Plot.

"Blood and Sand".

The story begins with an unnamed Thracian's involvement in a unit of Roman auxiliary in a campaign against the Getae (Thracian tribes that occupied the regions of the Lower Danube, in what today is Bulgaria and Romania, ancestors of Romanians) under the command of the "legatus", Claudius Glaber. In 72-71 BC, Roman general Marcus Terentius Varro Lucullus, proconsul of the Roman province of Macedonia, marched against the Getae, who were allies of Rome's enemy, Mithridates VI of Pontus. The Getae frequently raid the Thracians' lands, so the Thracians are persuaded by Glaber to enlist in the Romans' service as auxiliaries. Glaber is persuaded by his wife Ilithyia to seek greater glory, decides to break off attacking the Getae and directly confront the forces of Mithridates in Asia Minor. The Thracian, feeling betrayed, leads a mutiny against Glaber, and returns to find his village destroyed. The Thracian and his wife Sura are captured by Glaber the next day; the Thracian is condemned to die in the gladiator arena for his crime, while Sura is taken away, condemned to slavery. The Thracian is shipped to Capua in Italy, a center of gladiator training. Against all odds in the arena he slays the four gladiators appointed to execute him and becomes an instant sensation with the crowd. Senator Albinius commutes the punishment from death to slavery. The prisoner's true name unknown, Lentulus Batiatus, the owner of a "ludus" in Capua, suggests to name him "Spartacus", because he fought like the ferocious Thracian king of that name.  Noting well the Thracian's fierce raw talent and popularity with the masses, Batiatus purchases him for training within the walls of his "ludus" under the tutelage of Oenomaus, a former gladiator and fellow slave who is known to the gladiators as 'Doctore', meaning instructor. He is befriended by Varro, a Roman who sold himself into slavery in order to pay his debts and support his family. He is harassed by more senior gladiators, notably Crixus, an undefeated Gaul, and Barca, a Carthaginian. Spartacus soon learns that Sura was sold to a Syrian slave trader. Batiatus, who has been unable to control Spartacus during his first days of training, promises to find Sura and reunite them in exchange for the promising neophyte's cooperation in the arena.  After many near-fatal ordeals and much further training Spartacus attains the status of a living legend and is named the "Champion of Capua". Batiatus arranges the purchase of Sura, but she is delivered mortally wounded, supposedly having been waylaid by bandits en route. Her murder was secretly ordered by Batiatus to keep Spartacus loyal and focused. Spartacus casts off his heritage as a Thracian and forgets his dream of freedom, becoming content with life as champion. Meanwhile, Barca, wishing to buy his freedom, was slain by Batiatus with the help of slave and former gladiator, Ashur. Batiatus' wife, Lucretia, is conducting an affair with Crixus. She and Batiatus have been unable to conceive a child, though she later falls pregnant with Crixus' child.  The turning point comes when Spartacus is set to fight his only friend in the ludus, Varro, in an exhibition match celebrating the coming to manhood of the Capuan magistrate's son, Numerius. Ilithyia, who has hated Spartacus since he embarrassed her husband Glaber by his mutiny, seduces the young man and convinces him to demand death for the loser of the match. Spartacus wins (as expected), and when the young man gives the "thumbs down", Batiatus, wishing to ingratiate himself with the boy's powerful father, forces Spartacus to comply and kill Varro. While suffering from both his wound in this match and his remorse and sorrow over having to kill his friend, Spartacus has fever dreams that lead him to suspect that Batiatus arranged Sura's death. He is able to confirm this by forcing Batiatus' man, Aulus, to confess the act. Knowing that it is all or nothing when it comes to resistance of his enslavement, he resolves to "kill them all" and lead a revolt against the ruling house he once fought for.  In order to get his revenge, Spartacus enlists the help of Crixus and the rest of the gladiators to defeat the house of Batiatus once and for all. A battle to the death between Crixus and Spartacus is arranged for the Capuan elite at the ludus. Doctore (whom Batiatus refers to by his real name, Oenomaus) confronts Batiatus about Barca's death and Ashur's hand in it. Spartacus gains support from Mira, Batiatus' wife's slave, who is tasked with opening the gate to the villa from the training area. Crixus resists aiding Spartacus in hopes of reuniting with his lover Naevia; however, after learning he was weakened to ensure Spartacus' victory, at the last moment he joins with Spartacus. Doctore initially stops Spartacus from killing Batiatus. In the ensuing chaos of the gladiators' killing of the guards and some guests, Crixus persuades Doctore to join him with Spartacus, while Illithyia escapes and has her guards seal the door to the ludus from the outside. Doctore, making good on his word, tries to kill Ashur but his intended victim eludes him. Crixus grievously wounds Lucretia with a sword stab to her abdomen, piercing her womb and killing their unborn child. Varro's wife kills Numerius after revealing to him that Varro was her husband, and Spartacus finally kills Batiatus in front of the seriously wounded Lucretia. After the massacre, Spartacus vows to make "Rome tremble".

"Gods of the Arena".

The mini-series features the history of the House of Batiatus and the city of Capua before the arrival of Spartacus. The main story opens not long after Quintus Lentulus Batiatus becomes lanista, manager of the House's slaves and gladiators, when he takes over his father's ludus.  Batiatus is quickly discovered to have grand ambitions, beginning with the stepping out from his father's shadow. As the story progresses, Batiatus continues to seek greatness for the House under "his" leadership, as well as recognition for his own name. By his side stands his devoted wife, Lucretia, who is willing to help her husband achieve his goals regardless of the cost. Batiatus soon places all of his fortunes on one man whom he believes will bring fame and glory to the House of Batiatus, his best gladiator, the Celt Gannicus. Gannicus is a skilled warrior almost without equal, who wields his dual swords in the arena with great prowess. However, Batiatus' opponents would not sit idly and allow his ascent to greatness without challenge.  Purchased as an undisciplined and disheveled recruit in the first episode, Crixus the Gaul initially endures mockery and threats of death, before eventually rising to become a gladiator of skill and fame second only to Gannicus. As Batiatus fends off repeated attempts by his professional rival Tullius to obtain Gannicus, his relationships with his father Titus and friend Solonius begin to suffer the strain of Quintus' relentless ambition. Former champion gladiator Oenomaus reluctantly retires from combat to become Doctore, while Syrian recruits Ashur and Dagan become fierce enemies as Ashur tries to prove himself worthy of being a gladiator. Veteran gladiators Barca and Gannicus accept the rising star of Crixus but fear that their own careers will suffer, as the machinations of Batiatus and Lucretia to court Capua's elite end in tragedy for several members of the household. Against all of this, the city's splendid new arena nears completion and with it the opening games that will make men into gods. When the arena opens Solonius' and Batiatus' gladiators compete with each other, Batiatus' gladiators prevailing in the contest. Gannicus again proves himself to be the champion of Capua and the god of the arena and by virtue of his win against Solonius' gladiators, becomes the champion of Capua and gains his freedom.

"Vengeance".

After the bloody escape from the House of Batiatus that concluded "Spartacus: Blood and Sand", the gladiator rebellion begins to strike fear into the heart of the Roman Republic in "Spartacus: Vengeance". Praetor Claudius Glaber and his Roman troops are sent to Capua to crush Spartacus' growing band of freed slaves before they can inflict further damage. Spartacus is given a choice between satisfying his personal need for vengeance against the man who condemned his wife to slavery and eventual death, or making the larger sacrifices necessary to keep his budding army from breaking.

"War of the Damned".

This final season of Spartacus began airing January 25, 2013 and concluded April 12, 2013. This season follows the final struggle between Spartacus and Marcus Licinius Crassus. Crassus pursues Spartacus as he struggles to feed his ever-growing army of former slaves. Spartacus wins several victories against Crassus' forces and continues to frustrate the Romans. The series culminates in a direct all-out battle between Spartacus and Crassus.

Production.

