History.

The influence of rhythmic sounds and drums to enter altered states of consciousness is used in different indigenous tribes (see Shamanic music), as well as optical stimulation produced by the flickering light of camp fires or pressing lightly on the eyeballs. This “stroboscopic photo-stimulation produces "photic driving", the alpha type of brain electrical activity associated with an altered state in which people are susceptible to suggestion.” ( p. 12).  The first scientific observations were made by William Charles Wells in the 1790s who described different effects of binocular vision. His results were later transferred to be applied in binaural beats. Visual experiments with flickering lights were conducted in the 1940s by William Grey Walter who used stroboscopic light flashes to measure their effects on brain activity, assessed with EEG. He reported effect not just on visual areas but on the whole cortex.  During the '60s and '70s the interest in different methods to induce altered states without the use of drugs were rising. Some of the induced states by rhythmic light and sound combinations were even described as psychedelic-like, while these claims lack adequate measurements for their similarity. In this line, numerous nightclubs began using strobes to maximize the effects of the music for dancing.  The development of alpha EEG feedback (see neurofeedback) is an important starting point for biofeedback and its explicit use for entering altered states of consciousness. In these decades, Jack Schwarz built one of the first mind machines using rhythmic sounds and variable frequency lights in goggles to produce certain mental states. Enterprises started to produce different types of mind machines and some scientists followed the line of research to explore if and how these devices elicit effects on brain processes.  In the late 1980s and early 1990s Farley initiated an investigation concerning medical claims made by some manufacturers and sellers. The FDA concluded that Light and Sound Machines were not medical devices and did not warrant regulation. Sellers and manufacturers were given guidelines for how they could advertise these devices, and were required to include a disclaimer and cautionary document with each machine. Nowadays, mind machines are rediscovered by some teenage cultures as so called “digital drugs”, a legal way to enter altered states of consciousness.

Application.

Mind machines include flashing light devices, which are similar to the Brion Gysin dreamachine in that both produce a flickering visual field. Unlike flashing light devices, the dreamachine can be used by several people at once, but has few, if any, technical features.

Technical setting.

Mind machines typically consist of a control unit, a pair of headphones and/or strobe light goggles. The unit controls the sessions and drives the LEDs in the goggles. Professionally, they are usually referred to as Auditory Visual Stimulation Devices (AVS devices). Also mind machines are offered that can connect to the Internet for updates download of new session material.

Application rate.

One session normally takes between 15 and 60 minutes. During a session the user should lie relaxed, and place the glasses on the eyes, which should remain closed during the whole session. Many machines have pre-programmed sessions which vary in parameters like light brightness, audio pitch or beat frequency but also give the opportunity for the user to create a custom session. Typically one session consists of a series of segments. Within one segment parameters change in a constant way. Mind machines are often used together with biofeedback or neurofeedback equipment in order to adjust the frequency on the fly, while proof for their effectiveness is lacking.

Description of altered states.

Perceptual changes.

Light & sound mind machines can have various effects on the user. Most users describe seeing a flashing light, others perceive swirling patterns that have been compared to psychedelic light shows or fractals. A few users report seeing detailed, virtual reality like scenes. But also tactile and emotional changes are reported after a 6 Hz photic stimulation, as well as auditory hallucinations like binaural beats.

Changes in brain activity.

Sessions will typically aim to address the target frequencies which correspond to delta (1-3 hertz), theta (4–7 Hz), alpha (8–12 Hz) or beta brain waves (13–40 Hz). Those frequency bands can be adjusted by the user based on the desired effects.  For relaxation, often a reduction from beta waves to lower alpha or theta frequencies is observed. It is aimed to reach a level of “slow alpha” (8 Hz). Glickson (1986) states that an alpha frequency of 10 Hz is optimal for perceiving visual hallucinations. He assumes that it's the change in alpha activity and not the alpha activity itself that is facilitating an altered state of consciousness.

After effects.

Even if research is not sufficient so far, Michael Hutchison and other scientists report strong tranquilizing effects that lasted up to 3 days. After several sessions it is observed that users can produce a desired brain state with the associated brainwaves easier and deliberately.

Other information.

Therapeutic Use.

Clinical research has been done on the use of auditory and visual stimulation to improve cognitive abilities in learning-disabled children as well as in the treatment of autism and attention deficit hyperactivity disorder (ADHD). However, lacking enough evidence to conclude that these treatments show efficacy, several studies indicate the potential to increase effects on IQ and reading levels in primary students (cite Olmstead, 2005), but also improvements in inattention and impulsiveness in children with ADHD (cite Siever).

Safety.

Rapidly flashing lights may be dangerous for people with photosensitive epilepsy or other nervous disorders. It is thought that one out of 10,000 adults will experience a seizure while viewing such a device; about twice as many children will have a similar ill effect.

Regulation.

Mind machine devices are legally available throughout the United States from many sources.  With some exceptions, these devices commonly do not have FDA approval for medical applications in the USA. They have been found by a U.S. district court to be Class III medical devices, and consequentially require FDA pre-market approval for all medical uses. One company making medical claims for a possibly unsafe device has been shut down and seen their devices destroyed.

A mind machine (aka "brain machine" or "light and sound machine") uses pulsing rhythmic sound, flashing light, electrical or magnetic fields, or a combination of these, to alter the frequency of the user's brainwaves. Mind machines can induce deep states of relaxation, concentration, and in some cases altered states of consciousness, which have been compared to those obtained from meditation and shamanic exploration. Photic mind machines work with flickering lights embedded in sunglasses or a lamp that sits on a tripod above your head or facing you. You then "Watch" with your eyes closed.

Psychowalkman (audiovizuálně stimulační přístroj, AVS) je přístroj, který světelnými záblesky a zvukovými šumy stimuluje mozek a tak ovlivňuje jeho chování.


Etymology.

The name of the city, Graz, formerly spelled Gratz, most likely stems from the Slavic "gradec", which means "small castle". Some archaeological finds point to the erection of a small castle by Alpine Slavic people, which over time became a heavily defended fortification. In literary Slovene and Croatian, "gradec" still means "small castle", forming a hypocoristic derivative of Proto-West-South Slavic *gradьcъ, whichs descends via liquid metathesis from Common Slavic *gardьcъ and via the Slavic third palatalisation from Proto-Slavic *gardiku, originally denoting "small town, settlement". The name thus follows the common South Slavic pattern for naming settlements as "grad". The German name 'Graz' first appears in records in 1128.

Geography.

Graz is situated on both sides of the Mur river in southeast Austria. It is about southwest of Vienna ("Wien"). The nearest larger urban centre is Maribor ("Marburg") in Slovenia, which is about to the south. Graz is the state capital and largest city in Styria, a green and heavily forested region on the eastern edge of the Alps.

Neighbouring municipalities.

These towns and villages border Graz:

Districts.

Graz is divided into 17 municipal districts ("Stadtbezirke"):

History.

