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His research in geometry led to the abstract topological definition of homotopy and homology.

He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the fundamental group.

In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on).

They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions.

Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable.

In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies.

His work in this area was the first major achievement in celestial mechanics since Isaac Newton.

These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems.

Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid.

He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability.

In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations".

Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found.

Poincaré also developed a general theory of integral invariants and solutions of the variational equations.

For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions.

Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris.

He linked his way of thinking to how he made several discoveries.

The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation.

Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m..

He would read articles in journals later in the evening.

He was always in a rush and disliked going back for changes or corrections.

He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he consciously worked on another problem.

The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death.

Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré.

Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909).

The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory.

He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic.

Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic.

His views were similar to those of Immanuel Kant (Kolak, 2001, Folina 1992).

He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions.

However, Poincaré did not share Kantian views in all branches of philosophy and mathematics.

For example, in geometry, Poincaré believed that the structure of non-Euclidean space can be known analytically.

Poincaré held that convention plays an important role in physics.

His view (and some later, more extreme versions of it) came to be known as "conventionalism".

Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012).

He also believed that the geometry of physical space is conventional.

He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution.

However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.

In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.

Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.

Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.

Poincaré–Hopf theorem: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks.

Poincaré–Birkhoff–Witt theorem: an explicit description of the universal enveloping algebra of a Lie algebra.

Poincaré–Bjerknes circulation theorem: theorem about a conservation of quantity for the rotating frame.

Poincaré conjecture (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Poincaré–Miranda theorem: a generalization of the intermediate value theorem to n dimensions.

Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique, by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914.

Henri Poincaré, l'œuvre mathématique, by Vito Volterra.

For more information on the psychiatric side, including how psychiatry groups phobias such as agoraphobia, social phobia, or simple phobia, see phobia.

The following lists include words ending in -phobia, and include fears that have acquired names.

In some cases, the naming of phobias has become a word game, of notable example being a 1998 humorous article published by BBC News.

In some cases, a word ending in -phobia may have an antonym with the suffix -phil-, e.g.

Many -phobia lists circulate on the Internet, with words collected from indiscriminate sources, often copying each other.

Also, a number of psychiatric websites exist that at the first glance cover a huge number of phobias, but in fact use a standard text to fit any phobia and reuse it for all unusual phobias by merely changing the name.

Sometimes it leads to bizarre results, such as suggestions to cure "prostitute phobia".

Such practice is known as content spamming and is used to attract search engines.

Terms should strictly have a Greek prefix although many are irregularly formed with Latin or even English prefixes.

Many use inaccurate or imprecise prefixes, such as aerophobia (fear of air) for fear of flying.

The term is a piece of computer humor entered into the 1981 The Devil's DP Dictionary.

Anatidaephobia – the fictional fear that one is being watched by a duck.

The word comes from the name of the family Anatidae, and was used in Gary Larson's The Far Side.

Anoraknophobia – a portmanteau of "anorak" and "arachnophobia".

Arachibutyrophobia – fear of peanut butter sticking to the roof of the mouth, from Latin "peanut" and "butter".

The word is used by Charles M. Schulz in a 1982 installment of his Peanuts comic strip, and by Peter O'Donnell in his 1985 Modesty Blaise adventure novel Dead Man's Handle.

Hippopotomonstrosesquipedaliophobia – fear of long words, from the root word combined with monstrum and hippopotamus.

This was mentioned on the first episode of Brainiac Series Five as a Tickle's Teaser.

Nihilophobia – fear of nothingness, from Latin nihil and "nothing, none", as described by the Doctor in the Star Trek: Voyager episode "Night".

Voyager's morale officer and chef Neelix suffers from this condition, having panic attacks while the ship was traversing a dark expanse of space known as the Void.

Robophobia – irrational fear of robots and/or androids, also known as "Grimwade's Syndrome".

It was first used in "The Robots of Death", the fifth serial of the 14th season of the British science fiction television series Doctor Who.

Semaphobia – fear of average web developers to use Semantic Web technologies.

Venustraphobia – fear of beautiful women, according to a 1998 humorous article published by BBC News.

His mother was a drama teacher which influenced his upbringing as an actor.