After filming in early 2009 and promoting for some time, it was announced that Starz would premiere "Spartacus: Blood and Sand" on January 22, 2010. On December 22, 2009, a month before it premiered, it was announced that the show was renewed by Starz for a second season.  On March 9, 2010, IGN.com reported that production of Season 2 had been delayed due to star Whitfield being diagnosed with early-stage non-Hodgkin lymphoma. Due to the delay, Starz announced in May 2010 that it would be developing a six-episode prequel series, entitled "Spartacus: Gods of the Arena", to allow star actor Whitfield to seek medical treatment. The prequel featured both new and returning stars, headlined by John Hannah as Batiatus and Lucy Lawless as Lucretia. Star Andy Whitfield also briefly appeared in a voice-over role. Production began in New Zealand in the summer of 2010 and the prequel aired beginning January 2011.  In September 2010, Starz announced that Whitfield's cancer had returned and that he had decided not to return for the production of Season 2, then tentatively scheduled for September 2011. Starz announced that the show would nevertheless continue, and planned on recasting the role of Spartacus in the wake of Whitfield's exit. Whitfield gave his blessing for Starz to recast the role when he announced he would not return.  "Spartacus" series creator Steven S. DeKnight said in an interview, "There are a 'couple of very strong candidates' for the role of Spartacus, and season two should begin production in New Zealand in April 2011." DeKnight added that the "Spartacus" producers and Starz executives weren't always sure they would go forward without Andy Whitfield, who they said had brought "gravity and heart" to the role of the famous warrior. "It's unheard of to recast your titular character in a television show, and we did a lot of soul searching about whether we even wanted to try," DeKnight said. "And then Andy [Whitfield] said, 'I really think the show should go forward without me. I give you the blessing. I want this story told. On January 17, 2011, it was announced that Australian film and TV actor Liam McIntyre had been selected to replace Whitfield.  On February 26, 2011 interview with "Entertainment Weekly", DeKnight revealed that the second season was set to air "the end of January" 2012. Additionally, he revealed that Lesley-Ann Brandt, the actress who portrayed the slave Naevia, would also not be able to return for season 2 due to the delay in production. On August 1, 2011, Starz released a trailer indicating that the long-delayed second season would premiere in January 2012, under the new subtitle, "Spartacus: Vengeance".  Andy Whitfield died on September 11, 2011, from complications of non-Hodgkin lymphoma. He was 39 years old.  Over two months before the premiere, on November 7, 2011, Starz announced that it was renewing "Spartacus" for a third season–a second year with Liam McIntyre in the titular role; and fourth year on the air overall.  The second season eventually premiered on January 27, 2012.  On June 1, 2012 Starz released a teaser video of the next season on YouTube.  On June 4, 2012, Starz announced that the 4th season of Spartacus (War of the Damned) will be the final one for the series.  Starz released the first full trailer for "Spartacus: War of the Damned" on July 13, 2012.

Other media.

Board game.

In 2012 Gale Force Nine announced the creation of licensed board game based on the Spartacus series. The English language release of the game "Spartacus: A Game of Blood and Treachery" had a limited release at Gen Con 2012 and a general release to game and hobby stores on September 28, 2012.

Comics.

In 2009, Devil's Due published a four-part prequel comic series, titled "Spartacus – Blood And Sand". Each issue spotlighted a character from the upcoming television series, mostly the minor gladiator rivals of the main cast.  The series was adapted as a 4-part motion comic adaptation called "Spartacus – Blood and Sand – Motion Comic". Ray Park and Heath Freeman were cast. Kyle Newman was the director, and the producers were Andy Collen and Jeff Krelitz.

Novels.

In 2012 Titan Books announced the publication of a series of novels based on Spartacus: Blood and Sand. The first one, titled "Spartacus: Swords & Ashes", was written by J.M. Clements and released on January 3, 2012.  The second book in the series, "Spartacus: Morituri" by Mark Morris, was released in August 2012.

Video game.

In 2012, Ubisoft announced that they would be publishing a video game based on the series. The game, titled "Spartacus Legends", has been developed by Kung Fu Factory and was released on June 26, 2013 on Xbox Live and the PlayStation Network.

Home video releases.

The DVD and Blu-ray Disc sets of each season were released in various regions after their television broadcast.

Reception.

The premiere episode of the series set a record for Starz, with 553,000 viewers on their network, and another 460,000 on Encore, where the show was available only that weekend. For the rest of the season the show had an average of 1.285 million viewers. Critical reception of the first episode was mixed; the review aggregate website Metacritic which assigns a normalized average score out of 100 gave the show a score of 54% based on 22 reviews. Ken Tucker of "Entertainment Weekly" gave it the grade B+, saying it "might prove to be the not-at-all-guilty pleasure of the season." Barry Garron of "The Hollywood Reporter" suggested that with "such thin stories... it's small wonder that sex and violence are used to take up the slack." Robert Lloyd of the "Los Angeles Times" wrote that John Hannah as Batiatus "keeps the show grounded with a persuasive portrait of a man engaged in a stressful daily business." Mark Perigard of the "Boston Herald" gave the season finale a positive review, rating it a B+. He commented on the improvement of the series throughout its first season.

Broadcast.

The series aired in Canada on TMN beginning on January 25, 2010. RTL 5 announced in their January newsletter that "Spartacus: Blood and Sand" will debut in the Netherlands in March. In the United Kingdom, Bravo began airing the series on May 25, 2010. Following the closure of Bravo on UK television, Sky1 has now picked up the rights to the series and will air all subsequent seasons. In conjunction with the UK airings of Spartacus: War of the Damned on Monday nights at 10pm, a Tweet-a-long is held from the official Spartacus Twitter page, in which fans post tweets during the episode, using the hashtag #SpartacusWarOfTheDamned. The series is also scheduled to premiere in Poland on HBO Poland starting from June 19, 2010 and in Hungary on HBO Magyarország starting from June 1, 2010. The series airs in Ireland on TV3. In Brazil, the show airs on Globosat HD. In Turkey, the show releases on CNBC-E TV, while in Italy Sky Television is the broadcaster who gained the rights of the series. In India and Pakistan, the show is currently airing (as of June 2011) on HBO. In Slovenia, the series started airing on Kanal A on 2 January 2012, from Monday to Friday at 9.45 pm. In 2014, Syfy began to air an edited version of the series. In 2016 the series was made available on Netflix.

Spartacus is an American television series produced in New Zealand that premiered on Starz on January 22, 2010, and concluded on April 12, 2013. The fiction series was inspired by the historical figure of Spartacus, a Thracian gladiator who from 73 to 71 BC led a major slave uprising against the Roman Republic departing from Capua. Executive producers Steven S. DeKnight and Robert Tapert focused on structuring the events of Spartacus' obscure early life leading up to the beginning of historical records.

Spartakus (v anglickém originále Spartacus) je americký seriál televize Starz, jehož první epizoda byla odvysílána v USA v lednu 2010. Seriál je inspirován historickou postavou Spartaka (hraje Andy Whitfield), thráckým gladiátorem, který v letech 73–71 př. n. l. vedl největší povstání otroků proti Římské republice. Výkonní producenti Steven S. DeKnight a Robert Tapert se zaměřili na příběh Spartaka od časného života až do počátku historických záznamů. Objevuje se zde hodně násilí, silný sexuální obsah a hrubý jazyk.


Background.

The party was founded after Shmuel Flatto-Sharon had fled to Israel from France in 1976 after being charged with embezzling $60 million. In order to prevent his extradition, which France had requested, and despite hardly speaking Hebrew, he decided to run in the 1977 Knesset elections, hoping to gain parliamentary immunity.  Named after himself, Flatto-Sharon's party ran on right-wing populism (he was a supporter of the Gush Emunim settlement movement) and promises to provide apartments to young voters.  The new party won 2% of the vote, enough for two seats. However, as it was a one-man list, only one seat was taken. It has been suggested that the surprising level of support that the party won was a response to France's refusal to extradite Abu Daoud, who was wanted in Israel for the murder of 11 Israeli Olympic athletes in Munich. Once ensconced in the Knesset, Flatto-Sharon joined Menachem Begin's coalition, with one of his first acts being to vote in favour of a law that prohibited the extradition of Israeli citizens.  Later during the Knesset session he renamed the party "Development and Peace" and also helped in negotiations to free Israeli POWs. However, after serving for two years, Flatto-Sharon was convicted of vote buying, sentenced to nine months in prison and suspended from the Knesset. After several appeals, he was finally jailed for three months in 1984.  The party ran in the 1981 and 1984 elections, but did not pass the electoral threshold of 1% in either, and subsequently disappeared. Flatto-Sharon was never extradited to France to serve the five-year sentence given to him in his absence, as the five-year statute of limitations had expired. He was an established businessman and billionaire until his death in 2018.

Development and Peace (, "Pituah VeShalom"), originally known as Flatto-Sharon (Hebrew: פלאטו שרון) was a right wing one-man political party in Israel.

Pituach ve-šalom (: ; "Rozvoj a mír") byla izraelská politická strana existující v letech 1977-1984. Původně se jmenovala Flatto-Šaron (פלאטו שרון), podle jejího zakladatele, kterým byl Šmu'el Flatto-Šaron.


Statement.