The oldest settlement on the ground of the modern city of Graz dates back to the Copper Age. However, no historical continuity exists of a settlement before the Middle Ages.  During the 12th century, dukes under Babenberg rule made the town into an important commercial center. Later, Graz came under the rule of the Habsburgs and, in 1281, gained special privileges from King Rudolph I.  In the 14th century, Graz became the city of residence of the Inner Austrian line of the Habsburgs. The royalty lived in the Schlossberg castle and from there ruled Styria, Carinthia, most of today's Slovenia, and parts of Italy (Carniola, Gorizia and Gradisca, Trieste).  In the 16th century, the city's design and planning were primarily controlled by Italian Renaissance architects and artists. One of the most famous buildings representative of this style is the Landhaus, designed by Domenico dell'Allio, and used by the local rulers as a governmental headquarters.  The University of Graz was founded by Archduke Karl II in 1585, it's the city's oldest university. For most of its existence, it was controlled by the Catholic church, and was closed in 1782 by Joseph II in an attempt to gain state control over educational institutions. Joseph II transformed it into a lyceum where civil servants and medical personnel were trained. In 1827 it was re-established as a university by Emperor Franz I, and was named 'Karl-Franzens Universität' or 'Charles-Francis University' in English. More than 30,000 students are currently enrolled at this university.  The astronomer Johannes Kepler lived in Graz for a short period. He worked as a math teacher and was a professor of mathematics at the University of Graz, but still found time to study astronomy. He left Graz for Prague when Lutherans were banned from the city.  Ludwig Boltzmann was Professor for Mathematical Physics from 1869 to 1890. During that time, Nikola Tesla studied electrical engineering at the Polytechnic in 1875. Nobel Laureate Otto Loewi taught at the University of Graz from 1909 until 1938. Ivo Andric, the 1961 Nobel Prize for Literature Laureate obtained his doctorate at the University of Graz. Erwin Schrödinger was briefly chancellor of the University of Graz in 1936.  Graz is centrally located within today's "Bundesland" (state) of Styria, or "Steiermark" in German. "Mark" is an old German word indicating a large area of land used as a defensive border, in which the peasantry is taught how to organize and fight in the case of an invasion. With a strategic location at the head of the open and fertile Mur valley, Graz was historically a target of invaders, such as the Hungarians under Matthias Corvinus in 1481, and the Ottoman Turks in 1529 and 1532. Apart from the Riegersburg Castle, the Schlossberg was the only fortification in the region that never fell to the Ottoman Turks. Graz is home to the region's provincial armory, which is the world's largest historical collection of late medieval and Renaissance weaponry. It has been preserved since 1551, and displays over 30,000 items.  From the earlier part of the 15th century, Graz was the residence of the younger branch of the Habsburgs, which succeeded to the imperial throne in 1619 in the person of Emperor Ferdinand II, who moved the capital to Vienna. New fortifications were built on the Schlossberg at the end of the 16th century. Napoleon's army occupied Graz in 1797. In 1809, the city withstood another assault by the French army. During this attack, the commanding officer in the fortress was ordered to defend it with about 900 men against Napoleon's army of about 3,000. He successfully defended the Schlossberg against eight attacks, but they were forced to give up after the Grande Armée occupied Vienna and the Emperor ordered to surrender. Following the defeat of Austria by Napoleonic forces at the Battle of Wagram in 1809, the fortifications were demolished using explosives, as stipulated in the Peace of Schönbrunn of the same year. The belltower (Glockenturm) and the civic clock tower ("Uhrturm"), which is a leading tourist attraction and serves as a symbol for Graz, were spared after the citizens of Graz paid a ransom for their preservation.  Archduke Karl II of Inner Austria had 20,000 Protestant books burned in the square of what is now a mental hospital, and succeeded in returning Styria to the authority of the Holy See. Archduke Franz Ferdinand was born in Graz in what is now the Stadtmuseum (city museum).

Population development.

The more recent population figures do not give the whole picture as only people with principal-residence status are counted and people with secondary residence status are not. Most of the people with secondary residence status in Graz are students. At the end of 2016 there were 33,473 people with secondary residence status in Graz.

Climate.

Oceanic climate is the type found in the city, but due to the 0 °C isotherm, the same occurs in a humid continental climate with based in Köppen system (Cfb/Dfb borderline). Wladimir Köppen himself was in town and conducted studies to see how the climate of the past influenced the Continental Drift theory. Due to its position southeast of the Alps, Graz is shielded from the prevailing westerly winds that bring weather fronts in from the North Atlantic to northwestern and central Europe. The weather in Graz is thus influenced by the Mediterranean, and it has more hours of sunshine per year than Vienna or Salzburg and also less wind or rain. Graz lies in a basin that is only open to the south, causing the climate to be warmer than would be expected at that latitude. Plants are found in Graz that normally grow much further south.

Slovenes and Graz.

Politically, culturally, scientifically and religiously, Graz was an important centre for all Slovenes, especially from the establishment of the University of Graz in 1586 until the establishment of University of Ljubljana in 1919. In 1574, the was published in Graz, and in 1592, Hieronymus Megiser published in Graz the book "Dictionarium quatuor linguarum", the first multilingual dictionary of Slovene.  The Styrian Slovenes did not consider Graz a German-speaking city, but their own, a place to study while living at their relatives' homes and to fulfill one's career ambitions. The student associations in Graz were a crucible of the Slovene identity, and the Slovene students in Graz were more nationally aware than some others. This led to fierce anti-Slovene efforts of German-speaking nationalists in Graz before and during World War II.  Many Slovenian Styrians study there. Slovenes are among the professors at the Institute for Jazz in Graz. Numerous Slovenes have found employment there, while being formerly unemployed in Slovenia. For the Slovene culture, Graz remains permanently important due to its university and the Universalmuseum Joanneum archives containing numerous documents from the Slovenian Styria.  A symposium on the relation of Graz and the Slovenes was held in Graz in 2010, at the occasion of the 200th anniversary of the establishment of the first and oldest chair of Slovene. It was established at the Lyzeum of Graz in July 1811 on the initiative of. A collection of lectures on the topic was published. The Slovenian Post commemorated the anniversary with a stamp.

Main sights.

For the year that Graz was Cultural Capital of Europe, new structures were erected. The Graz Museum of Contemporary Art (German: Kunsthaus) was designed by Peter Cook and Colin Fournier and is situated next to the Mur river. The Island in the Mur is a floating platform made of steel. It was designed by American architect Vito Acconci and contains a café, an open-air theatre and a playground.

Historic city centre.

The historic centre was added to the UNESCO World Heritage List in 1999 due to the harmonious co-existence of typical buildings from different epochs and in different architectural styles. Situated in a cultural borderland between Central Europe, Italy and the Balkan States, Graz absorbed various influences from the neighbouring regions and thus received its exceptional townscape. Today the historic centre consists of over 1,000 buildings, their age ranging from Gothic to contemporary.  The most important sights in the historic centre are:

Culture.

During 2003 Graz held the title of "European Capital of Culture" and was one of the UNESCO "Cities of Design" in 2011.

Museums.

The most important museums in Graz are:

Architecture.

The Old Town and the adjacent districts are characterized by the historic residential buildings and churches found there. In the outer districts buildings are predominantly of the architectural styles from the second half of the 20th century.  In 1965 the "Grazer Schule" (School of Graz) was founded. Several buildings around the universities are of this style, for example the green houses by Volker Giencke and the "RESOWI" center by Günther Domenig.  Before Graz became the European Capital of Culture in 2003, several new projects were realized, such as the "Stadthalle", the "Kindermuseum" (museum for children), the "Helmut-List-Halle", the "Kunsthaus" and the "Murinsel".  Buildings in Graz which are at least 50m tall:

Sports.