He attended Bowdoin College in Brunswick, Maine because it was the most distant point in the continental United States from Fresno.

He also attended Fresno State College and UCLA, but did not graduate from any of them.

He was in the UCLA Drama Department at the same time as another dropout, Ray Manzarek of The Doors.

At KPFK he worked with other staffers David Ossman and Peter Bergman who hosted Radio Free Oz on that station.

Along with Bergman's friend Phil Proctor, they formed The Firesign Theatre.

Starting as live radio actors, the group would go on to record a series of surrealistic comedy albums that were a hit amongst an underground audience.

Austin played the group's best-known creation, private investigator Nick Danger.

Other prominent roles were as (Happy) Harry Cox, the narrator of Everything You Know Is Wrong and Bebop Loco/Lobo on Give Me Immortality or Give Me Death.

He had also served as the troupe's musician and record producer.

The cause of death was originally given as cardiac arrest, but this was later changed to an aneurysm.

When he died, his wife Oona mentioned that Austin also had been diagnosed with cancer months before.

Neo-fascism is a post-World War II ideology that includes significant elements of fascism.

Neo-fascism usually includes ultranationalism, racial supremacy, populism, authoritarianism, nativism, xenophobia and anti-immigration sentiment as well as opposition to liberal democracy, parliamentarianism, liberalism, Marxism, communism, and socialism.

Allegations that a group is neo-fascist may be hotly contested, especially when the term is used as a political epithet.

Some post–World War II regimes have been described as neo-fascist due to their authoritarian nature, and sometimes due to their fascination with and sympathy towards fascist ideology and rituals.

Europe then became both the myth and the utopia of the neo-fascists, who abandoned previous theories of racial inequalities within the white race to share a common euro-nationalist stance after World War II, embodied in Oswald Mosley's Europe a Nation policy.

The following chronology can therefore be delineated: an ideological gestation before 1919; the historical experience of fascism between 1919 and 1942, unfolded in several phases; and finally neo-fascism from 1942 onward.

Drawing inspiration from the Italian Social Republic, institutional neo-fascism took the form of the Italian Social Movement (MSI).

It became one of the chief reference points for the European far-right until the late 1980s, and "the best (and only) example of a Neofascist party", in the words of political scientist Cas Mudde.

At the initiative of the MSI, the European Social Movement was established in 1951 as a pan-European organization of like-minded neo-fascist groups and figures such as the Francoist Falange, Maurice Bardèche, Per Engdahl, and Oswald Mosley.

Other organizations like Jeune Nation called in the late 1950s for an extra-parliamentarian insurrection against the regime in what extents to a remnant of pre-war fascist strategies.

The main driving force of neo-fascist movements was what they saw as the defense of a Western civilization from the rise of both communism and the Third World, in some cases the loss of the colonial empire.

In 1961, Bardèche redefined the nature of fascism in a book deemed influential in the European far-right at large entitled Qu'est-ce que le fascisme?

In the spirit of Bardèche's strategy of disguise through framework change, the MSI had developed a policy of inserimento (insertion, entryism), which relied on gaining political acceptance via the cooperation with other parties within the democratic system.

In the political context of the Cold War, anti-communism began to replace anti-fascism as the dominant trend in liberal democracies.

Because intense nationalism is almost always a part of neo-fascism, the parties which make up this movement are not pan-European, but are specific to each country they arise in; other than this, the neo-fascist parties and other groups have many ideological traits in common.

While certainly fascistic in nature, it is claimed by some that there are differences between neo-fascism and what can be called "historical fascism", or the kind of neo-fascism which came about in the immediate aftermath of World War II.

In general, the anti-immigrant impetus is strong when the economy is weak or unemployment is high, and people fear that outsiders are taking their jobs.

Because of this, neo-Fascist parties have more electoral traction during hard economic times.

In the absence in post-war Europe of a strong socialist movement, this has the tendency to move the political center to the right overall.

While both historical fascism and contemporary neo-fascism are xenophobic, nativist and anti-immigrant, neo-fascist leaders are careful not to present these views in so strong a manner as to draw obvious parallels to historical events.

Both Jean-Marie Le Pen of France's National Front and Jörg Haider's Freedom Party of Austria, in the words of historian Tony Judt, "revealed [their] prejudices only indirectly".

