title: Claude Shannon

text: He is also considered the co-inventor of the first wearable computer along with Edward O. Thorp. The device was used to improve the odds when playing roulette.


title: Claude Shannon

text: Shannon married Norma Levor, a wealthy, Jewish, left-wing intellectual in January 1940. The marriage ended in divorce after about a year. Levor later married Ben Barzman.


title: Claude Shannon

text: Shannon met his second wife, Betty Shannon (née Mary Elizabeth Moore), when she was a numerical analyst at Bell Labs. They were married in 1949. Betty assisted Claude in building some of his most famous inventions. They had three children.


title: Claude Shannon

text: There are six statues of Shannon sculpted by Eugene Daub: one at the University of Michigan; one at MIT in the Laboratory for Information and Decision Systems; one in Gaylord, Michigan; one at the University of California, San Diego; one at Bell Labs; and another at AT&T Shannon Labs. The statue in Gaylord is located in the Claude Shannon Memorial Park. After the breakup of the Bell System, the part of Bell Labs that remained with AT&T Corporation was named Shannon Labs in his honor.


title: Claude Shannon

text: According to Neil Sloane, an AT&T Fellow who co-edited Shannon's large collection of papers in 1993, the perspective introduced by Shannon's communication theory (now called information theory) is the foundation of the digital revolution, and every device containing a microprocessor or microcontroller is a conceptual descendant of Shannon's publication in 1948: "He's one of the great men of the century. Without him, none of the things we know today would exist. The whole digital revolution started with him." The cryptocurrency unit shannon (a synonym for gwei) is named after him.


title: Claude Shannon

text: "A Mind at Play", a biography of Shannon written by Jimmy Soni and Rob Goodman, was published in 2017.


title: Claude Shannon

text: On April 30, 2016, Shannon was honored with a Google Doodle to celebrate his life on what would have been his 100th birthday.


title: Claude Shannon

text: "The Bit Player", a feature film about Shannon directed by Mark Levinson premiered at the World Science Festival in 2019. Drawn from interviews conducted with Shannon in his house in the 1980s, the film was released on Amazon Prime in August 2020.


title: Claude Shannon

text: Shannon's "The Mathematical Theory of Communication," begins with an interpretation of his own work by Warren Weaver. Although Shannon's entire work is about communication itself, Warren Weaver communicated his ideas in such a way that those not acclimated to complex theory and mathematics could comprehend the fundamental laws he put forth. The coupling of their unique communicational abilities and ideas generated the Shannon-Weaver model, although the mathematical and theoretical underpinnings emanate entirely from Shannon's work after Weaver's introduction. For layman Weaver's introduction better communicates "The Mathematical Theory of Communication", but Shannon's subsequent logic, mathematics, and expressive precision was responsible for defining the problem itself.


title: Claude Shannon

text: "Theseus", created in 1950, was a mechanical mouse controlled by an electromechanical relay circuit that enabled it to move around a labyrinth of 25 squares. The maze configuration was flexible and it could be modified arbitrarily by rearranging movable partitions. The mouse was designed to search through the corridors until it found the target. Having travelled through the maze, the mouse could then be placed anywhere it had been before, and because of its prior experience it could go directly to the target. If placed in unfamiliar territory, it was programmed to search until it reached a known location and then it would proceed to the target, adding the new knowledge to its memory and learning new behavior. Shannon's mouse appears to have been the first artificial learning device of its kind.


title: Claude Shannon

text: In 1949 Shannon completed a paper (published in March 1950) which estimates the game-tree complexity of chess, which is approximately 10. This number is now often referred to as the "Shannon number", and is still regarded today as an accurate estimate of the game's complexity. The number is often cited as one of the barriers to solving the game of chess using an exhaustive analysis (i.e. brute force analysis).


title: Claude Shannon

text: On March 9, 1949, Shannon presented a paper called "Programming a Computer for playing Chess". The paper was presented at the National Institute for Radio Engineers Convention in New York. He described how to program a computer to play chess based on position scoring and move selection. He proposed basic strategies for restricting the number of possibilities to be considered in a game of chess. In March 1950 it was published in "Philosophical Magazine", and is considered one of the first articles published on the topic of programming a computer for playing chess, and using a computer to solve the game.


title: Claude Shannon

text: His process for having the computer decide on which move to make was a minimax procedure, based on an evaluation function of a given chess position. Shannon gave a rough example of an evaluation function in which the value of the black position was subtracted from that of the white position. "Material" was counted according to the usual chess piece relative value (1 point for a pawn, 3 points for a knight or bishop, 5 points for a rook, and 9 points for a queen). He considered some positional factors, subtracting ½ point for each doubled pawn, backward pawn, and isolated pawn; "mobility" was incorporated by adding 0.1 point for each legal move available.


title: Claude Shannon

text: Shannon formulated a version of Kerckhoffs' principle as "The enemy knows the system". In this form it is known as "Shannon's maxim".


title: Claude Shannon

text: The Shannon centenary, 2016, marked the life and influence of Claude Elwood Shannon on the hundredth anniversary of his birth on April 30, 1916. It was inspired in part by the Alan Turing Year. An ad hoc committee of the IEEE Information Theory Society including Christina Fragouli, Rüdiger Urbanke, Michelle Effros, Lav Varshney and Sergio Verdú, coordinated worldwide events. The initiative was announced in the History Panel at the 2015 IEEE Information Theory Workshop Jerusalem and the IEEE Information Theory Society Newsletter.


title: Claude Shannon

text: A detailed listing of confirmed events was available on the website of the IEEE Information Theory Society.


title: Game Boy Advance SP

text: The Game Boy Advance SP (GBA SP), released in Japan on February 14, 2003, is a sixth-generation handheld game console developed, released, and marketed by Nintendo that served as an upgraded version of the original Game Boy Advance. The "SP" in the name stands for "Special". It is the penultimate console in the Game Boy Advance family before the Game Boy Micro, which was released in September 2005. The Game Boy Advance line was followed by the Nintendo DS family, starting with the release of the original Nintendo DS in November 2004.


title: Game Boy Advance SP

text: The GBA SP launched in Platinum Silver and Cobalt Blue, with the addition of Onyx in Europe and Japan. Later colors include: Flame, Pearl Pink, Pearl Blue, Graphite, Midnight Blue, Charizard Fire Red, Torchic Orange, Venusaur Leaf Green, NES classic design, and Pikachu Yellow. A limited gold edition with a Triforce and the Hyrule Royal Family crest was available in Europe which included a copy of "". In 2003, Toys "R" Us sold an exclusive gold edition (without any Zelda symbols) in the US starting on Black Friday of that year, initially with a "Super Mario Advance 4" game.


title: Game Boy Advance SP

text: In Japan, it was released in a variety of standard colors and special packages. In most other regions it was released in Platinum Silver and Charcoal Black. Later, a Flame Red version was released. Six special editions have also been released: an NES Classics model with the same color scheme as a classic NES controller (and designed to resemble an NES deck when closed), a SpongeBob SquarePants model, a Pikachu model, and a silver model with a tattoo design printed on it, known as the 'Tribal Edition'. In other regions, such as Europe, Asia, and the Middle East, additional colors have been released, such as Pearl Green and Starlight Gold.


title: Game Boy Advance SP

text: Nintendo was planning to have the SP be 3D-compatible, but the resolution of the LCD was too low, resulting in Nintendo scrapping it.


title: Game Boy Advance SP

text: Nintendo removed the TRS headphone jack from the SP, which had been included on all previous Game Boy models. Headphones designed specifically for the GBA SP can be purchased, or standard headphones can be attached with an optional, stereophonic adapter that plugs into the same port as the AC adapter. Since the Nintendo DS used the same charging port as the GBA SP, the headphone adapter works with said console.


title: Game Boy Advance SP

text: As both AC adapter and headphones use the same port, it is not possible to charge the SP and listen to headphones at the same time with the Nintendo brand adapter. There are, however, third-party adapters that "split" into two different cords; the power jack on one side, and a TS headphone jack on the other.


title: Game Boy Advance SP

text: In 2005, Nintendo released an improved version of the Game Boy Advance SP in North America, featuring a brighter backlit screen instead of the previous version's frontlit screen. This GBA SP was Nintendo's first internationally-released handheld system to feature an integrated backlight. (However, there had already been a backlit system years earlier with the Japan-only Game Boy Light, which was essentially a Game Boy Pocket with an electroluminescent backlit display.) The AGS-101 was never officially released in Japan, however, as the frontlit AGS-001 models were still available on the market until the end of production in late 2009.


title: Game Boy Advance SP

text: The North American backlit version comes in three standard colors: "Pearl Blue", "Pearl Pink" and "Graphite" (a greyer version of Onyx Black). There were also two Toys "R" Us exclusive backlit models; a "SpongeBob SquarePants" model and a "Limited Edition Pikachu" model.


title: Game Boy Advance SP

text: In 2006, the AGS-101 backlit model also saw a very limited release in Europe. Few models made it to market, likely due to the release of the Game Boy Advance-compatible Nintendo DS one year prior. The European version was released in "Surf Blue" as well as re-issued in "Pink" and "Tribal" editions.


title: Game Boy Advance SP

text: Unlike the North American release, the European box does not feature any prominent text to distinguish the backlit models from the older frontlit models. In addition, only the "Surf Blue" color was unique to the AGS-101, the other two colors "Pink" and "Tribal" had already been released as frontlit models - for these reasons it can be very difficult to identify a European backlit SP. Apart from the AGS-101 model number on the base of the unit, the only other obvious distinguishing feature of the European backlit model is the large picture of the Game Boy Advance SP featured on the front of the box. (The European frontlit models of "Pink" and "Tribal" only feature small pictures of the Game Boy Advance SP on the sides of the box and Flower/Tattoo patterns on the front respectively.)


title: Game Boy Advance SP

text: The AGS-101 Game Boy Advance SP was the final Nintendo handheld to have backwards compatibility with Game Boy and Game Boy Color games in North America and Europe.


