, a view factor, F A → B, is the proportion of the radiation which leaves surface A that strikes surface B. In a complex'scene' there can be any number of different objects, which can be divided in turn into more surfaces and surface segments. View factors are sometimes known as configuration factors, form factors, angle factors or shape factors; because radiation leaving a surface is conserved, the sum of all view factors from a given surface, S i, is unity: ∑ j = 1 n F S i → S j = 1 For example, consider a case where two blobs with surfaces A and B are floating around in a cavity with surface C. All of the radiation that leaves A must either hit B or C, or if A is concave, it could hit A. 100% of the radiation leaving A is divided up among A, B, C. Confusion arises when considering the radiation that arrives at a target surface. In that case, it does not make sense to sum view factors as view factor from A and view factor from B are different units. C may see 10% of A's radiation and 50% of B's radiation and 20% of C's radiation, but without knowing how much each radiates, it does not make sense to say that C receives 80% of the total radiation.

For a convex surface, no radiation can leave the surface and hit because radiation travels in straight lines. Hence, for convex surfaces, F A → A = 0 For concave surfaces, this doesn't apply, so for concave surfaces F A → A > 0 The superposition rule is useful when a certain geometry is not available with given charts or graphs. The superposition rule allows us to express the geometry, being sought using the sum or difference of geometries that are known. F 1 → = F 1 → 2 + F 1 → 3 The reciprocity theorem for view factors allows one to calculate F B → A if one knows F A → B. Using the areas of the two surfaces A A and A B, A A F A → B = A B F B → A Taking the limit of a small flat surface gives differential areas, the view factor of two differential areas of areas d A 1 and d A 2 at a distance s is given by: d F 1 → 2 = cos ⁡ θ 1 cos ⁡ θ 2 π s 2 d A 2 where θ 1 and θ 2 are the angle between the surface normals and a ray between the two differential areas; the view factor from a general surface A 1 to another general surface A 2 is given by: F 1 → 2 = 1 A 1 ∫ A 1 ∫ A 2 cos ⁡ θ 1 cos ⁡ θ 2 π s 2 d A 2 d A 1 The view factor is related to the etendue.

The crossed string rule allows calculation of radiation transfer between opposite sides of a quadrilateral, furthermore applies in some cases where there is partial obstruction between the objects. For a derivation and further details, see this article by G H Derrick. A geometrical picture that can aid intuition about the view factor was developed by Wilhelm Nusselt, is called the Nusselt analog; the view factor between a differential element dAi and the element Aj can be obtained projecting the element Aj onto the surface of a unit hemisphere, projecting that in turn onto a unit circle around the point of interest in the plane of Ai. The view factor is equal to the differential area dAi times the proportion of the unit circ

Jeff Astle

Jeffrey Astle was an English professional footballer who played the majority of his career as a centre forward for West Bromwich Albion. Nicknamed "the King" by the club's fans, he scored 174 goals, he won five caps for England without scoring. Born in Eastwood, Astle turned professional with Notts County when he was 17, his style was that of a classic centre forward. In 1964 he signed for West Brom for a fee of £25,000, he scored 174 goals in 361 games for the Baggies, including the only goal in the 1968 FA Cup Final, in which he completed the feat of scoring in every round of the competition. Two years Astle scored in Albion's 2–1 defeat by Manchester City in the League Cup final, becoming the first player to score in the finals of both of the major English cup competitions at Wembley, he had scored in the first leg of the 1966 League Cup Final four years but, at West Ham United's Upton Park. At the height of Astle's Albion career, the words "ASTLE IS THE KING" appeared in large white letters on the brickwork of Primrose Bridge, which carries Cradley Road over a canal in Netherton, in the heart of the Black Country.

The bridge became known locally as "the Astle Bridge". When the council removed the letters, they re-appeared a few days later. Following Astle's death in 2002, a campaign was launched to have the bridge named in his honour, but this has so far been rejected over fears of vandal attacks by supporters of rival teams, as the area has many Wolverhampton Wanderers fans. In 1969–1970 Astle was the leading scorer in Division One with 25 goals. In 1970, he was called up to the England squad for the World Cup finals tournament in Mexico, he won the fourth of his five caps, as a substitute, when England were a goal down against eventual champions Brazil. He missed a easy scoring chance. In subsequent years his fitness deteriorated through repeated injuries, in 1974 he left Albion to join the South African club Hellenic, his final bow came with a brief spell at the English non-league side Dunstable Town, where he teamed up with former Manchester United star George Best. After his retirement, Astle launched an industrial cleaning business, working around the Burton upon Trent area.

He made television comedy appearances with Frank Skinner and David Baddiel on Fantasy Football League. On 19 January 2002, Astle choked to death at his daughter's home aged 59; the cause of death was a degenerative brain disease that had first become apparent as much as five years earlier. He had been described as an exceptional header of the ball, the coroner found that the repeated minor traumas had been the cause of his death, as the leather footballs used in Astle's playing days were heavier than the plastic ones used especially when wet; this was not the first case of a footballer's illness or death being connected to heading old-fashioned footballs, another example being the former Tottenham Hotspur captain Danny Blanchflower who died of Alzheimer's disease in December 1993. A verdict of death by industrial injury was recorded. In 2014, the Justice for Jeff campaign was launched, calling for an independent inquiry into a possible link between degenerative brain disease and heading footballs.

Subsequently, he was confirmed as the first British footballer known to have died as a result of heading a football. In the same year it was claimed by a neurosurgeon that Astle had died as a result of chronic traumatic encephalopathy, a disease associated with boxers. On the day after his death, West Bromwich Albion held a minute's silence in honour of Astle, before their home match against Walsall. Albion striker Jason Roberts scored the only goal of the game and celebrated by removing his jersey to reveal a T-shirt bearing Astle's image, his funeral in Derbyshire was attended by hundreds of football fans. Fantasy Football League returned for a number of special editions after Astle's death. In November 2002, Astle's widow Laraine said. Astle had been worshipped as a hero by the Albion fans, who would sing: "Astle is the king, Astle is the king, the Brummie Road will sing this song, Astle is the king"; the chant is still heard at the Hawthorns. Following his death, a campaign was launched to fund a set of gates dedicated to his memory at the ground.

The gates, which are situated on the Birmingham Road, close to the Woodman Corner, were unveiled on 11 July 2003. In April 2003 Astle became the first person to have a Midland Metro tram named after him. In 2004, he was named as one of West Bromwich Albion's 16 greatest players, in a poll organised as part of the club's 125th anniversary celebrations. Astle was voted as one of Albion's five "FA Cup heroes", receiving the most votes for a striker in the poll organised by the club's official website in 2006. Since early 2013, for around a year, West Brom fans make it a point at every home game to applaud on the ninth minute of the game for the entire minute in tribute to Jeff Astle and in support of the Justice for Jeff campaign; this is. During the minute the screens at the Hawthorns would have an image of Jeff Astle with the caption'If in doubt, Sit them out.' Albion's fierce rivals Aston Villa and Birmingham City showed a picture of Astle on their screens when the Baggies played there that season.

It was announced on 26 March 2015 that the club would be holding "Astle Day" in his memory on 11 April at The Hawthorns, when the cl