Jews would not be castigated as a group, but a person would be specifically named as danger who just happened to be a Jew.

The public presentation of their leaders is one principle difference between the neo-Fascists and historical Fascists: their programs have been "finely honed and 'modernized'" to appeal to the electorate, a "'far-right ideology with a democratic veneer'".

Modern neo-fascists do not appear in "jackboots and brownshirts", but in suits and ties.

The choice is deliberate, as the leaders of the various groups work to differentiate themselves from the brutish leaders of historical fascism and also to hide whatever bloodlines and connections tie the current leaders to the historical Fascist movements.

When these become public, as they did in the case of Haider, it can lead to their decline and fall.

It was a more radical splinter group of the European Social Movement.

The NEO had its origins in the 1951 Malmö conference when a group of rebels led by René Binet and Maurice Bardèche refused to join the European Social Movement as they felt that it did not go far enough in terms of racialism and anti-communism.

As a result, Binet joined with Gaston-Armand Amaudruz in a second meeting that same year in Zurich to set up a second group pledged to wage war on communists and non-white people.

Several Cold War regimes and international neo-fascist movements collaborated in operations such as assassinations and false flag bombings.

Stefano Delle Chiaie, who was involved in Italy's Years of Lead, took part in Operation Condor; organizing the 1976 assassination attempt on Chilean Christian Democrat Bernardo Leighton.

Vincenzo Vinciguerra escaped to Franquist Spain with the help of the SISMI, following the 1972 Peteano attack, for which he was sentenced to life.

Michael Townley was sentenced in Italy to 15 years of prison for having served as intermediary between the DINA and the Italian neo-fascists.

The regimes of Francoist Spain, Augusto Pinochet's Chile and Alfredo Stroessner's Paraguay participated together in Operation Condor, which targeted political opponents worldwide.

During the Cold War, these international operations gave rise to some cooperation between various neo-fascist elements engaged in a "Crusade against Communism".

Anti-Fidel Castro terrorist Luis Posada Carriles was condemned for the Cubana Flight 455 bombing on 6 October 1976.

The PCI was expelled from power in May 1947, a month before the Paris Conference on the Marshall Plan, along with the French Communist Party (PCF).

In 1946, a group of Fascist soldiers founded the Italian Social Movement (MSI) to continue advocating the ideas of Benito Mussolini.

The leader of the MSI was Giorgio Almirante, who remained at the head of the party until his death in 1988.

Despite attempts in the 1970s towards a "historic compromise" between the PCI and the DC, the PCI did not have a role in executive power until the 1980s.

In December 1970, Junio Valerio Borghese attempted, along with Stefano Delle Chiaie, the Borghese Coup which was supposed to install a neo-fascist regime.

Neo-fascist groups took part in various false flag terrorist attacks, starting with the December 1969 Piazza Fontana massacre, for which Vincenzo Vinciguerra was convicted, and they are usually considered to have stopped with the 1980 Bologna railway bombing.

Neo-fascist parties in Italy include the Tricolour Flame (Fiamma Tricolore), the New Force (Forza Nuova), the National Social Front (Fronte Sociale Nazionale), and CasaPound.

The national-conservative Brothers of Italy, main heirs of MSI and AN, have some neo-fascist factions within their internal organization.

In 2013, after the murder of an anti-fascist musician by a person with links to Golden Dawn, the Greek government ordered the arrest of Golden Dawn's leader Nikolaos Michaloliakos and other Golden Dawn members on charges related to being associated with a criminal organization.

In October, 2020, the court declared Golden Dawn to be a criminal organization, convicting 68 members of various crimes including murder.

However, far-right politics continue to be strong in Greece, such as Ilias Kasidiaris' Greeks for the Fatherland, an Ultranationalist party.

In 2021, Greek neo-Nazi youth attacked a rival group at a school in Greece.

A report by the European Parliament defined the ideology of the New Order as revolutionary fascist and hyper-nationalist.

The group also had connections to Fuerza Nueva in Spain.

The Party's leader, Marian Kotleba, is a former neo-Nazi, who once wore a uniform modelled on that of the Hlinka Guard, the militia of the 1939–45 Nazi-sponsored Slovak State.