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:  This can be generalized to an arbitrary finite dimension:  A slightly more general version is as follows:  An even more general form is better known under a different name:

Importance of the pre-conditions.

The theorem holds only for sets that are "compact" (thus, in particular, bounded and closed) and "convex" (or homeomorphic to convex). The following examples show why the pre-conditions are important.

Boundedness.

Consider the function which is a continuous function from formula_9 to itself. As it shifts every point to the right, it cannot have a fixed point. The space formula_9 is convex and closed, but not bounded.

Closedness.

Consider the function which is a continuous function from the open interval (−1,1) to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. The space (−1,1) is convex and bounded, but not closed. The function "f" have a fixed point for the closed interval [−1,1], namely "f"(1) = 1.

Convexity.

Convexity is not strictly necessary for BFPT. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball formula_12. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).  The following example shows that BFPT doesn't work for domains with holes. Consider the function formula_13, which is 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 (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex). The function "f" have a fixed point for the unit disc, since it takes the origin to itself.  A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.

Notes.

The continuous function in this theorem is not required to be bijective or even surjective.

Illustrations.

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 (without ripping or tearing) 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 then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) 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 the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.  2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.  3. In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.

Intuitive approach.

Explanations attributed to Brouwer.

The theorem is supposed to have originated from Brouwer's observation of a cup of coffee. If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving. The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.  Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet." Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

One-dimensional case.

In one dimension, the result is intuitive and easy to prove. The continuous function "f" is defined on a closed interval ["a", "b"] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval ["a", "b"] which maps "x" to "x" (light green).  Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function "g" which maps "x" to "f"("x") - "x". It is ≥ 0 on "a" and ≤ 0 on "b". By the intermediate value theorem, "g" has a zero in ["a", "b"]; this zero is a fixed point.  Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."

History.

The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case "n" = 3 first was proved by Piers Bohl in 1904 (published in "Journal für die reine und angewandte Mathematik"). It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known.

Prehistory.

To understand the prehistory of Brouwer's fixed point theorem one needs to pass through differential equations. At the end of the 19th century, the old problem of the stability of the solar system returned into the focus of the mathematical community. Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge." He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".  He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow? Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle. Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval "t". If the area is a circular band, or if it is not closed, then this is not necessarily the case.  To understand differential equations better, a new branch of mathematics was born. Poincaré called it "analysis situs". The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group. This method can be used for a very compact proof of the theorem under discussion.  Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem. Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.

First proofs.

At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations. In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed.  It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially mathematical logic and topology. His initial interest lay in an attempt to solve Hilbert's fifth problem. In 1909, during a voyage to Paris, he met Henri Poincaré, Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point. These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem. The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as homotopy, the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."  Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension. In the context of this work, Brouwer also generalized the Jordan curve theorem to arbitrary dimension and established the properties connected with the degree of a continuous mapping. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became algebraic topology.

Reception.

The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of topological manifolds and is often used to prove other important results such as the Jordan curve theorem.  Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the "n"-dimensional sphere to R has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces. This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to multivalued functions. One also meets the theorem and its variants outside topology. It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. The theorem can also be found in existence proofs for the solutions of certain partial differential equations.  Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white. In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria (Hotelling's law), financial equilibria and incomplete markets.  Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are not constructive, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory. Brouwer disavowed his original proof of the fixed-point theorem. The first algorithm to approximate a fixed point was proposed by Herbert Scarf. A subtle aspect of Scarf's algorithm is that it finds a point that is by a function "f", but in general cannot find a point that is close to an actual fixed point. In mathematical language, if is chosen to be very small, Scarf's algorithm can be used to find a point "x" such that "f"("x") is very close to "x", i.e., formula_14. But Scarf's algorithm cannot be used to find a point "x" such that "x" is very close to a fixed point: we cannot guarantee formula_15 where formula_16 Often this latter condition is what is meant by the informal phrase "approximating a fixed point".

Proof outlines.

A proof using degree.

Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping. Modern accounts of the proof can also be found in the literature.  Let formula_17 denote the closed unit ball in formula_18 centered at the origin. Suppose for simplicitly that formula_19 is continuously differentiable. A regular value of formula_1 is a point formula_21 such that the Jacobian of formula_1 is non-singular at every point of the preimage of formula_23. In particular, by the inverse function theorem, every point of the preimage of formula_1 lies in formula_25 (the interior of formula_26). The degree of formula_1 at a regular value formula_21 is defined as the sum of the signs of the Jacobian determinant of formula_1 over the preimages of formula_23 under formula_1:  The degree is, roughly speaking, the number of "sheets" of the preimage "f" lying over a small open set around "p", with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of winding number to higher dimensions.  The degree satisfies the property of "homotopy invariance": let formula_1 and formula_34 be two continuously differentiable functions, and formula_35 for formula_36. Suppose that the point formula_23 is a regular value of formula_38 for all "t". Then formula_39.  If there is no fixed point of the boundary of formula_26, then the function  is well-defined, and  formula_42  defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so formula_34 also has degree one at the origin. As a consequence, the preimage formula_44 is not empty. The elements of formula_44 are precisely the fixed points of the original function "f".  This requires some work to make fully general. The definition of degree must be extended to singular values of "f", and then to continuous functions. The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature.

A proof using homology.

The proof uses the observation that the boundary of the "n"-disk "D" is "S", the ("n" − 1)-sphere. Suppose, for contradiction, that a continuous function "f" : "D" → "D" has "no" fixed point. This means that, for every point x in "D", the points "x" and "f"("x") are distinct. Because they're distinct, for every point x in "D", we can construct a unique ray from "f"("x") to "x" and follow the ray until it intersects the boundary "S" (see illustration). By calling this intersection point "F"("x"), we define a function "F" : "D" → "S" sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point "F"("x") must be "x".  Consequently, F is a special type of continuous function known as a retraction: every point of the codomain (in this case "S") is a fixed point of "F".  Intuitively it seems unlikely that there could be a retraction of "D" onto "S", and in the case "n" = 1, the impossibility is more basic, because "S" (i.e., the endpoints of the closed interval "D") is not even connected. The case "n" = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce an injective group homomorphism from the fundamental group of "S" to that of "D", but the first group is isomorphic to Z while the latter group is trivial, so this is impossible. The case "n" = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.  For "n" > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: the homology "H"("D") is trivial, while "H"("S") is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.

A proof using Stokes' theorem.

To prove that a map has fixed points, one can assume that it is smooth, because if a map has no fixed points then convolving it with a smooth function of sufficiently small support and integral equal to one produces a smooth function with no fixed points. As in the proof using homology, one is reduced to proving that there is no smooth retraction "F" from the ball "B" onto its boundary "∂B". If ω is a volume form on the boundary then by Stokes' Theorem, giving a contradiction.  More generally, this shows that there is no smooth retraction from any non-empty smooth orientable compact manifold onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology (or rather cohomology), because the form ω generates the de Rham cohomology group "H"("∂B") used in the cohomology proof.

A combinatorial proof.

The BFPT can be proved using Sperner's lemma. We now give an outline of the proof for the special case in which "f" is a function from the standard "n"-simplex, formula_47 to itself, where  For every point formula_49 also formula_50 Hence the sum of their coordinates is equal:  Hence, by the pigeonhole principle, for every formula_49 there must be an index formula_53 such that the formula_54th coordinate of formula_55 is greater than or equal to the formula_54th coordinate of its image under "f":  Moreover, if formula_55 lies on a "k"-dimensional sub-face of formula_47 then by the same argument, the index formula_54 can be selected from among the coordinates which are not zero on this sub-face.  We now use this fact to construct a Sperner coloring. For every triangulation of formula_47 the color of every vertex formula_55 is an index formula_54 such that formula_64  By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an "n"-dimensional simplex whose vertices are colored with the entire set of available colors.  Because "f" is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point formula_55 which satisfies the labeling condition in all coordinates: formula_66 for all formula_67  Because the sum of the coordinates of formula_55 and formula_69 must be equal, all these inequalities must actually be equalities. But this means that:  That is, formula_55 is a fixed point of formula_72

A proof by Hirsch.

There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map "f" can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem, for example. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.  R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point, "q", on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from "q" to a fixed point. It is an easy numerical task to follow such a path from "q" to the fixed point so the method is essentially computable. gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

A proof using oriented area.

