Base transceiver station.

The base transceiver station, or BTS, contains the equipment for transmitting and receiving radio signals (transceivers), antennas, and equipment for encrypting and decrypting communications with the base station controller (BSC). Typically a BTS for anything other than a picocell will have several transceivers (TRXs) which allow it to serve several different frequencies and different sectors of the cell (in the case of sectorised base stations).  A BTS is controlled by a parent BSC via the "base station control function" (BCF). The BCF is implemented as a discrete unit or even incorporated in a TRX in compact base stations. The BCF provides an operations and maintenance (O&M) connection to the network management system (NMS), and manages operational states of each TRX, as well as software handling and alarm collection.  The functions of a BTS vary depending on the cellular technology used and the cellular telephone provider. There are vendors in which the BTS is a plain transceiver which receives information from the MS (mobile station) through the Um air interface and then converts it to a TDM (PCM) based interface, the Abis interface, and sends it towards the BSC. There are vendors which build their BTSs so the information is preprocessed, target cell lists are generated and even intracell handover (HO) can be fully handled. The advantage in this case is less load on the expensive Abis interface.  The BTSs are equipped with radios that are able to modulate layer 1 of interface Um; for GSM 2G+ the modulation type is Gaussian minimum-shift keying (GMSK), while for EDGE-enabled networks it is GMSK and 8-PSK. This modulation is a kind of continuous-phase frequency shift keying. In GMSK, the signal to be modulated onto the carrier is first smoothed with a Gaussian low-pass filter prior to being fed to a frequency modulator, which greatly reduces the interference to neighboring channels (adjacent-channel interference).  Antenna combiners are implemented to use the same antenna for several TRXs (carriers), the more TRXs are combined the greater the combiner loss will be. Up to 8:1 combiners are found in micro and pico cells only.  Frequency hopping is often used to increase overall BTS performance; this involves the rapid switching of voice traffic between TRXs in a sector. A hopping sequence is followed by the TRXs and handsets using the sector. Several hopping sequences are available, and the sequence in use for a particular cell is continually broadcast by that cell so that it is known to the handsets.  A TRX transmits and receives according to the GSM standards, which specify eight TDMA timeslots per radio frequency. A TRX may lose some of this capacity as some information is required to be broadcast to handsets in the area that the BTS serves. This information allows the handsets to identify the network and gain access to it. This signalling makes use of a channel known as the Broadcast Control Channel (BCCH).

Sectorization.

By using directional antennas on a base station, each pointing in different directions, it is possible to sectorise the base station so that several different cells are served from the same location. Typically these directional antennas have a beamwidth of 65 to 85 degrees. This increases the traffic capacity of the base station (each frequency can carry eight voice channels) whilst not greatly increasing the interference caused to neighboring cells (in any given direction, only a small number of frequencies are being broadcast). Typically two antennas are used per sector, at spacing of ten or more wavelengths apart. This allows the operator to overcome the effects of fading due to physical phenomena such as multipath reception. Some amplification of the received signal as it leaves the antenna is often used to preserve the balance between uplink and downlink signal.

Base station controller.

The base station controller (BSC) provides, classically, the "intelligence" behind the BTSs. Typically a BSC has tens or even hundreds of BTSs under its control. The BSC handles allocation of radio channels, receives measurements from the mobile phones, and controls handovers from BTS to BTS (except in the case of an inter-BSC handover in which case control is in part the responsibility of the anchor MSC). A key function of the BSC is to act as a concentrator where many different low capacity connections to BTSs (with relatively low utilisation) become reduced to a smaller number of connections towards the mobile switching center (MSC) (with a high level of utilisation). Overall, this means that networks are often structured to have many BSCs distributed into regions near their BTSs which are then connected to large centralised MSC sites.  The BSC is undoubtedly the most robust element in the BSS as it is not only a BTS controller but, for some vendors, a full switching center, as well as an SS7 node with connections to the MSC and serving GPRS support node (SGSN) (when using GPRS). It also provides all the required data to the operation support subsystem (OSS) as well as to the performance measuring centers.  A BSC is often based on a distributed computing architecture, with redundancy applied to critical functional units to ensure availability in the event of fault conditions. Redundancy often extends beyond the BSC equipment itself and is commonly used in the power supplies and in the transmission equipment providing the A-ter interface to PCU.  The databases for all the sites, including information such as carrier frequencies, frequency hopping lists, power reduction levels, receiving levels for cell border calculation, are stored in the BSC. This data is obtained directly from radio planning engineering which involves modelling of the signal propagation as well as traffic projections.

Transcoder.

The transcoder is responsible for transcoding the voice channel coding between the coding used in the mobile network, and the coding used by the world's terrestrial circuit-switched network, the Public Switched Telephone Network. Specifically, GSM uses a regular pulse excited-long term prediction (RPE-LTP) coder for voice data between the mobile device and the BSS, but pulse code modulation (A-law or μ-law standardized in ITU G.711) upstream of the BSS. RPE-LPC coding results in a data rate for voice of 13 kbit/s where standard PCM coding results in 64 kbit/s. Because of this change in data rate "for the same voice call", the transcoder also has a buffering function so that PCM 8-bit words can be recoded to construct GSM 20 ms traffic blocks.  Although transcoding (compressing/decompressing) functionality is defined as a base station function by the relevant standards, there are several vendors which have implemented the solution outside of the BSC. Some vendors have implemented it in a stand-alone rack using a proprietary interface. In Siemens' and Nokia's architecture, the transcoder is an identifiable separate sub-system which will normally be co-located with the MSC. In some of Ericsson's systems it is integrated to the MSC rather than the BSC. The reason for these designs is that if the compression of voice channels is done at the site of the MSC, the number of fixed transmission links between the BSS and MSC can be reduced, decreasing network infrastructure costs.  This subsystem is also referred to as the "transcoder and rate adaptation unit" (TRAU). Some networks use 32 kbit/s ADPCM on the terrestrial side of the network instead of 64 kbit/s PCM and the TRAU converts accordingly. When the traffic is not voice but data such as fax or email, the TRAU enables its rate adaptation unit function to give compatibility between the BSS and MSC data rates.

Packet control unit.

The packet control unit (PCU) is a late addition to the GSM standard. It performs some of the processing tasks of the BSC, but for packet data. The allocation of channels between voice and data is controlled by the base station, but once a channel is allocated to the PCU, the PCU takes full control over that channel.  The PCU can be built into the base station, built into the BSC or even, in some proposed architectures, it can be at the SGSN site. In most of the cases, the PCU is a separate node communicating extensively with the BSC on the radio side and the SGSN on the Gb side.

The base station subsystem (BSS) is the section of a traditional cellular telephone network which is responsible for handling traffic and signaling between a mobile phone and the network switching subsystem. The BSS carries out transcoding of speech channels, allocation of radio channels to mobile phones, paging, transmission and reception over the air interface and many other tasks related to the radio network.

Systém základnových stanic (anglicky "Base Station S(ubs)ystem, BSS") je část sítě GSM, která je zodpovědná za přenos a příjem radiových signálů z mobilního telefonu. Systém základnových stanic provádí překódování hovorových kanálů, přidělování radiových kanálů mobilním telefonům, paging a mnoho dalších úkolů patřících k radiové síti.


Aims of the opening.

Common aims in opening play.

Whether they are trying to gain the upper hand as White, or to equalize as Black or to create dynamic imbalances, players generally devote a lot of attention in the opening stages to the following strategies:  Apart from these ideas, other strategies used in the middlegame may also be carried out in the opening. These include preparing pawn breaks to create counterplay, creating weaknesses in the opponent's pawn structure, seizing control of key squares, making favorable exchanges of minor pieces (e.g. gaining the bishop pair), or gaining a space advantage, whether in the center or on the flanks.

Top-level objectives.

At higher levels of competition, for many years the main objectives of opening play were to obtain a better position when playing as White and to equalize when playing as Black. The idea behind this is that playing first gives White a slight initial advantage; for example, White will be the first to attack if the game opens symmetrically (Black mirrors White's moves).  Since about the 1950s another objective has gradually become more dominant. According to IM Jeremy Silman, the purpose of the opening is to create dynamic imbalances between the two sides, which will determine the character of the middlegame and the strategic plans chosen by both sides. For example, in the main line of the of the French Defense (1.e4 e6 2.d4 d5 3.Nc3 Bb4 4.e5 c5 5.a3 Bxc3+ 6.bxc3), White will try to use his and advantage to mount an attack on Black's, while Black will seek simplifying exchanges (in particular, trading off one of White's bishops to blunt this advantage) and counterattack against the weakened pawns on White's ; both players accept different combinations of advantages and disadvantages. This idea was a doctrine of the Soviet school of chess.  A third objective, which is complementary to the previous ones and has been common since the 19th century, is to lure the opponent into positions with which the player is more familiar and comfortable than the opponent. This is usually done by transpositions, in which a game that apparently starts with one opening can reach a position that is normally produced by a different opening.

Opening repertoires.

Most players realize after a while that they play certain types of positions better than others, and that the amount of theory they can learn is limited. Therefore, most players specialize in certain openings where they know the theory and that lead to positions they favor. The set of openings a player has specialized in is called an opening repertoire. The main elements a player needs to consider in a repertoire are:  A very narrow repertoire allows for deeper specialization but also makes a player less flexible to vary against different opponents. In addition, opponents may find it easier to prepare against a player with a narrow repertoire.  The main openings in a repertoire are usually reasonably sound; that is, they should lead to playable positions even against optimal counterplay. Unsound gambits are sometimes used as surprise weapons, but are unreliable for a stable repertoire. Repertoires often change as a player develops, and a player's advancement may be stifled if the opening repertoire does not evolve. Some openings that are effective against amateur players are less effective at the master level. For example, Black obtains active play in return for a pawn in the Benko Gambit; amateur players may have trouble defending against Black's activity, while masters are more skilled at defending and making use of the extra pawn. Some openings played between grandmasters are so complex and theoretical that amateur players will have trouble understanding them. An example is the Perenyi Attack of the Sicilian Defense (see diagram), which yields an immensely complicated and tactical position that even strong players have difficulty handling, and that is beyond the comprehension of most amateurs.

