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Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer, it states that for any continuous function f mapping a compact convex set to itself there is a point x 0 such that f = x 0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem; this gives it a place among the fundamental theorems of topology. The theorem is used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.

It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu; the theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods; this work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Luitzen Egbertus Jan Brouwer; the theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: In the plane Every continuous function from a closed disk to itself has at least one fixed point; this can be generalized to an arbitrary finite dimension: In Euclidean space Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

A more general version is as follows: Convex compact set Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point. An more general form is better known under a different name: Schauder fixed point theorem Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point; the theorem holds only for sets that are convex. The following examples show. Consider the function f = x + 1, a continuous function from R to itself; as it shifts every point to the right, it cannot have a fixed point. The space R is not bounded. Consider the function f = x + 1 2, a continuous function from the open interval to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point; the space is not closed. The function f does have a fixed point for the closed interval, namely f = 1. Convexity is not necessary for BFPT; because the properties involved are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D n.

For the same reason it holds for every set, homeomorphic to a closed ball. The following example shows. Consider the function f = − x, a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point; the analogous example works for the n-dimensional sphere. The unit circle is closed and bounded; the function f does have a fixed point for the unit disc. A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem; the continuous function in this theorem is not required to be bijective or surjective. The theorem has several "real world" illustrations. Here are some examples. 1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will be at least one point of the crumpled sheet that lies directly above its corresponding point of the flat sheet.

This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to

Wang Noi District

Wang Noi is a district in Ayutthaya Province, Thailand, 50 km north of Bangkok. The population of Wang Noi were farmers but now this area is industrial; the district is in the southeastern part of Ayutthaya province. Neighboring districts are Nong Khae, Saraburi province, Nong Suea and Khlong Luan of Pathum Thani province, Bang Pa-in and Uthai of Ayutthaya province; the district was split off from Uthai District in 1907. At first named Uthai Noi, it was renamed Wang Noi in 1917; the district is the site of one of Thailand Post's automated sorting centers. As of 2018, THP is negotiating to purchase 100 rai of land to house the facility; the district is divided into 10 sub-districts. Lam Ta Sao is a sub-district municipality, which covers the whole sub-district Lam Ta Sao and parts of Bo Ta Lo, Lam Sai, Chamaep. There are a further nine tambon administrative organizations for the non-municipal areas.

Silvia ArderĂ­us

Silvia Arderius Martín is a Spanish handballer for Super Amara Bera Bera and the Spain national team. Super Amara Bera Bera Spanish División de Honor Femenina: Winner: 2017–18 Copa de la Reina de Balonmano: Runner-up: 2017–18 Women's World University Handball Championship: Winner: 2016 Trofeo Vicen Muñoz: Winner: 2013/14, 2016/17 Spanish División de Honor Femenina: Best centre back: 2016/17 Top Scorer: 2016/17 MVP: 2017/18 Silvia Arderíus profil at European Handball Federation

Smooth butterfly ray

The smooth butterfly ray is a species of cartilaginous fish in the family Gymnuridae. It is a member of the order Myliobatiformes, its natural habitats are shallow seas, subtidal aquatic beds, estuarine waters, coastal saline lagoons. Its common name is derived from its compressed body, pectoral fins that are wider than their length, overall diamond shape. Gymnura is derived from Greek roots and translates into'naked tail', they belong to a monophyletic group of Batoid fish. This group contains over 500 other elasmobranch fishes which includes electric rays, guitarfishes and stingrays, they are a part of Order Myliobatiformes and are characterized by their pectoral fins being expanded and fused to their heads. The family Gymnuridae contains 2 genera; these are broad diamond-shaped rays with a short tail that has low ventral fin folds. The tail has 3 to 4 dark lines; the edges of the disc are concave. The caudal fin is never present and a variable number of tubercles can be found on larger specimens.

The smooth butterfly rays have disc widths nearly twice the size of their body lengths and are flat-bodied. The width of the rays are between 16 and 22 centimeters when they are born and are about 50 cm when mature for a female and about 42 cm for a male. Females are bigger than their male counterparts, they have a maximum size of 120 cm. The ventral side is colored while the dorsal side is variable in color; the ventral side is white but can contain a rusty or bronze coloration. The dorsal side can be grey, light green and not uniform in color, they tend to use countershading to blend in with the bottom of their environments in order to hide from predators and to catch prey. The dorsal spine on the tail is absent. Smooth butterfly rays are found in the western and eastern parts of the Atlantic Ocean and the Gulf of Mexico, they are most found in neritic waters, but are known to enter brackish estuaries and hypersaline lagoons. They have a range that extends from the continental shelf to 40 meters deep in tropical and warm waters.