title: Game Boy Advance SP

text: M. Wiley of "IGN" called the Game Boy Advance SP "a step in the right direction for Nintendo", praising the system's new redesign over the original GBA and highlighting its inclusion of a backlit screen and rechargeable battery, although minor criticism went towards the system's omission of a headphone jack. "Engadget" gave it a global score of 84 out of 100, also praising the new features of the redesign while noting the system's lack of a headphone jack. Lawson Wong of "Fresh Gear" called it "darn close to perfection" aside from the missing headphone jack. Matthew D. Sarrel of "PC Magazine" suggested it to consumers as an improvement over the original model, praising the backlight and integration of a charger as well as improved visual quality, though he noted that players with larger hands may find difficulty with the system's design.


title: Game Boy Advance SP

text: As of June 30, 2009, the Game Boy Advance series has sold 81.48 million units worldwide, of which 43.52 million are Game Boy Advance SP units.


title: Punjab Legislative Assembly

text: The Punjab Legislative Assembly or the Punjab Vidhan Sabha is the unicameral legislature of the state of Punjab in India. The Sixteenth Punjab Legislative Assembly was constituted in March 2022. At present, it consists of 117 members, directly elected from 117 single-seat constituencies. The tenure of the Legislative Assembly is five years unless dissolved sooner. The Speaker of the sixteenth assembly is Kultar Singh Sandhwan. The meeting place of the Legislative Assembly since 6 March 1961 is the "Vidhan Bhavan" in Chandigarh.


title: Punjab Legislative Assembly

text: In the British Raj, an Executive Council was formed under The Indian Councils Act, 1861. It was only under the Government of India Act 1919 that a Legislative Council was set up in Punjab. Later, under the Government of India Act 1935, the Punjab Legislative Assembly was constituted with a membership of 175. It was summoned for the first time on 1 April 1937. In 1947, Punjab Province was partitioned into West Punjab and East Punjab and the East Punjab Legislative Assembly was formed, the forerunner of the current assembly consisting of 79 members.


title: Punjab Legislative Assembly

text: After the independence of India, on 15 July 1948, eight princely states of East Punjab grouped together to form a single state, Patiala and East Punjab States Union. The Punjab State Legislature was a bicameral house in April 1952, comprising the Vidhan Sabha (lower house) and Vidhan Parishad (upper house). In 1956 that state was largely merged into Punjab, the strength of the Vidhan Parishad of the new State of Punjab was enhanced from 40 seats to 46 seats and in 1957, it was increased to 51. Punjab was trifurcated in 1966 to form Haryana, Himachal Pradesh, and Punjab. The Vidhan Parishad was reduced to 40 seats and the Vidhan Sabha was grown by 50 seats to 104 seats. On 1 January 1970, the Vidhan Parishad was abolished leaving the state with a unicameral legislature.


title: Punjab Legislative Assembly

text: The legislature comprises the governor and the Punjab Legislative Assembly, which is the highest political organ in the state. The governor has the power to summon the assembly or to close the same. All members of the legislative assembly are directly elected, normally once in every five years by the eligible voters who are above 18 years of age. The current assembly consists of 117 elected members. The elected members select one of its own members as its Chairperson who is called the speaker of the assembly. The speaker is assisted by the deputy speaker who is also elected by the members. The conduct of a meeting in the house is the responsibility of the speaker.


title: Punjab Legislative Assembly

text: The main function of the assembly is to pass laws and rules. Every bill passed by the house has to be finally approved by the governor before it becomes applicable.


title: Punjab Legislative Assembly

text: The normal term of the legislative assembly is five years from the date appointed for its first meeting.<section end="Legislature"/>


title: DFB-Pokal

text: The DFB-Pokal ( is a German knockout football cup competition held annually by the German Football Association (DFB). Sixty-four teams participate in the competition, including all clubs from the Bundesliga and the 2. Bundesliga. It is considered the second-most important club title in German football after the Bundesliga championship. Taking place from August until May, the winner qualifies for the DFL-Supercup and the UEFA Europa League unless the winner already qualifies for the UEFA Champions League in the Bundesliga.


title: DFB-Pokal

text: The competition was founded in 1935, then called the "Tschammer-Pokal". The first titleholders were 1. FC Nürnberg. In 1937, Schalke 04 were the first team to win the double. The Tschammer-Pokal was suspended in 1944 due to World War II and disbanded following the demise of Nazi Germany. In 1952–53, the cup was reinstated in West Germany as the "DFB-Pokal", named after the DFB, and was won by Rot-Weiss Essen. (FDGB-Pokal, the East German equivalent, started in 1949 and operated through the 1991 season, when it merged with the DFB-Pokal).


title: DFB-Pokal

text: Bayern Munich have won a record 20 titles. The current holders are RB Leipzig, who beat SC Freiburg 4–2 on penalties in the 2022 final to win their first title. Fortuna Düsseldorf hold the record for most consecutive tournament game wins (18) between 1978 and 1981, winning the cup in 1979 and 1980.


title: DFB-Pokal

text: The DFB-Pokal begins with a round of 64 teams. The 36 teams of the Bundesliga and 2. Bundesliga, along with the top four finishers of the 3. Liga are automatically qualified for the tournament. Of the remaining slots 21 are given to the cup winners of the regional football associations, the "Verbandspokale". The three remaining slots are given to the three regional associations with the most men's teams. They may assign the slot as they see fit but usually give it to the runner-up in the association cup.


title: DFB-Pokal

text: As every team taking part in the German football league system is entitled to participate in local tournaments which qualify for the association cups, every team can, in principle, compete in the DFB-Pokal. The only exception is that reserve teams (e.g. Bayern Munich II) are ineligible to enter.


title: DFB-Pokal

text: For the first round, the 64 teams are split into two pots of 32. One pot contains the 18 teams from the previous season of the Bundesliga and the top 14 teams from the previous season of the 2. Bundesliga. The other pot contains the bottom 4 teams from the previous season of the 2. Bundesliga, the top 4 teams from the previous season of the 3. Liga and the 24 amateur teams that qualified through regional football tournaments. Teams from one pot are drawn against teams from the other pot. Since 1982, teams from the pot containing amateur teams have played the game at home.


title: DFB-Pokal

text: For the second round, the teams are again divided into two pots according to the same principles. Depending on the results of the first round, the pots might not be equal in terms of number. Teams from one pot are drawn against teams from the other pot until one pot is empty. The remaining teams are then drawn against each other with the team first drawn playing the game at home.


title: DFB-Pokal

text: For the remaining rounds, other than the final, the teams are drawn from one pot. Since 1985, the final has been held in the Olympic Stadium in Berlin.


title: DFB-Pokal

text: Extra time will be played if the scores are level after 90 minutes with a penalty shootout following if needed.


title: DFB-Pokal

text: Historically the number of participants in the main tournament has varied between four from 1956 until 1960 and 128 from 1973 through 1982 resulting in tournaments of two to seven rounds. Since the inception of the Bundesliga in 1963 all clubs from the Bundesliga are automatically qualified for the DFB-Pokal as are all clubs from the 2. Bundesliga since its inception in 1974. Reserve sides for most of the time were allowed to participate in the DFB-Pokal but have been excluded since 2008.


title: DFB-Pokal

text: The final has been held at the Olympic Stadium in Berlin every season since 1985. Before 1985, the host of the final was determined on short notice. In the decision, the German Football Association took into consideration that, due to the political situation between Germany and East Germany, Berlin was not chosen to be a venue for the UEFA Euro 1988.


title: DFB-Pokal

text: Originally the cup games were held over two 45 minute halves with two 15 minute overtime periods in case of a draw. If the score was still level after 120 minutes the game was replayed with the home field right reversed. In the 1939 "Tschammer-Pokal" the semi-final between Waldhof Mannheim and Wacker Wien was played to a draw three times before the game was decided by lot. The German Football Association decided to hold a penalty shootout if the replay was another draw after a similar situation arose in the 1970 cup, when the match between Alemannia Aachen and Werder Bremen had to be decided by lot after two draws.


title: DFB-Pokal

text: In 1971–72 and 1972–73, the matches were held over two legs. The second leg was extended by two additional 15-minute overtime periods if the aggregate was a draw after both legs. In case the extension brought no decision, a penalty shootout was held.


title: DFB-Pokal

text: In 1977, the final 1. FC Köln vs. Hertha BSC had to be replayed, leading to great logistical difficulties. In the aftermath, the DFB opted not to replay cup finals in the future, instead holding a penalty shootout after extra time. Eventually, this change was extended to all cup games in 1991.


title: DFB-Pokal

text: Since 1960, the winner of the DFB-Pokal qualified for the European Cup Winners' Cup. If the cup winner had already qualified for the European Club Champions Cup, the losing finalist moved into the Cup Winners' Cup instead. Following the abolition of the Cup Winners' Cup in 1999, the winner of the DFB-Pokal qualified for the UEFA Cup, known as the UEFA Europa League since 2009. If the DFB-Pokal winner or both finalists qualify through the Bundesliga for European cup competitions, the best placed team of the Bundesliga not already qualified for at least the Europa League receives the spot.


title: DFB-Pokal

text: The first German cup was held in 1935. It was then called "von Tschammer und Osten Pokal", or "Tschammerpokal" for short, named after "Reichssportführer" (Sports Chief of the Reich) Hans von Tschammer und Osten. The first final was contested between the two most successful clubs of that era, 1. FC Nürnberg and Schalke 04, with Nürnberg winning 2–0. After the last "Tschammerpokal" was held in 1943, the cup was not held for almost ten years, being re-introduced by the German Football Association (DFB) in 1952 under its current name, "DFB-Pokal". In 1965, the original trophy, "Goldfasanen-Pokal", was replaced by the trophy which is still awarded today, because the original reminded DFB president Peco Bauwens of the Nazi era.


title: DFB-Pokal

text: Originally, the DFB-Pokal was a competition open to clubs from the top divisions of German football only. This continued after the establishment of the Bundesliga in 1963. Semi-professional and amateur clubs could only enter the competition from 1974 onwards, when it was enlarged. Up until 2008, only the top two divisions of German football, the Bundesliga and 2. Bundesliga, were fully professional but from 2008, with the establishment of the 3. Liga, the third tier also became fully professional.