Opening nomenclature.

Major changes in the rules of chess in the late fifteenth century increased the speed of the game, consequently emphasizing the importance of opening study. Thus, early chess books, such as the 1497 text of Luis Ramirez de Lucena, present opening analysis, as do Pedro Damiano (1512), and Ruy López de Segura (1561). Ruy Lopez's disagreement with Damiano regarding the merits of 2...Nc6 led to 3.Bb5 (after 1.e4 e5 2.Nf3 Nc6) being named for him as the Ruy Lopez or "Spanish Opening". Opening theory was studied more scientifically from the 1840s on, and many opening variations were discovered and named in this period and later. Opening nomenclature developed haphazardly, and most names are historical accidents not based on systematic principles. In the early 1930s, the nascent FIDE embarked on a project to standardize opening nomenclature, culminating in the publication of a short booklet in 1933, but this had little impact.  The oldest openings tend to be named for geographic places and people. Many openings are named after nationalities of players who advocated them, for example Indian, English, Spanish, French, Dutch, Scotch, Russian, Italian, Scandinavian, and Sicilian, or places where important games featuring the opening were played such as Vienna, Berlin, and Wilkes-Barre. The Catalan System is named after the Catalonia region of Spain. Chess players' names are the most common sources of opening names. The name given to an opening is not always that of the first player to adopt it; often an opening is named for the player who was the first to popularize it or to publish analysis of it. Eponymic openings include the Ruy Lopez, Alekhine's Defense,, and the Réti Opening. Some opening names honor two people, such as the Caro–Kann. A few opening names are descriptive, such as Giuoco Piano (). More prosaic descriptions include Two Knights and Four Knights. Descriptive names are less common than openings named for places and people.  Some openings have been given fanciful names, often names of animals. This practice became more common in the 20th century. By then, most of the more common and traditional sequences of opening moves had already been named, so these tend to be unusual or recently developed openings like the Orangutan, Hippopotamus, Elephant, and Hedgehog.  Many terms are used for the opening as well. In addition to Opening, common terms include Game, Defense, Gambit, and Variation; less common terms are System, Attack, Counterattack, Countergambit, Reversed, and Inverted. To make matters more confusing, these terms are used very inconsistently. Consider some of the openings named for nationalities: Scotch Game, English Opening, French Defense, and Russian Game—the Scotch Game and the English Opening are both White openings (White chooses to play), the French is indeed a defense, but so is the Russian Game. Although these do not have precise definitions, here are some general observations about how they are used:  A small minority of openings are prefixed with "Anti-". These are openings intended to avoid a particular line otherwise available to one's opponent, for example the Anti-Marshall (against the Marshall (Counter) Attack in the Ruy Lopez) and the Anti-Meran Gambit (against the Meran Variation of the Semi-Slav Defense).

Classification of chess openings.

Chess openings are primarily categorized by move sequences. The beginning chess position offers White twenty possible first moves. Of these, 1.e4, 1.d4, 1.Nf3, and 1.c4 are by far the most popular as these moves do the most to promote rapid development and control of the center. A few other opening moves are considered reasonable but less consistent with opening principles than the four most popular moves. The Dunst Opening, 1.Nc3, develops a knight to a good square, but is somewhat inflexible because it blocks White's c-pawn; also, after 1...d5 the knight is liable to be driven to an inferior square by...d4. (Note that after 1.Nf3 the analogous 1...e5? loses a pawn.) Bird's Opening, 1.f4, addresses center control but not development and weakens the king position slightly. The Sokolsky Opening 1.b4 and the King's and Queen's fianchettos: Larsen's Opening 1.b3 and 1.g3 aid development a bit, but they only address center control peripherally and are slower than the more popular openings. The eleven remaining possibilities are rarely played at the top levels of chess. Of these, the best are merely slow such as 1.c3, 1.d3, and 1.e3. Worse possibilities either ignore the center and development such as 1.a3, weaken White's position (for instance, 1.f3 and 1.g4), or place the knights on poor squares (1.Na3 and 1.Nh3).  Black has twenty possible responses to White's opening move. Many of these are mirror images of the most popular first moves for White, but with one less tempo. Defenses beginning with 1...c6 and 1...e6, often followed by the center thrust 2...d5, are also popular. Defenses with an early...d6 coupled with a kingside fianchetto are also commonly played.  The most important scheme of classifying chess openings for serious players is by ECO code, a series of 500 opening codes assigned by the "Encyclopaedia of Chess Openings". Although these codes are invaluable for the serious study of the chess opening, they are not very practical for a broad survey of the chess opening as the codes obscure common structural features between related openings.  A simple descriptive categorization of the chess opening is King's Pawn Openings, Queen's Pawn Openings, and Others. Since these categories are still individually very large, it is common to divide each of them further. One reasonable way to group the openings is: The Indian systems (1.d4 Nf6) are the most important of the Semi-Closed Games, and warrant separate treatment.

Open games: 1.e4 e5.

White starts by playing 1.e4 (moving his king pawn two spaces). This is the most popular opening move and it has many strengths—it immediately works on controlling the center, and it frees two pieces (the queen and a bishop). The oldest openings in chess follow 1.e4. Bobby Fischer rated 1.e4 as "Best by test." On the downside, 1.e4 places a pawn on an undefended square and weakens d4 and f4; the Hungarian master Gyula Breyer melodramatically declared that "After 1.e4 White's game is in its last throes." If Black mirrors White's move and replies with 1...e5, the result is an open game.  The most popular second move for White is 2.Nf3 attacking Black's king pawn, preparing for a kingside castle, and anticipating the advance of the queen pawn to d4. Black's most common reply is 2...Nc6, which usually leads to the Ruy Lopez (3.Bb5), Scotch Game (3.d4), or Italian Game (3.Bc4). If Black instead maintains symmetry and counterattacks White's center with 2...Nf6 then the Petrov's Defense results. The Philidor Defense (2...d6) is not popular in modern chess because it allows White an easy space advantage while Black's position remains cramped and passive, although solid. Other responses to 2.Nf3 are not seen in master play.  The most popular alternatives to 2.Nf3 are the Vienna Game (2.Nc3), the Bishop's Opening (2.Bc4), and the King's Gambit (2.f4). These openings have some similarities with each other, in particular the Bishop's Opening frequently transposes to variations of the Vienna Game. The King's Gambit was extremely popular in the 19th century. White sacrifices a pawn for quick development and to pull a black pawn out of the center. The Vienna Game also frequently features attacks on the Black center by means of a f2–f4 pawn advance.  In the Center Game (2.d4) White immediately opens the center but if the pawn is to be recovered after 2...exd4, White must contend with a slightly premature queen development after 3.Qxd4. An alternative is to sacrifice one or two pawns, for example in the Danish Gambit.  Many other variations after 1.e4 e5 have been studied; see Open Game for details.

Semi-open games: 1.e4, Black plays other than 1...e5.

In the semi-open games White plays 1.e4 and Black breaks symmetry immediately by replying with a move other than 1...e5. The most popular Black defense to 1.e4 is the Sicilian (1...c5), but the French (1...e6, normally followed by 2.d4 d5) and the Caro–Kann (1...c6, normally followed by 2.d4 d5) are also very popular. The Pirc and the Modern are closely related openings that are also often seen, while the Alekhine and the Scandinavian have made occasional appearances in World Chess Championship games.  The Sicilian and French Defenses lead to unbalanced positions that can offer exciting play with both sides having chances to win. The Caro–Kann Defense is solid as Black intends to use his c-pawn to support his center (1.e4 c6 2.d4 d5). Alekhine's, the Pirc and the Modern are hypermodern openings in which Black tempts White to build a large center with the goal of attacking it with pieces.  Other semi-open games have been studied but are less common; see Semi-Open Game for details.

Closed games: 1.d4 d5.

The openings classified as closed games begin 1.d4 d5. The move 1.d4 offers the same benefits to development and center control as does 1.e4, but unlike with King Pawn openings where the e4-pawn is undefended after the first move, the d4-pawn is protected by White's queen. This slight difference has a tremendous effect on the opening. For instance, whereas the King's Gambit is rarely played today at the highest levels of chess, the Queen's Gambit remains a popular weapon at all levels of play. Also, compared with the King Pawn openings, transpositions among variations are more common and critical in the closed games.  The most important closed openings are in the Queen's Gambit family (White plays 2.c4). The Queen's Gambit is somewhat misnamed, since White can always regain the offered pawn if desired. In the Queen's Gambit Accepted, Black plays...dxc4, giving up the center for free development and the chance to try to give White an isolated queen pawn with a subsequent...c5 and...cxd4. White will get active pieces and possibilities for the attack. Black has two popular ways to decline the pawn, the Slav (2...c6) and the Queen's Gambit Declined (2...e6). Both of these moves lead to an immense forest of variations that can require a great deal of opening study to play well. Among the many possibilities in the Queen's Gambit Declined are the Orthodox Defense, Lasker's Defense, the Cambridge Springs Defense, the Tartakower Variation, and the Tarrasch and Semi-Tarrasch Defenses. Black replies to the Queen's Gambit other than 2...dxc4, 2...c6, and 2...e6 are uncommon.  The Colle System and Stonewall Attack are classified as Queen's Pawn Games because White plays d4 but not c4. They are also examples of "Systems", rather than specific opening variations. White develops aiming for a particular formation without great concern over how Black chooses to defend. Both systems are popular with club players because they are easy to learn, but are rarely used by professionals because a well-prepared opponent playing Black can equalize fairly easily. The Stonewall is characterized by the White pawn formation on c3, d4, e3, and f4, and can be achieved by several move orders and against many different Black setups. The diagram positions and the move sequences given below are typical.  Other closed openings have been studied but are less common; see Closed Game for details.