He opposes Romani people, immigrants, the Slovak National Uprising, NATO, the United States, and the European Union.

The party also endorses clerical fascist war criminal Slovak President Jozef Tiso.

In 2003, Kotleba founded the far-right political party Slovak Community (Slovak: Slovenská Pospolitosť).

In 2007, the Slovak interior ministry banned the party from running and campaigning in elections.

In spite of this ban, Kotleba's party got 8.04% of votes in the Slovak 2016 parliamentary elections, and voter support of the party continues to increase.

According to Turkish authorities, the organization carried out 694 murders during the late-1970s political violence in Turkey, between 1974 and 1980.

The nationalist political party MHP founded by Alparslan Türkeş is also sometimes described as neo-fascist.

In the 2009 European elections, it gained two members of the European Parliament (MEPs), including former party leader Nick Griffin.

Other British organisations described as fascist or neo-fascist include the National Front, Combat 18, the English Defence League, and Britain First.

Luis García Meza Tejada's regime took power during the 1980 Cocaine Coup in Bolivia with the help of Italian neo-fascist Stefano Delle Chiaie, Nazi war criminal Klaus Barbie and the Buenos Aires junta.

That regime has been accused of neo-fascist tendencies and of admiration for Nazi paraphernalia and rituals.

Malema and the party have frequently courted controversy for engaging in anti-White and anti-Indian racism.

In November 2019, the Professor of International Relations at University of the Witwatersrand, Vishwas Satgar, defined them as a manifestation of a new phenomena, 'Black Neofascism'.

The NSA views Adolf Hitler as its leader and often uses the slogan "Long live Hitler".

This has brought them condemnation from the Simon Wiesenthal Center, an international Jewish human rights center.

The first fascist party was the Partai Fasis Indonesia (PFI).

Sukarno admired Nazi Germany under Hitler and its vision of happiness for all: "It's in the Third Reich that the Germans will see Germany at the apex above other nations in this world," he said in 1963.

The Swedes were escorting a supply column of more than 4,500 wagons for their main army in Ukraine.

Peter I intercepted Lewenhaupt's column before it reached the safety of Charles XII, the Swedish king, with the intention of destroying it.

After eight hours of fighting, with heavy casualties, neither side stood as winner.

As the night approached the Russians decided to withdraw to the nearest forest where they would stay until next morning to continue the fight.

The Swedes however stayed in their battle formations for hours during the night, in case of a renewed attack.

With no sign of further combat and intelligence saying further Russian reinforcements had arrived, the Swedes in turn withdrew from the place of battle, in order to continue the march towards the main army.

Fearing a full-scale Russian pursuit, Lewenhaupt decided to burn or abandon most of the wagons and cannons in order to increase speed.

While doing this many of the Swedish soldiers decided to loot the abandoned wagons and get drunk, thousands got lost in the woods, many of whom fell victim to Russian irregular cavalry.

Lewenhaupt soon crossed the river of Sozh with the rest of his army, to find himself relatively safe.

After some days he met up with Charles XII at Rukova with very few wagons left and only half of his initial army.

Saxony, under Augustus II, invaded Swedish Livonia and quickly attacked the city of Riga.

Meanwhile, Denmark–Norway under Frederick IV of Denmark attacked the Swedish allied duchies of Holstein and Gottorp in order to secure his rear, before commencing with the planned invasion of Scania, which had been previously annexed by Sweden in the Treaty of Roskilde in 1658.

A short time later, Russia under Peter I swept into Swedish Ingria and besieged the strategic city of Narva.

Unprepared for these developments, the Swedes were forced into a war on three fronts.

Denmark–Norway was quickly knocked out of the war by a bold Swedish landing on Humlebæk resulting in the Peace of Travendal.

After this the Russians were forced to abandon their campaign in Ingria after their crushing defeat in the battle of Narva.

The Swedes next beat the Saxons, Poles and Russians at the battle of Düna.

Soon Sweden invaded the Polish–Lithuanian Commonwealth in order to remove Augustus from the Polish throne.

The subsequent conflict became known as the Swedish invasion of Poland.

After several defeats in the battles of Kliszów, Kraków, Pultusk and Toruń, Augustus was finally dethroned in favor of a monarch installed by the Swedes, Stanisław Leszczyński who was crowned king in 1705.