SK Sturm Graz is the main football club of the city, with three Austrian championships and five runner-up seasons. The Grazer AK also won an Austrian championship, but went into administration in 2007 and was excluded from the professional league system.  In ice hockey, the ATSE Graz was the Austrian Hockey League champion in 1975 and 1978. The EC Graz was runner-up in 1991–92, 1992–93 and 1993–94. The Graz 99ers plays in first division since 2000.  UBSC Raiffeisen Graz plays in the Austrian Basketball League.  The Graz Giants play in the Austrian Football League (American Football).  The city bid for the 2002 Winter Olympics in 1995, but lost the election to Salt Lake City. Nowadays there is a plan to bid for the 2026 Winter Olympics with some venues in Bavaria, Germany to cut costs with using existing venues around national borders. It's still facing referendum, meaning usually the end for many former olympic bids in Europe and North America since 1970 -century.

Styriarte.

Graz hosts the annual festival of classical music Styriarte, founded in 1985 to tie conductor Nikolaus Harnoncourt closer to his hometown. Events have been held at different venues in Graz and in the surrounding region.

Transport.

An extensive public transport network makes Graz an easy city to navigate without a car. The city has a comprehensive bus network, complementing the Graz tram network consisting of eight lines. Four lines pass through the underground tram stop at the central railway station (Hauptbahnhof) and on to the city centre before branching out. Furthermore, there are seven night-time bus routes, although these run only at weekends and on evenings preceding public holidays.  The Schlossbergbahn, a funicular railway, and the Schlossberg lift, a vertical lift, link the city centre to the Schlossberg.  From the central railway station (Hauptbahnhof), regional trains link to most of Styria. Direct trains run to most major cities nearby including Vienna, Salzburg, Innsbruck, Maribor and Ljubljana in Slovenia, Zagreb in Croatia, Budapest in Hungary, Prague and Brno in the Czech Republic, Zürich in Switzerland, as well as Munich, Stuttgart, Heidelberg, and Frankfurt in Germany. Trains for Vienna leave every hour. In recent years many railway stations within the city limits and in the suburbs have been rebuilt or modernised and are now part of the "S-Bahn Graz", a commuter train service connecting the city with its suburban area and towns nearby.  Graz airport is located about south of the city centre and is accessible by bus, railway, and car. Direct destinations include Amsterdam, Berlin, Düsseldorf, Frankfurt, Munich, Stuttgart, Istanbul, Vienna and Zurich.

Health.

In Graz there are seven hospitals, several private hospitals and sanatoriums, as well as 44 pharmacies.  The "University Hospital Graz" (LKH-Universitäts-Klinikum Graz) is located in eastern Graz and has 1,556 beds and 7,190 employees. The "Regional Hospital Graz II" (LKH Graz II) has two sites in Graz. The western site (LKH Graz II Standort West) is located in Eggenberg and has 280 beds and about 500 employees, the southern site (LKH Graz II Standort Süd) specializes in neurology and psychiatry and is located in Straßgang with 880 beds and 1,100 employees. The "AUVA Accident Hospital" (Unfallkrankenhaus der AUVA) is in Eggenberg and has 180 beds and a total of 444 employees.  The Albert Schweitzer Clinic in the western part of the city is a geriatric hospital with 304 beds, the Hospital of St. John of God (Krankenhaus der Barmherzigen Brüder) has two sites in Graz, one in Lend with 225 beds and one in Eggenberg with 260 beds. The Hospital of the Order of Saint Elizabeth (Krankenhaus der Elisabethinen) in Gries has 182 beds.  There are several private clinics as well: the Privatklinik Kastanienhof, the Privatklinik Leech, the Privatklinik der Kreuzschwestern, the Sanatorium St. Leonhard, the Sanatorium Hansa and the Privatklinik Graz-Ragnitz.  EMS in Graz is provided solely by the Austrian Red Cross. Perpetually two emergency doctor's cars ("NEF – Notarzteinsatzfahrzeug"), two NAWs ("Notarztwagen" – ambulances staffed with a physician in addition to regular personnel) and about 30 RTWs ("Rettungswagen" – regular ambulances) are on standby. Furthermore, several non-emergency ambulances ("KTW – Krankentransportwagen") and a Mobile Intensive Care Unit (MICU) are operated by the Red Cross to transport non-emergency patients to and between hospitals. In addition to the Red Cross, the Labourers'-Samaritan-Alliance (Arbeiter-Samariter-Bund Österreichs), the Austrian organisation of the Order of Malta Ambulance Corps (Malteser Hospitaldienst Austria) and the Green Cross ("Grünes Kreuz") operate ambulances ("KTW") for non-emergency patient transport. In addition to the cars, there's also the C12 air ambulance helicopter stationed at Graz airport, staffed with an emergency physician in addition to regular personnel.

International relations.

Twin towns and sister cities.

Graz is twinned with:

Notable residents.

The following are past and present notable residents of Graz.

External links.

Official websites  History  Further information

Graz (, ; ) is the capital city of Styria and Austria's second-largest city after Vienna. As of 1 January 2019, Graz had a population of 328,276 (292,269 of whom had principal-residence status). In 2015, the population of the Graz larger urban zone (LUZ) stood at 633,168, based on principal-residence status. Graz has a long tradition as a seat of higher education. It has four colleges and four universities with more than 60,000 students. Its historic centre "(Altstadt") is one of the best-preserved city centres in Central Europe.

Štýrský Hradec (německy "Graz", slovinsky "Gradec"), hlavní město Štýrska, je druhé největší město v Rakousku. Historické centrum Štýrského Hradce bylo zapsáno na Seznam světového dědictví UNESCO v roce 1999. Hlavním symbolem a jakýmsi poznávacím znakem Štýrského Hradce je Schloßberg – zámecký vrch s hodinovou věží a pevností, jejíž základy pocházejí již z 9. století. Graz je znám také jako město studentů, neboť na čtyřech univerzitách tu studuje na 50 tisíc vysokoškoláků. Ve městě panuje teplé klima, které sem ze středomoří přivádí vzdušné proudy, takzvané fény. Žije zde obyvatel.


Applications.

Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers. In a strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as her/his own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what she/he expects the others to do. Nash equilibrium requires that their choices be consistent: no player wishes to undo her/his decision given what the others are deciding.  The concept has been used to analyze hostile situations like wars and arms races (see prisoner's dilemma), and also how conflict may be mitigated by repeated interaction (see tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises (see coordination game). Other applications include traffic flow (see Wardrop's principle), how to organize auctions (see auction theory), the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations (see tragedy of the commons), natural resource management, analysing strategies in marketing, even penalty kicks in football (see matching pennies), energy systems, transportation systems, evacuation problems and wireless communications.

History.