Indian defenses: 1.d4 Nf6.

The Indian systems are asymmetrical defenses to 1.d4 that employ hypermodern chess strategy. Fianchettos are common in many of these openings. As with the closed games, transpositions are important and many of the Indian defenses can be reached by several different move orders. Although Indian defenses were championed in the 1920s by players in the hypermodern school, they were not fully accepted until Soviet players showed in the late 1940s that these systems are sound for Black. Since then, Indian defenses have been the most popular Black replies to 1.d4 because they offer an unbalanced game with chances for both sides. The usual White second move is 2.c4, grabbing a larger share of the center and allowing the move Nc3, to prepare for moving the e-pawn to e4 without blocking the c-pawn. Black's most popular replies are:  Advocated by Nimzowitsch as early as 1913, the Nimzo-Indian Defense was the first of the Indian systems to gain full acceptance. It remains one of the most popular and well-respected defenses to 1.d4 and White often adopts move orders designed to avoid it. Black attacks the center with pieces and is prepared to trade a bishop for a knight to weaken White's queenside with doubled pawns.  The King's Indian Defense is aggressive, somewhat risky, and generally indicates that Black will not be satisfied with a draw. Although it was played occasionally as early as the late 19th century, the King's Indian was considered inferior until the 1940s, when it was taken up by Bronstein, Boleslavsky, and Reshevsky. Despite being Fischer's favored defense to 1.d4, its popularity faded in the mid-1970s. Kasparov's successes with the defense restored the King's Indian to prominence in the 1980s.  Ernst Grünfeld debuted the Grünfeld Defense in 1922. Distinguished by the move 3...d5, Grünfeld intended it as an improvement to the King's Indian which was not considered entirely satisfactory at that time. The Grünfeld has been adopted by World Champions Smyslov, Fischer, and Kasparov.  The Queen's Indian Defense is considered solid, safe, and perhaps somewhat drawish. Black often chooses the Queen's Indian when White avoids the Nimzo-Indian by playing 3.Nf3 instead of 3.Nc3. Black constructs a sound position that makes no positional concessions, although sometimes it is difficult for Black to obtain good winning chances. Karpov is a leading expert in this opening.  The Modern Benoni is a risky attempt by Black to unbalance the position and gain active piece play at the cost of allowing White a pawn wedge at d5 and a central majority. Tal popularized the defense in the 1960s by winning several brilliant games with it, and Fischer occasionally adopted it, with good results, including a win in his 1972 world championship match against Boris Spassky. Often Black adopts a slightly different move order, playing 2...e6 before 3...c5 in order to avoid the sharpest lines for White.  The Benko Gambit is often played by strong players, and is very popular at lower levels. Black plays to open lines on the queenside where White will be subject to considerable pressure. If White accepts the gambit, Black's compensation is positional rather than tactical, and his initiative can last even after many piece exchanges and well into the endgame. White often chooses instead either to decline the gambit pawn or return it.  The Catalan Opening is characterized by White forming a pawn center at d4 and c4 and fianchettoing his king's bishop. It resembles a combination of the Queen's Gambit and Réti Opening. Since the Catalan can be reached from many different move orders, (one Queen's Gambit Declined-like move sequence is 1.d4 d5 2.c4 e6 3.Nf3 Nf6 4.g3), it is sometimes called the Catalan System.  The most important Indian Defenses are listed below, but many others have been studied and played; see Indian Defense for details.

Other Black responses to 1.d4.

Of the defenses to 1.d4 other than 1...d5 and 1...Nf6, the most important are the Dutch Defense and the Benoni Defense. The Dutch, an aggressive defense adopted for a time by World Champions Alekhine and Botvinnik, and played by both Botvinnik and challenger David Bronstein in their 1951 world championship match, is still played occasionally at the top level by Short and others. Another fairly common opening is the Benoni Defense, which may become very wild if it develops into the Modern Benoni, though other variations are more solid.  Several other uncommon semi-closed openings have been named and studied, see Semi-Closed Game for details.

Flank openings (including English, Réti, Bird's, and White fianchettos).

The flank openings are the group of White openings typified by play on one or both flanks. White plays in hypermodern style, attacking the center from the flanks with pieces rather than occupying it with pawns. These openings are played often, and 1.Nf3 and 1.c4 trail only 1.e4 and 1.d4 in popularity as opening moves.  If White opens with 1.Nf3, the game often becomes one of the d4 openings (closed games or semi-closed games) by a different move order (this is called "transposition"), but unique openings such as the Réti and King's Indian Attack are also common. The Réti itself is characterized by White playing 1.Nf3, fianchettoing one or both bishops, and not playing an early d4 (which would generally transpose into one of the 1.d4 openings).  The King's Indian Attack (KIA) is a system of development that White may use in reply to almost any Black opening moves. The characteristic KIA setup is 1.Nf3, 2.g3, 3.Bg2, 4.0-0, 5.d3, 6.Nbd2, and 7.e4, although these moves may be played in many different orders. In fact, the KIA is probably most often reached after 1.e4 when White uses it to respond to a Black attempt to play one of the semi-open games such as the Caro–Kann, French, or Sicilian, or even the open games which usually come after 1.e4 e5. Its greatest appeal may be that by adopting a set pattern of development, White can avoid the large amount of opening study required to prepare to meet the many different possible Black replies to 1.e4.  The English Opening (1.c4) also frequently transposes into a d4 opening, but it can take on independent character as well including the Symmetrical Variation (1.c4 c5) and the Reversed Sicilian (1.c4 e5).  Larsen's Opening (1.b3) and the Sokolsky Opening (1.b4) are occasionally seen in grandmaster play. Benko used 1.g3 to defeat both Fischer and Tal in the 1962 Candidates Tournament in Curaçao.  With Bird's Opening (1.f4) White tries to get a strong grip on the e5-square. The opening can resemble a Dutch Defense in reverse after 1.f4 d5, or Black may try to disrupt White by playing 1...e5!? (From's Gambit).

Unusual first moves for White.

First moves other than the king pawn (1.e4), queen pawn (1.d4), or flank openings (1.b3, 1.b4, 1.c4, 1.Nf3, 1.f4, or 1.g3) are not regarded as effective ways to exploit White's first-move advantage and thus are rarely played. Although some of these openings are not actually bad for White, each of the twelve remaining possible first moves suffers one or more of the following defects compared to the more popular choices:

A chess opening or simply an opening refers to the initial moves of a chess game. The term can refer to the initial moves by either side, White or Black, but an opening by Black may also be known as a defense. There are dozens of different openings, and hundreds of variants. "The Oxford Companion to Chess" lists 1,327 named openings and variants. These vary widely in character from to wild tactical play. In addition to referring to specific move sequences, the opening is the first phase of a chess game, the other phases being the middlegame and the endgame.

Zahájení představuje počáteční fázi šachové partie, během níž se obě strany snaží co nejlépe vyvinout figury a připravit se na následný boj ve střední hře. Obvykle se o zahájení mluví jako o prvních 15–20 tazích partie.


Definition.

A function formula_1 between two topological spaces is a homeomorphism if it has the following properties:  A homeomorphism is sometimes called a bicontinuous function. If such a function exists, formula_6 and formula_7 are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. "Being homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes.

Notes.

The third requirement, that formula_22 be continuous, is essential. Consider for instance the function formula_23 (the unit circle in formula_24) defined byformula_25. This function is bijective and continuous, but not a homeomorphism (formula_26 is compact but formula_27 is not). The function formula_22 is not continuous at the point formula_29, because although formula_22 maps formula_29 to formula_32, any neighbourhood of this point also includes points that the function maps close to formula_33, but the points it maps to numbers in between lie outside the neighbourhood.  Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms formula_34 forms a group, called the homeomorphism group of "X", often denoted formula_35. This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a topological group.  For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group.  Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, formula_36 is a torsor for the homeomorphism groups formula_35 and formula_38, and, given a specific homeomorphism between formula_6 and formula_7, all three sets are identified.

Informal discussion.

The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only a finite number of points, including a single point.  This characterization of a homeomorphism often leads to a confusion with the concept of homotopy, which is actually "defined" as a continuous deformation, but from one "function" to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space "X" correspond to which points on "Y"—one just follows them as "X" deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.  There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between the identity map on "X" and the homeomorphism from "X" to "Y".

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word "homeomorphism" comes from the Greek words "ὅμοιος" ("homoios") = similar or same and "μορφή" ("morphē") = shape, form, introduced to mathematics by Henri Poincaré in 1895.

Homeomorfismus (z řeckého "homeos" = stejný, "morphe" = tvar) je vzájemně jednoznačné zobrazení mezi topologickými prostory, které zachovává topologické vlastnosti. Homeomorfismus je tedy jiný název pro izomorfismus topologických prostorů. Dva prostory, mezi kterými je homeomorfismus se nazývají homeomorfní. Z pohledu topologie jsou stejné (mají stejné vlastnosti).


History.