During this time, the Russians had been able to capture several Swedish possessions in their Baltic Dominions, among others, the fortresses of Nöteborg, Nyenskans, Dorpat and Narva.

In 1705 the two sides prepared for a final confrontation in the Polish–Lithuanian Commonwealth, the Russians intervened with full force in order to put Augustus back on the throne.

After the battles of Gemauerthof, Warsaw, Grodno and Fraustadt the campaign was decided in favor of the Swedes who chased their enemies out of Poland in 1706 and subsequently invaded Saxony, where Augustus saw himself defeated and forced to make peace.

Seeing how only one major threat remained, the Swedes decided to invade Russia in 1707.

Here he was instructed to obtain a large amount of supplies and wagons that could be sufficient for the main army for about three weeks.

Once having collected the supplies Lewenhaupt would assemble as much men as possible from the area, without leaving the garrisons completely stripped.

Lewenhaupt would then use these troops to escort the convoy and rendezvous with Charles' main army at Mogilev, in early August.

In May same year, Lewenhaupt returned to Riga in order to complete the task, which proved far from easy.

The near lands had suffered many campaigns in the years of the Great Northern War and so much was drained of needed resources.

In early June, the column—of which Lewenhaupt was gathering—was ordered to start campaigning to reach Charles XII in Mogilev, according to schedule.

However, the convoy was nowhere ready to leave because of the difficulties assembling it.

Only in the beginning of July it was ready, having then suffered three to four weeks behind the schedule and a significant shortage of men as 20,000 men were expected, but only 13,000 soldiers proved able to march.

During this time Lewenhaupt was about 135 kilometers (90 miles) away from Charles and on September 28, he received new orders to rendezvous at Starodub and started marching south himself.

His convoy passed between Mogilev and Gorki heading for Propoisk on the river Sozh.

By October 3, Lewenhaupt had crossed the Dnieper and headed south, the crossing itself has to be considered a "military masterpiece".

Having observed these movements, Peter I dispatched an army under Boris Sheremetev after Charles and gathered a force of his own to intercept with Lewenhaupt.

The Russians made contact with Lewenhaupt's convoy on October 6, and immediately started harassing it, forcing the Swedes to march in defensive formation across difficult terrain while the numbers of shadowing Russian troops steady grew.

Peter I, who overestimated the Swedish force being 16,000 men strong, had gathered numbers far superior to those of Lewenhaupt and was eager to catch his convoy while it was still out of reach of Charles' main army and safety.

He planned to destroy the convoy before its crossing of the river Sozh where it would otherwise reach—as Peter thought—the protection of the main army (the Russians had misleading reports saying Charles was 25 kilometers away from Sozh and not 120 as they had previously presumed).

On October 7, the Russians in the area were large enough that they posed a considerable threat to the convoy and so the two sides confronted each other for some time at the village of Belitsa.

Subsequently, however, Lewenhaupt ordered a cavalry attack consisting of 4,000 men on the equally numbered Russian dragoons who were facing them, the Russian horse did not desire a fight and instead started retreating, persecuted by their enemies for a good four–kilometers step.

In this encounter losses amounted to more than 40 Russians killed and three to eleven captured, to four wounded Swedes, a real battle did not develop as both sides parted and the confrontation ended with the quick cavalry skirmish.

The Swedes however, received intelligence from the captured Russians saying the Tsar was following Lewenhaupt with a force of about 20,000 cavalry, 12,000 infantry and four guns.

Later during the day, the vanguard of the convoy reached the small village of Lesnaya and there made preparations to set camp for the rest of the army, accordingly.

The next morning, Lewenhaupt, who during this time stayed with his rearguard, would once again find himself confronted as the Russians stepped up their efforts to harass the back of the convoy as it made its crossing over at Dolgij Moch, towards Lesnaya.

Here the Russians under Mikhail Golitsyn and Alexander Menshikov attacked him on two fronts over the river of Resta.

The engagement concluded in a standoff after four hours of musket and artillery exchange in which the Swedes successfully denied every attempt made by the Russians to cross the river for the convoy.

Later the same day, Lewenhaupt reached the village of Lesnaya with most of his army and was within a day's march from Propoisk.