Nash equilibrium is named after American mathematician John Forbes Nash, Jr. The same idea was used in a particular application in 1838 by Antoine Augustin Cournot in his theory of oligopoly. In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally.  The modern game-theoretic concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions (rather than choosing a deterministic action to be played with certainty). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book "The Theory of Games and Economic Behavior". However, their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Just putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper, and a variant upon it in his 1951 paper used the Brouwer fixed-point theorem to prove that there had to exist at least one mixed strategy profile that mapped back into itself for finite-player (not necessarily zero-sum) games; namely, a strategy profile that did not call for a shift in strategies that could improve payoffs.  Since the development of the Nash equilibrium concept, game theorists have discovered that it makes misleading predictions (or fails to make a unique prediction) in certain circumstances. They have proposed many related solution concepts (also called'refinements' of Nash equilibria) designed to overcome perceived flaws in the Nash concept. One particularly important issue is that some Nash equilibria may be based on threats that are not 'credible'. In 1965 Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeated, or what happens if a game is played in the absence of complete information. However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others.

Definitions.

Nash Equilibrium.

Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing her/his strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"  If any player could answer "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium.  The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal.  The Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with non-rational moves. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

Strict/Weak Equilibrium.

Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?"  If every player's answer is "Yes", then the equilibrium is classified as a "strict Nash equilibrium".  If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives the exactly same payout (i.e. this player is indifferent between switching and not), then the equilibrium is classified as a "weak Nash equilibrium".  A game can have a pure-strategy or a mixed-strategy Nash equilibrium. (In the latter a pure strategy is chosen stochastically with a fixed probability).

Nash's Existence Theorem.

Nash proved that if we allow mixed strategies (where a player chooses probabilities of using various pure strategies), then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium (which might be a pure strategy for each player or might be a probability distribution over strategies for each player).  Nash equilibria need not exist if the set of choices is infinite and noncompact. An example is a game where two players simultaneously name a number and the player naming the larger number wins. Another example is where each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game). However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of the other players. An example, in which the equilibrium is a mixture of continuously many pure strategies, is a game where two players simultaneously pick a real number between 0 and 1 (inclusive) and player one's winnings (paid by the second player) equal the square root of the distance between the two numbers.

Examples.

Coordination game.

The "coordination game" is a classic (symmetric) two player, two strategy game, with an example payoff matrix shown to the right. The players should thus coordinate, both adopting strategy A, to receive the highest payoff; i.e., 4. If both players chose strategy B though, there is still a Nash equilibrium. Although each player is awarded less than optimal payoff, neither player has incentive to change strategy due to a reduction in the immediate payoff (from 2 to 1).  A famous example of this type of game was called the stag hunt; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat (4 utility units) than the latter (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, she/he will fail in hunting (0 utility units), whereas if they both hunt it they will split the payload (2, 2). The game hence exhibits two equilibria at (stag, stag) and (rabbit, rabbit) and hence the players' optimal strategy depend on their expectation on what the other player may do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they suspect that the other will hunt the rabbit, they should hunt the rabbit. This game was used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.  Another example of a coordination game is the setting where two technologies are available to two firms with comparable products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game.  Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:  In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%).

Prisoner's dilemma.

Imagine two prisoners held in separate cells, interrogated simultaneously, and offered deals (lighter jail sentences) for betraying their fellow criminal. They can "cooperate" (with the other prisoner) by not snitching, or "defect" by betraying the other. However, there is a catch; if both players defect, then they both serve a longer sentence than if neither said anything. Lower jail sentences are interpreted as higher payoffs (shown in the table).  The prisoner's dilemma has a similar matrix as depicted for the coordination game, but the maximum reward for each player (in this case, a minimum loss of 0) is obtained only when the players' decisions are different. Each player improves their own situation by switching from "cooperating" to "defecting", given knowledge that the other player's best decision is to "defect". The prisoner's dilemma thus has a single Nash equilibrium: both players choosing to defect.  What has long made this an interesting case to study is the fact that this scenario is globally inferior to "both cooperating". That is, both players would be better off if they both chose to "cooperate" instead of both choosing to defect. However, each player could improve their own situation by breaking the mutual cooperation, no matter how the other player possibly (or certainly) changes their decision.

Network traffic.

An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are "cars" traveling from A to D, what is the expected distribution of traffic in the network?  This situation can be modeled as a "game" where every traveler has a choice of 3 strategies, where each strategy is a route from A to D (either,, or ). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via experiences travel time of, where is the number of cars traveling on edge. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For the graph on the right, if, for example, 100 cars are travelling from A to D, then equilibrium will occur when 25 drivers travel via, 50 via, and 25 via. Every driver now has a total travel time of 3.75 (to see this, note that a total of 75 cars take the edge, and likewise, 75 cars take the edge).  Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via and the other 50 through, then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess's paradox.

Competition game.

This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give up two points to the other.  This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0,0), (1,1), (2,2), and (3,3).

Nash equilibria in a payoff matrix.

There is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.  We can apply this rule to a 3×3 matrix:  Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash equilibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.  This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair is the maximum of the row. If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria.

Stability.

The concept of stability, useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria.  A Nash equilibrium for a mixed-strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:  If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed.  In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage the expectation of the player who did the change, if the other player's mixed strategy is still (50%,50%)), then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).  Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.  The Nash equilibrium defines stability only in terms of unilateral deviations. In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto efficient. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.  A refined Nash equilibrium known as coalition-proof Nash equilibrium (CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core.  Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward induction and backward induction. In a game theory context stable equilibria now usually refer to Mertens stable equilibria.

Occurrence.

If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:

Where the conditions are not met.

Examples of game theory problems in which these conditions are not met:

Where the conditions are met.

In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with the objective of showing how equilibrium points  A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players:  For a formal result along these lines, see Kuhn, H. and et al., 1996, "The Work of John Nash in Game Theory," "Journal of Economic Theory", 69, 153–185.  Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics and evolutionary biology, the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.

NE and non-credible threats.

The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game. This eliminates all non-credible threats, that is, strategies that contain non-rational moves in order to make the counter-player change their strategy.  The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left(L) or right(R), which is followed by player two being called upon to be kind (K) or unkind (U) to player one, However, player two only stands to gain from being unkind if player one goes left. If player one goes right the rational player two would de facto be kind to her/him in that subgame. However, The non-credible threat of being unkind at 2(2) is still part of the blue (L, (U,U)) Nash equilibrium. Therefore, if rational behavior can be expected by both parties the subgame perfect Nash equilibrium may be a more meaningful solution concept when such dynamic inconsistencies arise.

Proof of existence.

Proof using the Kakutani fixed-point theorem.

Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). We give a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible).  To prove the existence of a Nash equilibrium, let formula_1 be the best response of player i to the strategies of all other players.  Here, formula_3, where formula_4, is a mixed-strategy profile in the set of all mixed strategies and formula_5 is the payoff function for player i. Define a set-valued function formula_6 such that formula_7. The existence of a Nash equilibrium is equivalent to formula_8 having a fixed point.  Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied.  Condition 1. is satisfied from the fact that formula_13 is a simplex and thus compact. Convexity follows from players' ability to mix strategies. formula_13 is nonempty as long as players have strategies.  Condition 2. and 3. are satisfied by way of Berge's maximum theorem. Because formula_5 is continuous and compact, formula_16 is non-empty and upper hemicontinuous.  Condition 4. is satisfied as a result of mixed strategies. Suppose formula_17, then formula_18. i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff.  Therefore, there exists a fixed point in formula_19 and a Nash equilibrium.  When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed-point theorem." (See Nasar, 1998, p. 94.)

