Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology.

Grothendieck solved this problem by defining the notion of properness of a scheme ( actually, of a morphism of schemes ), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.

The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.

According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes.

This is a fundamental idea, which first surfaced in algebraic topology.

Thus one can understand equations by a pure understanding of abstract topology or geometry — this idea is of importance in algebraic geometry.

* Descent ( category theory ), an idea extending the notion of " gluing " in topology

However, the node does not have any idea of the actual network topology, nor does the node need it.

Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology.

The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century ; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra.

Brennan's system implemented this idea in a partially automated fashion as follows: the operator was required to input a frontal drawing of the desired person having a standardized topology ( the number and ordering of lines for every face ).

The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold.

In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of ' gluing ' in topology.

The idea comes from the Boundary ( topology ).

The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.

Yet, after Rousseau had given the social contract a new twist with his notion of the General Will, the same philosophy, it may be said, became the idea source of the French Revolution also.

As he made plans for the new Taliesin, Wright also got on paper his conception of a cathedral of steel and glass to house a congregation of all faiths, and the idea for a planetarium with a sloping ramp.

He also bought a huge square of pegboard for hanging up his tools, and lumber for his workbench, sandpaper and glue and assorted nails, levels and T squares and plumb lines and several gadgets that he had no idea how to use or what they were for.

The dated poems also give us an idea of the degree to which Hardy drew upon past productions for his various volumes, and therefore probably are an indication of the amount of poetry he was writing at the time.

Shaw could also give the flyer a pretty good idea of area visibility by a visual check of the mountains to be seen from his station.

This was a continuation of a good idea which was first tried out Saturday night when the Eddie Stack group, also local talent, went on first.

Although they " were expecting to see activity in the brain's reward centers ", based on the idea that " people perform altruistic acts because they feel good about it ", what they found was that " another part of the brain was also involved, and it was quite sensitive to the difference between doing something for personal gain and doing it for someone else's gain ".

It is important to note that, while research supports the idea that altruistic acts bring about happiness, it has also been found to work in the opposite direction — that happier people are also kinder.

It also reveals that the principle of linguistic relativity was so named because because Whorf saw that idea of his as having similar implications as Einstein's principle of physical relativity.

The large amount of travel done by Christie and Mallowan has not only made for a great writing theme, as shown in her famous novel: The Murder on the Orient Express, but also tied into the idea of archaeology as an adventure that has become so important in today ’ s popular culture as described by Cornelius Holtorf in his book Archaeology is a Brand.

He also keeps a diary, full of his observations of life in Oran, which Rieux incorporates into the narrative .</ br > It is Tarrou who first comes up with the idea of organizing teams of volunteers to fight the plague.

Woolfson, a songwriter and composer, was working as a session pianist ; he had also composed material for a concept album idea based on the work of Edgar Allan Poe.

Ampère also provided a physical understanding of the electromagnetic relationship, theorizing the existence of an “ electrodynamic molecule ” ( the forerunner of the idea of the electron ) that served as the component element of both electricity and magnetism.

Scholars in support of the idea of Amos being from the North also say it makes more sense because of Amaziah's accusation of conspiracy found in chapter seven, verse 10.

( Hume 1974: 355-356 ) He also argues in brief against the idea that causes are mere occasions of the will of some god ( s ), a view associated with the philosopher Nicolas Malebranche.

One of the central themes of Acts, indeed of the New Testament ( see also Great Commission ) is the universality of Christianity — the idea that Jesus's teachings were for all humanity — Jews and Gentiles alike.

The idea has also some backing in German legend, for example the Gesta Treverorum ( a 12th century German medieval chronicle ) makes Trebeta son of Ninus the founder of Trier.

In 1950 he also proposed an idea for a controlled nuclear fusion reactor, the tokamak, which is still the basis for the majority of work in the area.

Sakharov also proposed the idea of induced gravity as an alternative theory of quantum gravity.

Alexander is also known for rejecting the idea that there are many things in God ’ s mind, instead claiming that it is more perfect to know just one thing.