Upon his appointment as Bishop of Wrocław in 1198, Jarosław of Opole, elder son of the Silesian duke Bolesław I the Tall, received the territories around the town of Nysa () from his father. After Jarosław's death in 1201, his half-brother Duke Henry I the Bearded of Silesia decided to leave Nysa, as well as the castellany of Otmuchów, under the control of the Diocese of Wrocław, while the rest of the late Jarosław's Upper Silesian lands were annexed by his uncle Mieszko IV Tanglefoot in the following year. In 1290 Henry IV Probus, duke of Lower Silesia at Wrocław, gave the bishops privileges of autonomy on their lands in Nysa, creating the legal basis for the Duchy of Nysa. Henry of Wierzbna, Bishop of Wrocław from 1302 to 1319, was the first to actually use the title of a Duke of Nysa.  The duchy in its original form only lasted until 1335, when Silesia passed to the Kingdom of Bohemia by the Treaty of Trentschin. In 1342 Bishop Przecław of Pogorzeli swore the oath of fealty to King John the Blind and afterwards played a vital role in the negotiations for the 1348 Treaty of Namslau between John's successor Charles IV and King Casimir III the Great of Poland. Pogorzeli also significantly enlarged his territory by acquisition of Grodków ("Grottkau") from Duke Bolesław III the Generous of Brzeg in 1344. For the next two hundred years, the Polish, Czech and German inhabitants of Nysa-Grodków lived in relative harmony, even during the beginnings of the Reformation as the control of the region switched between Protestant and Catholic rulers. This ended with the Thirty Years' War.  The episcopate was abandoned by the bishops during the Silesian Wars. Following the First Silesian War, the 1742 Treaty of Breslau divided the principality, with the larger northern part (1231 km2) including the town of Nysa becoming part of the Kingdom of Prussia and a smaller southern part (900 km2) remaining with the Austrian-ruled Kingdom of Bohemia. In 1810 the Prussian part was secularized, its autonomy removed and it was incorporated into the Prussian Silesia Province. The small part remaining in the Austrian Empire was likewise secularized to the crown land of Austrian Silesia in 1850 and is today part of Czech Silesia.

The Duchy of Nysa (, ) or Duchy of Neisse () was one of the duchies of Silesia with its capital at Nysa in Lower Silesia. Alongside the Duchy of Siewierz, it was the only ecclesiastical duchy in the Silesian region, as it was ruled by a bishop of the Catholic Church. Nowadays its territory is divided between Poland and the Czech Republic.

Niské knížectví ( "Księstwo Nyskie", "Fürstenum Neisse"), zkráceně Nisko, je historické území v Dolním Slezsku rozkládající se podél středního toku Kladské Nisy, původně zahrnující souvislé území kolem dnešních měst Nysa (dle něhož je knížectví pojmenováno), Otmuchów, Paczków, Głuchołazy, Vidnava a Jeseník, roce 1344 bylo ještě rozšířeno o Grodków. Dnes je Nisko rozděleno mezi Polsko (1231 km2) a Česko (900 km2).


Early life.

Michael Andrew Fox was born on June 9, 1961 in Edmonton, Alberta, Canada to Phyllis ("née" Piper), an actress/payroll clerk, and William Fox, a police officer and Canadian Forces member. He is of Irish, English and Scottish descent. His family lived in various cities and towns across Canada because of his father's career. Including North Bay, Ontario, where he attend Widdifield Secondary School where he took their Arts Program. They finally moved to Burnaby, a large suburb of Vancouver, British Columbia, when his father retired in 1971. His father died on January 6, 1990 from a heart attack. Fox attended Burnaby Central Secondary School, and now has a theatre named for him at Burnaby South Secondary. At age 15, Fox starred in the Canadian television series "Leo and Me", produced by the Canadian Broadcasting Corporation, and in 1979, at age 18, he moved to Los Angeles to further his acting career. Shortly after his marriage, he decided to move back to Vancouver. Fox is one of four members of the "Leo and Me" cast and crew who eventually developed Parkinson's disease in mid-life, an unusually high number that led to some investigation as to whether an environmental factor may have played a role.  Fox was discovered by producer Ronald Shedlo and made his American television debut in the television film "Letters from Frank", credited under the name "Michael Fox." He intended to continue to use the name, but when he registered with the Screen Actors Guild, which requires unique registration names to avoid credit ambiguities (and the possibility that royalty checks would be sent to the wrong actors), he discovered that Michael Fox, a veteran character actor, was already registered under the name. As he explained in his autobiography, "Lucky Man: A Memoir" and in interviews, he needed to come up with a different name. He did not like the sound of "Michael A. Fox" during a time when "fox" was coming to mean "attractive" and because his middle initial sounded too much like the Canadian "eh?" He also didn't like the sound of "Andrew" or "Andy," so he decided to adopt a new middle initial and settled on "J," as a tribute to actor Michael J. Pollard.

Acting career.

Early career.

Fox's first feature film roles were "Midnight Madness" (1980) and "Class of 1984" (1982), credited in both as Michael Fox. Shortly afterward, he began playing "Young Republican" Alex P. Keaton in the show "Family Ties", which aired on NBC for seven seasons from 1982–89. In an interview with Jimmy Fallon in April 2014, Fox stated he negotiated the role at a payphone at Pioneer Chicken. He received the role only after Matthew Broderick was unavailable. "Family Ties" had been sold to the television network using the pitch "Hip parents, square kids", with the parents originally intended to be the main characters. However, the positive reaction to Fox's performance led to his character becoming the focus of the show following the fourth episode. At its peak, the audience for "Family Ties" drew one-third of America's households every week. Fox won three Emmy awards for "Family Ties" in 1986, 1987, and 1988. He also won a Golden Globe Award in 1989.  Brandon Tartikoff, one of the show's producers, felt that Fox was too short in relation to the actors playing his parents, and tried to have him replaced. Tartikoff reportedly said that "this is not the kind of face you'll ever find on a lunch-box." After his later successes, Fox presented Tartikoff with a custom-made lunch-box with the inscription "To Brandon, this is for you to put your crow in. Love and Kisses, Michael J. Fox." Tartikoff kept the lunch-box in his office for the rest of his NBC career.  While filming "Family Ties", Fox met his wife Tracy Pollan, who portrayed his girlfriend, Ellen. When Fox left the TV series "Spin City", his final episodes made numerous allusions to "Family Ties": Michael Gross (who played Alex's father Steven) portrays Mike Flaherty's (Fox's character's) therapist, and there is a reference to an off-screen character named "Mallory". Also, when Flaherty becomes an environmental lobbyist in Washington, D.C., he meets a conservative senator from Ohio named Alex P. Keaton, and in one episode Meredith Baxter played Mike's mother.  As a result of working on "Family Ties", as well as his acting in "Teen Wolf" and "Back to the Future", Fox became a teen idol. The VH1 television series "The Greatest" later named him among their "50 Greatest Teen Idols."

Film career.

In January 1985, Fox was cast to replace Eric Stoltz as Marty McFly, a teenager who is accidentally sent back in time from 1985 to 1955 in "Back to the Future". Director Robert Zemeckis originally wanted Fox to play Marty, but Gary David Goldberg the creator of "Family Ties", which Fox was working on at the time, refused to allow Zemeckis even to approach Fox as he felt that as Meredith Baxter was on maternity leave at the time, Fox's character Alex Keaton was needed to carry the show in her absence. Eric Stoltz was cast and was already filming "Back to the Future", but Zemeckis felt that Stoltz was not giving the right type of performance for the humor involved. Zemeckis quickly replaced Stoltz with Fox, whose schedule was now more open with the return of Meredith Baxter. He was considered for the role of Mark Kendall in "Once Bitten", but it eventually went to Jim Carrey. During filming, Fox rehearsed for "Family Ties" from 10 a.m to 6 p.m, then rushed to the "Back to the Future" set where he would rehearse and shoot until 2:30 a.m. This schedule lasted for two full months. "Back to the Future" was both a commercial and critical success. The film spent 8 consecutive weekends as the number-one grossing movie at the US box office in 1985, and eventually earned a worldwide total of $381.11 million. "Variety" applauded the performances, opining that Fox and his co-star Christopher Lloyd imbued Marty and Doc Brown's friendship with a quality reminiscent of King Arthur and Merlin. The film was followed by two successful sequels, "Back to the Future Part II" (1989) and "Back to the Future Part III" (1990).  During and immediately after the "Back to the Future" trilogy, Fox starred in "Teen Wolf" (1985)," Light of Day" (1987), "The Secret of My Success" (1987), "Bright Lights, Big City" (1988), and "Casualties of War" (1989).  In "The Secret of My Success", Fox played a recent graduate from Kansas State University who moves to New York City, where he has to deal with the ups and downs of the business world. The film was successful at the box office, grossing $110 million worldwide. Roger Ebert in "The Chicago Sun Times" wrote; "Fox provides a fairly desperate center for the film. It could not have been much fun for him to follow the movie's arbitrary shifts of mood, from sitcom to slapstick, from sex farce to boardroom brawls."  In "Bright Lights, Big City", Fox played a fact-checker for a New York magazine, who spends his nights partying with alcohol and drugs. The film received mixed reviews, with Hal Hinson in "The Washington Post" criticizing Fox by claiming that "he was the wrong actor for the job". Meanwhile, Roger Ebert praised the actor's performance: "Fox is very good in the central role (he has a long drunken monologue that is the best thing he has ever done in a movie)". During the shooting of "Bright Lights, Big City", Fox co-starred again with Tracy Pollan, his on-screen girlfriend from "Family Ties".  Fox then starred in "Casualties of War", a dark and violent war drama about the Vietnam War, alongside Sean Penn. "Casualties of War" was not a major box office hit, but Fox, playing a private serving in Vietnam, was praised for his performance. Don Willmott wrote: "Fox, only one year beyond his "Family Ties" sitcom silliness, rises to the challenges of acting as the film's moral voice and sharing scenes with the always intimidating Penn."  In 1991, he starred in "Doc Hollywood", a romantic comedy about a talented medical doctor who decides to become a plastic surgeon. While moving from Washington, D.C. to Los Angeles, he winds up as a doctor in a small southern town in South Carolina. Michael Caton-Jones, from "Time Out", described Fox in the film as "at his frenetic best". "The Hard Way" was also released in 1991, with Fox playing an undercover actor learning from police officer James Woods. After being privately diagnosed with Parkinson's disease in 1991 and being cautioned he had "ten good working years left", Fox hastily signed a three-film contract, appearing in "For Love or Money" (1993), "Life With Mikey" (1993), and "Greedy" (1994). The mid-1990s saw Fox play smaller supporting roles in "The American President" (1995) and "Mars Attacks!" (1996).  His last major film role was in "The Frighteners" (1996), directed by Peter Jackson. "The Frighteners" tells the story of Frank Bannister (Fox), an architect who develops psychic abilities allowing him to see, hear, and communicate with ghosts. After losing his wife, he uses his new abilities by cheating customers out of money for his "ghost hunting" business. However, a mass murderer comes back from Hell, prompting Frank to investigate the supernatural presence. Fox's performance received critical praise, Kenneth Turan in the "Los Angeles Times" wrote; "The film's actors are equally pleasing. Both Fox, in his most successful starring role in some time, and [Trini] Alvarado, who looks rather like Andie MacDowell here, have no difficulty getting into the manic spirit of things."  He voiced the American Bulldog Chance in Disney's live-action film "" and its sequel "", the titular character in "Stuart Little" and its two sequels "Stuart Little 2" and "", and Milo Thatch in Disney's animated film "".