They prefer habitats that have either muddy bottoms. These rays only give birth to a few offspring, they use internal fertilization, the process of the male inserting his claspers into the female's cloaca to fertilize the eggs. The offspring take between four months to develop inside the mother, they use the young are histotrophs. The foraging strategy that these rays use is dependent on the abundance of prey in their environments, they either use opportunistic feeding where they eat what is available, or they use specialized feeding where they eat a specific organism. They tend to swallow them whole, they prey on Teleosts and crustaceans, but have been noted to consume bivalve mussels and polychaetes. They use; the lateral line contains neuromasts. The upper jaw consists of 6 to 120 teeth and the lower jaw has 52 to 106 teeth, they are hunted by larger predators, such as sharks. The great hammerhead specializes in feeding on butterfly rays and is their main predator in some areas. Gymnura micrura alter their swimming habits depending on where they are swimming in the water column.

They tend to change between an oscillation pattern. They use small amplitude undulations of their fins when they are swimming along the bottom, but switch to an oscillatory approach when they are swimming in the water; when swimming in the water column, they use a quick, powerful downstroke to increase their speed. They pause after each stroke and repeat; the species is classified as data deficient by the IUCN, as not information about population dynamics is known to assess its conservation status. It is taken as bycatch but released alive; the species is fished commercially and recreationally in parts of Australia and Asia

Bodenhoffs Plads

Bodenhodds Plads is an area located in the north-eastern part of Christianshavn, Denmark. The site is separated from Grønlandske Handels Plads to the west by Christianshavns Kanal and by Trangraven from Holmen to the north, it is connected to both areas by the three-way footbridge Trangravsbroen. The area was reclaimed by Andreas Bodenhoff from 1766 onwards and became known as Bodenhoffs Plads after him; the area was separated from the rest of Christianshavn by a canal, just like Bjørnsholm, on the other side of Christianshavn Canal, reclaimed some ten years prior by Andreas Bjørn. The site was used for storage of timber. In 1771, he established a shipyard at the site. Bodenhoff died in 1794; the site was in about 1830 acquired by Joseph Hambro and was from on known as Hambros Plads. He established a pig farm at the site; the rice huller was from 1930 powered by Denmark's firs tsteam engine. The chaft from the rice huller was used as feed for the pigs, he expanded the complex with Denmark's first canned food factory which made it possible to sell the meat to the many ships in the area.

He established a bakery which sold bread to the ships. Hambro's partner, Andreas Nicolai Hansen, played a leading role in the operations of the site, he established his own firm under the name A. n. Hansen & Co. in 1836. Joseph Hambro's son, Carl Joachim Hambro, moved to London where he founded Hambros Bank. In circa 1940, Joseph Hambro sold his share of Hambros Plads to Hansen. Hansen's two eldest sons and Harald Hansen, were made partners in the firm in 1850s. No. 8 and 10 were designed by Erik Schiødte and Rogert Møller. No. 12 is from 1904 and was designed by N. P. Larsen & G. Larsen; the northern part of the area was cleared by Dansk Totalenterprise in the early 1970s, except for a large warehouse from Islandske Handel. The Islands Plads development was built in 1975–78 for Lejerbo to design by BodThorvald Dreyer; the old warehouse contains 40 apartments. Bodenhodds Plads at Islands Plads at

Eastern Wu

Wu known as Dong Wu or Sun Wu, was one of the three major states that competed for supremacy over China in the Three Kingdoms period. It existed from 220–222 as a vassal kingdom nominally under Cao Wei, its rival state, but declared independence from Wei and became a sovereign state in 222, it became an empire in 229 after Sun Quan, declared himself emperor. Its name was derived from the place it was based in — the Jiangnan region, historically known as "Wu", it was referred to as "Dong Wu" or "Sun Wu" by historians to distinguish it from other Chinese historical states with similar names which were located in that region, such as the Wu state in the Spring and Autumn period and the Wuyue kingdom in the Five Dynasties and Ten Kingdoms period. It was called "Eastern Wu" because it occupied most of eastern China in the Three Kingdoms period, "Sun Wu" because the family name of its rulers was "Sun". During its existence, Wu's capital was at Jianye, but at times it was at Wuchang. Towards the end of the Han dynasty, Sun Ce, the eldest son of the warlord Sun Jian, his followers borrowed troops from the warlord Yuan Shu and embarked on a series of military conquests in the Jiangdong and Wu regions between 194 and 199, seizing several territories occupied by warlords such as Liu Yao, Yan Baihu and Wang Lang.