title: DFB-Pokal

text: From the start, the new match ups between Bundesliga and amateurs (most usually third division clubs) became a source of surprises. Often titled the "mother of all cup sensations" (), was Hamburger SV's second round loss to VfB Eppingen in 1974, the first instance of an amateur side knocking out a Bundesliga club. It took until 1990 for a fourth division side to achieve the same, when SpVgg Fürth took Borussia Dortmund out of the competition. Further milestones were the reserve side of Hertha BSC, Hertha BSC II, reaching the cup final in 1993, a first for a third division club and a reserve team. In 1997 Eintracht Trier proved too strong for both the UEFA Cup and Champions League winners, knocking Schalke 04 and Borussia Dortmund out of the competition. In 2000, 1. FC Magdeburg became the first fourth division side to eliminate two Bundesliga clubs in one season. Hannover 96, then playing in the 2. Bundesliga, became cup winners after eliminating several Bundesliga teams in the process. Kickers Offenbach won all matches including the semi final as a 2. Bundesliga team, but were promoted to the Bundesliga a week before they won the cup final.


title: DFB-Pokal

text: Surprise results in the cup attract strong media coverage in Germany and, at times, abroad. When TSV Vestenbergsgreuth eliminated Bayern Munich in 1994, who were then coached by the Italian Giovanni Trapattoni, Italian sports daily "La Gazzetta dello Sport" reported on its front page "Club di dilettanti elimina Trapattoni" ("Amateur club eliminate Trapattoni").


title: DFB-Pokal

text: Having won 20 titles, Bayern Munich has been the most successful team in the cup since they won their fourth title in 1969. Fortuna Düsseldorf established a record for consecutive German Cup match victories (18 straight victories between 1978 and 1981, taking the trophy in 1979 and 1980). Werder Bremen had won the most consecutive home games (37) from 1988 to 2019. Bayern Munich has won the most consecutive away games (33) from 2009 to 2020. Schalke 04 holds the record for the biggest win in a DFB-Pokal final, winning 5–0 against 1. FC Kaiserslautern in 1972 and 5–0 against MSV Duisburg in 2011.


title: DFB-Pokal

text: East Germany also had its own national cup: the FDGB Cup, the cup of the "Freie Deutsche Gewerkschaftsbund", the association of the East German trade unions. It was introduced in 1949 and awarded annually until 1991 after German reunification in 1990 led to the merger of the football leagues of the two Germanys.


title: DFB-Pokal

text: Since 1981 women's football clubs have competed for the DFB-Pokal Frauen. An East German women's cup was also held from 1987 to 1991.


title: V6 engine

text: A V6 engine is a six-cylinder piston engine where the cylinders share a common crankshaft and are arranged in a V configuration.


title: V6 engine

text: The first V6 engines were designed and produced independently by Marmon Motor Car Company, Deutz Gasmotoren Fabrik and Delahaye. Engines built after World War II include the Lancia V6 engine in 1950 for the Lancia Aurelia, and the Buick V6 engine in 1962 for the Buick Special. The V6 layout has become the most common layout for six-cylinder automotive engines.


title: V6 engine

text: Due to their short length, V6 engines are often used as the larger engine option for vehicles which are otherwise produced with inline-four engines, especially in transverse engine vehicles. A downside for luxury cars is that V6 engines produce more vibrations than straight-six engines. Some sports cars use flat-six engines instead of V6 engines, due to their lower centre of gravity (which improves the handling).


title: V6 engine

text: The displacement of modern V6 engines is typically between , though larger and smaller examples have been produced, such as the Mazda V6 used in the 1991–1998 Mazda MX-3, or the Mitsubishi V6 engine used in the 1992–1998 Mirage/Lancer, while the largest V6 built was the GMC V6 used in the 1962 GMC C/K series 6500.


title: V6 engine

text: All V6 engines with even firing spacing—regardless of the V-angle between the cylinder banks—are subject to a primary imbalance caused by each bank consisting of an inline-three engine, due to the odd number of cylinders in each bank. Straight-six engines and flat-six engines do not experience this imbalance. To reduce the vibrations caused by this imbalance, most V6 engines use a harmonic damper on the crankshaft and/or a counter-rotating balance shaft.


title: V6 engine

text: Six-cylinder designs have less pulsation in the power delivery than four-cylinder engines, due to the overlap in the power strokes of the six-cylinder engine. In a four-cylinder engine, only one piston is on a power stroke at any given time. Each piston comes to a complete stop and reverses direction before the next one starts its power stroke, which results in a gap between power strokes, especially at lower engine speeds (RPM). In a six-cylinder engine with an even firing interval, the next piston starts its power stroke 60° before the previous one finishes, which results in smoother delivery of power to the flywheel.


title: V6 engine

text: Comparing engines on a dynamometer, a V6 engine shows instantaneous torque peaks of 154% above mean torque and valleys of 139% below mean torque, with a small amount of negative torque (engine torque reversals) between power strokes. In the case of a four-cylinder engine, the peaks are approximately 270% above mean torque and 210% below mean torque, with 100% negative torque being delivered between strokes. However, a V6 with uneven firing intervals of 90° and 150° shows large torque variations of 185% above and 172% below mean torque.


title: V6 engine

text: Since 1991, Volkswagen has produced narrow angle "VR6" engines with V-angles of 10.5 and 15 degrees. These engines use a single cylinder head shared by both banks of cylinders, in a design similar to the 1922-1976 Lancia V4 engine. The VR6 engines were used in transverse engine front-wheel drive cars which were originally designed for inline-four engines. Due to the minimal extra length and width of the VR6 engine, it could be fitted to the engine compartments relatively easily, in order to provide a displacement increase of 50 percent.


title: V6 engine

text: Since there is no room in the V between the cylinder banks for an intake system, all the intakes are on one side of the engine, and all the exhausts are on the other side. It uses a firing order of 1-5-3-6-2-4 (which is the firing order used by most straight-six engines), rather than the common V6 firing order of 1-2-3-4-5-6 or 1-6-5-4-3-2.


title: V6 engine

text: A V-angle of 60 degrees is the optimal configuration for V6 engines regarding engine balance. When individual crank pins are used for each cylinder (i.e. using a six-throw crankshaft), an even firing interval of 120 degrees can be used. This firing interval is a multiple of the 60 degree V-angle, therefore the combustion forces can be balanced through use of the appropriate firing order.


title: V6 engine

text: The inline-three engine that forms each cylinder bank, however, produces unbalanced rotating and reciprocal forces. These forces remain unbalanced in all V6 engines, often leading to the use of a balance shaft to reduce the vibration.


title: V6 engine

text: The 1950 Lancia V6 engine was pioneering in its use of a six-throw crankshaft in order to reduce vibration. More recent designs often use a three-throw crankshaft with 'flying arms' between the crankpins to allow an even firing interval of 120 degrees to be achieved. A pair of counterweights on the crankshaft can then be used to almost perfectly cancel out the primary forces and reduce the secondary vibrations to acceptable levels. The engine mounts can be designed to absorb these remaining vibrations.


title: V6 engine

text: A 60 degree V-angle results in a narrower engine overall than V6 engines with larger V-angles. This angle often results in the overall engine size being a cube shape, making the engine easier to fit either longitudinally or transversely in the engine compartment.


title: V6 engine

text: Many manufacturers, particularly American ones, built V6 engines with a V-angle of 90 degrees based on their existing 90-degree V8 engines. Such configurations were easy to design by removing two cylinders and replacing the V8 engine's four-throw crankshaft with a three-throw crankshaft. This reduced design costs, allowed the new V6 to share components with the V8 engine, and sometimes allowed manufacturers to build the V6 and V8 engines on the same production line.


title: V6 engine

text: The downsides of a 90 degree design are a wider engine which is more vibration-prone than a 60 degree V6. The initial 90 degree V6 engines (such as the Buick Fireball V6 engine) had three shared crankpins arranged at 120 degrees from each other, due to their origins from the V8 engines. This resulted in an uneven firing order, with half of the cylinders using a firing interval of 90 degrees and other half using an interval of 150 degrees. The uneven firing intervals resulted in rough-running engines with "unpleasant" vibrations at low engine speeds.


title: V6 engine

text: Several modern 90 degree V6 engines reduce the vibrations using split crankpins offset by 30 degrees between piston pairs, which creates an even firing interval of 120 degrees for all cylinders. For example, the 1977 Buick 231 "even-fire" V6 engine was an upgraded version of the Buick Fireball engine with a split-pin crankshaft to reduce vibration by achieving an even firing order. Such a 'split' crankpin is weaker than a straight one, but modern metallurgical techniques can produce a crankshaft that is adequately strong.


title: V6 engine

text: A balance shaft and/or crankshaft counterweights can be used to reduce vibrations in 90 degree V6 engines.


title: V6 engine

text: At first glance, 120 degrees might seem to be the optimal V-angle for a V6 engine, since pairs of pistons in alternate banks can share crank pins in a three-throw crankshaft and the combustion forces are balanced by the firing interval being equal to the angle between the cylinder banks. A 120 degree configuration, unlike the 60 degree or 90 degree configurations, would not require crankshafts with flying arms, split crankpins, or seven main bearings to be even-firing. However, the primary imbalance caused by odd number of cylinders in each bank still remains in a 120 degree V6 engine. This differs from the perfect balance achieved by a 90 degree V8 engine with a commonly used crossplane crankshaft, because the inline-four engine in each bank of the V8 engine does not have this primary imbalance.


title: V6 engine

text: A 120 degree design also results in a large width for the engine, being only slightly narrower than a flat-six engine (which does not have the balance problems of the V6 engine). Therefore, the flat-six engine has been used in various automobiles, whereas use of the 120 degree V6 engine has been limited to a few truck and racing car engines, with the exception of McLaren Automotive's M630 V6 engine, which uses a 120 degree bank angle with a single balance shaft to eliminate all primary couples. The McLaren M630 engine also takes advantage of the wide angle by placing the turbochargers inside the vee, commonly referred to as a 'hot vee' configuration. The Ferrari 296 GTB is the first Ferrari road car to sport a V6 turbo with a vee angle of 120 degrees between the cylinder banks.


title: V6 engine

text: Other angle V6 engines are possible but can suffer from severe vibration problems unless very carefully designed. Notable V-angles include:


title: V6 engine

text: In 1906, a few years after 4 cylinder engines and V8 engines had come into existence, the first known V6 engine was built. This V6 engine was a single prototype automotive engine built by Marmon Motor Car Company in the United States. The engine did not reach production. Similarly, a single prototype engine was produced by Buick in 1918.


title: V6 engine

text: The Lancia V6 engine was introduced in the 1950 Lancia Aurelia. Lancia had been producing V4 engines for approximately 30 years, and one of the key goals was to reduce the vibrations compared with the V4 engine. The V6 engine used a 60 degree V-angle and six crankpins, resulting in an evenly-spaced firing order to reduce vibrations.