Later career.

"Spin City" ran from 1996 to 2002 on American television network ABC. The show was based on a fictional local government running New York City, originally starring Fox as Mike Flaherty, a Fordham Law School graduate serving as the Deputy Mayor of New York. Fox won an Emmy award for "Spin City" in 2000, three Golden Globe Awards in 1998, 1999, and 2000, and two Screen Actors Guild Awards in 1999 and 2000. During the third season of "Spin City", Fox made the announcement to the cast and crew of the show that he had Parkinson's disease. During the fourth season, he announced his retirement from the show. He announced that he planned to continue to act and would make guest appearances on "Spin City" (he made three more appearances on the show during the final season). After leaving the show, he was replaced by Charlie Sheen, who portrayed the character Charlie Crawford. Altogether, 145 episodes were produced. Fox also served as an executive producer during his time on the show, alongside co-creators Bill Lawrence and Gary David Goldberg.  In 2004, Fox guest starred in two episodes of the comedy-drama "Scrubs" as Dr. Kevin Casey, a surgeon with severe obsessive-compulsive disorder. The series was created by "Spin City" creator Bill Lawrence. In 2006, he appeared in four episodes of "Boston Legal" as a lung cancer patient. The producers brought him back in a recurring role for season three, beginning with the season premiere. Fox was nominated for an Emmy Award for best guest appearance.  In 2009, he appeared in five episodes of the television series "Rescue Me" which earned him an Emmy for Outstanding Guest Actor in a Drama Series. Since 2000 Fox has released three books, "Lucky Man: A Memoir" (2002), "Always Looking Up: The Adventures of an Incurable Optimist" (2009), and "A Funny Thing Happened on the Way to the Future: Twists and Turns and Lessons Learned" (2010). Starting in 2010, Fox played a recurring role in the US drama "The Good Wife" as crafty attorney Louis Canning and earned Emmy nominations for three consecutive years. In 2011, Fox was featured as himself in the eighth season of the Larry David vehicle, "Curb Your Enthusiasm". David's character (also himself) becomes a temporary resident of the New York City apartment building that Fox resides in and a conflict arises between the two, whereby David believes that Fox is using his condition (Parkinson's disease) as a manipulative tool. Fox returned in 2017 for a brief appearance, referencing his prior time on the show. On August 20, 2012, NBC announced "The Michael J. Fox Show", loosely based on his life. It was granted a 22-episode commitment from the network and premiered on NBC on September 26, 2013.  Fox also made several appearances in other media. At the 2010 Winter Olympics closing ceremony in Vancouver, British Columbia, Canada, he delivered comedy monologues, along with William Shatner and Catherine O'Hara, in the "I am Canadian" part of the show. Along with Tatjana Patitz, Fox appears in the 2011 Carl Zeiss AG calendar, photographed by Bryan Adams in New York City in the summer of 2010.  Despite a sound-alike, A.J Locascio, voicing his character of Marty McFly in, Fox lent his likeness to the in-game version of Marty alongside Christopher Lloyd. Fox made a special guest appearance in the final episode of the series as an elder version of Marty, as well as his great-grandfather Willie McFly. In 2018, Fox was cast in the recurring role of Ethan West on the second season of the ABC political drama "Designated Survivor".

Personal life.

Fox married actress Tracy Pollan on July 16, 1988 at West Mountain Inn in Arlington, Vermont. The couple have four children. He holds dual Canadian-US citizenship. He provided a light-hearted segment during the 2010 Winter Olympics' closing ceremony in Vancouver, British Columbia on February 28, 2010 when he expressed how proud he is to be Canadian. On June 4, 2010, the city of Burnaby, British Columbia granted him the Freedom of the City. Fox and his family live primarily in Manhattan. The family owns a second home in Quogue, New York.

Health and activism.

Fox started displaying symptoms of early-onset Parkinson's disease in 1991 while shooting the movie "Doc Hollywood", and was diagnosed shortly after. After his diagnosis, Fox started drinking heavily and suffered from depression. He eventually sought help and stopped drinking altogether. In 1998, he decided to go public about his Parkinson's disease, and since then, he has been a strong advocate of Parkinson's disease research. His foundation, The Michael J. Fox Foundation, was created to help advance every promising research path to curing Parkinson's disease, including embryonic stem cell studies.  Fox manages the symptoms of his Parkinson's disease with the drug carbidopa/levodopa, and he also had a thalamotomy in 1998.  His first book, "Lucky Man", focused on how, after seven years of denial of the disease, he set up the Michael J. Fox Foundation, stopped drinking and began to be an advocate for people living with Parkinson's disease. In "Lucky Man", Fox wrote that he did not take his medication prior to his testimony before the Senate Appropriations Subcommittee in 1999 (full C-SPAN video clip). In an interview with NPR in April 2002, Fox explained what he does when he becomes symptomatic during an interview:  In 2006, Fox starred in a campaign ad for then State Auditor of Missouri Claire McCaskill in her successful 2006 Senate campaign against incumbent Jim Talent, expressing her support for embryonic stem cell research. In the ad, he visibly showed the effects of his Parkinson's disease:  "The New York Times" called it "one of the most powerful and talked about political advertisements in years" and polls indicated that the commercial had a measurable impact on the way voters voted, in an election that McCaskill won. His second book, "Always Looking Up: The Adventures of an Incurable Optimist", describes his life between 1999 and 2009, with much of the book centred on how Fox got into campaigning for stem-cell research. On March 31, 2009, Fox appeared on "The Oprah Winfrey Show" with Mehmet Oz to publicly discuss his condition as well as his book, his family and his prime-time special which aired May 7, 2009, ("Michael J. Fox: Adventures of an Incurable Optimist").  His work led him to be named one of the 100 people "whose power, talent or moral example is transforming the world" in 2007 by Time magazine. On March 5, 2010, Fox received an honorary doctorate in medicine from Karolinska Institutet for his contributions to research in Parkinson's disease. He has also received an honorary doctorate of laws from the University of British Columbia.  On May 31, 2012, he received an honorary degree of Doctor of Laws from the Justice Institute of British Columbia to recognize his accomplishments as a performer as well as his commitment to raising research funding and awareness for Parkinson's disease. Fox recalled performing in role-playing simulations as part of police recruit training exercises at the Institute early in his career.  In 2016, his organization the Michael J. Fox Foundation for Parkinson's Research created a raffle to raise awareness for Parkinson's disease and raised $6.75 million, with the help of Nike via two auctions, one in Hong Kong and the other in London.

Michael Andrew Fox (born June 9, 1961), known professionally as Michael J. Fox, is a Canadian-American actor, comedian, author, film producer and activist with a film and television career spanning from the 1970s. He starred in the "Back to the Future" trilogy in which he portrayed Marty McFly. Other notable roles have included his portrayal of Alex P. Keaton on the American sitcom "Family Ties" (1982–1989) and Mike Flaherty on the ABC sitcom "Spin City" (1996–2000). He has won five Primetime Emmy Awards, four Golden Globe Awards, a Grammy Award, and two Screen Actors Guild Awards.

Michael Andrew Fox (* 9. června 1961 Edmonton) je kanadský herec, známý zejména rolí v trilogii "Návrat do budoucnosti". Jeho původní jméno je Michael Andrew Fox, prostřední iniciálu „J“ přijal jako poctu herci Michaelovi J. Pollardovi.


Geography.

Saxony-Anhalt is one of 16 "Bundesländer" of Germany. It is located in the western part of eastern Germany. By size, it is the 8th largest state in Germany and by population, it is the 11th largest.  It borders four other "Bundesländer": Brandenburg to the north-east, Saxony to the south-east, Thuringia to the south-west and Lower Saxony to the north-west.  In the north, the Saxony-Anhalt landscape is dominated by the flat expanse of the North German Plain. The old Hanseatic towns Salzwedel, Gardelegen, Stendal and Tangermünde are located in the sparsely populated Altmark. The Colbitz-Letzlingen Heath and the Drömling near Wolfsburg mark the transition between the Altmark region and the Elbe-Börde-Heath region with its fertile, sparsely wooded Magdeburg Börde. Notable towns in the Magdeburg Börde are Haldensleben, Oschersleben (Bode), Wanzleben, Schönebeck (Elbe), Aschersleben and the capital Magdeburg, from which the Börde derives its name.  The Harz mountains are located in the south-west, comprising the Harz National Park, the Harz Foreland and Mansfeld Land. The highest mountain of the Harz (and of Northern Germany) is Brocken, with an elevation of 1,141 meters (3,735 ft). In this area, one can find the towns of Halberstadt, Wernigerode, Thale, Eisleben and Quedlinburg.  The wine-growing area Saale-Unstrut and the towns of Zeitz, Naumburg (Saale), Weißenfels and Freyburg (Unstrut) are located on the rivers Saale and Unstrut in the south of the state.  The metropolitan area of Halle (Saale) forms an agglomeration with Leipzig in Saxony. This area is known for its highly developed chemical industry (the "Chemiedreieck" – chemical triangle), with major production plants at Leuna, Schkopau ("Buna-Werke") and Bitterfeld. Finally, in the east, Dessau-Roßlau and Wittenberg are situated on the Elbe (as is the capital Magdeburg) in the Anhalt-Wittenberg region.