A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If formula_73 is a smooth retraction, one considers the smooth deformation formula_74 and the smooth function Differentiating under the sign of integral it is not difficult to check that ""("t") = 0 for all "t", so "φ" is a constant function, which is a contradiction because "φ"(0) is the "n"-dimensional volume of the ball, while "φ"(1) is zero. The geometric idea is that "φ"("t") is the oriented area of "g"("B") (that is, the Lebesgue measure of the image of the ball via "g", taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter "t" passes form 0 to 1 the map "g" transforms continuously from the identity map of the ball, to the retraction "r", which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of "r" is necessarily 0, as its image is the boundary of the ball, a set of null measure.

A proof using the game hex.

A quite different proof given by David Gale is based on the game of Hex. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed-point theorem for dimension 2. By considering "n"-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.

A proof using the Lefschetz fixed-point theorem.

The Lefschetz fixed-point theorem says that if a continuous map "f" from a finite simplicial complex "B" to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number and in particular if the Lefschetz number is nonzero then "f" must have a fixed point. If "B" is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero homology group is :formula_77 and "f" acts as the identity on this group, so "f" has a fixed point.

A proof in a weak logical system.

In reverse mathematics, Brouwer's theorem can be proved in the system WKL, and conversely over the base system RCA Brouwer's theorem for a square implies the weak König's lemma, so this gives a precise description of the strength of Brouwer's theorem.

Generalizations.

The Brouwer fixed-point theorem forms the starting point of a number of more general fixed-point theorems.  The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space l of square-summable real (or complex) sequences, consider the map "f" : l → l which sends a sequence ("x") from the closed unit ball of l to the sequence ("y") defined by It is not difficult to check that this map is continuous, has its image in the unit sphere of l, but does not have a fixed point.  The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.  There is also finite-dimensional generalization to a larger class of spaces: If formula_79 is a product of finitely many chainable continua, then every continuous function formula_80 has a fixed point, where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement formula_81, such that formula_82 if and only if formula_83. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.  The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in R, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.  The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of "D".

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function formula_1 mapping a compact convex set to itself there is a point formula_2 such that formula_3. The simplest forms of Brouwer's theorem are for continuous functions formula_1 from a closed interval formula_5 in the real numbers to itself or from a closed disk formula_6 to itself. A more general form than the latter is for continuous functions from a convex compact subset formula_7 of Euclidean space to itself.

Browerova věta o pevném bodu je v matematice tvrzení, že pro každé spojité zobrazení "f" z uzavřené koule do sebe existuje bod "x" takový, že formula_1. Takový bod se nazývá pevný bod "f".


Description.

A computer with a von Neumann architecture stores program data and instruction data in the same memory; a computer with a Harvard architecture has separate memories for storing program and data. Both are stored-program designs.  "Stored-program computer" is sometimes used as a synonym for von Neumann architecture, however Professor Jack Copeland considers that it is "historically inappropriate, to refer to electronic stored-program digital computers as 'von Neumann machines'". Hennessy and Patterson write that the early Harvard machines were regarded as "reactionary by the advocates of stored-program computers".

History.

The stored-program computer idea can be traced back to the 1936 theoretical concept of a universal Turing machine. Von Neumann was aware of this paper, and he impressed it on his collaborators as well.  Many early computers, such as the Atanasoff–Berry computer, were not reprogrammable. They executed a single hardwired program. As there were no program instructions, no program storage was necessary. Other computers, though programmable, stored their programs on punched tape, which was physically fed into the machine as needed.  In 1936 Konrad Zuse anticipated in two patent applications that machine instructions could be stored in the same storage used for data.  The University of Manchester's Baby is generally recognized as world's first electronic computer that ran a stored program—an event that occurred on 21 June 1948. However the Baby was not regarded as a full-fledged computer, but more a proof of concept predecessor to the Manchester Mark 1 computer, which was first put to research work in April 1949. On 6 May 1949 the EDSAC in Cambridge ran its first program, making it another electronic digital stored-program computer. It is sometimes claimed that the IBM SSEC, operational in January 1948, was the first stored-program computer; this claim is controversial, not least because of the hierarchical memory system of the SSEC, and because some aspects of its operations, like access to relays or tape drives, were determined by plugging. The first stored-program computer to be built in continental Europe was the MESM, completed in the Soviet Union in 1951.

The first stored-program computer.

Several computers could be considered the first stored-program computer, depending on the criteria.

Charles "Lucky" Luciano (, ; born Salvatore Lucania ; November 24, 1897 – January 26, 1962) was an Italian-born gangster, who operated mainly in the United States. Luciano started his criminal career in the Five Points gang and was instrumental in the development of the National Crime Syndicate. Luciano is considered the father of modern organized crime in the United States for the establishment of The Commission in 1931, after he abolished the boss of bosses title held by Salvatore Maranzano following the Castellammarese War. He was also the first official boss of the modern Genovese crime family.

Eulerova charakteristika je v matematice celé číslo, které charakterizuje nějaký topologický prostor, geometrický útvar, graf, mnohostěn a podobně. Toto číslo je "topologický" a dokonce i "homotopický invariant".


Playing career.

Junior.

Crawford was drafted 52nd overall in the 2003 NHL Entry Draft by the Chicago Blackhawks. He spent his pre-junior career playing for the Midget AAA Gatineau Intrépides, before spending his junior career with the Moncton Wildcats of the Quebec Major Junior Hockey League (QMJHL). Crawford was the goaltender for the Wildcats in the 2003–04 QMJHL playoffs, when they ousted the Rimouski Océanic in the semi-finals to advance to the President's Cup final, but lost that series to the Gatineau Olympiques. Crawford currently holds the Wildcats record for lowest goals against average (GAA) (2.47 in 2004–05), most wins (35 in 2003–04) and is tied with Simon Lajeunesse for most shutouts in a season (six in 2004–05). He was twice named to the QMJHL's Second All-Star team (2003–04, 2004–05).

Professional.

