The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

Thus, V ( S ) is " the same as " the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals ; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.

An algebro-geometric variant of this construction is applied to the category of algebraic varieties ; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves ( or coherent sheaves ) on X.

Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism ; Serre duality was recovered as the case of the morphism of a non-singular projective variety ( or complete variety ) to a point.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

It is undisputed that Galois was more than qualified ; however, accounts differ on why he failed.

The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel – Ruffini theorem on the algebraic insolubility of quintic equations.

The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings ; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Waterhouse showed that every profinite group is isomorphic to one arising from the Galois theory of some field K ; but one cannot ( yet ) control which field K will be in this case.

There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981 ; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 ( when victory was initially declared ), but was only generally agreed to be finished in 2004., work on improving the proofs and understanding continues ; see for 19th century history of simple groups.

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group ; and states that these L-functions are identical to certain Dirichlet L-series or more general series ( that is, certain analogues of the Riemann zeta function ) constructed from Hecke characters.

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Not all closed-form expressions have closed-form antiderivatives ; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed form antiderivatives.

Galois theory can be used to determine which algebraic numbers can be expressed using roots, and to prove the Abel-Ruffini theorem, which states that a general polynomial equation of degree five or higher cannot be solved using roots alone ; this result is also known as " the insolubility of the quintic ".

In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K ; and associated to A a Galois group G which will be a pro-finite group, so a compact topological group, and also abelian.

Bill Ruger's long-awaited Deerstalker ( under $110 ) is a new rifle action in a caliber that upsets all the modern theory of high-velocity fans ; ;

Later in the 1960s and 1970s, Edmund Leach and his students Mary Douglas and Nur Yalman, among others, introduced French structuralism in the style of Lévi-Strauss ; while British anthropology has continued to emphasize social organization and economics over purely symbolic or literary topics, differences among British, French, and American sociocultural anthropologies have diminished with increasing dialogue and borrowing of both theory and methods.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

( Despite common belief, he did not take a day from February ; see the debunked theory on month lengths ) According to a Senatus consultum quoted by Macrobius, he chose this month because it was the time of several of his great triumphs, including the conquest of Egypt.

In one such theory, put forward by Sir E. B. Tylor, early humans initially, through mere observation, recognized what might be called a soul, life-force, spirit, breath or animus within themselves ; that which was present in the body in life and absent in death.

Germain Morin broke new ground by suggesting in 1899 that the writer was Isaac, a converted Jew and writer of a tract on the Trinity and Incarnation, who was exiled to Spain in 378-380 and then relapsed to Judaism ; but he afterwards abandoned this theory of the authorship in favour of Decimus Hilarianus Hilarius, proconsul of Africa in 377.

Supporting literature includes: the work of social impact theory, which discusses persuasion in part through the number of persons engaging in influence ; as well as studies made on the relative influence of communicator credibility in different kinds of persuasion ; and examinations of the trustworthiness of the speaker.

According to the theory, the human experience of moral obligations was the result of evolutionary pressures, which attached a sense of morality to human psychology because it was useful for moral development ; this entail that moral values do not exist independently of the human mind.

Alexander Grothendieck (; ; born 28 March 1928 ) is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry.

AA generally avoids discussing the medical nature of alcoholism ; nonetheless AA is regarded as a proponent and popularizer of the disease theory of alcoholism.

The challenge to the assumption that beauty was central to art and aesthetics, thought to be original, is actually continuous with older aesthetic theory ; Aristotle was the first in the Western tradition to classify " beauty " into types as in his theory of drama, and Kant made a distinction between beauty and the sublime.

In three papers which were published in 1952 – 53, Bohr and Mottelson demonstrated close agreement between theory and experiment ; for example, showing that the energy levels of certain nuclei could be described by a rotation spectrum.

The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space.

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

Consequently the Gaussian theory only supplies a convenient method of approximating to reality ; and no constructor would attempt to realize this unattainable ideal.

Sir Isaac Newton was probably the discoverer of astigmation ; the position of the astigmatic image lines was determined by Thomas Young ( A Course of Lectures on Natural Philosophy, 1807 ); and the theory was developed by Allvar Gullstrand.

The social skills that are impaired by alcohol abuse include impairments in perceiving facial emotions, prosody perception problems and theory of mind deficits ; the ability to understand humour is also impaired in alcohol abusers.

John Miles Foley held, specifically with reference to the Beowulf debate, that while comparative work was both necessary and valid, it must be conducted with a view to the particularities of a given tradition ; Foley argued with a view to developments of oral traditional theory that do not assume, or depend upon, finally unverifiable assumptions about composition, and that discard the oral / literate dichotomy focused on composition in favor of a more fluid continuum of traditionality and textuality.

According to bundle theory, an object consists of its properties and nothing more: thus neither can there be an object without properties nor can one even conceive of such an object ; for example, bundle theory claims that thinking of an apple compels one also to think of its color, its shape, the fact that it is a kind of fruit, its cells, its taste, or at least one other of its properties.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

In the 1920s and 1930s almost every major cosmologist preferred an eternal steady state Universe, and several complained that the beginning of time implied by the Big Bang imported religious concepts into physics ; this objection was later repeated by supporters of the steady state theory.

It was dark and, I sensed, very large ; ;

The slaves never shared in their profits, while they did share, in a very real sense, in the profits of the slave-owners: they were fed, clothed, doctored, and so forth ; ;

To escape from a prison camp required a very special state of mind ; ;

It was very widely read, too ; ;

The very firmness of her convictions and logical clearness of her arguments captivated and stimulated him to make greater efforts ; ;

Not all recent science fiction, however, is dystopian, for the optimistic strain is still very much alive in Mission Of Gravity and Childhood's End, as we have seen, as well as in many other recent popular novels and stories like Fred Hoyle's The Black Cloud ( 1957 ) ; ;

The girls, very fetching in their uniforms, are shown firing rockets from a launcher mounted on a dump truck ; ;

As we know, the Soviet peasant today still very largely thrives on being able to sell the produce grown on his private plot ; ;

He could now but the weakness is very definite ; ;

There were two liquor saloons not very far from the church, one white, that is conducted for white people with a side entrance for Negroes ; ;

On May 15, a very large increase occurred with Af of mass between Af and Af ; ;

It is very easy to establish by induction on R that if F is in Af then Af ; ;

I realized, now, that she had been showing me, in what impressed me as being a very accurate way, something her mother had once said to her ; ;