Cantor also mentioned the idea in his letters to Richard Dedekind ( text in square brackets not present in original ):

* The idea of replicant models being mass-produced, and in particular a woman identical to Rachael existing, is also from Do Androids Dream of Electric Sheep?

Beauty ( also called prettiness, loveliness or comeliness ) is a characteristic of a person, animal, place, object, or idea that provides a perceptual experience of pleasure or satisfaction.

When the idea and reality of the National Debt came about during the 18th century this was also managed by the bank.

However, as has been pointed out by Alexis Manaster Ramer and Paul Sidwell ( 1997 ), Strahlenberg actually opposed the idea of a closer relationship between the languages which later became known as " Altaic "..

Among well known ( generally non-kabbalist or anti-kabbalist ) Rabbis who rejected the idea of reincarnation are Saadia Gaon, David Kimhi, Hasdai Crescas, Yedayah Bedershi ( early 14th century ), Joseph Albo, Abraham ibn Daud, the Rosh and Leon de Modena.

Other major ideas in the book of Amos include: social justice and concern for the disadvantaged ; the idea that Israel's covenant with God did not exempt them from accountability for sin ; God is God of all nations ; God is judge of all nations ; God is God of moral righteousness ; God made all people ; God elected Israel and then liberated Israel so that He would be known throughout the world ; election by God means that those elected are responsible to live according to the purposes clearly outlined to them in the covenant ; if God destroys the unjust, a remnant will remain ; and God is free to judge whether to redeem Israel.

Among Schoenberg's teaching was the idea that the unity of a musical composition depends upon all its aspects being derived from a single basic idea ; this idea was later known as developing variation.

The idea that immediately suggested itself was that the star's declination varied because of short-term changes in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as nutation.

But it is not known if any experiment based on this idea has been ever achieved.

Whorf is widely known as an advocate for the idea that because of linguistic differences in grammar and usage, speakers of different languages conceptualize and experience the world differently.

Russian physicist Lev Landau used the idea for the Fermi liquid theory wherein low energy properties of interacting fermion systems were given in terms of what are now known as Landau-quasiparticles.

The idea of a clade did not exist in pre-Darwinian Linnaean taxonomy, which was based by necessity only on internal or external morphological similarities between organisms – although as it happens, many of the better known animal groups in Linnaeus ' original Systema Naturae ( notably among the vertebrate groups ) do represent clades.

The idea, or possibility, of female centaurs was certainly known in early modern times, as evidenced by Shakespeare's King Lear, Act IV, Scene vi, ln. 124 – 125:

The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction.

He is best known as the author of Principles of Geology, which popularised James Hutton's concepts of uniformitarianism – the idea that the earth was shaped by the same processes still in operation today.

He adumbrated a school of thought that is known as Critical Rationalism with a central tenet being the rejection of the idea that knowledge can ever be justified in the strong form that is sought by most schools of thought.

The idea of the Erdős number was created by friends as a humorous tribute to the enormous output of Erdős, one of the most prolific modern writers of mathematical papers, and has become well known in scientific circles as a tongue-in-cheek measurement of mathematical prominence.

He is widely known for use of the term " central dogma " to summarize an idea that genetic information flow in cells is essentially one-way, from DNA to RNA to protein.

This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

Henri Gautier, a French Royal Engineer, recognized the " natural slope " of different soils in 1717, an idea later known as the soil's angle of repose.

The observed charge quantization, namely the fact that all known elementary particles carry electric charges which appear to be exact multiples of of the " elementary " charge, has led to the idea that hypercharge interactions and possibly the strong and weak interactions might be embedded in one Grand Unified interaction described by a single, larger simple symmetry group containing the Standard Model.

The idea was to use Latin vocabulary, since this was widely known, but simplify the grammar as much as possible and remove all irregular and anomalous forms to make it easier to learn.

Proponents of this idea, also known as the barefoot horse movement, argue that with proper care, horses may never need shoes at any time once they have been properly transitioned.

Fantasy author Philip Pullman echoes this idea in the fantasy series His Dark Materials, in which the characters finally come to the conclusion that people should make life better on Earth rather than wait for heaven ( this idea is known as the Republic of Heaven ).