Crawford made his NHL debut with the Blackhawks on January 22, 2006, against the Minnesota Wild. He relieved Adam Munro in the third period, made seven saves and did not allow any goals. He recorded his first career start against the St. Louis Blues on February 2, 2006. He finished the evening with five goals allowed and 29 saves in a 6–5 shootout loss. He recorded his first career win and shutout against the Anaheim Ducks on March 5, 2008. Crawford also recorded a strong performance against the Detroit Red Wings, stopping 45 of 47 shots in a 3–1 loss on March 11. The Blackhawks re-signed Crawford to a one-year deal on July 21, 2008. Crawford was recalled from the American Hockey League (AHL) on November 28 to take the place of Nikolai Khabibulin, who had been injured two nights earlier. Crawford made his first Stanley Cup playoffs appearance on May 24, 2009, in the second period of the Western Conference Finals against the Detroit Red Wings, replacing Cristobal Huet. He stopped six of seven shots before being replaced by Huet after the second period.  Due to the departures of Antti Niemi to the San Jose Sharks and Cristobal Huet to the Swiss National League A, Crawford was promoted to back-up goaltender behind Marty Turco for the 2010–11 season. Later in the season, Crawford became the team's starting goaltender, replacing the veteran Turco. He had a two-game shutout streak later in the season and compiled a 176:09 shutout streak from January 7 to 15, the longest by a Blackhawks netminder since Tony Esposito from January 16 to 30, 1972. From February 20 until March 5, 2011, Crawford had an eight-game win streak, setting a new record for the longest such streak for a rookie. The streak was also one game short of the team record set by Glenn Hall.  Crawford earned his 30th win of his rookie season on March 28, 2011, at Detroit, becoming the first Blackhawks goaltender to reach the 30-win milestone since Jocelyn Thibault did it with 33 wins in the 2001–02 season. In his first playoff series against the Presidents' Trophy-winning Vancouver Canucks, he recorded a 36-save shutout in game five of the 2011 Western Conference Quarter-finals, marking the first time he recorded a shutout in the NHL playoffs, and the Blackhawks' largest margin of victory (5–0) in franchise history in a playoff game. The Blackhawks ultimately lost the series to the Canucks in seven games, however, when Alexandre Burrows scored the series-winning overtime goal for Vancouver. On May 19, 2011, Crawford signed a three-year, $8 million contract extension with the Blackhawks. After the 2010–11 season, he was named to the NHL All-Star Rookie Team.  Following the 2010–11 regular season, Crawford entered the year as the Blackhawks' number one netminder. After starting the year off strong, Crawford struggled for a majority of the season and was often benched in favour of veteran backup Ray Emery, who played extensively down the stretch. However, Emery could not solidify his hold on the starter's role and as a result, Crawford reclaimed the team's starting goaltender position. After going 8–1–2 over his last 11 games, Crawford was confirmed as the starter for the Blackhawks heading into the 2012 Stanley Cup playoffs. In all, Crawford finished his second full season playing in 57 games and posting a 30–17–7 record with a.903 save percentage and 2.72 GAA with no shutouts. He became the first Blackhawks goalie to win at least 30 games in back-to-back seasons since Ed Belfour did it in the 1992–93 and 1993–94 seasons. Entering the lock-out-shortened 2012–13 season, there were doubts as to whether Crawford could return to his stellar form from his rookie season. The team re-signed Ray Emery to give him a push for the starting job. The Blackhawks started off the season with a point streak of 24 consecutive games (21–0–3), with Crawford getting off to a strong 11–0–3 start before struggling in a loss to the Colorado Avalanche in which he was pulled midway through the game after allowing three goals on 16 shots. He finished the season with a record of 19–5–5, with a 1.94 GAA, and a.926 save percentage with three shutouts. Due to his efforts, he was awarded the William M. Jennings Trophy along with Ray Emery, who finished with a 1.94 GAA as well. Crawford was confirmed as the starter for the 2013 Stanley Cup playoffs and started every game as Chicago eventually won the Stanley Cup, prevailing over the Boston Bruins in six games. On September 2, 2013, Crawford signed a six-year, $36 million contract extension with the Blackhawks through the 2019–20 season.  Crawford started the 2014–15 season with a 12–5–1 record. He injured his foot, however, on December 1, 2014, while attending a Rise Against concert. Due to this, Crawford was expected to miss two-to-three weeks. Crawford was selected to play in the 2015 NHL All-Star Game, one of six players selected via fan-vote (the others being his teammates Duncan Keith, Brent Seabrook, Patrick Kane and Jonathan Toews, as well as the Buffalo Sabres' Zemgus Girgensons). On March 23, 2015, Crawford became the fourth goalie in Blackhawks franchise history to have four 30-win seasons. Crawford and Carey Price of the Montreal Canadiens, won the William M. Jennings Trophy for the 2014–15 season. He finished the season with 32–20–5, recording a 2.27 GAA, a.924 SV % and two shut outs. After allowing nine goals in two first round playoff games against the Nashville Predators, Crawford was benched in favour of rookie goaltender Scott Darling, who posted three wins and one loss against the Predators. Crawford relieved Darling in game six of the series after the Predators scored three goals in the first period. Crawford stopped all 13 shots he faced en route to a 4–3 win to take the series. Crawford was named the starter over Darling for the second round against Minnesota. Crawford started all four games in the second round sweep over Minnesota, allowing 7 goals in those 4 games. Crawford also performed well in the following round versus the Anaheim Ducks, leading Chicago to a come-from-behind four games to three series win. On June 15, 2015, Crawford collected his second shutout in a 2–0 victory over the Tampa Bay Lightning in Game 6 of the 2015 Stanley Cup Finals to give the Blackhawks their third Stanley Cup in the last six years. The win was also the second Stanley Cup for Crawford in his career.  During the 2015–16 NHL season, Crawford appeared in 58 games for the Blackhawks. He posted a 2.37 goals-against average, a.924 save percentage, 35 wins, and seven shutouts. In March 2016, The Blackhawks announced that Crawford would miss indefinite amount of time due to an unspecified upper-body injury. Crawford finished the year with a career high 35 wins and tied his career high save percentage at.924. He also had a career high seven shutouts, which also led the NHL among goalies. He broke Tony Esposito's franchise playoff record for wins, with his 46th playoff win after stopping 29 shots in a 3-2 Game 2 victory over the St. Louis Blues. The Blackhawks would be eliminated from the playoffs by the Blues in seven games.  Crawford underwent emergency appendectomy surgery on December 3, 2016, in Philadelphia on the morning before a road game against the Flyers. He returned to the Blackhawks lineup on December 23, 2016. Crawford was selected to play 2017 NHL All-Star Game. Crawford finished the season a 32–18 record, while recording a 2.55 GAA and a.918 save percentage. The Blackhawks won the Central Division, but were swept by the Nashville Predators in the first round of the playoffs.  Crawford started the 2017–18 NHL season ranking seventh in the NHL with a 2.29 goals-against average and fourth with a.930 save percentage in 21 appearances. On December 1, 2017, the Blackhawks placed Crawford on IR after he sustained a lower-body injury the previous night. Crawford returned after missing three games, but sustained an undisclosed upper-body injury after playing in seven games. The Blackhawks stated Crawford would be out indefinitely, but were hopeful that he could return before the end of the season. However, the Blackhawks fell into last place in the Central Division, and were eliminated from playoff contention in early March. The Blackhawks left Crawford on IR for the remainder of the season.  Crawford returned to the Blackhawks on October 18, 2018 after missing 52 games. He collected his first win of the 2018–19 Chicago Blackhawks season on October 20, in a 4–1 victory over the Columbus Blue Jackets. Crawford suffered another serious concussion on December 16 after San Jose Sharks forward Evander Kane shoved teammate Dylan Strome into him. The Blackhawks placed Crawford on IR again. Crawford recorded a.902 save percentage and 3.28 goals-against average up to that point in the season. On February 25, 2019, The Blackhawks activated Crawford from IR after missing 28 games. His strongest performance of the campaign came on March 16, when made a career-high 48 saves en route to a 2–0 win against the Montreal Canadiens. Crawford posted a 7–4–3 record in his next 15 starts and finished the 2019-18 season with a.908 save percentage and 2.77 goals against average. Crawford missed the Blackhawks' season finale after sustaining a groin injury during the team's penultimate game.

Personal life.

A native of Châteauguay, Quebec, Crawford speaks both French and English fluently.  He and his wife, Kristy, have a two sons: Cooper, born in 2017, and Camden Emery, born in 2020.

Corey Crawford (born December 31, 1984) is a Canadian professional ice hockey goaltender currently playing for the Chicago Blackhawks of the National Hockey League (NHL). He was selected in the second round, 52nd overall, by the Blackhawks in the 2003 NHL Entry Draft. His teammates and fans often refer to him as "Crow". Crawford has won two Stanley Cup championships with the Blackhawks in 2013 and 2015.

Corey Crawford (* 31. prosince 1984, Montréal, Québec, Kanada) je kanadský hokejový brankář hrající v týmu Chicago Blackhawks v severoamerické lize NHL, který dovedl tým Chicaga Blackhawks ke zisku Stanley Cupu v letech 2013 a 2015. Byl součástí výběru Kanady, který vyhrál Světový pohár v roce 2016.


Early life.

Schumacher was born in New York City, the son of Marian (née "Kantor") and Francis Schumacher. His mother was a Swedish Jew, whereas his father was a Baptist from Knoxville, Tennessee, who died when Joel was four years old. Schumacher studied at Parsons The New School for Design and the Fashion Institute of Technology in New York. After first working in the fashion industry, he realized his true love was in filmmaking. He moved to Los Angeles, where he began his media work as a costume designer in films such as Woody Allen's "Sleeper" and "Interiors" and developed his skills with television work while earning an MFA from UCLA.  Schumacher's first screenplay was for the musical drama "Sparkle" in 1976, which Schumacher had developed with Howard Rosenman before moving to Los Angeles. He also wrote the screenplays for the 1976 low-budget hit movie "Car Wash", 1978's "The Wiz"—an adaptation of the stage play of the same name—and a number of other minor successes. His film directorial debut was "The Incredible Shrinking Woman" in 1981, which starred Lily Tomlin.

Career.

The Brat Pack.

The Brat Pack films "St. Elmo's Fire" and "The Lost Boys" were two of Schumacher's biggest hits. Their style impressed audiences and their financial success allowed studios to trust him with ever-larger projects. He states in the director's commentary for "St. Elmo's Fire" that he resents the "Brat Pack" label, as he feels it misrepresents the group.

John Grisham.

Schumacher has directed two adaptations of John Grisham's novels: "The Client" (1994) and "A Time to Kill" (1996). Grisham personally requested that Schumacher return to direct "A Time to Kill".

Batman.