title: V6 engine

text: Other manufacturers took note and soon other V6 engines were designed. In 1959, the GMC V6 engine was introduced in the form of a 60-degree petrol engine used in pickup trucks and carryalls. The Buick V6 engine was introduced in 1962 and was based on the all-alloy Buick 215 V8, which shared its 90° bank angle, but unlike the Buick V8, used all-cast iron construction. Initially an uneven-firing engine, Buick later redesigned the crankshaft to a "split-pin" configuration to create an "even-firing" version. After it became the 3800 V6 in 1990, the engine gained a reputation as a reliable, powerful, fuel-efficient workhorse that became a mainstay of GM's FWD mid-size and full-size cars. It was discontinued in 2008. Over 25 million units had been built, making it one of the most-produced engines in history, and it was on Ward's 10 Best Engines of the 20th century list.


title: V6 engine

text: Ford introduced its European road car engines in 1965 with the German division's Cologne V6, and the Ford Essex V6 engine, introduced by Ford's United Kingdom division in 1966; both engines used a 60-degree V-angle. The 1967 Dino 206 GT was Ferrari's first V6 road car, which had a 65-degree V-angle.


title: V6 engine

text: The 1979-2005 Alfa Romeo V6 engine was introduced in the Alfa Romeo Alfa 6 luxury sedan and later used in many other Alfa Romeo models. This engine used a 60 degree V-angle, an all-aluminium construction and two valves per cylinder. A turbocharged version was introduced in 1991 and a four valve per cylinder version was introduced in 1997. Also in 1970, the Citroën SM grand tourer was introduced, powered by a 90-degree V6 built by Maserati. The Chevrolet 90° V6 engine was introduced in 1978 and produced for 36 years.


title: V6 engine

text: The first mass-produced Japanese V6 engine was the Nissan VG engine, a 60-degree design which was produced from 1983 to 2004. The Honda C engine was introduced in 1985, followed by the Mitsubishi 6G7 engine and the Mazda J engine in 1986, the Toyota VZ engine in 1988, and the Isuzu V engine in 1992. Hyundai introduced the first South Korean Hyundai Sigma engine based on technology shared from the Mitsubishi unit in 1995.


title: V6 engine

text: German car manufacturers were relatively slow to adopt V6 engines, because engineers believed that they lacked the smoothness of an inline-6 engine. Eventually, the first German V6 engine was a 2.8 liter 90° V6 that was launched in the 1990 Audi 100, and the narrow-angle VR6, which was introduced across Volkswagen's mid-size and sports car lineup in the 1990s. In 1998, Mercedes-Benz introduced the M112, its first V6 engine, while BMW has continued to use inline-6 engines. Mercedes-Benz discontinued its V6 engines in 2017, and has since returned to making inline-6 engines.


title: V6 engine

text: The first independently designed British V6 engine was the Rover KV6 engine, which replaced the Honda C engine that was previously used in the Rover 800. Jaguar used the Ford-based AJ-V6 engine until 2011 in their smaller cars, but also shared a V6 version of the 90° AJ-V8 engine with Land Rover for use in the XE, XF, XJ, F-Type and the F-Pace. Land Rover used it in the Range Rover, Range Rover Sport, Range Rover Velar, and the Discovery 4.


title: V6 engine

text: The 90° V6 engine was discontinued in 2020, and Jaguar Land Rover replaced it with the new Ingenium engine, which has an inline-6 variant for JLR's bigger cars and SUVs.


title: V6 engine

text: By the mid-1990s, the V6 layout was the most common configuration for six-cylinder automotive engines, with V6 engines having replaced most of the straight-six engines. Today, it is being progressively replaced across the car industry by turbocharged 4-cylinder engines, which can produce similar power, but in a smaller package that produces cleaner emissions, has better fuel economy, and are less expensive to produce.


title: V6 engine

text: The Lancia Aurelia (the first series production car with a V6 engine) was also successful in motor racing. Four of the "Aurelia B20 Coupes" were entered in the 1951 Mille Miglia with the best placed cars finishing second and fourth. A tuned version of the Lancia V6 engine producing was used in the Lancia D24. The D24 competed in sports car racing and won the 1953 Carrera Panamericana with Juan Manuel Fangio at the wheel.


title: V6 engine

text: The initial version of the Ferrari Dino engine was a racing engine used in Formula Two racing in the 1957 season. It had a V-angle of 65 degrees and dual overhead camshafts. The Dino V6 underwent several evolutions, including a version used in the 1958 Ferrari 246 Formula One racing car. A few years later, the 1961-1964 Ferrari 156 Formula One car used a new V6 engine with a V-angle of 120 degrees and a displacement of . This engine was shorter and lighter than the Ferrari Dino engine, and the simplicity and low center of gravity of the engine was an advantage in racing. It won a large number of races between and . However, Ferrari's founder had a personal dislike of the 120 degree layout, preferring a 65 degree layout, and after that time it was replaced by other engines. The Dino engine was also used in the Lancia Stratos, which was a highly successful rally car that won the World Rally Championship in 1974, 1975 and 1976.


title: V6 engine

text: A notable racing use of the Alfa Romeo V6 engine was the Alfa Romeo 155 V6 TI, designed for the 1993 DTM season and equipped with a engine making a peak power of at 11,900 rpm.


title: V6 engine

text: The "Renault-Gordini CH1" was a 90 degree V6 engine with an iron block. It was introduced in the 1973 "Alpine -Renault A440" sportscar racing car. This engine won the European 2 L prototype championship in 1974 and several European Formula Two Championships. A turbocharged version was used in the Renault Alpine A442, which won the 24 Hours of Le Mans in 1978.


title: V6 engine

text: A turbocharged version of the "Renault-Gordini CH1" engine was introduced in the 1977 Renault RS01 Formula One car. Renault struggled with reliability issues in 1977 and 1978, however the 1979 season saw some good results at a few races. In 1981, the Ferrari 126C Formula One car used a turbocharged V6 engine. Ferrari won the Formula One constructors' championship with turbocharged V6 engines in 1982 and 1983. Initial versions used a 120 degree V-angle, before switching to a 90 degree V-angle for the 1987 Ferrari F1/87 racing car. Other successful turbocharged V6 Formula One cars in the era of 1982-1988 were the McLaren MP4/2, McLaren MP4/3, McLaren MP4/4, Williams FW10, Williams FW11, Williams FW12, Lotus 95T, Lotus 97T, Lotus 98T, Lotus 99T and Lotus 100T.


title: V6 engine

text: The Nissan GTP ZX-Turbo and Nissan NPT-90 competed in the IMSA sports car prototype category from 1985 to 1994 and used a turbocharged V6 engine loosely based on the Nissan VG30ET production car engine. The Nissan 300ZX used a similar engine to compete in the 1996-1997 All Japan Grand Touring Car Championship (now known as the 'Super GT' championship).


title: V6 engine

text: The first V6 engine to reach production was built from 1908 to 1913 by the Deutz Gasmotoren Fabrik in Germany. These V6 engines were used as the generator for gasoline-electric railway engines.


title: V6 engine

text: The Laverda V6 was a racing motorcycle which was unveiled at the 1977 Milan show. It entered the 1978 Bol d'Or 24 hour endurance race, however it retired with mechanical issues after approximately 8 hours.


title: Lada Niva

text: The Lada Niva Legend, formerly called the Lada Niva, VAZ-2121, VAZ-2131, and Lada 4×4 (), is a series of four-wheel drive, small (hatchback), and compact (wagon and pickup) off-road cars designed and produced by AvtoVAZ since 1977. Initially aimed at the rural market, later models also targeted urban users. The three- and later five-door 4×4 hatchbacks were sold under the Lada marque in many markets, and have been in continuous production since 1977.


title: Lada Niva

text: In the 1990s, three and five-door wagons on a longer wheelbase and an extra-long wheelbase pick-up were added to the range. After the original Land Rover and its successor, the Land Rover Defender, were discontinued in 2016, the Niva became the longest production run off-road light vehicle still manufactured in its original form. By the end of 2020, an estimated 650,000 Lada Nivas had been sold globally.


title: Lada Niva

text: The Lada Niva is the world's first mass-produced off-road vehicle with a unibody construction (fully integrated body and frame). It is the predecessor of current crossover SUVs, most of which are built similarly. It was part inspiration for the Suzuki Vitara.


title: Lada Niva

text: In August 2020, Lada took over production of the 2003 Chevrolet Niva and rebranded it the "new" Lada Niva. In December 2020, the new Niva was further rebranded as the Lada Niva Travel, while the old model was renamed Lada Niva Legend in January 2021.


title: Lada Niva

text: Outside Russia, the Niva primarily competes with the Suzuki Jimny, a similarly designed off-road mini SUV.


title: Lada Niva

text: Нива (Niva) is a Russian word that literally means "(corn) field", but also "area of practice". The name "Niva "(НИВА) was formed as an acronym from the initials of Niva's chief designers' children: Pyotr Prusov's two daughters and Vladimir Solovyev's two sons.


title: Lada Niva

text: The Lada Niva was formerly called the Lada 4×4, or the VAZ-2121 in the domestic Russian market. The name was changed after the brand was transferred to General Motors, though AvtoVAZ retained the rights to the equivalent Cyrillic name: Нива. It was also marketed as the Lada Sport in Iceland, Lada Taiga in Austria, Bognor Diva in Uruguay, and Lada Cossack in the United Kingdom. In 2014, it was sold as the LADA 4×4 in Russia. Since 2021, it is sold as the Lada Niva Legend in Russia.