Alternate proof using the Brouwer fixed-point theorem.

We have a game formula_20 where formula_21 is the number of players and formula_22 is the action set for the players. All of the action sets formula_23 are finite. Let formula_24 denote the set of mixed strategies for the players. The finiteness of the formula_23s ensures the compactness of formula_26.  We can now define the gain functions. For a mixed strategy formula_27, we let the gain for player formula_28 on action formula_29 be  The gain function represents the benefit a player gets by unilaterally changing their strategy. We now define formula_31 where  for formula_33. We see that  Next we define:  It is easy to see that each formula_36 is a valid mixed strategy in formula_37. It is also easy to check that each formula_36 is a continuous function of formula_39, and hence formula_40 is a continuous function. As the cross product of a finite number of compact convex sets, formula_26 is also compact and convex. Applying the Brouwer fixed point theorem to formula_40 and formula_26 we conclude that formula_40 has a fixed point in formula_26, call it formula_46. We claim that formula_46 is a Nash equilibrium in formula_48. For this purpose, it suffices to show that  This simply states that each player gains no benefit by unilaterally changing their strategy, which is exactly the necessary condition for a Nash equilibrium.  Now assume that the gains are not all zero. Therefore, formula_50 and formula_29 such that formula_52. Note then that  So let  Also we shall denote formula_55 as the gain vector indexed by actions in formula_23. Since formula_46 is the fixed point we have:  Since formula_59 we have that formula_60 is some positive scaling of the vector formula_61. Now we claim that  To see this, we first note that if formula_52 then this is true by definition of the gain function. Now assume that formula_64. By our previous statements we have that  and so the left term is zero, giving us that the entire expression is formula_66 as needed.  So we finally have that  where the last inequality follows since formula_60 is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore, formula_46 is a Nash equilibrium for formula_48 as needed.

Computing Nash equilibria.

If a player A has a dominant strategy formula_71 then there exists a Nash equilibrium in which A plays formula_71. In the case of two players A and B, there exists a Nash equilibrium in which A plays formula_71 and B plays a best response to formula_71. If formula_71 is a strictly dominant strategy, A plays formula_71 in all Nash equilibria. If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy.  In games with mixed-strategy Nash equilibria, the probability of a player choosing any particular strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy. In order for a player to be willing to randomize, their expected payoff for each strategy should be the same. In addition, the sum of the probabilities for each strategy of a particular player should be 1. This creates a system of equations from which the probabilities of choosing each strategy can be derived.

Examples.

In the matching pennies game, player A loses a point to B if A and B play the same strategy and wins a point from B if they play different strategies. To compute the mixed-strategy Nash equilibrium, assign A the probability "p" of playing H and (1−"p") of playing T, and assign B the probability "q" of playing H and (1−"q") of playing T.  Thus a mixed-strategy Nash equilibrium, in this game, is for each player to randomly choose H or T with p = 1/2 and q = 1/2.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

Nashova rovnováha je v teorii her taková situace, kdy žádný z hráčů nemůže jednostrannou změnou zvolené strategie vylepšit svoji situaci. Současně se jedná i o koncept řešení nekooperativních her více hráčů. Své jméno získala po Johnu Nashovi, který dokázal, že každá konečná hra má alespoň jedno takové řešení.


Phrases and clauses.

An "infinitive phrase" is a verb phrase constructed with the verb in infinitive form. This consists of the verb together with its objects and other complements and modifiers. Some examples of infinitive phrases in English are given below – these may be based on either the full infinitive (introduced by the particle "to") or the bare infinitive (without the particle "to").  Infinitive phrases often have an implied grammatical subject making them effectively clauses rather than phrases. Such "infinitive clauses" or "infinitival clauses", are one of several kinds of non-finite clause. They can play various grammatical roles like a constituent of a larger clause or sentence; for example it may form a noun phrase or adverb. Infinitival clauses may be embedded within each other in complex ways, like in the sentence: Here the infinitival clause "to get married" is contained within the finite dependent clause "that Brett Favre is going to get married"; this in turn is contained within another infinitival clause, which is contained in the finite independent clause (the whole sentence).  The grammatical structure of an infinitival clause may differ from that of a corresponding finite clause. For example, in German, the infinitive form of the verb usually goes to the end of its clause, whereas a finite verb (in an independent clause) typically comes in second position.

Clauses with subject in the accusative case.

Following certain verbs or prepositions, infinitives commonly "do" have an expressed subject, e.g., As these examples illustrate, the subject of the infinitive is in the objective case (them, him) in contrast to the nominative case that would be used with a finite verb, e.g., "They ate their dinner." Such accusative and infinitive constructions are present in Latin and Ancient Greek, as well as many modern languages. The unusual case for the subject of an infinitive is an example of exceptional case-marking, where the infinitive clause's role being an object of a verb or preposition (want, for) overpowers the pronoun's subjective role within the clause.

Marking for tense, aspect and voice.

In some languages, infinitives may be marked for grammatical categories like voice, aspect, and to some extent tense. This may be done by inflection, as with the Latin perfect and passive infinitives, or by periphrasis (with the use of auxiliary verbs), as with the Latin future infinitives or the English perfect and progressive infinitives.  Latin has present, perfect and future infinitives, with active and passive forms of each. For details see.  English has infinitive constructions that are marked (periphrastically) for aspect: perfect, progressive (continuous), or a combination of the two (perfect progressive). These can also be marked for passive voice (as can the plain infinitive): Further constructions can be made with other auxiliary-like expressions, like "(to) be going to eat" or "(to) be about to eat", which have future meaning. For more examples of the above types of construction, see.  Perfect infinitives are also found in other European languages that have perfect forms with auxiliaries similarly to English. For example, "avoir mangé" means "(to) have eaten" in French.

English.

Regarding English, the term "infinitive" is traditionally applied to the unmarked form of the verb (the "plain form") when it forms a non-finite verb, whether or not introduced by the particle "to". Hence "sit" and "to sit", as used in the following sentences, would each be considered an infinitive:  The form without "to" is called the "bare infinitive"; the form introduced by "to" is called the "full infinitive" or "to-infinitive".  The other non-finite verb forms in English are the gerund or present participle (the "-ing" form), and the past participle – these are not considered infinitives. Moreover, the unmarked form of the verb is not considered an infinitive when it forms a finite verb: like a present indicative ("I "sit" every day"), subjunctive ("I suggest that he "sit""), or imperative (""Sit" down!"). (For some irregular verbs the form of the infinitive coincides additionally with that of the past tense and/or past participle, like in the case of "put".)  Certain auxiliary verbs are defective in that they do not have infinitives (or any other non-finite forms). This applies to the modal verbs ("can", "must", etc.), as well as certain related auxiliaries like the "had" of "had better" and the "used" of "used to". (Periphrases can be employed instead in some cases, like "(to) be able to" for "can", and "(to) have to" for "must".) It also applies to the auxiliary "do", like used in questions, negatives and emphasis like described under "do"-support. (Infinitives are negated by simply preceding them with "not". Of course the verb "do" when forming a main verb can appear in the infinitive.) However, the auxiliary verbs "have" (used to form the perfect) and "be" (used to form the passive voice and continuous aspect) both commonly appear in the infinitive: "I should have finished by now"; "It's thought to have been a burial site"; "Let him be released"; "I hope to be working tomorrow."  Huddleston and Pullum's "Cambridge Grammar of the English Language" (2002) does not use the notion of the "infinitive" ("there is no form in the English verb paradigm called 'the infinitive'"), only that of the "infinitival clause", noting that English uses the same form of the verb, the "plain form", in infinitival clauses that it uses in imperative and present-subjunctive clauses.  A matter of controversy among prescriptive grammarians and style writers has been the appropriateness of separating the two words of the "to"-infinitive (as in "I expect "to" happily "sit" here"). For details of this, see split infinitive. Opposing linguistic theories typically do not consider the "to"-infinitive a distinct constituent, instead regarding the scope of the particle "to" as an entire verb phrase; thus, "to buy a car" is parsed like "<nowiki>to [buy [a car]]</nowiki>", not like "<nowiki>[to buy] [a car]</nowiki>".

