In 1976, while other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University of Illinois announced that they had proven the theorem.

Kenneth Ira Appel ( born October 8, 1932 in Brooklyn, New York ) is a mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem.

In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem.

Haken delivered an invited address at the 1978 International Congress of Mathematicians in Helsinki.

First mentioned by Teutonic Order in macher colony documents in 1429 and 1497, the settlement was originally 5 km south of today's position near the Hohe Düne ( high dune ) at Grabscher Haken ( Old Prussian grabis = hill ).

In 1994, the Trevor Haken was summonsed for an examination at the New South Wales Crime Commission.

The Math Department at UIUC celebrated the new primes with a postal meter cancellation stamp — until Kenneth Appel | Appel and Wolfgang Haken | Haken proved the Four color theorem | 4-color theorem in 1976.

The four color theorem was finally proved by Kenneth Appel and Wolfgang Haken at the University of Illinois, with the aid of a computer.

The Continuum Fingerboard or Haken Continuum is a music performance controller developed by Lippold Haken, a professor of Electrical and Computer Engineering at the University of Illinois, and sold by Haken Audio, located in Champaign, Illinois.

Kenneth Appel and Wolfgang Haken finally proved this in 1976.

The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken.

Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property.

Appel and Haken concluded that no smallest counterexamples existed because any must contain, yet not contain, one of these 1, 936 maps.

To dispel remaining doubt about the Appel – Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas.

Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.

In 1986, Appel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof.

In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic time algorithm, improving on a quartic algorithm based on Appel and Haken ’ s proof (; ).

At first, The New York Times refused as a matter of policy to report on the Appel – Haken proof, fearing that the proof would be shown false like the ones before it.

A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of " discharging " developed by Heesch.

These include the gesture-controlled Buchla Thunder, sonomes such as the C-Thru Music Axis, which rearrange the scale tones into an isometric layout, and Haken Audio's keyless, touch-sensitive Continuum playing surface.

Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos attacked the conjecture.

These were the first such examples ; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken.

To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

Other examples are given by the Seifert – Weber space, or " sufficiently complicated " Dehn surgeries on links, or most Haken manifolds.

The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.

A second route to Geometrization is the method of Bessières et al., which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm for 3-manifolds.

In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.

Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.

A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken.

The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.

ADTs are often implemented as modules: the module's interface declares procedures that correspond to the ADT operations, sometimes with comments that describe the constraints.

In a Spin magazine article written about the genre, Steven Blush declares that " the man often credited " for dubbing the style grindcore was Shane Embury, Napalm Death's bassist since 1987.

In most wrestling promotions, a championship cannot change hands as a result of a disqualification, unless the on-screen authority figure declares that the championship via disqualification which is good for only at least one match, often referred to as the " champion's advantage ".

For example, in the British sitcom Blackadder Goes Forth, set in 1917 ( 27 years before the word was first used ), the character General Melchett declares that he likes the term and orders his batman to take note of it because he wants to " use it more often in conversation ".

An often quoted example of the effects on the Welsh language is the first section of the 1535 Act, which states: " the people of the same dominion have and do daily use a speche nothing like ne consonant to the naturall mother tonge used within this Realme " and then declares the intention " utterly to extirpe alle and singular sinister usages and customs " belonging to Wales.

According to Bulgarian law, a Bulgarian president is not allowed to be a member of a political party, thus Parvanov is independent and he left the Socialist Party after his success in the presidential elections, although he identified as a socialist, Parvanov often declares being a ' social president '.

It is now better known for the more shocking final scene ( often a concert-piece for dramatic sopranos ), where Salome declares her love to – and kisses – the severed head of John the Baptist.

Of the daughter of the irritatingly grand Lady Catherine De Bourgh in Jane Austen's Pride and Prejudice, the lickspittle Mr Collins declares, " she is perfectly amiable, and often condescends to drive by my humble abode in her little phaeton and ponies.

In the case of a business it is often a statement that declares a party ’ s policy on how it collects, stores, and releases personal information it collects.

" Holden declares prophetically that the man has arrived at the saloon for " the dance "— in other words, the dance of violence, war, and bloodshed that the judge so often praised.