title: Lada Niva

text: The Niva was described by its designers as a "Renault 5 put on a Land Rover chassis". Development began in 1971 after the 24th Congress of the CPSU, in which Alexei Kosygin (the then-Premier of the Soviet Union) gave the designers at VAZ and AZLK the task of creating a car suitable for rural areas (specifically for the villagers and farmers of the Soviet Union), since the usual Zhiguli, Moskvitch, and Zaporozhets, intended primarily for ordinary people, were not much of use in the isolated areas that made up a large part of the USSR. In the same year, a team of VAZ designers led by Solovyev began competing with AZLK to work on a "civilized" four-wheel drive vehicle. The new car was partly inspired by the IZh-14 prototype of 1974. It was VAZ's first model that was not based on Fiat, though many of its mechanics were carried over from the earlier Fiat 124- or 125-based Zhiguli models (mostly the VAZ-2103 and 2106); the body, four-wheel drive system, and front suspension were all designed by VAZ. The first prototype appeared in 1971 and was officially designated the E-2121 and nicknamed "krokodil" (the crocodile) due to its distinctive frontal section, but was rejected for being too utilitarian, so doors and a hardtop were added. This version (designated the 2E-2121) debuted in 1973 and deviated from the off-road vehicles of the period, as it used a modern hatchback body. The design choice was inspired by the prototype known as the VAZ-1101 (itself derived from the Fiat 127), and was created by designer Valery Pavlovitch.


title: Lada Niva

text: The influence of the Fiat 127 is obvious from the Niva's distinctive "clamshell" hood design and its rear three-quarter section. The 1974 prototype (designated the VAZ-2121M) was derived from an existing model, the VAZ-2103 (then the newest model in VAZ arsenal). It used the VAZ-2103's 1,452 cc engine and shared some of its features, like chrome-plated bumpers, headlights and taillights, instruments, seats, and steering wheel. In the following year, two samples of the prototype were modified to install a 1,478 cc UZAM-412 engine, which originally powered the Moskvich 1500. Testing later revealed that the engine was incompatible with the newly designed car. The Moskvich engine was abandoned, and the older engine was reused. The 1974 prototype (2121M) was ready to be mass produced in 1975, but the appearance of the new VAZ-2106 at the end of the year resulted in a delay because the company wanted to develop another prototype based on the model in 1976, which became the sixth and final (serial production) prototype.


title: Lada Niva

text: Before its production, the Niva was tested over a period of years by a team led by Vadim Kotlyarov in the most difficult terrain of the Soviet Union, such as the Ural Mountains and Siberia, the deserts of the Kazakh SSR, and the Pamir Mountains in the Tajik SSR, where it was compared with its military counterpart, the UAZ-469, and some Western off-road vehicles—the British-made Land Rover Series and the Range Rover Classic. Its off-road capabilities were demonstrated for the very first time, which were based on its effective permanent four-wheel drive system featuring a transfer case and central differential lock, with the combination of a short wheelbase (only 2,2m), relatively low weight (just over a ton), independent suspension, small dimensions (3,74×1,68 m), and high ground clearance (about 265 mm). Its large but narrow wheels (175/80-R16 in dimensions), originally featuring domestically designed "Voltyre" VLI-5 high-tread tires, offered relatively strong ground pressure, which lowered the chances of slipping or getting bogged down off-road. For example, in the 1973 and 1974 trials, the Niva climbed a 58% slope, and crossed of water and of mud and snow. Solovyev died the following year and was replaced by engineer Pyotr Prusov, who took credit for creating the car.


title: Lada Niva

text: In 1976, the final prototype appeared. The engine was changed to a one, the most powerful engine in VAZ during the Soviet era, taken from their VAZ-2106 (from which the production model of the first Niva generation was generally derived). Like the previous prototype, it featured a permanent four-wheel drive and a transfer case with a lockable central differential. The VAZ prototype displayed better design and real-world performance than its AZLK counterpart, the Moskvitch 416, so it was approved for production in March of the same year, after the 25th Congress of the CPSU. The car was named Niva 1600 (Field) for exports and the VAZ-2121 (alternatively "Niva 2121" or "VAZ Niva") for the domestic market. For domestic customers, it was described as "[a] high-speed car with the improved capabilities and comfort of all VAZ models," and also as "[a] combination of the speed and comfort of a Zhiguli with the capabilities of a UAZ." Production began the following year on 5 April 1977, while the first export models appeared in 1978 at the Paris Motor Salon and quickly took over at least 40% of Europe's market for four-wheel drive vehicles, making it Lada's top-selling export. It was the only Soviet car that was ever sold in Japan, starting from the early 1980s, inspiring their Suzuki Samurai. Because of export demand and the higher priority given to exports, domestic customers faced long wait times, despite the car being developed primarily for Soviet citizens.


title: Lada Niva

text: The original Niva has a normally aspirated 1.6-L overhead-cam four-cylinder petrol engine producing 56 kW (76 hp) and 126 N·m (93 lb·ft) at 5,000 rpm, a four-speed manual transmission, and a full-time four-wheel drive. The drive system uses three differentials: center, front, and rear. The transfer case involves a high/low range selector lever and another to lock the central differential. Like usual gear shifting, both of the aforementioned features naturally require the use of clutch to be properly selected. However, while the central differential can usually be locked while in motion, the transfer case cannot and requires a complete stop. The original Niva has a maximum speed of .


title: Lada Niva

text: Coil springs are located at each of the four wheels, and the suspension is independent in front, whereas the rear axle is a five-link live-type, with ratios between 3.90 and 4.30 depending on the model and market. Ground clearance is 235 mm (9.25 in), and go as far as 510 mm (20 in) deep in water.


title: Lada Niva

text: The brakes are servo-assisted dual-circuit style and the clutch is hydraulic. The turning circle is 36.25 ft. Cargo space is 0.48 m³ (17 ft³), or 1.33 m³ (47 ft³) with the rear seats folded down and, like the classic Zhiguli, the front seats can be set up to make two berths. A spare tire can be stored in the engine bay under the bonnet.


title: Lada Niva

text: Additional equipment for the basic model was similar to other Lada and Eastern Bloc cars of the period; it included headlight wipers, a rear fog lamp, a right external mirror (for domestic units until the end of the 1980s), rear seat belts, a rear window wiper, a rear window heater, and a radio set. On the Niva 1.7 or VAZ-21213 from 1994, all of the above features except headlight wipers and the rear fog lamp (which were discontinued) became default and air conditioning, an antilock braking system, and a hydraulic servomotor for the steering column were made optional, although they became standard equipment on the newest models.


title: Lada Niva

text: Foreign (usually Western) local Lada importers often offered their own additional equipment before buying the car, such as front and rear bull bars and roll bars, side rocker rails, roof rails, a winch, fender plastics, additional headlights for front bumper and for roof rails, and aftermarket aluminium rims, which can be often seen mounted on many Nivas.


title: Lada Niva

text: The existing model was slightly restyled during first half of the 1980s when VAZ discontinued the VAZ-2103 (which had slowly been replaced by the VAZ-2106 since 1975). Due to high prices and production complexity, some of the chrome-plated elements disappeared in later models, such as the chrome-plated mask and rocker panel edges. Other elements were changed, such as the side mirrors, which became black metallic. The background of the front Lada badge was changed from red to white (and later black on the new model) on all models. The side rear retroreflectors (or turn signals on some models, depending on market) were replaced by Lada (left) and Niva (right) labels, and the rear mudflaps received the brand logo. For easier access to certain areas of the engine compartment, the window washer reservoir was moved from the left to the right section of the compartment (right in front of the battery). The interior was only minimally changed, along with some minor changes to the instrument panel and choke knob.


title: Lada Niva

text: Pre-1985 models are visually recognized by the aforementioned details, and are rare today (almost nonexistent in a well-preserved condition), so they are highly valued by various collectors of classic cars. A mint-condition 1980 unit in Russia reached a price of five million RUB and ultimately sold for 3,2 million in 2016 (approximately €38,570 or $44,570), which remains the most expensive Niva ever sold.


title: Lada Niva

text: The VAZ-2121 Niva or Lada Niva 1600 was in production until 1993 (it existed in parallel production with the new model until the end of 1993) when the newer and more powerful four-cylinder petrol engine was introduced, replacing the previously-used VAZ-2106 engine. The standard Weber carburetor was replaced by a single-point fuel injection initially supplied by General Motors on the 21214/1.7i model, and an improved Solex carburetor on the 21213/1.7 model. The transmission was changed from four to five speeds, mechanical ignition was replaced, and the electronics and suspension received some relatively minor changes. The exhaust system was also slightly redesigned, and on newer models, the drive shafts received homokinetic joints.


title: Lada Niva

text: On the exterior, the rear section of the body was redesigned with the tailgate extended and license plate on it. These changes prompted new vertical taillights that replaced the former VAZ-2106 horizontal ones and the VAZ-2102 chromed tailgate knob, which was replaced by a simple plastic handle. On the newest models, the old hand-adjustable, square black-metallic (previously chrome-plated) side mirrors, which originated from the VAZ-2103 were replaced with larger plastic ones that could be automatically adjusted.. The previous front Lada badge was replaced with the larger ellipsoid design. The Fiat 125 chrome-plated doorknobs are still present, though they are plastic on the newest models. Although the rest of the exterior has remained generally unchanged (except for the Lada 4×4 Urban and Bronto), the interior was almost completely redesigned and shares only a few interchangeable parts with the previous one: the archaic partially chrome-plated VAZ-2106 steering wheel was replaced with the more modern and thick one from the VAZ-2107 and later, even newer models, as well as its VAZ-2103/2106 instruments, which were replaced with the dashboard from the VAZ-21099. The black artificial leather seats that originated from the VAZ-2106 were replaced by higher, more modern ones from the VAZ-2108, with the front surface made of polyester, while the rear seat was improved for easier folding. The Fiat 124 swiveling windows from the side doors disappeared and were replaced with air conditioning, which was installed with the ABS and servomotor for the steering column as additional equipment on request, while the headlight wipers and rear fog lamp disappeared. It received completely new front and door panels (with interior door knobs relocated above the hand rests instead of under like on previous generation), various plastics and even new floor mats. The hand openings for the side-door windows was replaced with an automatic opening on the newest models (and on the Lada 4×4 Urban) starting from 2014. Multipoint fuel injection designed by Bosch has been used since 2004, replacing the previously mentioned single-point injection from General Motors.


title: Lada Niva

text: The lengthened model of the Niva appeared in 1993. It was known as the VAZ-2129 and was produced very briefly, only for about a year, so it is very rare. It was also never exported. Although it has a longer wheelbase, it kept three doors like the basic hatchback model. In design, it was a hybrid of old and new generations, since it featured the old VAZ-2121 (Niva 1600) body and interiors combined with new VAZ-21213 (Niva 1.7) 1680-cc engine and 5-speed transmission (although later, 1994 production models featured even new body and interiors).