Uses of the infinitive.

The bare infinitive and the "to"-infinitive have a variety of uses in English. The two forms are mostly in complementary distribution – certain contexts call for one, and certain contexts for the other; they are not normally interchangeable, except in occasional instances like after the verb "help", where either can be used.  The main uses of infinitives (or infinitive phrases) are like follows:  The infinitive is also the usual dictionary form or citation form of a verb. The form listed in dictionaries is the bare infinitive, although the "to"-infinitive is often used in referring to verbs or in defining other verbs: "The word 'amble' means 'to walk slowly'"; "How do we conjugate the verb "to go"?"  For further detail and examples of the uses of infinitives in English, see Bare infinitive and "To"-infinitive in the article on uses of English verb forms.

Other Germanic languages.

The original Proto-Germanic ending of the infinitive was "-an", with verbs derived from other words ending in "-jan" or "-janan".  In German it is "-en" ("sagen"), with "-eln" or "-ern" endings on a few words based on -l or -r roots ("segeln", "ändern"). The use of "zu" with infinitives is similar to English "to", but is less frequent than in English. German infinitives can form nouns, often expressing abstractions of the action, in which case they are of neuter gender: "das Essen" means "the eating", but also "the food".  In Dutch infinitives also end in "-en" ("zeggen" — "to say"), sometimes used with "te" similar to English "to", e.g., "Het is niet moeilijk te begrijpen" → "It is not hard to understand." The few verbs with stems ending in "-a" have infinitives in -n ("gaan" — "to go", "slaan" — "to hit"). Afrikaans has lost the distinction between the infinitive and present forms of verbs, with the exception of the verbs "wees" (to be), which admits the present form "is", and the verb "hê" (to have), whose present form is "het".  In North Germanic languages the final "-n" was lost from the infinitive as early as 500–540 AD, reducing the suffix to "-a". Later it has been further reduced to "-e" in Danish and some Norwegian dialects (including the written majority language bokmål). In the majority of Eastern Norwegian dialects and a few bordering Western Swedish dialects the reduction to "-e" was only partial, leaving some infinitives in "-a" and others in "-e" (å laga vs. å kaste). In northern parts of Norway the infinitive suffix is completely lost (å lag’ vs. å kast’) or only the "-a" is kept (å laga vs. å kast’). The infinitives of these languages are inflected for passive voice through the addition of "-s" or "-st" to the active form. This suffix appearance in Old Norse was a contraction of "mik" (“me”, forming "-mk") or "sik" (reflexive pronoun, forming "-sk") and was originally expressing reflexive actions: (hann) "kallar" (“(he) calls”) + "-sik" (“himself”) > (hann) "kallask" (“(he) calls himself”). The suffixes "-mk" and "-sk" later merged to "-s", which evolved to "-st" in the western dialects. The loss or reduction of "-a" in active voice in Norwegian did not occur in the passive forms ("-ast", "-as"), except for some dialects that have "-es". The other North Germanic languages have the same vowel in both forms.

Latin and Romance languages.

The formation of the infinitive in the Romance languages reflects that in their ancestor, Latin, almost all verbs had an infinitive ending with "-re" (preceded by one of various thematic vowels). For example, in Italian infinitives end in "-are", "-ere", "-rre" (rare), or "-ire" (which is still identical to the Latin forms), and in "-arsi", "-ersi", "-rsi", "-irsi" for the reflexive forms. In Spanish and Portuguese, infinitives end in "-ar", "-er", or "-ir" (Spanish also has reflexive forms in "-arse", "-erse", "-irse"), while similarly in French they typically end in "-re", "-er", "oir", and "-ir". In Romanian, both short and long-form infinitives exist; the so-called "long infinitives" end in "-are, -ere, -ire" and in modern speech are used exclusively as verbal nouns, while there are a few verbs that cannot be converted into the nominal long infinitive. The "short infinitives" used in verbal contexts (e.g., after an auxiliary verb) have the endings "-a","-ea", "-e", and "-i" (basically removing the ending in "-re"). In Romanian, the infinitive is usually replaced by a clause containing the conjunction "să" plus the subjunctive mood. The only verb that is modal in common modern Romanian is the verb "a putea", to be able to. However, in popular speech the infinitive after "a putea" is also increasingly replaced by the subjunctive.  In all Romance languages, infinitives can also form nouns.  Latin infinitives challenged several of the generalizations about infinitives. They did inflect for voice ("amare", "to love", "amari", to be loved) and for tense ("amare", "to love", "amavisse", "to have loved"), and allowed for an overt expression of the subject ("video Socratem currere", "I see Socrates running"). See.  Romance languages inherited from Latin the possibility of an overt expression of the subject (as in Italian "vedo Socrate correre"). Moreover, the "inflected infinitive" (or "personal infinitive") found in Portuguese and Galician inflects for person and number. These, alongside Sardinian, are the only Indo-European languages that allow infinitives to take person and number endings. This helps to make infinitive clauses very common in these languages; for example, the English finite clause "in order that you/she/we have..." would be translated to Portuguese like "para teres/ela ter/termos..." (Portuguese is a null-subject language). The Portuguese personal infinitive has no proper tenses, only aspects (imperfect and perfect), but tenses can be expressed using periphrastic structures. For instance, "even though you sing/have sung/are going to sing" could be translated to "apesar de cantares/teres cantado/ires cantar".  Other Romance languages (including Spanish, Romanian, Catalan, and some Italian dialects) allow uninflected infinitives to combine with overt nominative subjects. For example, Spanish "al abrir yo los ojos" ("when I opened my eyes") or "sin yo saberlo" ("without my knowing about it").

Hellenic languages.

Ancient Greek.

In Ancient Greek the infinitive has four tenses (present, future, aorist, perfect) and three voices (active, middle, passive). Present and perfect have the same infinitive for both middle and passive, while future and aorist have separate middle and passive forms.  Thematic verbs form present active infinitives by adding to the stem the thematic vowel and the infinitive ending, and contracts to, e.g.,. Athematic verbs, and perfect actives and aorist passives, add the suffix instead, e.g.,. In the middle and passive, the present middle infinitive ending is, e.g., and most tenses of thematic verbs add an additional between the ending and the stem, e.g.,.

Modern Greek.