Administrative subdivisions.

The capital of Saxony-Anhalt is Magdeburg. It is the second-largest city in the state, closely after Halle (Saale). From 1994 to 2003, the state was divided into three regions ("Regierungsbezirke"), Dessau, Halle and Magdeburg and, below the regional level, 21 districts ("Landkreise"). Since 2004, however, this system has been replaced by 11 rural districts and three urban districts.  The counties are:  The independent cities are:

Largest towns.

The largest towns in Saxony-Anhalt according to a 31 December 2017 estimate:

History.

In April 1945 the US Army took control of most of the western and northern area of the future Saxony-Anhalt. The "U.S. Group Control Council, Germany" (a precursor of the OMGUS) appointed the first non-Nazi officials in leading positions in the area. So Erhard Hübener, put on leave by the Nazis, was reappointed Landeshauptmann (state governor). By early July the US Army withdrew from the former Prussian Province of Saxony to make way for the Red Army to take it as part of the Soviet occupation zone, as agreed by the London Protocol in 1944.  On 9 July the Soviet SVAG ordered the merger of the Free State of Anhalt, Halle-Merseburg, the governorate of Magdeburg (in its then borders), Allstedt (before Thuringia) and some Brunswickian eastern exclaves and salients (Calvörde and the eastern part of the former Blankenburg district) with the Province of Saxony. The previously Saxon Erfurt governorate had become a part of Thuringia.  "Anhalt" takes its name from Anhalt Castle near Harzgerode; the origin of the name of the castle remains unknown.  The SVAG appointed Hübener as president of the provincial Saxon administration, a newly created function. The administration was seated in Halle an der Saale, which became the capital, also of later Saxony-Anhalt until 1952. On 3 September 1945 the new administration enacted by Soviet-inspired ordinance the mass expropriations, mostly hitting holders of large real estates, often of noble descent.  On the occasion of the first (and one and only) election in the Soviet zone, allowing parties truly to compete for seats in provincial and state parliaments, on 20 October 1946, the Province of Saxony was renamed as the Province of Saxony-Anhalt (""), taking the prior merger into account. On 3 December 1946 the members of the new provincial parliament elected Hübener the first minister-president of Saxony-Anhalt with the votes of CDU and Liberal Democratic Party of Germany (LDPD). Thus he became the only governor in the Soviet zone, who was not a member of the communist Socialist Unity Party of Germany (SED). He was an inconvenient governor for the Soviet rulers.  After the official Allied decision to dissolve the Free State of Prussia, which had remained in limbo since the Prussian coup of 1932, its former provinces, in as far as they still existed, achieved statehood, thus the province emerged into the "State of Saxony-Anhalt" on 6 October 1947. It became part of the German Democratic Republic (East Germany) in 1949. From 1952 to 1990 the East German states were dissolved and Saxony-Anhalt's territory was divided into the East German districts of Halle and Magdeburg except territory around Torgau was in Leipzig. In 1990, in the course of German reunification, the districts were reintegrated as a state. But, territory around Torgau did not return to the state and joined Saxony. Now, Torgau is the centre of Nordsachsen district (since 2008).  In 2015 the skeletal remains of an ancient inhabitant of Karsdorf dated from the Early Neolithic (7200 BP) were analysed; he turned out to belong to the paternal T1a-M70 lineage and maternal lineage H1.

Demographics.

Since German reunification, there has been a continuous downward trend in the population of Saxony-Anhalt. This is partly due to outward migration and partly because the death rate exceeds the birth rate. Although the birth rate has been steady since 1994, the net reproduction rate is only approximately 70%. However, the total fertility rate reached 1.50 in 2014, the highest value since 1990.

Religion.

The region has historically been associated with the Lutheran faith, but under Communist rule, church membership was strongly discouraged and much of the population disassociated itself from any religious body. Saxony-Anhalt contains many sites tied to Martin Luther's life, including Lutherstadt Eisleben and Lutherstadt Wittenberg.  In 2018, the majority of citizens in Saxony-Anhalt were irreligious and more were leaving the churches than entering them – in fact, Saxony-Anhalt is the most irreligious state in Germany. 15.2% of the Saxon-Anhaltish adhered to the major denominations of Christianity (11.9% were members of the Evangelical Church in Germany and 3.3% were Catholics), 2% were members of other religions (mostly Islam, Judaism, the New Apostolic Church and Mandeism). 82.8% of the citizens of Saxony-Anhalt were religiously unaffiliated.

Foreigners.

The percentage of foreigners in Saxony-Anhalt was 4.9 percent by 31 December 2018, the third lowest among the 16 states of Germany (together with Saxony and Thuringia).  The largest foreign resident groups by 31 December 2019 were:

Politics.

13 March 2016 state election.

< 2011 Next >!style="background-color:#E9E9E9" align=left colspan="2" rowspan="2" width=400 |Party!style="background-color:#E9E9E9" align=center colspan="3" |Popular vote!style="background-color:#E9E9E9" align=center colspan="3" |Seats!style="background-color:#E9E9E9" align=right width=60|Votes!style="background-color:#E9E9E9" align=right width=40|%!style="background-color:#E9E9E9" align=right width=50|+/–!style="background-color:#E9E9E9" align=right width=30|Seats!style="background-color:#E9E9E9" align=right width=30|+/–  Minister-president Reiner Haseloff (CDU) retained his position in a coalition with former partner SPD and the Greens.

Economy.

The Gross domestic product (GDP) of the state was 62.7 billion euros in 2018, accounting for 1.9% of German economic output. GDP per capita adjusted for purchasing power was 26,000 euros or 86% of the EU27 average in the same year. The GDP per employee was 88% of the EU average. The GDP per capita was the second lowest of all German states.

Development.

Saxony-Anhalt was part of the communist German Democratic Republic. After the breakdown of communism and the German reunification in 1990, the collapse of non-competitive former GDR industries temporarily caused severe economic problems. In 2000, Saxony-Anhalt had the highest unemployment rate of all German states, at 20.2%.  However, the process of economic transformation towards a modern market economy seems to be completed. Massive investments in modern infrastructure have taken place since 1990, and the remaining and newly created businesses are highly competitive. For example, the industry has doubled its share of international revenue from 13 percent in 1995 to 26 percent in 2008. Meanwhile, the unemployment rate has fallen considerably. By 2010 the GDP of Saxony-Anhalt was almost two and a half times higher than it was in 1991.  Even though part of this recovery was induced by the quite good performance of the Germany economy, Saxony-Anhalt did not only follow the national trend, but clearly outperformed other German states. For example, it got ahead of three German states in terms of unemployment (10.8%, as of September 2011): the German capital and city-state of Berlin (12.7%), the city-state Free Hanseatic City of Bremen (11.3%) and Mecklenburg-Western Pomerania (11%).  The unemployment began to fall under 10% in 2016 and stood at 7.1% in October 2018.

Tourism.

Saxony-Anhalt has seven World Heritage Sites, the highest number of all states in Germany.

Saxony-Anhalt ( (; Low German: ) is a state of Germany, bordering the states of Brandenburg, Saxony, Thuringia and Lower Saxony. It covers an area of and has a population of 2.2 million inhabitants, making it the 8th-largest state in Germany by area and the 11th-largest by population. Its capital is Magdeburg and its largest city is Halle (Saale).

Sasko-Anhaltsko (německy "Sachsen-Anhalt") je jedna ze 16 spolkových zemí Německa. Jejím hlavním městem je Magdeburg. Na severozápadě a západě hraničí s německou spolkovou zemí Dolním Saskem, na jihozápadě a jihu s německou spolkovou zemí Durynskem, na jihovýchodě s německou spolkovou zemí Saskem, na východě a severovýchodě pak s německou spolkovou zemí Braniborskem. V současných hranicích a rozloze existuje od 3. října 1990. Územím Saska-Anhaltska protékají řeky Labe, Sála, Unstruta a další. Na jihozápadě sem zasahuje pohoří Harz.


Description.

His face is that of a lion. "When he looks at someone, it is the look of death." "Humbaba's roar is a flood, his mouth is death and his breath is fire! He can hear a hundred leagues away any [rustling?] in his forest! Who would go down into his forest!" In various examples, his face is scribed in a single coiling line like that of the coiled entrails of men and beasts, from which omens might be read.  Another description from Georg Burckhardt's translation of the "Epic of Gilgamesh" says, "he had the paws of a lion and a body covered in thorny scales; his feet had the claws of a vulture, and on his head were the horns of a wild bull; his tail and phallus each ended in a snake's head."  However, another description in a tablet sold to a museum in Sulaymaniyah in 2011 is more positive about Humbaba:  In this version of the story, Humbaba is beloved of the gods and a kind of king in the palace of the forest. Monkeys are his heralds, birds his courtiers, and his entire throne room breathes with the aroma of cedar resin. The tablet goes on to portray Gilgamesh as an aggressor who destroys a forest unnecessarily, and Humbaba's death is lamented by Enkidu.