title: Lada Niva

text: The lengthened estate model with a longer wheelbase and five doors was made in 1995. It was originally known as the VAZ-2131 Niva, but commercially, it has no difference in name from the standard three-door model, and it is often referred as the Niva five-door. Due to its increased weight from the longer wheelbase, it can use the 1774-cc inline four petrol engine. Although it was still in production like the basic model, it had limited use and was infrequently exported, mainly due to the longer wheelbase and increased weight, and it was slightly higher priced compared to the standard model. In 2015, a five-door model also appeared for the Lada 4×4 Urban.


title: Lada Niva

text: In 1999, a diesel-engine model of the Niva known as the VAZ-21215 Niva, or commercially as the Lada Niva 1.9, was released. It was produced until 2007 and sold only in a few markets. It was powered by a 1905-cc Peugeot XUD 9SD inline four diesel engine, and it was less powerful than the standard VAZ-21213/21214 1680-cc petrol engine. However, the VAZ-21215 was a relatively short-lived model despite lower fuel consumption, because it was not popular in the domestic market, Peugeot stopped producing XUD engines in 2001 (although existing supplies were available until 2007, when the last VAZ-21215 came off the production line), and VAZ did not make another contract for diesel engine production for the Niva.


title: Lada Niva

text: In 2006, the model with the carbureted engine of the new generation (VAZ-21213/Lada Niva 1.7) was discontinued, and the name Niva was formally dropped in favor of the new official name Lada 4×4, as General Motors received naming rights. The Niva was discontinued under the old Soviet VAZ-2121 domestic market classification.


title: Lada Niva

text: In 2014, the new, restyled model called the Lada 4×4 Urban was released. It has more modern, ordinary looks, including a new mask, plastic bumpers, new side mirrors, different aluminium rims, new tires, a central brake light, heated front seats, and a slightly restyled interior. As of 2019, no version of the Lada 4x4 has airbags.


title: Lada Niva

text: In 2017, the Lada 4×4 Bronto, intended mainly for off-road use, is released. It features characteristic mask and plastic bumpers with integrated front and rear fog lamps, plastic roof rails, new, larger wheels with black aluminium rims and 235/75-R16 Bontyre Stalker off-road tires, even higher ground clearance (increased another 35 mm) and plastic rust-protected body edges (fenders and rocker panels). Internally, it is similar to the Lada 4×4 URBAN, also featuring heated front seats. It also includes a front winch, roof headlights, and additional fog lights on the front bumper. With all the mentioned equipment included, the Bronto is currently the most expensive Lada 4×4 model, with a current price of 720 thousand RU in the domestic market (around €10,130). The slightly different pickup model designed by VIS was also revealed (featuring an integrated metallic grille, different bumpers, and a snorkel), called the Lada Bronto AMC.


title: Lada Niva

text: On 5 April at the car's 40th anniversary, a special, limited edition of the classic Lada 4×4 called the Lada 4×4 40th-Anniversary was presented, which used the 1977 design. It features new aluminium rims similar to the Urban, five new color schemes, "40th Anniversary" metallic labels on the fenders, a tailgate, a glove compartment cover, front-seat backrests, and floor mats. Also, it has a slightly restyled interior featuring genuine leather seat upholstery and a steering wheel cover, and sills made out of stainless steel. The model is available from June 2017 in both domestic and export markets.


title: Lada Niva

text: A convertible version appeared back in 1983, but was neither serially produced nor designed by VAZ. It was a modification of the original Niva 1600 bodied by French coachbuilder Wassermann for the local market. It was known as the Niva Plein Soleil (Sunlight) and it was one of many Niva modifications made by local importers for their markets.


title: Lada Niva

text: In recent years, changes included better lights, more comfortable seats, an instrument panels with LCD displays, alloy wheels, electric windows and mirrors, heated seats and air conditioning.


title: Lada Niva

text: During the 1980s, local Lada importers in various markets made their own upgrades to help compete with more modern SUVs. In the UK, the Cossack model featured large body decals, roof rails, running boards, 15-in alloy wheels, and on some versions, a sunroof, steel bullbars, spotlights, a rear-mounted spare tire, and semi-bucket seats. Other markets' importers made similar upgrades and many were also called Cossack.


title: Lada Niva

text: In 1995, Lada UK introduced a face-lifted version of the Niva Cossack and renamed the basic model as the "Hussar". Whereas the Hussar had the original 1977 trim, the new UK Cossack featured a new Rover-designed grille and other body kit items, and gained soft nudge-bars at the front in deference to public opinion against bullbars. Both models received the same new 1.7-litre engine and a new deeper tailgate which extended the rear opening to the level of the bumper – a vast improvement over the original model's high lip. Official Niva imports to the UK ceased in 1997 due to the importers having difficulty in sourcing the GM fuel injection unit required to satisfy ever-tightening UK emissions regulations.


title: Lada Niva

text: Several attempts were made to reintroduce Ladas into the UK. In May 2010, the basic Niva became available again through an independent importer. Aimed largely at the agricultural market, three models were made available: the three-door, four-seat hatchback; a two-seat commercial van; and a two-door, four-seat pickup.


title: Lada Niva

text: Although VAZ was meant exclusively to produce passenger cars, the Soviet military was impressed by its off-road capabilities and ordered VAZ to create a military version, to replace the UAZ-469 as the standard military 4×4 vehicle. In 1976, when VAZ-2121 development was almost finished, engineers from Tolyatti began working on an amphibian off-roader, which became known as VAZ-2122 Reka (River). A simplified and more rugged body was combined with the Niva chassis. It weighed about the same as the usual Niva, but instead of the 1600-cc VAZ-2106 engine, it used an older and less powerful 1300-cc engine derived from the VAZ-2101. Internally, it featured a different metal dashboard, though it used the same Niva instruments along with the same steering wheel and artificial leather seats.


title: Lada Niva

text: Six different evolution prototypes were built. They were tested in an experimental military unit, on the proving grounds of the Ministry of Defense, and in the Karakum Desert in Turkmenistan. The VAZ-2122 showed better results than the UAZ-469 (especially the fact that it featured amphibious capabilities, which UAZ did not have) and received very positive feedback; the military was satisfied with its capabilities, reliability, firmness and durability. By 1987, VAZ had completed development work, and the car was ready for mass production and military service. Nevertheless, the results were not enough to convince the military to start mass production.


title: Lada Niva

text: Longer wheelbase versions – the five-door VAZ-2131, three-door VAZ-2129, and VAZ-2329 pickup were also produced from the 1990s, but not generally exported.


title: Lada Niva

text: The Lada Niva was formally renamed Lada 4×4 back in 2006 and has a low-range gearbox, differential lock, ABS, airbags, heated front seats, air conditioning, servomotor for the steering column, underbody protection. Prices for the model started at €10,990. It is available in five body styles: three- and five-door hatchbacks, two- and four-door pickups, and a two-door van. Two petrol engines are available for the model – the standard 1.7-litre and the 1.8-litre (only for five-door model) along with a five-speed manual transmission.


title: Lada Niva

text: Starting with the 2014 model year, the Niva nameplate was discontinued for the Russian market, and the vehicle is branded Lada 4×4 for both short and long wheelbases to make way for the upcoming restart of the Chevrolet Niva series, which was scheduled to start production by March 2015. Due to Russia–West economic relations, AvtoVAZ analysts suggested that the production of Chevrolet Niva may be delayed further into 2015.


title: Lada Niva

text: GM-AvtoVAZ, a joint venture between AvtoVAZ and General Motors, produces the Chevrolet Niva. It features an updated body and 1.7-litre gasoline engine with fuel injection. Although the body and the interiors are new, it is still based on the old VAZ-2121 engine, transmission, and most mechanicals. Its off-road ability is exemplary compared with many modern budget SUVs, having been designed for tough tundra territory. In August 2020 Lada took over production of the Chevy Niva rebranding it as the Lada Niva.


title: Lada Niva

text: In 2019 Lada acquired General Motors' stake in GM-AvtoVAZ and then announced that the Chevrolet Niva would be produced under Lada's name. The rebranded Lada Niva was revealed by AvtoVAZ in July 2020, with the firm saying that there was still strong demand for the 20 year old line of cars. The 2020 model is produced at the Togliatti factory with minor design updates (but no technical changes) and joins the Lada 4×4 in the firm's range.


title: Lada Niva

text: In December 2020, following a new generation of the line, the new Lada Niva was further rebranded as Lada Niva Travel, while the old Lada 4x4 was rebranded as the Lada Niva Legend in January 2021.


title: Lada Niva

text: In 2002, the Lada Niva was awarded zero stars out of a possible four by the Russian ARCAP safety-assessment program. The reviewer noted the very rugged body of the car as the only positive aspect in terms of safety. During the safety test, the passenger dummy was hit by the glove compartment hard enough to risk traumatic brain injury.


title: Lada Niva

text: The reviewers noted that they did not expect a high rating and that the result was natural, as "it would be naive to believe that a 30-year-old design complies with modern requirements for passive safety". They added that a complete modernization of the car would be required to comply with modern safety standards.


title: Lada Niva

text: The Niva was the first wheeled vehicle to spend more than 10 years in Antarctica, where the classic VAZ-2121 Niva (Lada Niva 1600) was used by the Soviet Antarctic Expedition for transportation of personnel and goods, communication between Antarctic stations, and towing boats. It covered more than 40,000 km (24,855 miles) in 1990–2001 and operated at temperatures as low as −54 ℃. A Niva also reached the North Pole in 1998, when the lengthened VAZ-2131 Niva was dropped by parachute on ice and successfully completed its route, operating at an average temperature of −30 ℃, becoming the first wheeled vehicle to spend time there. It also set the world record of the highest point ever reached by a motorized vehicle, when another VAZ-2131 climbed to the 5200-m-high base camp on Mount Everest in 1998 and even higher just a year later, on 16 September 1999, when a similar Niva belonging to the Saint Petersburg extreme expedition team reached a height of 5725 m on a Tibet mountain during off-roading.


title: Lada Niva

text: Automeccanica assembled different versions of the Lada Niva in the late 1980s and early 1990s, including an in-house developed convertible version.