Schumacher replaced Tim Burton as the director of the Batman film franchise when he directed "Batman Forever" in 1995. Val Kilmer replaced Michael Keaton in the title role. Despite mixed to negative critical reception, the film scored the highest-grossing opening weekend of 1995. It finished as the second-highest-grossing film of the year in North America, and sixth-highest worldwide.  After this success, Warner Bros. hired Schumacher to direct a sequel, "Batman & Robin", which was released in 1997. The film did not perform as well at the box office as its predecessors, and was critically panned; it is frequently considered to be one of the worst films ever made. Warner Bros. subsequently put the Batman film series on hiatus for several years, canceling Schumacher's next planned "Batman" movie, "Batman Unchained". On the DVD commentary, Schumacher has admitted that his movie disappointed fans of darker "Batman" adaptations, saying that the film was made intentionally marketable (or "toyetic") and kid-friendly. He claims to have been under heavy pressure from the studio to do so; however, he admits full responsibility and, at one point, apologized to any fans who were disappointed. Schumacher is a devoted Batman fan himself, and has said he would have preferred to work on an adaptation of the comic "".  Schumacher also served as the director for the music videos of two songs appearing in the franchise: "Kiss from a Rose", by Seal, and "The End Is the Beginning Is the End", by The Smashing Pumpkins, which he co-directed with Jonathan Dayton and Valerie Faris.

Post-Batman career.

After back-to-back Grisham and Batman films, Schumacher decided to reinvent his career with darker, lower-budget fare like "8mm" with Nicolas Cage, and "Flawless" with Robert De Niro. "8mm" was entered into the 49th Berlin International Film Festival.  In 1999, Schumacher also directed the music video for "Letting the Cables Sleep" by English rock band Bush. In 2000, Schumacher directed the Vietnam-era boot camp drama "Tigerland", which introduced Hollywood to a young Colin Farrell. Kirk Honeycutt of "The Hollywood Reporter" praised the film. He wrote: ""Tigerland" lands squarely in the top tier of best movies about America's Vietnam experience."  Schumacher returned to big-budget Hollywood with "Bad Company" starring Anthony Hopkins and Chris Rock. The film was originally slated to be released in November 2001, but after the September 11 attacks it was pushed back to the summer of 2002 because of its theme about terrorist attacks in New York City. The film was panned by most critics and was a box office failure. In 2003, he released the controversial "Phone Booth", in which he once again worked with Farrell. The film—about an unseen gunman tormenting a publicist—was also delayed for months due to the Beltway sniper attacks. It received generally positive reviews, earning a 71 percent "Fresh" rating on the review aggregator Rotten Tomatoes. Buoyed by Farrell's recently found fame, the film earned $98.7 million worldwide.  In 2002, he directed Cate Blanchett in the Jerry Bruckheimer-produced biopic "Veronica Guerin". The film is about the eponymous Irish journalist, who was murdered by drug dealers in 1996.  Schumacher directed a film version of the musical "The Phantom of the Opera" in 2004, an adaptation of Andrew Lloyd Webber's original stage musical. Despite mixed reviews, the film earned $154.6 million worldwide (Schumacher's biggest hit of the 21st century to date) and was nominated for three Academy Awards, as well as three Golden Globes, including Best Motion Picture – Musical or Comedy.  Schumacher directed "The Number 23" in 2007, which was a critical flop but a moderate financial success. His next project was the vampire thriller "Blood Creek", which was filmed in the spring of 2007 in rural Romania. It had a limited release.  In August 2008, Schumacher directed the music video for American rock band Scars on Broadway, for their single "World Long Gone".  In December 2006, Schumacher was attached to direct the film version of "The Minds of Billy Milligan". The projected release date was supposed to be in 2008, but any film based on the book wasn't released and its author Daniel Keyes died in 2014.  In October 2011, Schumacher released his latest film, "Trespass". The action-thriller reunited Schumacher with stars Nicole Kidman and Nicolas Cage.  He was next slated to direct the film "The Hive" (retitled "The Call" for release), but left the project for an undisclosed reason, replaced by Brad Anderson.  Joel Schumacher is friends with David Fincher, and directed two episodes of the first season of "House of Cards", which Fincher produced.

Personal life.

Schumacher has been openly gay throughout most of his career. According to Schumacher, this fact has been purposely reflected as a statement in many of his films. Schumacher claims that he has had sex with up to 20,000 men.  When asked by the BBC if he believed in God, Schumacher said: Politically, Schumacher donated thousands of dollars to Democratic candidates.

Recurring collaborators.

Schumacher often casts the same actors in different films. Nicole Kidman, Jim Carrey, Michael Paul Chan, Nicolas Cage, John Diehl, Julia Roberts, Kiefer Sutherland, Colin Farrell, and Shea Whigham are among his more frequent acting collaborators.  Harry Gregson-Williams often composes the music for his films and Mark Stevens often serves as editor.

Joel T. Schumacher (; born August 29, 1939) is an American filmmaker. Schumacher rose to fame after directing three hit films: "St. Elmo's Fire" (1985), "The Lost Boys" (1987), and "Flatliners" (1990). He later went on to direct the John Grisham adaptations "The Client" (1994) and "A Time to Kill" (1996). His films "Falling Down" (1993) and "8mm" (1999) competed for Palme d'Or and Golden Bear, respectively.

Joel Schumacher (* 29. srpna 1939, New York, USA) je americký filmový scenárista, režisér a producent, původním povoláním oděvní návrhář.


Playing career.

Amateur.

As a youth, Quick played in the 2000 Quebec International Pee-Wee Hockey Tournament with the New York Rangers minor ice hockey team. He later played for the Mid Fairfield youth hockey association out of Darien Ice Rink. He carried his team to win a national championship and he was single time during his midget major year. His team also featured forwards Sean Backman, John Mori and Augie DiMarzo, Joey Sides, Joe DeBello, Chris Davis, Cam Atkinson who all went on to play professional hockey.  Quick played at Hamden High School in Hamden, Connecticut, before transferring to Avon Old Farms, where he was named to the 2002 New Haven Register All-Area Ice Hockey Team. Quick's number 32 Kings jersey is displayed in the Lou Astorino Ice Arena of Hamden. He led Avon Old Farms to two-straight New England Prep Championships in his junior and senior seasons. He had nine shutouts during his senior year.  Quick played collegiate hockey for the University of Massachusetts Amherst out of Hockey East (HE). As a freshman, he earned his first victory by a score of 4–2 on October 15, 2005, in his first start against Clarkson University. He also appeared in one playoff game as a freshman, which the Minutemen lost 4–1 to Boston University. Quick scored his first goal January 6, 2007, against Merrimack College. In the 4–2 win, the unassisted goal came during a delayed penalty into an empty net at 9:31 of the second period; it was the game-winning goal. While playing for the Minutemen, he led the team to their first-ever NCAA Men's Ice Hockey Championship appearance. In his first NCAA Tournament game, Quick shut-out Clarkson University, stopping all 33 shots he faced, in a 1–0 overtime victory. He appeared in five playoff games and posted a.944 save percentage.

Professional.