Demise.

Humbaba is first mentioned in Tablet II of the "Epic of Gilgamesh". After Gilgamesh and Enkidu become friends following their initial fight, they set out on an adventure to the Cedar Forest beyond the seventh mountain range, to slay Humbaba (Huwawa): "Enkidu," Gilgamesh vows, "since a man cannot pass beyond the final end of life, I want to set off into the mountains, to establish my renown there." Gilgamesh tricks the monster into giving away his seven "radiances" by offering his sisters as wife and concubine. When Humbaba's guard is down, Gilgamesh punches him and captures the monster. Defeated, Humbaba appeals to a receptive Gilgamesh for mercy, but Enkidu convinces Gilgamesh to slay Humbaba. In a last effort, Humbaba tries to escape but is decapitated by Enkidu, or in some versions by both heroes together; his head is put in a leather sack, which is brought to Enlil, the god who set Humbaba as the forest's guardian. Enlil becomes enraged upon learning this and redistributes Humbaba's seven splendors (or in some tablets "auras"):  No vengeance was laid upon the heroes, though Enlil says, "He should have eaten the bread that you eat, and should have drunk the water that you drink! He should have been honored."  As each gift was given by Gilgamesh, he received a "terror" (= "radiance") in exchange, from Humbaba. The seven gifts successively given by Gilgamesh were:  While Gilgamesh thus distracts and tricks this spirit of the cedar forest, the fifty unmarried young men he has brought on the adventure are felling cedar timber, stripping it of its branches and laying it "in many piles on the hillside," ready to be taken away. Thus the adventure reveals itself in the context of a timber raid, bringing cedar wood to timberless Mesopotamia.  As his death approaches, and Gilgamesh is oppressed with his own mortality, the gods remind him of his great feats: "...having fetched cedar, the unique tree, from its mountains, having killed Humbaba in the forest..."  The iconography of the apotropaic severed head of Humbaba, with staring eyes, flowing beard and wild hair, is well documented from the First Babylonian dynasty, continuing into Neo-Assyrian art and dying away during the Achaemenid rule. The severed head of the monstrous Humbaba found a Greek parallel in the myth of Perseus and the similarly employed head of Medusa, which Perseus placed in his leather sack. Archaic Greek depictions of the gorgoneion render it bearded, an anomaly in the female Gorgon. Judith McKenzie detected Humbaba heads in a Nabatean tomb frieze at Petra.

In Ancient Mesopotamian religion, Humbaba ( Assyrian spelling), also spelled Huwawa ( Sumerian spelling) and surnamed "the Terrible", was a monstrous giant of immemorial age raised by Utu, the Sun. Humbaba was the guardian of the Cedar Forest, where the gods lived, by the will of the god Enlil, who "assigned [Humbaba] as a terror to human beings. Gilgamesh and Enkidu defeated this great enemy."

Chuvava (též Chumbaba, Huvava, Chuwawa i Huwawa; řecky Kombabos) je cedrový obr, vyskytující se v sumerské, akkadské a chetitské mytologii. Jeho jméno je pravděpodobně odvozeno od elamského boha Embany.


Definitions.

The Linux Information Project defines source code as: Source code (also referred to as source or code) is the version of software as it is originally written (i.e., typed into a computer) by a human in plain text (i.e., human readable alphanumeric characters). The notion of source code may also be taken more broadly, to include machine code and notations in graphical languages, neither of which are textual in nature. An example from an article presented on the annual IEEE conference and on Source Code Analysis and Manipulation:  For the purpose of clarity "source code" is taken to mean any fully executable description of a software system. It is therefore so construed as to include machine code, very high level languages and executable graphical representations of systems.  Often there are several steps of program translation or minification between the original source code typed by a human and an executable program. While some, like the FSF, argue that an intermediate file "is not real source code and does not count as source code", others find it convenient to refer to each intermediate file as the source code for the next steps.

History.

The earliest programs for stored-program computers were entered in binary through the front panel switches of the computer. This first-generation programming language had no distinction between source code and machine code.  When IBM first offered software to work with its machine, the source code was provided at no additional charge. At that time, the cost of developing and supporting software was included in the price of the hardware. For decades, IBM distributed source code with its software product licenses, until 1983.  Most early computer magazines published source code as type-in programs.  Occasionally the entire source code to a large program is published as a hardback book, such as "Computers and Typesetting", vol. B: "TeX, The Program" by Donald Knuth, "PGP Source Code and Internals" by Philip Zimmermann, "PC SpeedScript" by Randy Thompson, and "μC/OS, The Real-Time Kernel" by Jean Labrosse.

Organization.

The source code which constitutes a program is usually held in one or more text files stored on a computer's hard disk; usually these files are carefully arranged into a directory tree, known as a source tree. Source code can also be stored in a database (as is common for stored procedures) or elsewhere. The source code for a particular piece of software may be contained in a single file or many files. Though the practice is uncommon, a program's source code can be written in different programming languages. For example, a program written primarily in the C programming language, might have portions written in assembly language for optimization purposes. It is also possible for some components of a piece of software to be written and compiled separately, in an arbitrary programming language, and later integrated into the software using a technique called library linking. In some languages, such as Java, this can be done at run time (each class is compiled into a separate file that is linked by the interpreter at runtime).  Yet another method is to make the main program an interpreter for a programming language, either designed specifically for the application in question or general-purpose, and then write the bulk of the actual user functionality as macros or other forms of add-ins in this language, an approach taken for example by the GNU Emacs text editor.  The code base of a computer programming project is the larger collection of all the source code of all the computer programs which make up the project. It has become common practice to maintain code bases in version control systems. Moderately complex software customarily requires the compilation or assembly of several, sometimes dozens or maybe even hundreds, of different source code files. In these cases, instructions for compilations, such as a Makefile, are included with the source code. These describe the programming relationships among the source code files and contain information about how they are to be compiled.

Purposes.

Source code is primarily used as input to the process that produces an executable program (i.e., it is compiled or interpreted). It is also used as a method of communicating algorithms between people (e.g., code snippets in books).  Computer programmers often find it helpful to review existing source code to learn about programming techniques. The sharing of source code between developers is frequently cited as a contributing factor to the maturation of their programming skills. Some people consider source code an expressive artistic medium.  Porting software to other computer platforms is usually prohibitively difficult without source code. Without the source code for a particular piece of software, portability is generally computationally expensive. Possible porting options include binary translation and emulation of the original platform.  Decompilation of an executable program can be used to generate source code, either in assembly code or in a high-level language.  Programmers frequently adapt source code from one piece of software to use in other projects, a concept known as software reusability.

Legal aspects.

The situation varies worldwide, but in the United States before 1974, software and its source code was not copyrightable and therefore always public domain software.  In 1974, the US Commission on New Technological Uses of Copyrighted Works (CONTU) decided that "computer programs, to the extent that they embody an author's original creation, are proper subject matter of copyright".  In 1983 in the United States court case "Apple v. Franklin" it was ruled that the same applied to object code; and that the Copyright Act gave computer programs the copyright status of literary works.  In 1999, in the United States court case "Bernstein v. United States" it was further ruled that source code could be considered a constitutionally protected form of free speech. Proponents of free speech argued that because source code conveys information to programmers, is written in a language, and can be used to share humor and other artistic pursuits, it is a protected form of communication.

Licensing.

An author of a non-trivial work like software, has several exclusive rights, among them the copyright for the source code and object code. The author has the right and possibility to grant customers and users of his software some of his exclusive rights in form of software licensing. Software, and its accompanying source code, can be associated with several licensing paradigms; the most important distinction is open source vs proprietary software. This is done by including a copyright notice that declares licensing terms. If no notice is found, then the default of "All rights reserved" is implied.  Generally speaking, software is "open source" if the source code is free to use, distribute, modify and study, and "proprietary" if the source code is kept secret, or is privately owned and restricted. One of the first software licenses to be published and to explicitly grant these freedoms was the GNU General Public License in 1989; the BSD license is another early example from 1990.  For proprietary software, the provisions of the various copyright laws, trade secrecy and patents are used to keep the source code closed. Additionally, many pieces of retail software come with an end-user license agreement (EULA) which typically prohibits decompilation, reverse engineering, analysis, modification, or circumventing of copy protection. Types of source code protection—beyond traditional compilation to object code—include code encryption, code obfuscation or code morphing.

Quality.

The way a program is written can have important consequences for its maintainers. Coding conventions, which stress readability and some language-specific conventions, are aimed at the maintenance of the software source code, which involves debugging and updating. Other priorities, such as the speed of the program's execution, or the ability to compile the program for multiple architectures, often make code readability a less important consideration, since code "quality" generally depends on its "purpose".

In computing, source code is any collection of code, possibly with comments, written using a "human-readable" programming language, usually as plain text. The source code of a program is specially designed to facilitate the work of computer programmers, who specify the actions to be performed by a computer mostly by writing source code. The source code is often transformed by an assembler or compiler into binary machine code that can be executed by the computer. The machine code might then be stored for execution at a later time. Alternatively, source code may be interpreted and thus immediately executed.

Zdrojový kód (též "zdrojový text", slangově "zdroják", ) je v informatice označení zápisu počítačového programu nebo jeho části v nějakém programovacím jazyce. Kompletní zdrojový kód softwarového projektu je obvykle uložen v jednom nebo více textových souborech. Zdrojový kód může být buď přímo prováděn ("interpretován") interpretem nebo "přeložen" kompilátorem, který z něj vytvoří spustitelný soubor (případně doplněný knihovnami) obsahující strojový kód, který může operační systém zavést do paměti a spustit, takže je prováděn procesorem počítače. Běžný uživatel počítače obvykle se zdrojovým kódem nepřijde do styku.