title: Lada Niva

text: In August 2014 at the Moscow International Automobile Salon, GM-AvtoVAZ introduced the next generation Niva. The car had a longitudinal mounted engine, full-time four-wheel-drive, two-gear transfer case and rigid-axle rear suspension. The production model was supposed to get a 1.8-liter PSA Peugeot Citroën EC8 engine (135 hp) paired with a 5-speed manual gearbox. The production was expected to start in 2016, but GM had no interest in the model.


title: Lada Niva

text: In May 2018 AvtoVAZ patented the new concept car called the Lada 4x4 Vision. At a Moscow International Automobile Salon in August 2018, AvtoVAZ presented the concept, built in honour of the original Niva's 40th anniversary. According to preliminary information, the car would be based on Renault Duster and production is scheduled to start in 2022 at the earliest.


title: Lada Niva

text: A version of the base three-door car, it sports updated bumpers, grille, steering wheel, and some additional options: power and heated mirrors, power windows, air conditioning, and alloy wheels. The car has been produced by VIS-Auto since October 2014.


title: Lada Niva

text: Bronto March is one of the few mass-produced vehicles on ultra-large-diameter pneumatic wheels. A production is being made by the PSA Bronto in Tolyatti since 1997. In the first 8 years of production, about 350 cars were produced.


title: Lada Niva

text: A modification based on the VAZ-21214, it was planned for series production at the plant "Pishchemash" in the city of Argun, Chechen Republic. This short run is made by LLC "APAL" since 2003.


title: Lada Niva

text: Vladimir Putin bought a Niva in May 2009. He showed his camouflage-painted Opel-engined offroader in the Russian media, allegedly to support domestic car producers despite the financial crisis.


title: Lada Niva

text: Serbian football player Aleksandar Kolarov received a certificate for a Lada 4×4/Niva as a reward for scoring a goal in his team's match against Costa Rica during the Russia 2018 World Cup.


title: No Way Out (1987 film)

text: No Way Out is a 1987 American neo-noir political action thriller film directed by Roger Donaldson and starring Kevin Costner, Gene Hackman, Will Patton and Sean Young. Howard Duff, George Dzundza, Jason Bernard, Fred Thompson, and Iman appear in supporting roles. The film is based on the 1946 novel "The Big Clock" by Kenneth Fearing, previously filmed as "The Big Clock" (1948) and "Police Python 357" (1976).


title: No Way Out (1987 film)

text: In the opening scene of the movie, set in a house in Washington, near the Pentagon, Lt. Cdr. Tom Farrell of the Office of Naval Intelligence is shown being debriefed by 2 other men. He is tired and bloodied. His interrogators press him on how Farrell came to meet the Secretary of Defense David Brice.


title: No Way Out (1987 film)

text: The story flashes back 6 months, showing Farrell attending an inaugural ball, invited by Scott Pritchard, a his college buddy who intends to introduce him to Secretary of Defense David Brice. There, Farrell also meets Susan Atwell, and the two begin an affair. Brice and Pritchard, his second-in-command, later hire Farrell to get secret information from other government agencies, such as the CIA, and pass it on to Brice. Farrell finds that he may be at times working with Sam Hesselman, an old friend now working as a programmer/analyst in the Pentagon's new computer center. Atwell eventually tells Farrell that she is Brice's mistress.


title: No Way Out (1987 film)

text: After Atwell and Farrell return from a romantic weekend, Brice visits her unexpectedly and becomes suspicious that she has another lover. When Atwell tells Brice to leave, he becomes enraged and accidentally pushes her to her death over an upstairs railing. Brice confesses what has happened to Pritchard, who suggests that if Atwell's other lover were alleged to be a suspected KGB sleeper agent code-named "Yuri", then investigating her death could be made a matter of national security and "Yuri" could be killed "in the line of duty" by operatives under Pritchard's control. At Atwell's house, Pritchard discovers the negative of a photograph she had earlier taken of Farrell. The negative is blurry and does not show a recognizable face, but Hesselman attempts to have the image enhanced by computer, a process that may take days.


title: No Way Out (1987 film)

text: Army CID officers, commanded by Major Donovan, scour Atwell's apartment for evidence. Meanwhile, as his initial shock begins to wear away, Farrell realizes that Brice is the real murderer and that Pritchard is helping him cover up the crime. At the same time, Farrell also becomes aware that the evidence gathered so far makes him the prime suspect. Farrell determines to play along with the bogus investigation until he can develop evidence linking Brice to Atwell, so that he can defend himself against being charged with both murder and espionage (as "Yuri").


title: No Way Out (1987 film)

text: Farrell learns that one piece of evidence is a Moroccan jewel box, a gift to Atwell from Brice. As any foreign gift must be registered with the State Department, Farrell gets Hesselman to "raid" State's computerized registry of such items, which should link the gift to Brice. However, the plan begins unraveling when Pritchard finds Atwell's address book, which Pritchard uses to track down and question her friend Nina. She pretends not to recognize Farrell, but reveals that she knows that Atwell was dating Brice. Pritchard sends two former CIA assassins to eliminate her. Overhearing this, Farrell delays the assassins while warning Nina, who goes into hiding. This activity raises Pritchard's suspicions of Farrell's loyalty to Brice and himself, as well as his motives for attempting to disrupt the coverup.


title: No Way Out (1987 film)

text: Farrell convinces Hesselman to delay the photo enhancement by confiding to him that Farrell is the person in the photo, that he was in love with Atwell, and that Brice killed Atwell. The CID begins a search of the Pentagon on grounds that "Yuri" is somewhere in the building, but Farrell eludes the search by climbing into a ceiling vent. Believing that Farrell is delusional, Hesselman informs Pritchard about what he has told him. Hesselman is then killed by Pritchard. Knowing that Farrell has a printout of the gift registry data connecting him to Atwell, Brice improvises a different story: Pritchard, who is gay, killed Atwell out of jealousy of Brice's relationship with her. The devastated Pritchard commits suicide and, when guards break in, Brice identifies Pritchard as "Yuri," concluding the search for the spy.


title: No Way Out (1987 film)

text: Farrell quietly sends the printout by courier to the Director of the CIA, an enemy of Brice, then leaves the Pentagon as the finished image enhancement of the photograph positively reveals Farrell as Atwell's lover. Later, Farrell is picked up by two men while sitting despondently at Atwell's grave, the men shown at the beginning of the story. They begin to interrogate Farrell about why things were so "poorly handled."


title: No Way Out (1987 film)

text: Farrell angrily confronts the chief interrogator who is revealed to be, not American, but Russian, and that Farrell "has" actually been "Yuri" the entire time, a deep cover Soviet agent passing as an American and working as a high level mole inside the Pentagon. The KGB ordered Farrell to seduce Brice's mistress to gather intelligence from her. Farrell's handler, his landlord, tells "Yuri" that America is no longer safe for him and that it is time for him to return to the Soviet Union. Revealing that he genuinely loved Susan Atwell, Farrell refuses and tells his handlers that he is finished being a spy. Though the other KGB are ready to kill Farrell, his handler orders to them to allow Farrell to escape, believing that Farrell will return on his own. In the final scenes, Farrell is seen driving away from the house, to meet an uncertain fate.


title: No Way Out (1987 film)

text: Exteriors were shot on location in Baltimore, Annapolis, Arlington, Washington, D.C., and Auckland, New Zealand, between April 1986 and June 1986. The film is dedicated to the memory of its director of photography John Alcott who died after principal photography had wrapped in July 1986, over a year prior to the film's eventual release.


title: No Way Out (1987 film)

text: The film debuted at number 2 at the US box office after "Stakeout" with $4.3 million. The film's budget was an estimated $15 million; its total U.S. gross was $35.5 million.


title: No Way Out (1987 film)

text: On Rotten Tomatoes the film holds an approval rating of 91% based on 47 reviews, with an average rating of 7.4/10. The website's critics consensus states: "Roger Donaldson's modern spin on the dense, stylish suspense films of the 1940s features fine work from Gene Hackman and Sean Young, as well as the career-making performance that made Kevin Costner a star." On Metacritic, the film has a weighted average score of 77 out of 100, based on 18 critics, indicating "generally favorable reviews". Audiences polled by CinemaScore gave the film an average grade of "B+" on an A+ to F scale.


title: No Way Out (1987 film)

text: Roger Ebert gave the film 4 out of 4 stars, calling it "truly labyrinthine and ingenious." Richard Schickel of "Time" wrote, "Viewers who arrive at the movie five minutes late and leave five minutes early will avoid the setup and payoff for the preposterous twist that spoils this lively, intelligent remake of 1948's "The Big Clock"." Desson Thomson of "The Washington Post" wrote, "The film makes such good use of Washington and builds suspense so well that it transcends a plot bordering on ridiculous."


title: Softmax function

text: The softmax function, also known as softargmax or normalized exponential function, converts a vector of real numbers into a probability distribution of possible outcomes. It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes, based on Luce's choice axiom.


title: Softmax function

text: The softmax function takes as input a vector of real numbers, and normalizes it into a probability distribution consisting of probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some vector components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval formula_1, and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities.


title: Softmax function

text: In simple words, it applies the standard exponential function to each element formula_5 of the input vector formula_6 and normalizes these values by dividing by the sum of all these exponentials; this normalization ensures that the sum of the components of the output vector formula_7 is 1.


title: Softmax function

text: Instead of , a different base can be used. If , smaller input components will result in larger output probabilities, and decreasing the value of will create probability distributions that are more concentrated around the positions of the smallest input values. Conversely, if , larger input components will result in larger output probabilities, and increasing the value of will create probability distributions that are more concentrated around the positions of the largest input values. Writing formula_8 or formula_9 (for real ) yields the expressions:


title: Softmax function

text: In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter is varied.


title: Softmax function

text: The name "softmax" is misleading; the function is not a smooth maximum (a smooth approximation to the maximum function), but is rather a smooth approximation to the arg max function: the function whose value is "which" "index" has the maximum. In fact, the term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", but the term "softmax" is conventional in machine learning. This section uses the term "softargmax" to emphasize this interpretation.


title: Softmax function

text: Formally, instead of considering the arg max as a function with categorical output formula_11 (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg):


title: Softmax function

text: where the output coordinate formula_13 if and only if formula_14 is the arg max of formula_15, meaning formula_5 is the unique maximum value of formula_17. For example, in this encoding formula_18 since the third argument is the maximum.