After losing in his first professional start, with the ECHL's Reading Royals, Quick made a statistically unlikely goal during a shutout in his second start, scoring an empty-net goal against the Pensacola Ice Pilots on October 27, 2007, at 19:25 in the third period.  Quick began the 2008–09 season with the Los Angeles Kings' American Hockey League (AHL) affiliate, the Manchester Monarchs, sharing time with Jonathan Bernier. He was called up to the Kings on December 16 after goaltender Erik Ersberg suffered a groin injury. Quick played his first NHL game on December 6, 2007, against the Buffalo Sabres in an 8–2 win. He earned his first career NHL shutout on December 23, 2008, defeating the Columbus Blue Jackets in Columbus. He also received the NHL's Third Star of the Week honor for the week ending December 28. In three games, Quick posted a 2–1–0 record, with both victories coming via shutout, posting a 0.67 goals against average (GAA) and a.958 save percentage. On February 9, 2009, he was awarded the First Star of the Week by the NHL after stopping 95 of 100 shots in a three-game road win streak, which launched the Kings back into the 2009 playoff picture. He finished the season with a 21–18–2 record, with a 2.48 GAA and.914 save percentage. He earned his first career NHL Stanley Cup playoff shutout on April 17, 2011, against the San Jose Sharks, stopping all 34 shots for a 4–0 shutout win.  On April 25, 2012, after the end of the 2011–12 regular season, Quick was nominated as a Vezina Trophy finalist, along with Henrik Lundqvist and Pekka Rinne, the former of which ultimately won the award. Quick was also named a 2011–12 NHL Second Team All-Star. He led the League with ten shutouts (a Kings franchise record), had the NHL's second-lowest GAA (1.95) and posted a 35–21–13 final record. Quick gained much attention during the 2012 Stanley Cup playoffs, as the eighth-seeded Kings defeated the top-seeded and Presidents' Trophy-winning Vancouver Canucks in five games, followed by the Kings' first-ever four-game playoff series sweep against the second-seeded St. Louis Blues.  On May 30, 2012, Quick set a new Stanley Cup playoff record after winning his 11th consecutive road game dating back to the 2011 postseason, surpassing the old mark of ten set by Billy Smith of the New York Islanders, and continued adding to the record. The streak ran from April 16, 2011, at San Jose to June 2, 2012, at New Jersey (12 games). Los Angeles qualified for the Western Conference Final for the second time in its 45-year history, and first time since the 1993 playoffs. The Kings ended its 19-year drought and made it to the Stanley Cup Finals after defeating the third-seeded Phoenix Coyotes in Game 5 of the Western Conference Final. Quick was awarded the Conn Smythe Trophy following the Kings' first Stanley Cup victory over the New Jersey Devils, 6–1, taking the series four games to two. Quick posted a 16–4 record with a 1.41 GAA, a.946 save percentage and three shutouts along the way.  On June 28, 2012, Quick signed a ten-year contract extension that would last until the end of the 2022–23 season.  In November 2012, Quick joined the AHL's Manchester Monarchs to continue rehabilitation from an off-season back surgery. Quick won his second career Stanley Cup on June 14, 2014, after the Kings defeated the New York Rangers. He had two playoff shutouts, including one in Game 3 of the Finals, while also posting a save percentage of.911.  Following the 2013–14 season, Quick was awarded the William M. Jennings Trophy, which is presented annually "to the goalkeeper(s) having played a minimum of 25 games for the team with the fewest goals scored against it." Because Ben Scrivens and Martin Jones had not played the required number of games to qualify as co-recipients of the honor, Quick was named the sole winner of the trophy. Over the course of the season, Quick had compiled a 27–17–4 record with a 2.07 GAA,.915 save percentage and six shutouts in 49 appearances.  On October 16, 2014, Quick posted 43 saves on all 43 shots he faced in a 1–0 shootout victory over the St. Louis Blues. It was Quick's largest save percentage in a shootout win to date. On October 23, Quick surpassed Rogie Vachon as the Kings' all-time leader in shutouts by earning his 33rd after a 2–0 victory over the Buffalo Sabres.  In the 2014–15 season Quick would go on to play 72 games, starting 71 of them. Quick would finish with a record of 36–22–13, with a GAA of 2.24 and a Save Percentage of.918 and 6 shutouts. The Kings would go on to miss the playoffs, losing the last playoff spot to the Calgary Flames by 2 points. After the 5-0 shutout win against the Chicago Blackhawks on March 14, 2016, he is now the all-time leader in shutouts out of American born goaltenders.  On April 27, 2016, Jonathan was nominated for the Vezina Trophy, awarded to the league's best goalie, for the second time.  During the 2016–17 season, Quick suffered a lower-body injury in the season's first game on October 12, 2016 against the San Jose Sharks. He chose to not undergo surgery for the injury. Quick was out for four and a half months (59 games in all) and returned with the team on February 25, 2017 in win over the Anaheim Ducks 4–1.  Quick was invited to the 2018 NHL All-Star Game however he declined his invitation due to a nagging injury. Per NHL policy, any player who is invited, but declines, an All-Star invitation is required to sit out the next regular season game as a consequence. He was required by the league to sit out the following regular season game, which was on January 30, 2018. At the conclusion of the regular season, Quick was awarded his second Jennings Trophy for allowing the fewest goals against all season. With this, Quick helped the Kings qualify for the 2018 Stanley Cup playoffs, where they were subsequently swept by the Vegas Golden Knights in four games. Quick was however impressive, allowing only 7 goals over the four games, posting a 1.55 GAA and a.947 SV %.

International play.

On January 1, 2010, Quick was named to Team USA for the 2010 Winter Olympics in Vancouver as the team's third goaltender. He received a silver medal after the U.S. lost to Canada 3–2 in the gold medal game.  After his selection to Team USA for the 2014 Winter Olympics in Sochi, Head Coach Dan Bylsma named Quick the team's starting goaltender over teammates Ryan Miller and Jimmy Howard. He would go on to start five out of the U.S.' six games (Miller would start the other). However, the U.S. did not medal, as Quick finished the tournament with a 3–2 record, a 2.17 GAA and a.923 save percentage.

Personal life.

Quick is married to Jaclyn (née Backman), and they have a daughter and son together. Jaclyn's sister Alicia is the wife of former Buffalo Sabres forward Matt Moulson.  Quick was born in Milford but raised in Hamden, Connecticut, a major hockey town in the northeast and attended Hamden High School until eventually transferring to Avon Old Farms, a preparatory school known for its hockey program. His family still resides in Hamden today. Ray and Mike's Deli, a popular local deli in Hamden, named the "Quickwich" sandwich in Quick's honor after his 2012 Stanley Cup victory.

Jonathan Douglas Quick (born January 21, 1986) is an American professional ice hockey goaltender for the Los Angeles Kings of the National Hockey League (NHL). Quick was selected in the third round, 72nd overall, by Los Angeles at the 2005 NHL Entry Draft.

Jonathan Douglas Quick (* 21. ledna 1986, Milford, Connecticut, USA) je americký hokejový brankář hrající v týmu Los Angeles Kings v severoamerické lize NHL.


Biography.

Schmundt was born in Metz and served as a lieutenant for the German Army during World War I. In World War II he attained the rank of General of the Infantry on 1 September 1944, and became the Chief of the Personnel Department of the German Army.  Throughout the war, Schmundt was one of Adolf Hitler's many adjutants, and flew with Erwin Rommel in early 1941, just before the Afrika Korps was created. Schmundt was one of the casualties of the failed 20 July plot, planned to kill the German dictator Adolf Hitler. One of the conspirators, Colonel Claus von Stauffenberg, placed a bomb in a briefcase beside Hitler. Colonel Heinz Brandt moved it behind a heavy table leg and unwittingly saved Hitler's life, but as a consequence, he lost his own. Severely injured in the assassination attempt, Schmundt initially made a promising recovery, but ultimately died of complications resulting from his injuries on 1 October 1944.  After Schmundt's death, all current Generals and Field Marshals were summoned by Hitler to attend a funeral service at the Tannenberg Memorial, in east Prussia. As reported by Hauptmann Alexander Stahlberg (aide to Field Marshal Von Manstein) in his book "Bounden Duty", the group were entrained back to Berlin and Schmundt was buried, on Hitler's orders, in Invalids' Cemetery. Hitler did not attend either ceremony.  Schmundt was posthumously awarded the German Order on 7 October 1944. He was replaced as the Chief of the Personnel Department by General Wilhelm Burgdorf, the Deputy Chief.

Rudolf Schmundt (13 August 1896 – 1 October 1944) was a German officer and adjutant to Adolf Hitler. Between 1942 and 1944, he was chief of the German Army Personnel Office. Schmundt was injured during the 20 July 1944 assassination attempt on Hitler and died a few months later from his wounds.

Rudolf Schmundt (13. srpna 1896, Méty, Německé císařství - 1. října 1944, Rastenburg, nacistické Německo) byl německý vojenský důstojník a pozdější generál pěchoty ("General der Infanterie") za druhé světové války. Nejvíce známým se stal díky svému působení jako vrchní armádní pobočník německého vůdce a kancléře, Adolfa Hitlera, a jako přednosta personálního oddělení německé armády. Zároveň se jedná o jednu z obětí atentátu na Hitlera z 20. července 1944.


Examples.

Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. Some of the more important examples are described below.  The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be the projective plane, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.

The projective sphere.

Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane:  If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R.  The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop "AB" from the figure above to be the generator.

The projective hemisphere.

Because the sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are similarly identified.

Boy's surface – an immersion.

The projective plane can be immersed (local neighbourhoods of the source space do not have self-intersections) in 3-space. Boy's surface is an example of an immersion.  Polyhedral examples must have at least nine faces.

Roman surface.

Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap. A polyhedral representation is the tetrahemihexahedron, which has the same general form as Steiner's Roman Surface, shown here.

Hemi polyhedra.

Looking in the opposite direction, certain abstract regular polytopes – hemi-cube, hemi-dodecahedron, and hemi-icosahedron – can be constructed as regular figures in the "projective plane;" see also projective polyhedra.

Planar projections.

Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping:  Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below.

Cross-capped disk.