History.

Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.  Later, the plane was described in a so-called "Cartesian coordinate system", a coordinate system that specifies each point uniquely in a plane by a pair of numerical "coordinates", which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a "coordinate axis" or just "axis" of the system, and the point where they meet is its "origin", usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.  The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' "La Géométrie" was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.  Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

In geometry.

Coordinate systems.

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled "x" and "y". Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis.  Another widely used coordinate system is the polar coordinate system, which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray.

Polytopes.

In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below:

Convex.

The Schläfli symbol {p} represents a regular "p"-gon.

Degenerate (spherical).

The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere, 2-torus, or right circular cylinder.

Non-convex.

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.  In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {"n"/"m"} for all "m" such that "m" < "n"/2 (strictly speaking {"n"/"m"} = {"n"/("n" − "m")}) and "m" and "n" are coprime.

Circle.

The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere ("S") because it is a one-dimensional manifold. In a Euclidean plane, it has the length 2π"r" and the area of its interior is where formula_2 is the radius.

Other shapes.

There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola.

In linear algebra.

Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors.

Dot product, angle, and length.

The dot product of two vectors and is defined as:  A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by formula_4. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B.  The dot product of a vector A by itself is which gives the formula for the Euclidean length of the vector.

In calculus.

Gradient.

In a rectangular coordinate system, the gradient is given by

Line integrals and double integrals.

For some scalar field "f" : "U" ⊆ R → R, the line integral along a piecewise smooth curve "C" ⊂ "U" is defined as where r: [a, b] → "C" is an arbitrary bijective parametrization of the curve "C" such that r("a") and r("b") give the endpoints of "C" and formula_10.  For a vector field F : "U" ⊆ R → R, the line integral along a piecewise smooth curve "C" ⊂ "U", in the direction of r, is defined as  where · is the dot product and r: [a, b] → "C" is a bijective parametrization of the curve "C" such that r("a") and r("b") give the endpoints of "C".  A double integral refers to an integral within a region "D" in R of a function formula_12 and is usually written as:

Fundamental theorem of line integrals.

The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.  Let formula_14. Then

Green's theorem.

Let "C" be a positively oriented, piecewise smooth, simple closed curve in a plane, and let "D" be the region bounded by "C". If "L" and "M" are functions of ("x", "y") defined on an open region containing "D" and have continuous partial derivatives there, then  where the path of integration along C is counterclockwise.

In topology.

In topology, the plane is characterized as being the unique contractible 2-manifold.  Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected.

In graph theory.

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a "plane graph" or "planar embedding of the graph". A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

2D či 2−D je zkratka výrazu „dvoudimenzionální“, „dvourozměrný“ a označuje svět, který je možné popsat dvěma rozměry; předměty ve dvourozměrném světě mají obsah a např. délku a šířku, avšak nemají objem. 2D obrazec je ten, jehož body se nacházejí v jedné rovině.


Life and career.

Youth.

Hochhuth was born in Eschwege, and was descended from a Protestant Hessian middle class family. His father was the owner of a shoe‐factory, which became bankrupt in the Depression. During World War II, he was a member of the Deutsches Jungvolk, a subdivision of the Hitler Youth. In 1948 he did an apprenticeship as a bookseller. Between 1950 and 1955 he worked in bookshops in Marburg, Kassel and Munich. At the same time he attended university lectures as a guest student and began with early attempts at writing fiction. Between 1955 and 1963 he was an editor at a major West-German publishing house.

"The Deputy".

Hochhuth's drama, "The Deputy" (1963), was originally entitled "Der Stellvertreter. Ein christliches Trauerspiel" ("The Deputy, a Christian Tragedy", translated by Richard and Clara Winston, 1964), the play caused a great deal of controversy because of its criticism of Pope Pius XII's role in World War II. The play was subsequently published in the UK in Robert David MacDonald's translation as "The Representative" (1965).  Its publisher Ed Keating and journalist Warren Hinckle, who themselves considered it "dramaturgically flawed," organized a committee to defend the play as a matter of free speech. In 2007, Ion Mihai Pacepa, a former Romanian spymaster, alleged that the play was part of a KGB campaign to discredit Pius XII. A leading German newspaper opined "that Hochhuth did not require any KGB assistance for his one-sided presentation of history."  The unedited version of the play would have run some eight or nine hours. As a result, each production adapted the text in its own way. No audience saw it in its original form. It includes the true story of Kurt Gerstein. Gerstein, a devout Protestant and later a member of the SS, wrote an eyewitness report about the gas chambers and, after the war, died as a POW.  The play was first performed in Berlin on 20 February 1963 under the direction of Erwin Piscator. It received its first English production in London by the Royal Shakespeare Company at the Aldwych Theatre in 1963 in a translation by Robert David MacDonald. It was directed by Clifford Williams with Alan Webb or Eric Porter as Pius XII, Alec McCowen as Father Fontana and Ian Richardson. In the United Kingdom it has since been revived at the Citizens Theatre, Glasgow, in 1986, and at the Finborough Theatre, London, in 2006.  An abridged version opened on Broadway on 26 February 1964 at the Brooks Atkinson Theatre, with Emlyn Williams as Pius XII and Jeremy Brett as Father Fontana. The play ran for 316 performances.  "The Deputy" was made into a film "Amen" by Costa Gavras in 2002, which focused more on the story of Kurt Gerstein.

"Soldiers" and Sikorski crash theories.

Hochhuth's next play, "Soldiers, An obituary for Geneva" (1967), alleged that Winston Churchill was responsible for the death of the Prime Minister of the Polish Government in Exile, General Władysław Sikorski, in an airplane crash in 1943, contradicting the official version of events as an accident, and implying that General Sikorski had been murdered on Churchill's orders. Unbeknownst to Hochhuth, the pilot of the plane was still alive and he won a libel case that seriously affected the London theater which staged the play.  That aspect of the play has overshadowed Hochhuth's conceit that the play would contribute to a debate on the ethics of the area bombing of civilian areas by the Royal Air Force during the Second World War, with particular reference to Operation Gomorrah, the Royal Air Force raids on Hamburg in 1943, and culminating in a lengthy and invented debate between Winston Churchill and the pacifist George Bell, Bishop of Chichester. The play partially drew on the work of British author David Irving, later known as a Holocaust denier. Irving and Hochhuth remained long-standing friends.  Controversy arose in Britain in 1967 when the intended premiere at Britain's National Theatre Company was cancelled, due to the intervention of the National Theatre board, despite the support for the play by literary manager Kenneth Tynan and Laurence Olivier, under pressure from Joan Plowright, his wife. At the time of the controversy in Britain, Irving was the only figure who gave his "unequivocal" support for Hochhuth's thesis; others consulted by Tynan considered it highly improbable. The play was produced shortly afterwards in the West End with John Colicos in the cast. The English translation was again by Robert David MacDonald. In the UK, the play was seen on tour in the early 1990s and was revived most recently at the Finborough Theatre, London, in 2004.

"Sommer 14 - A Dance of Death".

"Sommer 14 - A Dance of Death" had its UK premiere (and world premiere in English) in August 2014 at the Finborough Theatre in London. It was directed by Christopher Loscher for Cerberus Theatre.

"A Love in Germany" and the Filbinger Affaire.

In 1978, his novel "A Love in Germany" about an affair between a Polish POW and a German woman in World War II stirred up a debate about the past of Hans Filbinger, Minister-President of Baden-Württemberg, who had been a Navy lawyer and judge at the end of World War II. The affair culminated in Filbinger's resignation.  For "A Love in Germany", Hochhuth was awarded the Geschwister-Scholl-Preis in 1980. In 1983 Andrzej Wajda, who would later win an Honorary Oscar for Lifetime Achievement, made the story into the film "Eine Liebe in Deutschland".

"Alan Turing".

His 1987 drama "Alan Turing" featured one of the fathers of modern computer science, who had made significant contributions to breaking German ciphers during World War II. The play also covered Turing's homosexuality, discovery of which resulted in his loss of career, court-ordered chemical castration, depression, and suicide.

McKinsey controversy.

In 2004, he again caused controversy with the play "McKinsey is Coming", which raises the questions of unemployment, social justice and the "right to work". A passage in which he put the chairman of the Deutsche Bank in one line with leading businessmen who had been murdered by left-wing terrorists and also with Gessler, the villainous bailiff killed by William Tell, was widely seen as advocating, or at least excusing, violence against leading economic figures. Hochhuth vigorously denied this.

Anti-Semitism allegations.

In March 2005, Hochhuth became embroiled in controversy when, during an interview with the German weekly "Junge Freiheit", he defended Holocaust denier David Irving, describing him as a "pioneer of modern history who has written magnificent books" and an "historian to equal someone like Joachim Fest". When asked about Irving's statement that "more women died on the back seat of Edward Kennedy's car at Chappaquiddick than ever died in a gas chamber in Auschwitz", Hochhuth dismissed it as provocative black humour.  Paul Spiegel, President of the Central Council of Jews in Germany, argued that with these statements Hochhuth himself was denying the Holocaust. After weeks of uproar, Hochhuth issued an apology.

Works.

Hochhuth also collaborated with scripts for cinema and television:

Rolf Hochhuth (; 1 April 1931 – 13 May 2020) was a German author and playwright, best known for his 1963 drama "The Deputy", which insinuates Pope Pius XII's indifference to Hitler's extermination of the Jews, and he remained a controversial figure both for his plays and other public comments and for his 2005 defense of Holocaust denier David Irving.

Rolf Hochhuth (1. dubna 1931, Eschwege, Německo -13. května 2020, Berlín) byl německý spisovatel a dramatik, který žil převážně ve Švýcarsku.