title: Softmax function

text: This can be generalized to multiple arg max values (multiple equal formula_5 being the maximum) by dividing the 1 between all max args; formally where is the number of arguments assuming the maximum. For example, formula_20 since the second and third argument are both the maximum. In case all arguments are equal, this is simply formula_21 Points with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points.


title: Softmax function

text: With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as , softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input as , formula_22 However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, formula_23 but formula_24 and formula_25 for all inputs: the closer the points are to the singular set formula_26, the slower they converge. However, softargmax does converge compactly on the non-singular set.


title: Softmax function

text: Conversely, as , softargmax converges to arg min in the same way, where here the singular set is points with two arg "min" values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization".


title: Softmax function

text: It is also the case that, for any fixed , if one input is much larger than the others "relative" to the temperature, formula_27, the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1:


title: Softmax function

text: However, if the difference is small relative to the temperature, the value is not close to the arg max. For example, a difference of 10 is small relative to a temperature of 100:


title: Softmax function

text: As , temperature goes to zero, formula_30, so eventually all differences become large (relative to a shrinking temperature), which gives another interpretation for the limit behavior.


title: Softmax function

text: In probability theory, the output of the softargmax function can be used to represent a categorical distribution – that is, a probability distribution over different possible outcomes.


title: Softmax function

text: In statistical mechanics, the softargmax function is known as the Boltzmann distribution (or Gibbs distribution): the index set formula_31 are the microstates of the system; the inputs formula_5 are the energies of that state; the denominator is known as the partition function, often denoted by ; and the factor is called the coldness (or thermodynamic beta, or inverse temperature).


title: Softmax function

text: The softmax function is used in various multiclass classification methods, such as multinomial logistic regression (also known as softmax regression) , multiclass linear discriminant analysis, naive Bayes classifiers, and artificial neural networks. Specifically, in multinomial logistic regression and linear discriminant analysis, the input to the function is the result of distinct linear functions, and the predicted probability for the th class given a sample vector and a weighting vector is:


title: Softmax function

text: This can be seen as the composition of linear functions formula_34 and the softmax function (where formula_35 denotes the inner product of formula_36 and formula_37). The operation is equivalent to applying a linear operator defined by formula_37 to vectors formula_36, thus transforming the original, probably highly-dimensional, input to vectors in a -dimensional space formula_40.


title: Softmax function

text: The standard softmax function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression.


title: Softmax function

text: Since the function maps a vector and a specific index formula_14 to a real value, the derivative needs to take the index into account:


title: Softmax function

text: Here, the Kronecker delta is used for simplicity (cf. the derivative of a sigmoid function, being expressed via the function itself).


title: Softmax function

text: In order to achieve stable numerical computations of the derivative, one often subtracts a constant from the input vector. In theory, this does not change the output, and neither the derivative. But it is more stable as it can control explicitly the largest value computed in each exponent.


title: Softmax function

text: If the function is scaled with the parameter formula_45, then these expressions must be multiplied by formula_45.


title: Softmax function

text: In the field of reinforcement learning, a softmax function can be used to convert values into action probabilities. The function commonly used is:


title: Softmax function

text: where the action value formula_48 corresponds to the expected reward of following action a and formula_49 is called a temperature parameter (in allusion to statistical mechanics). For high temperatures (formula_50), all actions have nearly the same probability and the lower the temperature, the more expected rewards affect the probability. For a low temperature (formula_51), the probability of the action with the highest expected reward tends to 1.


title: Softmax function

text: In neural network applications, the number of possible outcomes is often large, e.g. in case of neural language models that predict the most likely outcome out of a vocabulary which might contain millions of possible words. This can make the calculations for the softmax layer (i.e. the matrix multiplications to determine the formula_5, followed by the application of the softmax function itself) computationally expensive. What's more, the gradient descent backpropagation method for training such a neural network involves calculating the softmax for every training example, and the number of training examples can also become large. The computational effort for the softmax became a major limiting factor in the development of larger neural language models, motivating various remedies to reduce training times.


title: Softmax function

text: Approaches that reorganize the softmax layer for more efficient calculation include the hierarchical softmax and the differentiated softmax. The hierarchical softmax (introduced by Morin and Bengio in 2005) uses a binary tree structure where the outcomes (vocabulary words) are the leaves and the intermediate nodes are suitably selected "classes" of outcomes, forming latent variables. The desired probability (softmax value) of a leaf (outcome) can then be calculated as the product of the probabilities of all nodes on the path from the root to that leaf. Ideally, when the tree is balanced, this would reduce the computational complexity from formula_53 to formula_54. In practice, results depend on choosing a good strategy for clustering the outcomes into classes. A Huffman tree was used for this in Google's word2vec models (introduced in 2013) to achieve scalability.


title: Softmax function

text: A second kind of remedies is based on approximating the softmax (during training) with modified loss functions that avoid the calculation of the full normalization factor. These include methods that restrict the normalization sum to a sample of outcomes (e.g. Importance Sampling, Target Sampling).


title: Softmax function

text: Geometrically the softmax function maps the vector space formula_40 to the boundary of the standard formula_56-simplex, cutting the dimension by one (the range is a formula_57-dimensional simplex in formula_58-dimensional space), due to the linear constraint that all output sum to 1 meaning it lies on a hyperplane.


title: Softmax function

text: Along the main diagonal formula_59 softmax is just the uniform distribution on outputs, formula_60: equal scores yield equal probabilities.


title: Softmax function

text: More generally, softmax is invariant under translation by the same value in each coordinate: adding formula_61 to the inputs formula_62 yields formula_63, because it multiplies each exponent by the same factor, formula_64 (because formula_65), so the ratios do not change:


title: Softmax function

text: Geometrically, softmax is constant along diagonals: this is the dimension that is eliminated, and corresponds to the softmax output being independent of a translation in the input scores (a choice of 0 score). One can normalize input scores by assuming that the sum is zero (subtract the average: formula_67 where formula_68), and then the softmax takes the hyperplane of points that sum to zero, formula_69, to the open simplex of positive values that sum to 1formula_70, analogously to how the exponent takes 0 to 1, formula_71 and is positive.


title: Softmax function

text: The standard logistic function is the special case for a 1-dimensional axis in 2-dimensional space, say the "x"-axis in the plane. One variable is fixed at 0 (say formula_74), so formula_71, and the other variable can vary, denote it formula_76, so formula_77 the standard logistic function, and formula_78 its complement (meaning they add up to 1). The 1-dimensional input could alternatively be expressed as the line formula_79, with outputs formula_80 and formula_81


title: Softmax function

text: The softmax function was used in statistical mechanics as the Boltzmann distribution in the foundational paper , formalized and popularized in the influential textbook .


title: Softmax function

text: The use of the softmax in decision theory is credited to , who used the axiom of independence of irrelevant alternatives in rational choice theory to deduce the softmax in Luce's choice axiom for relative preferences.


title: Softmax function

text: In machine learning, the term "softmax" is credited to John S. Bridle in two 1989 conference papers, : and :


title: Softmax function

text: If we take an input of , the softmax of that is . The output has most of its weight where the "4" was in the original input. This is what the function is normally used for: to highlight the largest values and suppress values which are significantly below the maximum value. But note: softmax is not scale invariant, so if the input were (which sums to 1.6) the softmax would be . This shows that for values between 0 and 1 softmax, in fact, de-emphasizes the maximum value (note that 0.169 is not only less than 0.475, it is also less than the initial proportion of ).


title: Orbit of the Moon

text: The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days (a tropical month and sidereal month) and one revolution relative to the Sun in about 29.53 days (a synodic month). Earth and the Moon orbit about their barycentre (common centre of mass), which lies about from Earth's centre (about 73% of its radius), forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.


title: Orbit of the Moon

text: With a mean orbital velocity of 1.022 km/s (0.635 miles/s, 2,286 miles/h), the Moon covers a distance approximately its diameter, or about half a degree on the celestial sphere, each hour. The Moon differs from most satellites of other planets in that its orbit is close to the ecliptic plane instead of to its primary's (in this case, Earth's) equatorial plane. The Moon's orbital plane is inclined by about 5.1° with respect to the ecliptic plane, whereas the Moon's equatorial plane is tilted by only 1.5°.


title: Orbit of the Moon

text: The properties of the orbit described in this section are approximations. The Moon's orbit around Earth has many variations (perturbations) due to the gravitational attraction of the Sun and planets, the study of which (lunar theory) has a long history.


title: Orbit of the Moon

text: The orbit of the Moon is a nearly circular ellipse about the Earth (the semimajor and semiminor axes are 384,400 km and 383,800 km, respectively: a difference of only 0.16%). The equation of the ellipse yields an eccentricity of 0.0549 and perigee and apogee distances of 362,600 km and 405,400 km respectively (a difference of 12%).


title: Orbit of the Moon

text: Since nearer objects appear larger, the Moon's apparent size changes as it moves toward and away from an observer on Earth. An event referred to as a "supermoon" occurs when the full Moon is at its closest to Earth (perigee). The largest possible apparent diameter of the Moon is the same 12% larger (as perigee versus apogee distances) than the smallest; the apparent area is 25% more and so is the amount of light it reflects toward Earth.


title: Orbit of the Moon

text: The variance in the Moon's orbital distance corresponds with changes in its tangential and angular speeds, as stated in Kepler's second law. The mean angular movement relative to an imaginary observer at the Earth–Moon barycentre is ° per day to the east (J2000.0 epoch).


title: Orbit of the Moon

text: The Moon's elongation is its angular distance east of the Sun at any time. At new moon, it is zero and the Moon is said to be in conjunction. At full moon, the elongation is 180° and it is said to be in opposition. In both cases, the Moon is in syzygy, that is, the Sun, Moon and Earth are nearly aligned. When elongation is either 90° or 270°, the Moon is said to be in quadrature.


title: Orbit of the Moon

text: The orientation of the orbit is not fixed in space but rotates over time. This orbital precession is called apsidal precession and is the rotation of the Moon's orbit within the orbital plane, i.e. the axes of the ellipse change direction. The lunar orbit's major axis – the longest diameter of the orbit, joining its nearest and farthest points, the perigee and apogee, respectively – makes one complete revolution every 8.85 Earth years, or 3,232.6054 days, as it rotates slowly in the same direction as the Moon itself (direct motion) – meaning precesses eastward by 360°. The Moon's apsidal precession is distinct from the nodal precession of its orbital plane and axial precession of the moon itself.