A closed surface is obtained by gluing a disk to a cross-cap. This surface can be represented parametrically by the following equations:  where both "u" and "v" range from 0 to 2"π".  These equations are similar to those of a torus. Figure 1 shows a closed cross-capped disk.  A cross-capped disk has a plane of symmetry which passes through its line segment of double points. In Figure 1 the cross-capped disk is seen from above its plane of symmetry "z" = 0, but it would look the same if seen from below.  A cross-capped disk can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points. The result is shown in Figure 2.  Once this exception is made, it will be seen that the sliced cross-capped disk is homeomorphic to a self-intersecting disk, as shown in Figure 3.  The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are:  where "u" ranges from 0 to 2"π" and "v" ranges from 0 to 1.  Projecting the self-intersecting disk onto the plane of symmetry ("z" = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself).  The plane "z" = 0 cuts the self-intersecting disk into a pair of disks which are mirror reflections of each other. The disks have centers at the origin.  Now consider the rims of the disks (with "v" = 1). The points on the rim of the self-intersecting disk come in pairs which are reflections of each other with respect to the plane "z" = 0.  A cross-capped disk is formed by identifying these pairs of points, making them equivalent to each other. This means that a point with parameters ("u", 1) and coordinates formula_4 is identified with the point ("u" + π, 1) whose coordinates are formula_5. But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this is how a real projective plane is formed out of a disk. Therefore the surface shown in Figure 1 (cross-cap with disk) is topologically equivalent to the real projective plane "RP".

Homogeneous coordinates.

The points in the plane can be represented by homogeneous coordinates. A point has homogeneous coordinates ["x" : "y" : "z"], where the coordinates ["x" : "y" : "z"] and ["tx" : "ty" : "tz"] are considered to represent the same point, for all nonzero values of "t". The points with coordinates ["x" : "y" : 1] are the usual real plane, called the finite part of the projective plane, and points with coordinates ["x" : "y" : 0], called points at infinity or ideal points, constitute a line called the line at infinity. (The homogeneous coordinates [0 : 0 : 0] do not represent any point.)  The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane in R has the homogeneous coordinates ("a" : "b" : "c"). Thus, these coordinates have the equivalence relation ("a" : "b" : "c") = ("da" : "db" : "dc") for all nonzero values of "d". Hence a different equation of the same line "dax" + "dby" + "dcz" = 0 gives the same homogeneous coordinates. A point ["x" : "y" : "z"] lies on a line ("a" : "b" : "c") if "ax" + "by" + "cz" = 0. Therefore, lines with coordinates ("a" : "b" : "c") where "a", "b" are not both 0 correspond to the lines in the usual real plane, because they contain points that are not at infinity. The line with coordinates (0 : 0 : 1) is the line at infinity, since the only points on it are those with "z" = 0.

Points, lines, and planes.

A line in P can be represented by the equation "ax" + "by" + "cz" = 0. If we treat "a", "b", and "c" as the column vector l and "x", "y", "z" as the column vector x then the equation above can be written in matrix form as:  Using vector notation we may instead write x ⋅ l = 0 or l ⋅ x = 0.  The equation "k"(xl) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in R and "k"("x") sweeps out a line, again going through zero. The plane and line are linear subspaces in R, which always go through zero.

Ideal points.

In P the equation of a line is and this equation can represent a line on any plane parallel to the "x", "y" plane by multiplying the equation by "k".  If we have a normalized homogeneous coordinate. All points that have "z" = 1 create a plane. Let's pretend we are looking at that plane (from a position further out along the "z" axis and looking back towards the origin) and there are two parallel lines drawn on the plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the "z" axis, (still looking backwards towards the origin), we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the adjacent image we have divided by 2 so the "z" value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at ). Lines on the plane when are ideal points. The plane at is the line at infinity.  The homogeneous point is where all the real points go when you're looking at the plane from an infinite distance, a line on the plane is where parallel lines intersect.

Duality.

In the equation there are two column vectors. You can keep either constant and vary the other. If we keep the point x constant and vary the coefficients l we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon x as a point, because the axes we are using are "x", "y", and "z". If we instead plotted the coefficients using axis marked "a", "b", "c" points would become lines and lines would become points. If you prove something with the data plotted on axis marked "x", "y", and "z" the same argument can be used for the data plotted on axis marked "a", "b", and "c". That is duality.

Lines joining points and intersection of lines (using duality).

The equation calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. In P, the line between the points x and x may be represented as a column vector l that satisfies the equations and, or in other words a column vector l that is orthogonal to x and x. The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation. The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines,.

Embedding into 4-dimensional space.

The projective plane embeds into 4-dimensional Euclidean space. The real projective plane P(R) is the quotient of the two-sphere  by the antipodal relation. Consider the function given by. This map restricts to a map whose domain is S and, since each component is a homogeneous polynomial of even degree, it takes the same values in R on each of any two antipodal points on S. This yields a map. Moreover, this map is an embedding. Notice that this embedding admits a projection into R which is the Roman surface.

Higher non-orientable surfaces.

By gluing together projective planes successively we get non-orientable surfaces of higher demigenus. The gluing process consists of cutting out a little disk from each surface and identifying ("gluing") their boundary circles. Gluing two projective planes creates the Klein bottle.  The article on the fundamental polygon describes the higher non-orientable surfaces.

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R passing through the origin.

Reálná projektivní rovina je topologický prostor, který je modelem pro nejjednodušší typ projektivní roviny. Skládá se ze všech přímek procházejících počátkem v třírozměrném Euklidově prostoru formula_1.


Biography.

Moshe Baram was born in Zdolbuniv in the Russian Empire (today in Ukraine) in 1911, attending school in Kovno (now Kaunas in Lithuania). In his youth, he joined the HeHalutz Movement and "Freiheit-Dror", which was affiliated with the Poale Zion party. In 1931 he made aliyah to Mandate Palestine. After arriving in the country, he worked in the construction industry and joined the Haganah.

Political career.

In 1934 he began working for the Jewish Agency, and in 1938 became a member of the Secretariat of the Jerusalem Branch of Mapai, and in 1943 was made the Secretary. He also served as a delegate to the Assembly of Representatives between 1944 and 1949. In 1948 Baram was elected Secretary of the Jerusalem Workers Council, and served as a Member of the Jerusalem City Council from 1955 until 1959.  In 1959 he was elected to the Knesset on Mapai's list. In 1965 Mapai became part of the Alignment, with Baram remaining a Knesset Member until 1977. In 1974, towards the end of his political career, Baram was made Minister of Labour, also taking over the Welfare Ministry portfolio shortly before the 1977 elections. The elections saw the Alignment's parliamentary representation halved, and Baram lost his seat. However, his son, Uzi Baram did win a seat on the Alignment's list, and went on to become Minister of Tourism and Minister of Internal Affairs before retiring in 2001.  Moshe Baram is the author of the book "Lo BeTelem" ("Not in a Furrow"), published in 1981.

Moshe Baram (, 17 March 1911 – 5 December 1986) was an Israeli politician who served as a member of the Knesset between 1969 and 1977, and also as Minister of Labour and Minister of Welfare.

Moše Bar'am (: משה ברעם, žil 17. března 1911 - 5. prosince 1986) byl izraelský politik a poslanec Knesetu za strany Mapaj Ma'arach, Izraelská strana práce a opět Ma'arach.


Biography.

Born in Odessa in the Russian Empire (today in Ukraine), Shekhterman made aliyah to Mandatory Palestine in 1924. He attended high school in Tel Aviv and in 1927 he joined Betar, later becoming one of its commanders. He studied civil engineering at Ghent University, where he chaired the Jewish Students Organisation, as well as working as a Betar instructor in Antwerp.  He helped establish Kfar Yavetz in 1932. In 1936 he joined the Betar faction of the Haganah, later becoming an Irgun member after it broke away from the Haganah. In 1938 he became political editor of the "HaMashkif" newspaper, the publication of the Revisionist Party.  Shekhterman was involved in promoting aviation in Palestine. In preparation for establishing a Jewish air force, he used his contacts to convince the British Mandate's Department of Civilian Aviation to approve a flight school at an airfield in Lod.  Between 1939 and 1945 he chaired the Tel Aviv Pilots Club.

Political career.

Shekhterman served as a member of the Assembly of Representatives. In 1955 he was elected to Tel Aviv city council, remaining a member until 1965 and serving as deputy mayor between 1957 and 1959. In 1969 he was elected to the Knesset on the Gahal list. He was re-elected in 1973, by which time Gahal had merged into Likud, but lost his seat in the 1977 elections.

Business career.

Between 1943 and 1946 he managed the Potash Company at the Dead Sea.

Avraham Shekhterman (, born 1910, died 7 December 1986) was an Israeli politician who served as a member of the Knesset for Gahal and Likud between 1969 and 1977. He was the older brother of Herzl Shafir, a former IDF general.

Avraham Šechterman (: אברהם שכטרמן, žil 3. prosince 1910 – 7. prosince 1986) byl izraelský politik a bývalý poslanec Knesetu za strany Gachal a Likud.