The infinitive "per se" does not exist in Modern Greek. To see this, consider the ancient Greek "ἐθέλω γράφειν" “I want to write”. In modern Greek this becomes "θέλω να γράψω" “I want that I write”. In modern Greek, the infinitive has thus changed form and function and is used mainly in the formation of periphrastic tense forms and not with an article or alone. Instead of the Ancient Greek infinitive system "γράφειν, γράψειν, γράψαι, γεγραφέναι", Modern Greek uses only the form "γράψει", a development of the ancient Greek aorist infinitive "γράψαι". This form is also invariable. The modern Greek infinitive has only two forms according to voice: for example, "γράψει" for the active voice and "γραφ(τ)εί" for the passive voice (coming from the ancient passive aorist infinitive "γραφῆναι").

Balto-Slavic languages.

The infinitive in Russian usually ends in "-t’" (ть) preceded by a thematic vowel, or "-ti" (ти), if not preceded by one; some verbs have a stem ending in a consonant and change the "t" to "č’", like "*mogt’ → moč’" (*могть → мочь) "can". Some other Balto-Slavic languages have the infinitive typically ending in, for example, "-ć" (sometimes "-c") in Polish, "-t’" in Slovak, "-t" (formerly "-ti") in Czech and Latvian (with a handful ending in -s on the latter), "-ty" (-ти) in Ukrainian, -ць ("-ts"') in Belarusian. Lithuanian infinitives end in -"ti", Slovenian end on -"ti" or -"či", and Croatian on -"ti" or -"ći".  Serbian officially retains infinitives -"ti" or -"ći", but is more flexible than the other slavic languages in breaking the infinitive through a clause. The infinitive nevertheless remains the dictionary form.  Bulgarian and Macedonian have lost the infinitive altogether except in a handful of frozen expressions where it is the same as the 3rd person singular aorist form. Almost all expressions where an infinitive may be used in Bulgarian are listed here; neverthess in all cases a subordinate clause is the more usual form. For that reason, the present first-person singular conjugation is the dictionary form in Bulgarian, while Macedonian uses the third person singular form of the verb in present tense.

Hebrew.

Hebrew has "two" infinitives, the infinitive absolute and the infinitive construct. The infinitive construct is used after prepositions and is inflected with pronominal endings to indicate its subject or object: "bikhtōbh hassōphēr" "when the scribe wrote", "ahare lekhtō" "after his going". When the infinitive construct is preceded by ("lə-", "li-", "lā-", "lo-") "to", it has a similar meaning to the English "to"-infinitive, and this is its most frequent use in Modern Hebrew. The infinitive absolute is used for verb focus and emphasis, like in "mōth yāmūth" (literally "a dying he will die"; figuratively, "he shall indeed/surely die"). This usage is commonplace in the Bible, but in Modern Hebrew it is restricted to high-register literary works.  Note, however, that the "to"-infinitive of Hebrew is not the dictionary form; that is the third person singular perfect form.

Finnish.

The Finnish grammatical tradition includes many non-finite forms that are generally labeled as (numbered) infinitives although many of these are functionally converbs. To form the so-called first infinitive, the strong form of the root (without consonant gradation or epenthetic 'e') is used, and these changes occur:  As such, it is inconvenient for dictionary use, because the imperative would be closer to the root word. Nevertheless, dictionaries use the first infinitive.  There are also four other infinitives, plus a "long" form of the first: Note that all of these must change to reflect vowel harmony, so the fifth infinitive (with a third-person suffix) of "hypätä" "jump" is "hyppäämäisillään" "he was about to jump", not "*hyppäämaisillaan".

Seri.

The Seri language of northwestern Mexico has infinitival forms used in two constructions (with the verb meaning 'want' and with the verb meaning 'be able'). The infinitive is formed by adding a prefix to the stem: either "iha-" (plus a vowel change of certain vowel-initial stems) if the complement clause is transitive, or "ica-" (and no vowel change) if the complement clause is intransitive. The infinitive shows agreement in number with the controlling subject. Examples are: "icatax ihmiimzo" 'I want to go', where "icatax" is the singular infinitive of the verb 'go' (singular root is "-atax"), and "icalx hamiimcajc" 'we want to go', where "icalx" is the plural infinitive. Examples of the transitive infinitive: "ihaho" 'to see it/him/her/them' (root "-aho"), and "ihacta" 'to look at it/him/her/them' (root "-oocta").

Translation to languages without an infinitive.

In languages without an infinitive, the infinitive is translated either as a "that"-clause or as a verbal noun. For example, in Literary Arabic the sentence "I want to write a book" is translated as either "urīdu an aktuba kitāban" (lit. "I want that I write a book", with a verb in the subjunctive mood) or "urīdu kitābata kitābin" (lit. "I want the writing of a book", with the "masdar" or verbal noun), and in Levantine Colloquial Arabic "biddi aktub kitāb" (subordinate clause with verb in subjunctive).  Even in languages that have infinitives, similar constructions are sometimes necessary where English would allow the infinitive. For example, in French the sentence "I want you to come" translates to "Je veux que vous veniez" (lit. "I want that you come", with "come" being in the subjunctive mood). However, "I want to come" is simply "Je veux venir", using the infinitive, just as in English. In Russian, sentences such as "I want you to leave" do not use an infinitive. Rather, they use the conjunction чтобы "in order to/so that" with the past tense form (most probably remnant of subjunctive) of the verb: "Я хочу, чтобы вы ушли" (literally, "I want so that you left").

Infinitive (abbreviated ) is a linguistics term referring to certain verb forms existing in many languages, most often used as non-finite verbs. As with many linguistic concepts, there is not a single definition applicable to all languages. The word is derived from Late Latin "[modus] infinitivus", a derivative of "infinitus" meaning "unlimited".

Infinitiv (také neurčitek) je neurčitý tvar slovesa, který nevyjadřuje osobu, číslo, způsob ani čas. Často se pokládá za základní (reprezentativní) tvar slovesa, který je uváděn ve slovnících.


Pierre Teilhard de Chardin's theory.

Etymology.

Teilhard de Chardin was a paleontologist and Roman Catholic priest in the Jesuit order. In France in the 1920s, he began incorporating his theories of the universe into lectures that placed Catholicism and evolution in the same conversation. Because of these lectures, he was suspected by the Holy Office of denying the doctrine of original sin. This caused Teilhard to be exiled to China and banned from publication by Church authorities. It was not until one year after his death, in 1955, that his writings were published for the world to read. His works were also supported by the writings of a group of Catholic thinkers, which includes Pope Benedict XVI. His book, "The Phenomenon of Man", has been dissected by astrophysicists and cosmologists to be a theological or philosophical theory that cannot be scientifically proven. Teilhard, who was not a cosmologist, opens his books with the statement:

Evolution.

According to Teilhard, Evolution does not end with mankind and Earth's biosphere evolved before humans existed. He described evolution as a progression that begins with inanimate matter to a future state of Divine consciousness through Earth's "hominization". He also maintained that one-cell organisms develop into metazoans, or animals, but some of the members of this classification develop organisms with complex nervous systems. This group has the capability to acquire intelligence. When "Homo sapiens" inhabited Earth through evolution, a noosphere, the cognitive layer of existence, was created. As evolution continues, the noosphere gains coherence. Teilhard explained that this noosphere can be moved toward or constructed to be the Omega Point or the final evolutionary stage with the help of science. Teilhard refers to this process as "planetization". Eventually, the noosphere gains total dominance over the biosphere and reaches a point of complete independence from tangential energy forming a metaphysical being, coined the Omega Point.