Sun Ce broke off relations with Yuan Shu around 196-197 after the latter declared himself emperor — an act deemed as treason against Emperor Xian, the figurehead ruler of the Han dynasty. The warlord Cao Cao, the de facto head of government in the Han imperial court, asked Emperor Xian to grant Sun Ce the title of "Marquis of Wu". Sun Ce was succeeded by his younger brother, Sun Quan. Sun Quan, like his elder brother paid nominal allegiance to Emperor Xian while maintaining autonomous rule over the Wu territories. In 208, Sun Quan allied with the warlord Liu Bei and they combined forces to defeat Cao Cao at the Battle of Red Cliffs. Sun Quan and Liu Bei maintained their alliance against Cao Cao after the battle for the next ten years or so, despite having some territorial disputes over Jing Province. In 219, Sun Quan severed ties with Liu Bei when he sent his general Lü Meng to invade Liu's territories in Jing Province. Guan Yu, defending Liu Bei's assets in Jing Province, was captured and executed by Sun Quan's forces.

After that, the boundaries of Sun Quan's domain extended from beyond the Jiangdong region to include the southern part of Jing Province, which covered present-day Hunan and parts of Hubei. In 220, Cao Cao's son and successor, Cao Pi, ended the Han dynasty by forcing Emperor Xian to abdicate in his favour and established the state of Cao Wei. Sun Quan agreed to submit to Wei and was granted the title of a vassal king, "King of Wu", by Cao Pi. A year Liu Bei declared himself emperor and founded the state of Shu Han. In 222, Liu Bei launched a military campaign against Sun Quan to take back Jing Province and avenge Guan Yu, leading to the Battle of Xiaoting. However, Liu Bei suffered a crushing defeat at the hands of Sun Quan's general Lu Xun and was forced to retreat to Baidicheng, where he died a year later. Liu Bei's successor, Liu Shan, his regent, Zhuge Liang, made peace with Sun Quan and reaffirmed their previous alliance. Sun Quan declared independence from Wei in 222, but continued to rule as "King of Wu" until 229, when he declared himself "Emperor of Wu".

His legitimacy was recognised by Shu. Sun Quan ruled for over his long reign resulted in stability in southern China. During his reign, Wu engaged Wei in numerous wars, including the battles of Ruxu and Hefei. However, Wu never managed to gain any territory north of the Yangtze River while Wei never succeeded in conquering the lands south of the Yangtze. A succession struggle broke out between Sun Quan's sons in the part of his reign — Sun Quan instated Sun He as the crown prince in 242 after his former heir apparent, Sun Deng, died in 241, but Sun He soon became involved in a rivalry with his younger brother, Sun Ba; the conflict resulted in the emergence of two rivalling factions, each supporting either Sun He or Sun Ba, in Sun Quan's imperial court. Sun Quan deposed Sun He and forced Sun Ba to commit suicide, while Lu Xun and many other ministers who took either Sun He's or Sun Ba's side in the struggle met with unhappy ends. Sun Quan appointed Sun Liang, as the crown prince after the incident.

Sun Quan was succeeded by Sun Liang, with Zhuge Ke and Sun Jun serving as regents. In 253, Zhuge Ke was assassinated in a coup launched by Sun Jun, the state power of Wu fell into Sun Jun's hands and was passed on to his cousin, Sun Chen, after his death. During Sun Liang's reign, two rebellions broke out in the Wei garrison at Shouchun in 255 and 257–258. Sun Jun and Sun Chen led Wu forces to support the rebels in the first and second rebellions in the hope of making some territorial gains in Wei, but both revolts were suppressed and the Wu forces retreated after suffering many losses. Sun Liang was deposed in 258 by Sun Chen, who installed Sun Xiu, another son of Sun Quan, on the throne. Sun Xiu killed Sun Chen in a coup with the help of Zhang Bu and Ding Feng. Sun Xiu died of illness in 264, a year. At the time, Wu was experiencing internal turmoil because rebellions had broken out in Jiaozhi in the south; the ministers Puyang Xing, Wan Yu and Zhang Bu decided to install Sun He's son, Sun Hao, on the throne.

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