Energy.

Energy exists in two basic modes:  Teilhard defines Radial Energy as becoming more concentrated and available as it is a critical element in man's evolution. The theory applies to all forms of matter, concluding that everything with existence has some sort of life. In regard to Teilhard's "The Phenomenon of Man", Peter Medawar wrote, "Teilhard's radial, spiritual, or psychic energy may be equated to 'information' or 'information content' in the sense that has been made reasonably precise by communication engineers."

Formal properties.

Teilhard's theory is maintained by four formal properties:

Forces of compression.

Teilhard calls the contributing universal energy that generates the Omega Point "forces of compression". Unlike the scientific definition, which incorporates gravity and mass, Teilhard's forces of compression sources from communication and contact between human beings. This value is limitless and directly correlated with entropy. It suggests that as humans continue to interact, consciousness evolves and grows. For the theory to occur, humans must also be bound to the finite earth. Creation of this boundary forces the world's convergence upon itself which he theorizes to result in time ending in communion with the Omega Point-God. This portion of Teilhard's thinking shows his lack of expectation for humans to engage in space travel and transcend past the borders of the planet.

The Omega Point cosmology.

Mathematical physicist Frank Tipler generalizes Teilhard's term "Omega Point" to describe what he maintains is the ultimate fate of the universe required by the laws of physics: roughly, Tipler argues that quantum mechanics is inconsistent unless the future of every point in spacetime contains an intelligent observer to collapse the wavefunction, and that the only way for this to happen is if the Universe is closed (that is, it will collapse to a single point) and yet contains observers with a "God-like" ability to perform an unbounded series of observations in finite time. However, scientists such as Lawrence Krauss have stated that Tipler's reasoning is erroneous on multiple levels, possibly to the point of being nonsensical pseudoscience.

Theological controversy.

Pierre Teilhard de Chardin's life (1881–1955) falls directly in between the First Vatican Council (1869) and the Second Vatican Council (1965), a time period where increasing global acceptance of evolution was forming a disconnect between modern humanity and the Roman Catholic Church. His time came shortly after Charles Darwin's 1859 book, "On the Origin of Species", a time when the intersection between the claims of scientific theories and the claims of traditional theological teachings became an enormous focus of the Vatican's agenda.  Pope Pius XII stated his concern on the theory of evolution, albeit without condemning it: Teilhard's theory was a personal attempt in creating a new Christianity in which science and theology coexist. The outcome was that his theory of the Omega Point was not perfectly scientific as examined by physicists, and not perfectly Christian either. By 1962, The Society of Jesus had strayed from Spanish Jesuit Priest Francisco Suarez's philosophies on Man in favor of "Teilhardian evolutionary cosmogenesis". Teilhard's Christ is the "Cosmic Christ" or the "Omega" of revelation. He is an emanation of God which is made of matter, and experienced the nature of evolution by being born into this world and dying. His resurrection from the dead was not to heaven, but to the noosphere, the area of convergence of all spirituality and spiritual beings, where Christ will be waiting at the end of time. When the earth reaches its Omega Point, everything that exists will become one with divinity.  Teilhard reaffirms the role of the Church in the following letter to Auguste Valensin. It is important to note that he defines evolution as a scientific phenomenon set in motion by God – that science and the divine are interconnected and acting through one another:

Related concepts.

Accelerating expansion of the universe.

In 1998, a value measured from observations of Type Ia supernovae seemed to indicate that what was once assumed to be temporary cosmological expansion was actually accelerating. The apparent acceleration has caused many to dismiss Tipler's Omega Point out of hand, since the necessity of a final big crunch singularity is key to the Omega Point's workability. However, Tipler himself believes that the Omega Point is still workable and has explained on multiple occasions why a big crunch/ final singularity is still required under many current universal models.

Technological singularity.

The technological singularity is the hypothetical advent of artificial general intelligence theoretically capable of recursive self-improvement, resulting in a runaway effect to an intelligence explosion. Eric Steinhart, a proponent of "Christian transhumanism", argues there is significant overlap of ideas between the secular singularity and Teilhard's religious Omega Point. Steinhart quotes Ray Kurzweil, one of the most prominent singularitarians, who stated that "evolution moves inexorably toward our conception of God, albeit never reaching this ideal." Like Kurzweil, Teilhard predicts a period of rapid technological change that results in a merger of humanity and technology. He believes that this marks the birth of the noosphere and the emergence of the "spirit of the Earth", but the Teilhardian Singularity comes later. Unlike Kurzweil, Teilhard's singularity is marked by the evolution of human intelligence reaching a critical point in which humans ascend from "transhuman" to "posthuman". He identifies this with the Christian "parousia".

The Omega Point in popular culture.

The Spanish painter Salvador Dali was fascinated by Teilhard, de Chardin, and the Omega Point theory. His 1959 painting "The Ecumenical Council" is said to represent the "interconnectedness" of the Omega Point. "Point Omega" by Don DeLillo takes its name from the theory and involves a character who is studying Teilhard de Chardin. Flannery O'Connor's acclaimed collection of short stories taps the Omega Point theory in its title, "Everything That Rises Must Converge", and science fiction writer Frederik Pohl references Frank Tipler and the Omega Point in his 1998 short story "The Siege of Eternity". Scottish writer / counterculture figure Grant Morrison has used the Omega Point as a plot line in several of his Justice League of America and Batman stories. Arthur C. Clarke and Stephen Baxter's "The Light of Other Days" references Teilhard de Chardin and includes a brief explanation of the Omega Point. Italian writer Valerio Evangelisti has used the Omega Point as main theme of his "Il Fantasma di Eymerich" novel.

The Omega Point is the subject of a belief that everything in the universe is fated to spiral towards a final point of unification. The term was coined by the French Jesuit Catholic priest Pierre Teilhard de Chardin (1881–1955). Teilhard argued that the Omega Point resembles the Christian Logos, namely Christ, who draws all things into himself, who in the words of the Nicene Creed, is "God from God", "Light from Light", "True God from true God", and "through him all things were made". In the Book of Revelation, Christ describes himself thrice as "the Alpha and the Omega, the beginning and the end". The idea of the Omega Point is developed in later writings, such as those of John David Garcia (1971), Paolo Soleri (1981), Frank Tipler (1994), and David Deutsch (1997).

Bod Omega je symbolické označení pro „druhý pól“ evoluce Vesmíru. V biblické Apokalypse říká oslavený Kristus: „Já jsem alfa i omega, počátek i konec.“ Tento symbol použil Pierre Teilhard de Chardin k označení budoucího pólu vyvíjejícího se Vesmíru, jakožto protějšku k minulému pólu, „bodu Alfa“ čili počátku. Vesmír tedy v čase směřuje od bodu Alfa k bodu Omega, který ovšem nelze chápat předmětně jako geometrické místo nebo určitý stav. Teilhard proto říká, že bod Omega je „ohnisko osobní povahy“, jež zprvu rozbíhavé linie vesmírného vývoje soustřeďuje a přitahuje k sobě.